# On the number of real zeroes of a homogeneous differential polynomial and a
generalization of the Hawaii conjecture

Olga Katkova  Department of Mathematics, University of Massachusetts Boston
<EMAIL_ADDRESS>,  Mikhail Tyaglov  Moscow Center for Fundamental and
Applied Mathematics, 119991, Moscow, Russia  School of Mathematical Sciences
and MOE-LSC, Shanghai Jiao Tong University<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Anna Vishnyakova  Department of Mathematics, Holon
Institute of Technology, Holon, Israel<EMAIL_ADDRESS>
(Date: September 1, 2024)

###### Abstract.

For a given real polynomial $p$ we study the possible number of real roots of
a differential polynomial
$H_{\varkappa}[p](x)=\varkappa\left(p^{\prime}(x)\right)^{2}-p(x)p^{\prime\prime}(x),\varkappa\in\mathbb{R}.$
In the special case when all real zeros of the polynomial $p$ are simple, and
all roots of its derivative $p^{\prime}$ are real and simple, the distribution
of zeros of $H_{\varkappa}[p]$ is completely described for each real
$\varkappa.$ We also provide counterexamples to two Boris Shapiro’s
conjectures about the number of zeros of the function $H_{\frac{n-1}{n}}[p].$

###### Key words and phrases:

The Hawaii conjecture, the Laguerre inequality, real-rooted polynomials,
hyperbolic polynomials, Rolle’s type theorems; Chebyshev polynomials of the
first kind

###### 1991 Mathematics Subject Classification:

26C10; 30C15.

##  1\. Introduction

To our dear teacher and friend Victor Katsnelson, an outstanding mathematician
and a bright person, with love and gratitude.

In this paper, we investigate the possible number of real zeros of special
homogeneous differential polynomials.

######  Notation 1.1.

Let $p$ be a real polynomial and $\varkappa$ be a complex number. We denote by
$Q_{\varkappa}[p]$ the following function associated with $p$

$Q_{\varkappa}[p](x)=p^{1-\varkappa}(x)\cdot\dfrac{d}{dx}\left(\dfrac{p^{\varkappa}(x)}{p^{\prime}(x)}\right)=\dfrac{\varkappa\left(p^{\prime}(x)\right)^{2}-p(x)p^{\prime\prime}(x)}{\left(p^{\prime}(x)\right)^{2}}.$
(1.1)

By $H_{\varkappa}[p]$ we denote the following function:

$H_{\varkappa}[p](x)\stackrel{{\scriptstyle
def}}{{=}}\varkappa\left(p^{\prime}(x)\right)^{2}-p(x)p^{\prime\prime}(x)=\left(Q_{\varkappa}[p](x)\right)(p^{\prime}(x))^{2}.$
(1.2)

We study the number of real roots of $Q_{\varkappa}[p]$ and
$H_{\varkappa}[p]$, and we need the following notation.

######  Notation 1.2.

For a real polynomial $p$, by $Z_{\mathbb{C}}(p)$ we denote the number of non-
real zeroes of $p$ counting multiplicities. If $f$ is a real rational
function, then $Z_{\mathbb{R}}(f)$ will denote the number of real zeroes of
$f$ counting multiplicities. Generally, the number of zeroes of a real
rational function $f$ on a set $X\subset\mathbb{R}$ counting multiplicities
will be denoted by $Z_{X}(f)$. In particular,
$Z_{\mathbb{R}}(f)=Z_{(-\infty,+\infty)}(f).$

Our interest is concentrated on bounding the number of real zeroes of
$Q_{\varkappa}[p]$ and $H_{\varkappa}[p]$, that is on bounding
$Z_{\mathbb{R}}(Q_{\varkappa}[p])$ and $Z_{\mathbb{R}}(H_{\varkappa}[p])$ for
$\varkappa\in\mathbb{R}.$

Note that if a polynomial $p$ doesn’t have multiple zeros, then

$Z_{\mathbb{R}}(Q_{\varkappa}[p])=Z_{\mathbb{R}}\left(H_{\varkappa}[p]\right).$
(1.3)

The following statement belongs to E. Laguerre.

######  Theorem A.

If a real polynomial $p$ has only real and simple zeros, then
$H_{1}[p](x)=\left(p^{\prime}(x)\right)^{2}-p(x)p^{\prime\prime}(x)>0$ for all
$x\in\mathbb{R}.$

The Laguerre inequality plays an important role in the study of distribution
of zeros of real entire functions and in the understanding the nature of the
Riemann $\xi-$function (see [1], [2], [4]–[8] and the references therein for
the generalization of the Laguerre inequality, as well).

The following generalization of the Laguerre inequality is known as the Hawaii
conjecture.

######  Conjecture 1.3 (the Hawaii Conjecture, [3]).

If all real roots of a real polynomial $p$ are simple, then

$Z_{\mathbb{R}}\left(H_{1}[p]\right)\leq Z_{\mathbb{C}}(p).$ (1.4)

The Hawaii conjecture was posed by T. Craven, G. Csordas, and W. Smith in [3]
in 1987, and was proved by Mikhail Tyaglov in 2011 ([15]).

A different situation when the polynomials $H_{\varkappa}[p]$ appear was
discovered by J.L.W.V. Jensen in [10]. Consider the function

$\Phi_{p}(x,y)=\left|p(x+iy)\right|^{2}.$ (1.5)

The function $\Phi_{p}(x,y)$ is real-analytic non-negative function in
$(x,y)$, whose zeros are zeros of $p(z)=p(x+iy).$ Assume that $\deg p=n$ and
expand $\Phi_{p}(x,y)$ in the variable $y$. We obtain

$\Phi_{p}(x,y)=\sum_{k=0}^{n}P_{k}(x)\frac{y^{2k}}{(2k)!},$ (1.6)

where

$P_{k}(x)=\sum_{j=0}^{k}(-1)^{j+k}\binom{2k}{j}p^{(j)}(x)p^{(2k-j)}(x).$ (1.7)

In particular, $P_{0}(x)=p^{2}(x)$ and
$P_{1}(x)=\left(p^{\prime}(x)\right)^{2}-p(x)p^{\prime\prime}(x).$ Jensen
obtained the following sufficient and necessary conditions for a polynomial to
have only real simple roots: the polynomials $P_{k}(x)$ are strictly positive
for all $k=1,2,\ldots,n.$ While studying unpublished notes left after Jensen’s
death in 1925, G. Pólya discovered a different criterion of real-rootedness.

######  Theorem B (G. Pólya).

A real polynomial $p$ of degree $n$ has only real simple zeros if and only if
the polynomials

$G_{k}[p](x)=(n-k)\left(p^{(k)}(x)\right)^{2}-(n-k+1)p^{(k-1)}(x)p^{(k+1)}(x)$

are strictly positive for all $k=1,2,\ldots,n.$

Inspired by the Hawaii conjecture and motivated by these two criteria, Boris
Shapiro formulated the following two conjectures.

######  Conjecture 1.4 (Boris Shapiro, [13]).

For any real polynomial $p$ of degree $n\geq 2$ all whose real roots are
simple

$Z_{\mathbb{R}}\left[(n-1)\left(p^{\prime}\right)^{2}-npp^{\prime\prime}\right]\leq
Z_{\mathbb{C}}\left(p\right).$ (1.8)

If this fact was true, the Hawaii conjecture would hold for $G_{1}.$

######  Conjecture 1.5 (Boris Shapiro, [13]).

For any real polynomial $p$ of even degree $n\geq 2$,

$Z_{\mathbb{R}}\left[(n-1)\left(p^{\prime}\right)^{2}-npp^{\prime\prime}\right]+Z_{\mathbb{R}}(p)>0.$
(1.9)

The latter inequality is trivially satisfied for odd degree polynomials.

##  2\. Main results

It was observed in [14] that the value $\varkappa=\dfrac{n-1}{n}$ is of
special significance for functions $Q_{\varkappa}[p].$ Although for
polynomials all whose zeroes are real the properties of
$Q_{\tfrac{n-1}{n}}[p]$ are close to the ones of $Q_{1}[p],$ there are
polynomials of degree $n\geqslant 4$ with non-real zeros not satisfying
Conjecture 1.4. In [14] there was given the example of such a polynomial of
degree $4$ having two real and two non-real zeros.

######  Theorem C (M. Tyaglov, M.J. Atia, [14]).

If

$p(x)=(x^{2}+a^{2})(x+a^{2})(x-1),\qquad
a\in\mathbb{R}\backslash\\{-1,0,1\\},$ (2.1)

then

$Z_{\mathbb{R}}\left(H_{\frac{3}{4}}[p]\right)=4.$

It is obvious that $Z_{\mathbb{C}}(p)=2<4.$ Therefore, Theorem C gives a
counterexample to the Conjecture 1.4.

However, the question still remains: whether or not counterexamples given by
polynomials of higher degrees exist. In the present paper, for each
$n=5,6,\ldots,$ we construct a real polynomial $p$ of degree $n$ that doesn’t
satisfy the Conjecture 1.4.

######  Proposition 2.1.

If $n\geq 5$ and

$p(x)=(x-1)^{n}+(x+1)^{n},$ (2.2)

then

$Z_{\mathbb{R}}\left(H_{\frac{n-1}{n}}[p]\right)=2n-4,$

while

$Z_{\mathbb{C}}(p)=2\left\lfloor\dfrac{n}{2}\right\rfloor<Z_{\mathbb{R}}\left(H_{\frac{n-1}{n}}[p]\right).$

It follows from Theorem C and Proposition 2.1 that for
$\varkappa=\frac{n-1}{n},\ \ n=4,5,6,\ldots,$ the generalization of the Hawaii
conjecture

$Z_{\mathbb{C}}\left(Q_{\varkappa}[p]\right)\leq Z_{\mathbb{R}}(p)$ (2.3)

is not valid.

There arises a natural question: for which values of $\varkappa$ the Hawaii
conjecture continues to be true?

We investigate the number of real zeros of the function $Q_{\varkappa}[p],\
\varkappa\in\mathbb{R},$ in the case when the derivative $p^{\prime}$ has only
real simple zeros. The statement bellow can be considered as a generalization
of the Hawaii conjecture in this case.

######  Theorem 2.2.

Let $\varkappa\geq\frac{n-1}{n}.$ For any real polynomial $p$ of degree $n\geq
2$ all whose real zeros are simple, and all zeros of whose derivative
$p^{\prime}$ are real and simple,

$Z_{\mathbb{R}}\left(H_{\varkappa}[p]\right)=Z_{\mathbb{C}}(p).$ (2.4)

The following two statements give estimations for the number
$Z_{\mathbb{R}}\left(H_{\varkappa}[p]\right)$ when $\varkappa<\frac{1}{2}.$

######  Theorem 2.3.

Let $\varkappa\leq 0.$ For any real polynomial $p$ of degree $n\geq 2$ all
whose real zeros are simple, and all zeros of whose derivative $p^{\prime}$
are real and simple,

$Z_{\mathbb{R}}\left(H_{\varkappa}[p]\right)=n+Z_{\mathbb{R}}(p)-2.$ (2.5)

######  Theorem 2.4.

Let $0<\varkappa<1/2.$ For any real polynomial $p$ of degree $n\geq 2$ all
whose real zeros are simple, and all zeros of whose derivative $p^{\prime}$
are real and simple,

$Z_{\mathbb{C}}(p)-2\leq Z_{\mathbb{R}}\left(H_{\varkappa}[p]\right)\leq
n+Z_{\mathbb{R}}(p)-2.$ (2.6)

We will show that both estimations in Theorem 2.4 are sharp. In particular, we
prove the following theorem.

######  Theorem 2.5.

For each $\varkappa,0<\varkappa<\frac{1}{2},$ and each $n\geq 3$ there exists
a polynomial $p_{n}$ of degree $n$ all whose real zeros are simple, and all
zeros of whose derivative $p^{\prime}$ are real and simple, such that

$Z_{\mathbb{R}}\left(H_{\varkappa}[p_{n}]\right)=n+Z_{\mathbb{R}}(p_{n})-2.$

Since

$n+Z_{\mathbb{R}}(p)-2\leq Z_{\mathbb{C}}(p)\ \Leftrightarrow\
Z_{\mathbb{R}}(p)\leq 1,$

Theorem 2.3 and Theorem 2.5 say that, in general, the Hawaii conjecture is not
valid for $\varkappa<\frac{1}{2}.$

In [14] M. Tyaglov and M.J. Atia found the upper and lower bounds of
$Z_{\mathbb{R}}\left(H_{\varkappa}[p]\right)$ for all real $\varkappa$ when a
polynomial $p$ has only real zeros. Their result can be reformulated as
follows.

######  Theorem D (M. Tyaglov, M.J. Atia, [14]).

Let $\frac{k-1}{k}\leq\varkappa<\frac{k}{k+1},\ k=2,3,\ldots,n-1.$ If a real
polynomial $p$ of degree $n\geq 2$ has only real and simple zeros, then

$2\leq Z_{\mathbb{R}}\left(H_{\varkappa}[p]\right)\leq 2n-2k.$ (2.7)

We will show that all estimations in Theorem D are sharp.

Analyzing the statements of Theorems 2.2 – 2.5 and TheoremD, one can guess
that for any real polynomial $p$ of degree $n$ the Hawaii conjecture (2.3)
holds when $\varkappa>\frac{n-1}{n}.$ The statement below, which is easy to
check by direct calculations, shows that this conjecture is not true.

######  Proposition 2.6.

Let $n\geq 3,$ and

$p(x)=x^{n}+ax^{n-2},\ \,a>0.$ (2.8)

Then for all $\varkappa\in\left[\frac{n-1}{n},\
\frac{(2n-3)^{2}}{4n(n-2)}\right]$

$Z_{\mathbb{R}}\left(Q_{\varkappa}[p]\right)=4.$ (2.9)

######  Remark 2.7.

Note that $\frac{(2n-3)^{2}}{4n(n-2)}>\frac{n-1}{n}.$ So, in general, the
number $\frac{n-1}{n}$ cannot serve as the infimum for the set of all real
$\varkappa$ for which (2.3) is valid.

######  Remark 2.8.

It is worth noting that the polynomial in the above example has $0$ as a root
of multiplicity $(n-2).$ Any polynomial of the form
$p_{2n,a,C}(x)=x^{2n}+ax^{2n-2}+C,\ \ C\neq 0,$ has either two or zero real
roots, and any polynomial $p_{2n+1,a,C}(x)=x^{2n+1}+ax^{2n-1}+C,\ \ C\neq 0,$
has only one real root. So, if $\deg p\geq 5,$ even a small shift of the
polynomial in Proposition 2.6 gives $Z_{\mathbb{C}}(p)\geq 4,$ which makes
(2.9) consistent with (2.3). Probably, the question about the infimum for the
set of all real $\varkappa$ for which (2.3) is valid will make more sense for
polynomials, all whose real roots are simple.

Unfortunately, in the general case, if there aren’t any restrictions on roots
of the derivative $p^{\prime}$ of a real polynomial $p,$ we could find only a
lower bound for the number of real zeroes of the function $Q_{\varkappa}[p]$
for an arbitrary real polynomial $p$ whenever $\varkappa\in\mathbb{R}$.

######  Notation 2.9.

We denote by $\mathring{Z}_{X}(f)$ the number of zeros of the function $f$ on
the set $X$ without counting their multiplicities.

######  Theorem 2.10.

Let $p$ be a real polynomial of degree $n\geqslant 2$. The following
inequalities hold

$Z_{\mathbb{R}}\left(Q_{\varkappa}[p]\right)\geqslant\mathring{Z}_{\mathbb{R}}(p^{\prime})+1-\mathring{Z}_{\mathbb{R}}(p),\qquad\text{for}\qquad\varkappa>\dfrac{n-1}{n},$
(2.10)
$Z_{\mathbb{R}}\left(Q_{\varkappa}[p]\right)\geqslant\mathring{Z}_{\mathbb{R}}(p^{\prime})-1-\mathring{Z}_{\mathbb{R}}(p),\qquad\text{for}\qquad
0<\varkappa\leqslant\dfrac{n-1}{n}.$ (2.11)

If $p(x)\neq(x-\alpha)^{n},$ then

$Z_{\mathbb{R}}\left(Q_{\varkappa}[p]\right)\geqslant\mathring{Z}_{\mathbb{R}}(p^{\prime})-1+\mathring{Z}_{\mathbb{R}}(p),\qquad\text{for}\qquad\varkappa\leqslant
0.$ (2.12)

Now let us return to Boris Shapiro’s Conjecture 1.5. In their recent preprint
[12], Lande Ma and Zhaokun Ma discussed this conjecture for entire
transcendental functions. For any even number $n\geq 4$ we give an example of
a real polynomial $p$ of degree $n$ that doesn’t satisfy Conjecture 1.5 .

######  Proposition 2.11.

Let $n\geq 2,$ and

$p_{2n}(x)=\frac{x^{2n}}{2n}+\frac{x^{2}}{2}+1.$ (2.13)

Then

$Z_{\mathbb{R}}\left[(2n-1)(p^{\prime}(x))^{2}-2np(x)p^{\prime\prime}(x)\right]=0.$

The following theorem describes a sufficiently broad class of polynomials
satisfying Conjecture 1.5.

######  Theorem 2.12.

Let $p$ be a real polynomial of even degree $n\geq 4.$ Assume that there
exists $k=1,2,\ldots,n-2,$ such that all the zeros of the derivative $p^{(k)}$
are real. Then

$Z_{\mathbb{R}}\left(H_{\frac{n-1}{n}}[p]\right)+Z_{\mathbb{R}}(p)>0.$ (2.14)

The paper has the following structure. In Section 3 we discuss counterexamples
to Boris Shapiro’s conjectures. In Section 4 we introduce a function $M[p],$
which plays a major role in all our constructions. As an example of
applications of $M[p],$ in this Section we prove two theorems: Theorem 2.10 on
a lower bound for the number $Z_{\mathbb{R}}\left(Q_{\varkappa}[p]\right),$
and Theorem 2.12 describing polynomials that satisfy Boris Shapiro’s
conjecture 1.5. In the rest of our paper we investigate the zero distribution
of the functions $Q_{\varkappa}[p]$, where $p$ are real polynomials, all whose
real zeros are simple and whose derivative $p^{\prime}$ has only real and
simple roots. Sections 5–9 are devoted to the case $\varkappa>\frac{n-1}{n}.$
We develop an idea we found in [9, p.106, Problem 725] and prove that in this
case the Hawaii conjecture is true. The result from [9] relates to the case
when $\varkappa=0$ and all roots of $p$ are real and simple. In Section 10 we
obtain a result (Lemma 10.2) similar to Rolle’s theorem that gives a relation
between roots of $H_{\varkappa}[p]$ and
$H_{2-\frac{1}{\varkappa}}[p^{\prime}].$ This Lemma allows to investigate the
zero distribution of the function $Q_{\varkappa}[p]$ in the case
$-\infty<\varkappa<\frac{n-1}{n}$ (Sections 11–13). In Section 14 we discuss
the accuracy of estimates in Theorem 2.4. Sections 15 contains some additional
results about the zero distribution of $Q_{\varkappa}[p]$ in the case when
$\frac{1}{2}\leq\varkappa<\frac{n-1}{n}$ and all roots of $p$ are real. In
Section 16 we describe possible distribution of roots of $Q_{\varkappa}[p]$
when $\frac{1}{2}\leq\varkappa<\frac{n-1}{n}$ in the general case.

##  3\.  Counterexamples to Boris Shapiro’s Conjectures: Proof of Theorem C,
and Propositions 2.1 and 2.11

The fact that Conjecture 1.4 is not true for $n=\deg p=4$ was established in
[14]. We provide a proof of Theorem C here for completeness and convenience of
readers.

Proof of Theorem C. The polynomial of the fourth degree

$p(x)=(x^{2}+a^{2})(x+a^{2})(x-1),\qquad
a\in\mathbb{R}\backslash\\{-1,0,1\\},$ (3.15)

has two distinct real zeroes, $-a^{2}$ and $1$, and two non-real zeroes $\pm
ia$, so $Z_{C}(p)=2$. The function $H_{\tfrac{3}{4}}[p]$ of this polynomial
satisfies the following equality.

$\frac{4}{3}H_{\tfrac{3}{4}}[p](x)=(a^{2}-1)^{2}x^{4}-8a^{2}(a^{2}-1)x^{3}-2a^{2}(a^{4}-10a^{2}+1)x^{2}$
$+8a^{4}(a^{2}-1)x+a^{4}(a^{2}-1)^{2}$
$=\left[(a-1)x-a(a+1)\right]^{2}\cdot\left[(a+1)x+a(a-1)\right]^{2}.$ (3.16)

So, whenever $a\in\mathbb{R}\backslash\\{-1,1\\}$ it has four real zeroes.
Thus,

$Z_{\mathbb{R}}\left(H_{\tfrac{3}{4}}[p]\right)=4>2=Z_{\mathbb{C}}(p),$ (3.17)

and Conjecture 1.4 fails.

Now we will prove Propositions 2.1 which gives a counterexample to Conjecture
1.4 for each $n=\deg p\geq 5.$

Proof of Proposition 2.1. Let $n\geqslant 5$. Consider the polynomial

$p(x)=(x-1)^{n}+(x+1)^{n}.$ (3.18)

The numbers $z_{k}=i\cot\left(\frac{\pi}{2n}+\frac{\pi k}{n}\right),\
k=0,1,\ldots,n-1,$ are the zeros of this polynomial. So, it has exactly
$2\left\lfloor\dfrac{n}{2}\right\rfloor$ pure imaginary zeroes.

For the polynomial $H_{\tfrac{n-1}{n}}[p]$, we have

$H_{\tfrac{n-1}{n}}[p](x)=-4n(n-1)(x-1)^{n-2}(x+1)^{n-2}.$ (3.19)

So, we have

$Z_{\mathbb{R}}\left(H_{\tfrac{n-1}{n}}[p]\right)=2n-4>2\left\lfloor\dfrac{n}{2}\right\rfloor=Z_{\mathbb{C}}(p)$
(3.20)

for any $n\geqslant 5$. Thus, we proved that Conjecture 1.4 is not true for
any $n\geqslant 4$.

At the end of this Section we will prove that the other Boris Shapiro’s
conjecture 1.5 is not valid. Let us prove Proposition 2.11.

Proof of Proposition 2.11. Let $n\geq 2,$ and

$p_{2n}(x)=\frac{x^{2n}}{2n}+\frac{x^{2}}{2}+1.$

It is obvious that $Z_{\mathbb{R}}(p)=0.$ For the polynomial $p_{2n}$ we have

$-2nH_{\tfrac{2n-1}{2n}}[p](x)=x^{2n}(2n^{2}-5n+3)+2n(2n-1)x^{2n-2}-(n-1)x^{2}+2n.$
(3.21)

Let us show that the polynomial (3.21) doesn’t have real zeros. First, assume
that $|x|\geq 1.$ Since $n\geq 2,$ in this case $x^{2n-2}\geq x^{2},$ and the
following estimation is true

$-2nH_{\tfrac{2n-1}{2n}}[p](x)\geq(n-1)(2n-3)x^{2n}+(2n(2n-1)-(n-1))x^{2}+2n>0.$
(3.22)

If $|x|<1,$ then $x^{2}<1,$ and

$-2nH_{\tfrac{2n-1}{2n}}[p](x)\geq(n-1)(2n-3)x^{2n}+2n(2n-1)x^{2n-2}+n+1>0.$
(3.23)

It follows from (3.22) and (3.23) that

$Z_{\mathbb{R}}\left(H_{\tfrac{2n-1}{2n}}[p](x)\right)=0.$

Therefore,

$Z_{\mathbb{R}}\left(H_{\tfrac{2n-1}{2n}}[p](x)\right)+Z_{\mathbb{R}}(p)=0,$

and Boris Shapiro’s conjecture 1.5 is not valid.

##  4\.  The function $M[p](x).$ Proof of Theorem 2.10 and Theorem 2.12

Without loss of generality we assume that $p$ is a monic polynomial, that is

$p(x)=x^{n}+a_{n-1}x^{n-1}+\ldots$ (4.24)

Let us introduce the following function

$M[p](x)=\frac{p(x)p^{\prime\prime}(x)}{(p^{\prime}(x))^{2}}.$ (4.25)

By virtue of (1.1)

$Q_{\varkappa}[p](x)=\varkappa-M[p](x).$ (4.26)

It is clear that $Z_{\mathbb{R}}\left(Q_{\varkappa}[p]\right)$ is the same as
the number of real roots of the equation

$M[p](x)=\varkappa.$ (4.27)

Note that if $p^{\prime}(x)$ doesn’t have real roots, all the statements of
Theorem 2.10 and Theorem 2.12 are obviously true. Let us denote by

$\xi_{1}<\xi_{2}<\cdots<\xi_{m-1},\ \ m<n,$ (4.28)

different real poles of the function $M[p].$

######  Remark 4.1.

Note that if $\xi$ is a root of multiplicity $j$ of the polynomial $p,$ then

$\lim_{x\to\xi}M[p](x)=\lim_{x\to\xi}\frac{p(x)p^{\prime\prime}(x)}{(p^{\prime}(x))^{2}}=\frac{j-1}{j}.$
(4.29)

So, the poles $\\{\xi_{k}\\}_{k=1}^{m-1}$ from 4.28 are those zeros of
$p^{\prime}$ that are not roots of $p.$

By virtue of Rolle’s theorem each of the intervals below contains at most one
zero of the polynomial $p$

$I_{1}=(-\infty,\xi_{1}),\ I_{2}=(\xi_{1},\xi_{2}),\ldots,\
I_{m}=(\xi_{m-1},\infty).$ (4.30)

Let us divide all the intervals $(I_{k})_{k=1}^{m}$ into two groups.

######  Definition 4.2.

We will say that an interval

  * •

$I_{k},\ k=1,2,\ldots,m,$ is an interval of the first type if it contains a
root of the polynomial $p$;

  * •

$I_{k},\ k=1,2,\ldots,m,$ is an interval of the second type if it doesn’t
contain roots of the polynomial $p.$

Assume that $\xi$ is a root of the derivative $p^{\prime}$ which is not a root
of $p.$ Then

$p(x)=p(\xi)+\frac{p^{(s)}(\xi)}{s!}(x-\xi)^{s}+\ldots,\ \ \mbox{where}\ \
p^{(s)}(\xi)\neq 0.$ (4.31)

If we take up the position that the right behavior of a polynomial $p$ is to
have a root in the interval $I_{k},$ while not to have a root in the interval
is a wrong behavior, we will come to the following classification of the
points $\\{\xi_{k}\\}_{k=1}^{m-1}$ (see 4.31).

######  Definition 4.3.

Assume that $\xi$ is a pole of $M[p]$ (see (4.25)).

  * •

$\xi$ is a right-hand right point if

$p(\xi)p^{(s)}(\xi)(x-\xi)^{s}<0,\ \mbox{when}\ x\ \mbox{is close to}\ \xi\
\mbox{and}\ x>\xi.$

Thus,

$\lim_{x\to\xi^{+}}M[p](x)=-\infty.$ (4.32)

  * •

$\xi$ is a left-hand right point if

$p(\xi)p^{(s)}(\xi)(x-\xi)^{s}<0,\ \mbox{when}\ x\ \mbox{is close to}\ \xi\
\mbox{and}\ x<\xi.$

Thus,

$\lim_{x\to\xi^{-}}M[p](x)=-\infty.$ (4.33)

  * •

$\xi$ is a right-hand wrong point if

$p(\xi)p^{(s)}(\xi)(x-\xi)^{s}>0,\ \mbox{when}\ x\ \mbox{is close to}\ \xi\
\mbox{and}\ x>\xi.$

Thus,

$\lim_{x\to\xi^{+}}M[p](x)=+\infty.$ (4.34)

  * •

$\xi$ is a left-hand wrong point if

$p(\xi)p^{(s)}(\xi)(x-\xi)^{s}>0,\ \mbox{when}\ x\ \mbox{is close to}\ \xi\
\mbox{and}\ x<\xi.$

Thus,

$\lim_{x\to\xi^{-}}M[p](x)=+\infty.$ (4.35)

######  Remark 4.4.

The above definition says the following about the end-points of the intervals
$I_{k}.$

  * •

If a finite interval $I_{k}=(\xi_{k-1},\xi_{k})$ is an interval of the first
type, then $\xi_{k-1}$ is a right-hand right point, and $\xi_{k}$ is a left-
hand right point.

  * •

If a finite interval $I_{k}=(\xi_{k-1},\xi_{k})$ is an interval of the second
type, then either $\xi_{k-1}$ is a right-hand right point, and $\xi_{k}$ is a
left-hand wrong point, or $\xi_{k-1}$ is a right-hand wrong point, and
$\xi_{k}$ is a left-hand right point.

  * •

If an infinite interval $I_{1}=(-\infty,\xi_{1})$ is an interval of the first
type, its finite endpoint $\xi_{1}$ is a left-hand right point. If an infinite
interval $I_{m}=(\xi_{m-1},\infty)$ is an interval of the first type, its
finite endpoint $\xi_{m-1}$ is a right-hand right point.

  * •

If an infinite interval $I_{1}=(-\infty,\xi_{1})$ is an interval of the second
type, its finite endpoint $\xi_{1}$ is a left-hand wrong point. If an infinite
interval $I_{m}=(\xi_{m-1},\infty)$ is an interval of the second type, its
finite endpoint $\xi_{m-1}$ is a right-hand wrong point.

So, using (4.26), (4.27), (4.32)-(4.35) we come to the following conclusion
about the number of real roots of the function $Q_{\varkappa}[p]$ on finite
intervals $I_{k}.$

######  Lemma 4.5.

Let $p$ be a real polynomial of degree $n\geq 2$. Then

  * •

If a finite interval $I_{k}$ is an interval of the first type, then for any
real $\varkappa$ the number $Z_{I_{k}}\left(Q_{\varkappa}[p]\right)$ is even.

  * •

If a finite interval $I_{k}$ is an interval of the second type, then for any
real $\varkappa$ the number $Z_{I_{k}}\left(Q_{\varkappa}[p]\right)$ is odd.

Let us discuss the behavior of the function $M[p]$ at infinity. Note that by
(4.25) we have

$\lim_{x\to\pm\infty}M[p](x)=\frac{n-1}{n}.$ (4.36)

The statement below follows from (4.36), two last statements of Remark 4.4 and
(4.32)-(4.35). It gives an estimation for the number of real roots of the
function $Q_{\varkappa}[p]$ on infinite intervals $I_{k}.$ As we noted in the
beginning of this section, the case when $p^{\prime}$ doesn’t have real roots
is trivial for Theorem 2.10 and Theorem 2.12. That’s why we do not consider it
in the lemma below.

######  Lemma 4.6.

Let $p$ be a real polynomial of degree $n\geq 2$, and its derivative
$p^{\prime}$ has at least one real root. Then

  * •

If an infinite interval $I_{k},\ k=1\ \mbox{or}\ m,$ is an interval of the
first type, and $\varkappa\geq\frac{n-1}{n},$ then the number
$Z_{I_{k}}\left(Q_{\varkappa}[p]\right)$ is even.

  * •

If an infinite interval $I_{k},\ k=1\ \mbox{or}\ m,$ is an interval of the
first type, and $\varkappa<\frac{n-1}{n},$ then the number
$Z_{I_{k}}\left(Q_{\varkappa}[p]\right)$ is odd.

  * •

If an infinite interval $I_{k},\ k=1\ \mbox{or}\ m,$ is an interval of the
second type, and $\varkappa>\frac{n-1}{n},$ then the number
$Z_{I_{k}}\left(Q_{\varkappa}[p]\right)$ is odd.

  * •

If an infinite interval $I_{k},\ k=1\ \mbox{or}\ m,$ is an interval of the
second type, and $\varkappa\leq\frac{n-1}{n},$ then the number
$Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=Z_{I_{k}}\left(H_{\varkappa}[p]\right)$
is even.

Now we can prove Theorem 2.10 on lower bounds for the number of real roots of
$Q_{\varkappa}[p].$ We will use the following standard notation.

######  Notation 4.7.

Let $S$ be a finite set. Then by $\\#S$ we will denote the number of elements
in the set $S$.

Proof of Theorem 2.10. Let $\varkappa>\frac{n-1}{n}.$ We need to estimate from
below $Z_{\mathbb{R}}\left(Q_{\varkappa}[p]\right)$. By virtue of Lemma 4.5
and Lemma 4.6, if we consider a contribution of each
$Z_{I_{k}}\left(Q_{\varkappa}[p]\right)$ into
$Z_{\mathbb{R}}\left(Q_{\varkappa}[p]\right)$ as minimum possible, we obtain

$Z_{\mathbb{R}}\left(Q_{\varkappa}[p]\right)\ \geq\ \\#\\{I_{k}|\ I_{k}\
\mbox{is an interval of the second type}\\}\ $ $=m\ -\ \\#\\{I_{k}|\
I_{k}\mbox{ is an interval of the first type}\\}.$

Recall that by (4.28), (4.29), (4.30) the number of pairwise unequal roots of
the polynomial $p^{\prime}$ that are not roots of $p$ equals $m-1.$ So,

$m\ =\ \mathring{Z}_{\mathbb{R}}(p^{\prime})\ +1\ -\ \\#\\{\xi|\ \xi\ \mbox{is
a multiple root of $p$}\\}.$ (4.37)

Note that

$\\#\\{I_{k}|\ I_{k}\mbox{ is an interval of the first type}\\}\ =$
$\mathring{Z}_{\mathbb{R}}(p)\ -\ \\#\\{\xi|\ \xi\ \mbox{is a multiple root of
$p$}\\}.$ (4.38)

The first statement of Theorem 2.10 follows immediately from (4.37) and
(4.38).

Now the second statement of Theorem 2.10 is obvious. Counting the number of
the intervals of the second type we need just to omit two infinite intervals.

In order to prove the third statement of Theorem 2.10 we need the following
lemma.

######  Lemma 4.8.

Let $p$ be a real polynomial of degree $n\geq 2$. If $I_{k}$ is a finite
interval of the first type, and $\varkappa\leq 0,$ then

$Z_{I_{k}}\left(Q_{\varkappa}[p](x)\right)\geq 2.$ (4.39)

Proof of Lemma 4.8. By virtue of Rolle’s theorem inside each finite interval
$I_{k}$ there is a root $\gamma$ of $p^{\prime\prime}.$ Thus, by (4.25) if
$I_{k}$ is an interval of the first type, it contains at least two roots of
the function $M[p].$ Therefore, taking into account formulas (4.32), (4.33)
and (4.26), we obtain

$Z_{I_{k}}\left(\varkappa-M[p](x)\right)=Z_{I_{k}}\left(Q_{\varkappa}[p](x)\right)\geq
2.$

Lemma 4.8 is proved. $\square$

It follows from Lemma 4.5, Lemma 4.6 and Lemma 4.8 that

$Z_{\mathbb{R}}\left(Q_{\varkappa}[p](x)\right)\geq\\#\\{I_{k}|\ I_{k}\
\mbox{is a finite interval of the second type}\\}\ +$ $+\ 2\cdot\\#\\{I_{k}|\
I_{k}\ \mbox{is a finite interval of the first type}\\}\ +$ $\\#\\{I_{k}|\
I_{k}\ \mbox{is an infinite interval of the first type}\\}\ =$ $\\#\\{I_{k}|\
I_{k}\ \mbox{is a finite interval}\\}+\\#\\{I_{k}|\ I_{k}\ \mbox{is an
interval of the first type}\\}=$ $m-2+Z_{\mathbb{R}}(p).$

The third statement of Theorem 2.10 follows from (4.37) and (4.38). $\square$

Proof of Theorem 2.12. Note that if a polynomial $p$ has real roots, the
statement is obvious.

If $\xi$ is a multiple real root of the polynomial $p^{\prime},$ then

$p(x)=a_{0}+a_{m}(x-\xi)^{m}q(x),\ \ m\geq 3,$

and whence $\xi$ is a root of
$H_{\frac{n-1}{n}}[p](x)=\frac{n-1}{n}(p^{\prime}(x))^{2}-p(x)p^{\prime\prime}(x).$
So, in this case theorem is also obvious.

Assume now that $p$ doesn’t have real roots, all real roots of $p^{\prime}$
are simple, and $p(x)=x^{n}+ax^{n-1}+\ldots$ Let us expand $p$ about
$-\frac{a}{n}.$

$p(x)=x^{n}+ax^{n-1}+\ldots=\left(x+\frac{a}{n}\right)^{n}+\
b\left(x+\frac{a}{n}\right)^{n-2}+\ldots$ (4.40)

Thus,

$p^{\prime}(x)=n\left(x+\frac{a}{n}\right)^{n-1}+\
b(n-2)\left(x+\frac{a}{n}\right)^{n-3}+\ldots$ (4.41)

and

$p^{\prime\prime}(x)=n(n-1)\left(x+\frac{a}{n}\right)^{n-2}+\
b(n-2)(n-3)\left(x+\frac{a}{n}\right)^{n-4}+\ldots$ (4.42)

It follows from (4.40), (4.41) and (4.42) that

$M[p](x)-\frac{n-1}{n}=\frac{2bn\left(x+\frac{a}{n}\right)^{2n-4}+\ldots}{\left(n\left(x+\frac{a}{n}\right)^{n-1}+b(n-2)\left(x+\frac{a}{n}\right)^{n-3}+\ldots\right)^{2}}.$
(4.43)

It is given that for some $k=1,2,\ldots,n-2,$ all the zeros of the derivative

$p^{(k)}(x)=\frac{n!}{(n-k)!}\left(x+\frac{a}{n}\right)^{n-k}+b\
\frac{(n-2)!}{(n-k-2)!}\left(x+\frac{a}{n}\right)^{n-k-2}+\cdots$

are real. Therefore, the coefficient $b$ in (4.40) must be negative (see [11,
p. 434, Lemma 3]). Since $b<0,$ by (4.43) there exists such a value $x_{0}$
that for $|x|>x_{0}>0$ the following is true

$M[p](x)-\frac{n-1}{n}<0.$ (4.44)

It means that the graph of the function $y=M[p](x)$ approaches its asymptote
$y=\frac{n-1}{n}$ from below.

Therefore, the statement of Theorem 2.12 follows immediately from the last
statement of Remark 4.4 and formulas (4.34), (4.35). $\square$

##  5\.  The functions $\varphi_{\lambda}(x)$ and $\Phi_{\lambda}(x)$

From now on we will assume that all real roots of the polynomial $p$ are
simple, and all roots of its derivative $p^{\prime}$ are real and simple. So,
in this case the number of intervals $I_{k}$ is equal to $n.$ The following
six sections are devoted to the proof of Theorem 2.2. First we are going to
study the case of $\varkappa>\frac{n-1}{n}.$ Put

$\lambda=\varkappa-1.$ (5.45)

Consider the function

$\varphi_{\lambda}(x)=\frac{p(x)}{p^{\prime}(x)}+\lambda x.$ (5.46)

Note that

$\varphi_{\lambda}^{\prime}(x)=\frac{(p^{\prime}(x))^{2}-p(x)p^{\prime\prime}(x)}{(p^{\prime}(x))^{2}}+\lambda=\frac{\varkappa(p^{\prime}(x))^{2}-p(x)p^{\prime\prime}(x)}{(p^{\prime}(x))^{2}}=Q_{\varkappa}[p](x).$
(5.47)

Therefore,

$Z_{\mathbb{R}}(\varphi_{\lambda}^{\prime}(x))=Z_{\mathbb{R}}\left(Q_{\varkappa}[p](x)\right).$
(5.48)

If $(\xi_{k},p(\xi_{k}))$ is a local maximum, then $p^{\prime}$ changes sign
from positive to negative, and if $(\xi_{k},p(\xi_{k}))$ is a local minimum,
then $p^{\prime}$ changes sign from negative to positive as $x$ increases
through $\xi_{k}$. Therefore, according to definition 4.3 and (5.46), if
$\xi_{k}$ is a right point, then

$\lim_{x\to\xi_{k}^{+}}\varphi_{\lambda}(x)=-\infty\ \ \mbox{and}\ \
\lim_{x\to\xi_{k}^{-}}\varphi_{\lambda}(x)=+\infty,$ (5.49)

and if $\xi_{k}$ is a wrong point, then

$\lim_{x\to\xi_{k}^{+}}\varphi_{\lambda}(x)=+\infty\ \ \mbox{and}\ \
\lim_{x\to\xi_{k}^{-}}\varphi_{\lambda}(x)=-\infty.$ (5.50)

The formula below describes asymptotic behavior of the function
$\varphi_{\lambda}(x)$ at infinity

$\varphi_{\lambda}(x)=\frac{p(x)}{p^{\prime}(x)}+\lambda
x=\left(\frac{1}{n}+\lambda\right)x+o(x),\ \ x\to\pm\infty.$ (5.51)

Given a real number $\lambda,$ we will evaluate the number of all real roots
of the equation

$\varphi_{\lambda}(x)=\frac{p(x)}{p^{\prime}(x)}+\lambda x=\mu$ (5.52)

in each interval $I_{k}$ for any real value of the parameter $\mu.$

The lemma below follows immediately from (5.49), (5.50), (5.51) and Remark
4.4.

######  Lemma 5.1.

The following statements are true.

  1. (1)

For all real values of the parameters $\mu,\lambda,$ and every
$k=2,3,\ldots,n-1$

     * •

$Z_{I_{k}}(\varphi_{\lambda}(x)-\mu)$ is odd if $I_{k}$ is an interval of the
first type.

     * •

$Z_{I_{k}}(\varphi_{\lambda}(x)-\mu)$ is even if $I_{k}$ is an interval of the
second type.

  2. (2)

For all real values of the parameter $\mu$, for every $\lambda>-\frac{1}{n},$
and $k=1$ or $k=n$

     * •

$Z_{I_{k}}(\varphi_{\lambda}(x)-\mu)$ is odd if $I_{k}$ is an interval of the
first type.

     * •

$Z_{I_{k}}(\varphi_{\lambda}(x)-\mu)$ is even if $I_{k}$ is an interval of the
second type.

  3. (3)

For all real values of the parameter $\mu$, for every $\lambda<-\frac{1}{n},$
and $k=1$ or $k=n$

     * •

$Z_{I_{k}}(\varphi_{\lambda}(x)-\mu)$ is even if $I_{k}$ is an interval of the
first type.

Further we will need the following obvious lemma that describes some
properties of a monic polynomial $p,$ all roots of whose derivative
$p^{\prime}$ are real and simple.

######  Lemma 5.2.

Let $p$ be a real monic polynomial, and all zeros of $p^{\prime}$ are real and
simple. Then

  1. (1)

$(-1)^{n-k-1}p^{\prime\prime}(\xi_{k})>0,\ \ k=1,2,\ldots,n-1.$

  2. (2)

$(-1)^{n-k}p^{\prime}(x)>0,\ \ x\in I_{k}.$

  3. (3)

$\lim_{x\to\infty}p^{\prime}(x)=\lim_{x\to-\infty}(-1)^{n-1}p^{\prime}(x)=+\infty.$

In order to estimate the number of real roots of (5.52) in intervals $I_{k}$
we will consider a function

$\Phi_{\lambda}(x,\mu)=(\varphi_{\lambda}(x)-\mu)p^{\prime}(x)=p(x)+\lambda
xp^{\prime}(x)-\mu p^{\prime}(x),$ (5.53)

and its derivative

$\Phi_{\lambda}^{\prime}(x,\mu)=(1+\lambda)p^{\prime}(x)+(\lambda
x-\mu)p^{\prime\prime}(x).$ (5.54)

##  6\.  Proof of Theorem 2.2 in the Partial Case of $\lambda=0$

Let us prove Theorem 2.2 in the partial case of $\lambda=0.$ In a sense, this
is the base case in the proof of this theorem.

If $\lambda=0,$ then (5.46) has the form

$\varphi_{0}(x)=\frac{p(x)}{p^{\prime}(x)}.$ (6.55)

The statement below describes monotonicity of the function $\varphi_{0},$
provides estimations for the number of real roots of the derivative
$\varphi^{\prime}_{0}$ on the intervals $I_{k},$ and as a result gives the
proof of Theorem 2.2 in the case of $\lambda=0.$

######  Lemma 6.1.

The following statements are true.

  1. (1)

The function $\varphi_{0}(x)=\frac{p(x)}{p^{\prime}(x)}$ is increasing on any
interval $I_{k}$ of the first type.

  2. (2)

$Z_{I_{k}}(\varphi_{0}^{\prime}(x))=Z_{I_{k}}((p^{\prime})^{2}-pp^{\prime\prime})=0$
if $I_{k}$ is an interval of the first type, and

  3. (3)

$Z_{I_{k}}(\varphi_{0}^{\prime}(x))=Z_{I_{k}}((p^{\prime})^{2}-pp^{\prime\prime})=1$
if $I_{k}$ is an interval of the second type.

  4. (4)

$Z_{\mathbb{R}}(\varphi_{0}^{\prime}(x))=Z_{\mathbb{R}}((p^{\prime})^{2}-pp^{\prime\prime})=Z_{\mathbb{C}}(p).$

Proof of Lemma 6.1. If $\lambda=0,$ then (5.53) and (5.54) can be rewritten in
the form

$\Phi_{0}(x,\mu)=(\varphi_{0}(x)-\mu)p^{\prime}(x)=p(x)-\mu p^{\prime}(x),$
(6.56)

and

$\Phi_{0}^{\prime}(x,\mu)=p^{\prime}(x)-\mu p^{\prime\prime}(x).$ (6.57)

Since $\xi_{k}$ is a root of the polynomial $p^{\prime}$ the following
identity is true

$(-1)^{n-k}\Phi_{0}^{\prime}(\xi_{k},\mu)=(-1)^{n-k+1}\mu
p^{\prime\prime}(\xi_{k}),\ \ k=1,2,\ldots,n-1.$ (6.58)

It follows from the statement 1 of Lemma 5.2 that

$(-1)^{n-k}\Phi_{0}^{\prime}(\xi_{k},\mu)>0,\ \ \mbox{whenever}\ \ \mu>0,$
(6.59)

and

$(-1)^{n-k}\Phi_{0}^{\prime}(\xi_{k},\mu)<0,\ \ \mbox{whenever}\ \ \mu<0.$
(6.60)

By the statement 3 of Lemma 5.2

$\lim_{x\to\infty}\Phi_{0}^{\prime}(x,\mu)=\lim_{x\to-\infty}(-1)^{n-1}\Phi_{0}^{\prime}(x,\mu)=+\infty.$
(6.61)

Since $\deg\Phi_{0}^{\prime}(x,\mu)=n-1,$ it follows from (6.59) and (6.61)
that for any $\mu>0$

$Z_{I_{k}}(\Phi_{0}^{\prime}(x,\mu))=1,\ \ \mbox{when}\ \ k=2,3,\ldots,n,$
(6.62)

and from (6.60) and (6.61) that for any $\mu<0$

$Z_{I_{k}}(\Phi_{0}^{\prime}(x,\mu))=1,\ \ \mbox{when}\ \ k=1,2,\ldots,n-1.$
(6.63)

Thus, for any real $\mu\neq 0$ and $k=1,2,\ldots,n,$ we obtain

$Z_{I_{k}}(\Phi_{0}(x,\mu))=Z_{I_{k}}(\varphi_{0}(x)-\mu)\leq 2.$ (6.64)

If $\mu=0,$ then according to $(\ref{le2})$

$\Phi_{0}(x,0)=\varphi_{0}(x)=p(x).$

Hence, by virtue of Rolle’s Theorem

$Z_{I_{k}}(\Phi_{0}(x,0))=Z_{I_{k}}(\varphi_{0}(x))=Z_{I_{k}}(p(x))\leq 1,$

and (6.64) is obviously true.

It follows from (6.64), and the statement 1 of Lemma 5.1 that for any real
$\mu$ and each interval $I_{k}$ of the first type we have

$Z_{I_{k}}(\Phi_{0}(x,\mu))=Z_{I_{k}}(\varphi_{0}(x)-\mu)=1.$ (6.65)

According to the Remark 4.4, both endpoints of a finite interval of the first
type $I_{k}$ are right points; and the finite endpoint of an infinite interval
of the first type $I_{k}$ is a right point. So, if $I_{k}$ is an interval of
the first type, the relation (6.65) together with (5.49) in the case of a
finite interval $I_{k},$ and (5.49), (5.51) in the case of an infinite
interval $I_{k},$ provide the fact that the function $\varphi_{0}(x)$ is
increasing on each interval $I_{k}$ of the first type. The statement 1 of
Lemma 6.1 is proved.

Since $\varphi_{0}(x)$ is monotone on each interval $I_{k}$ of the first type,
we have

$Z_{I_{k}}(\varphi_{0}^{\prime}(x))=Z_{I_{k}}((p^{\prime}(x))^{2}-p(x)p^{\prime\prime}(x))=0,$
(6.66)

where $I_{k}$ is an interval of the first type. It follows from the statement
2 of Lemma 5.1 and (6.64) that for any real $\mu$ and each interval $I_{k}$ of
the second type

$Z_{I_{k}}(\Phi_{0}(x,\mu))=Z_{I_{k}}(\varphi_{0}(x)-\mu)=0\ \mbox{or}\ 2.$
(6.67)

Using the statements of the Remark 4.4 related to intervals of the second
type, taking into account (5.49) and (5.50) in the case of a finite interval
$I_{k},$ and (5.50)(5.51) in the case of an infinite interval $I_{k},$ we
obtain that for each interval $I_{k}$ of the second type

$Z_{I_{k}}(\varphi_{0}^{\prime})=Z_{I_{k}}((p^{\prime})^{2}-pp^{\prime\prime})=1.$
(6.68)

The statement 2 of Lemma 6.1 is proved.

The formulas (6.66) and (6.68) show that the number of real zeros of the
polynomial $(p^{\prime})^{2}-pp^{\prime\prime}$ coincides with the total
number of the intervals of the second type, that is

$Z_{\mathbb{R}}((p^{\prime})^{2}-pp^{\prime\prime})=n-Z_{\mathbb{R}}(p)=Z_{\mathbb{C}}(p).$
(6.69)

Lemma 6.1 is proved. $\square$

##  7\. The function $\Psi_{\alpha}(x,\nu)$

Now we assume that $\lambda\neq 0.$ So, we can rewrite (5.54) in the form

$\frac{1}{\lambda}\Phi_{\lambda}^{\prime}(x,\mu)=\frac{1+\lambda}{\lambda}p^{\prime}(x)+\left(x-\frac{\mu}{\lambda}\right)p^{\prime\prime}(x),$
(7.70)

where $\mu$ is any real number.

Denote by

$\alpha=\frac{1+\lambda}{\lambda},$ (7.71)

and by

$\nu=\frac{\mu}{\lambda}.$ (7.72)

Note that $\nu$ takes any real value, while

$\alpha>1,\ \ \mbox{when}\ \ \lambda>0,$ (7.73)

and

$\alpha<-(n-1),\ \ \mbox{when}\ \ 1/n<\lambda<0.$ (7.74)

We introduce the function

$\Psi_{\alpha}(x,\nu)=:\alpha
p^{\prime}(x)+\left(x-\nu\right)p^{\prime\prime}(x).$ (7.75)

Obviously,

$\Psi_{\frac{1+\lambda}{\lambda}}\left(x,\frac{\mu}{\lambda}\right)=\frac{1}{\lambda}\Phi_{\lambda}^{\prime}(x,\mu).$
(7.76)

The lemma below describes behavior of the function $\Psi_{\alpha}(x,\nu)$ for
several important values of $x$ and at infinity.

######  Lemma 7.1.

The following statements are true.

  1. (1)

For all real values of the parameter $\nu$ and every $k=1,2,\ldots,n-1$

     * •

$(-1)^{n-k}\Psi_{\alpha}(\xi_{k},\nu)<0,$ whenever $\xi_{k}>\nu.$

     * •

$(-1)^{n-k}\Psi_{\alpha}(\xi_{k},\nu)>0,$ whenever $\xi_{k}<\nu.$

  2. (2)

If $\nu\in I_{k}=(\xi_{k-1},\xi_{k}),\ \ k=2,\ldots,n-1$, then

     * •

$(-1)^{n-k}\Psi_{\alpha}(\nu,\nu)>0,$ whenever $\alpha>0.$

     * •

$(-1)^{n-k}\Psi_{\alpha}(\nu,\nu)<0,$ whenever $\alpha<0.$

  3. (3)

For all real values of the parameter $\nu$

     * •

$\lim_{x\to+\infty}\Psi_{\alpha}(x,\nu)=\lim_{x\to-\infty}(-1)^{n-1}\Psi_{\alpha}(x,\nu)=+\infty,$
whenever $\alpha>-(n-1)$.

     * •

$\lim_{x\to+\infty}\Psi_{\alpha}(x,\nu)=\lim_{x\to-\infty}(-1)^{n-1}\Psi_{\alpha}(x,\nu)=-\infty,$
whenever $\alpha<-(n-1)$.

  4. (4)

For all real values of the parameter $\alpha$ and any $k=1,2,\ldots,n-1$

$\Psi_{\alpha}(x,\xi_{k})=(x-\xi_{k})h(x),$

where

$h(\xi_{k})=(\alpha+1)p^{\prime\prime}(\xi_{k}).$

Proof of Lemma 7.1. Since $\xi_{k}$ is a root of $p^{\prime},$ by virtue of
(7.75) we have

$(-1)^{n-k}\Psi_{\alpha}(\xi_{k},\nu)=(-1)^{n-k-1}p^{\prime\prime}(\xi_{k})(\nu-\xi_{k}).$
(7.77)

Now the first statement of Lemma 7.1 follows immediately from the first
statement of Lemma 5.2.

The following identity can also be derived from (7.75)

$(-1)^{n-k}\Psi_{\alpha}(\nu,\nu)=(-1)^{n-k}\alpha p^{\prime}(\nu).$ (7.78)

The second statement of Lemma 7.1 is a direct corollary of this identity and
the second statement of Lemma 5.2.

Using definition (7.75) of the function $\Psi_{\alpha}$ and the third
statement of Lemma 5.2 we easily obtain the third statement of Lemma 7.1.

Let us prove the fourth statement of Lemma 7.1. Since $\xi_{k}$ is a root of
the polynomial $p^{\prime},$ there exists a polynomial $g$ such that

$p^{\prime}(x)=(x-\xi_{k})g(x),$ (7.79)

and

$g(\xi_{k})=\lim_{x\to\xi_{k}}\frac{p^{\prime}(x)-p^{\prime}(\xi_{k})}{x-\xi_{k}}=p^{\prime\prime}(\xi_{k}).$
(7.80)

It follows from (7.75) with $\nu=\xi_{k}$ and (7.79) that

$\Psi_{\alpha}(x,\xi_{k})=\alpha
p^{\prime}(x)+(x-\xi_{k})p^{\prime\prime}(x)=(x-\xi_{k})(\alpha
g(x)+p^{\prime\prime}(x))=:(x-\xi_{k})h(x),$ (7.81)

where

$h(\xi_{k})=(\alpha+1)p^{\prime\prime}(\xi_{k}).$ (7.82)

Lemma 7.1 is proved. $\square$

##  8\. Estimations for $Z_{I_{k}}\left(\Psi_{\alpha}(x,\nu)\right)$

The following Lemma provides the main technical tools for proving Theorem 2.2.

######  Lemma 8.1.

The following statements are true.

  1. (1)

If $\alpha\geq 0,$ then for all real values of the parameter $\nu$ we have

     * •

$Z_{I_{k}}(\Psi_{\alpha}(x,\nu))\leq 2$ when $k=2,3,\ldots,n-1.$

     * •

$Z_{I_{1}}(\Psi_{\alpha}(x,\nu))\leq 1$ and
$Z_{I_{n}}(\Psi_{\alpha}(x,\nu))\leq 1.$

  2. (2)

If $\alpha<-(n-1),$ then for all real values of the parameter $\nu$ and all
$k=1,2,\ldots,n$ we have $Z_{I_{k}}(\Psi_{\alpha}(x,\nu))\leq 1.$

Proof of Lemma 8.1. First, consider the case of $\alpha=0.$ Using (7.75) we
obtain

$\Psi_{\alpha}(x,\nu))=(x-\nu)p^{\prime\prime}(x),$

and the statement 1 of Lemma 8.1 follows from Rolle’s theorem.

By definition (7.75) of the function $\Psi_{\alpha}(x,\nu),$ for all real
values of the parameter $\nu$ and $\alpha\neq-(n-1)$ we have

$\deg\Psi_{\alpha}(x,\nu))=n-1.$ (8.83)

Now assume that $\alpha>0.$ Using the first statement of Lemma 7.1 in the case
of a finite interval $I_{k},$ and additionally the third statement of Lemma
7.1 in the case of the infinite intervals $I_{1}$ and $I_{n},$ we conclude
that the relations below are valid

$Z_{I_{k}}(\Psi_{\alpha}(x,\nu))=1,\ \ k=2,3,\ldots,n,\ \ \mbox{whenever}\ \
\nu>\xi_{n-1},$ (8.84)

and

$Z_{I_{k}}(\Psi_{\alpha}(x,\nu))=1,\ \ k=1,2,\ldots,n-1,\ \ \mbox{whenever}\ \
\nu<\xi_{1}.$ (8.85)

If $\xi_{j-1}<\nu<\xi_{j},$ then applying the first and the second statements
of Lemma 7.1 we obtain the following relations.

$Z_{I_{k}}(\Psi_{\alpha}(x,\nu))=1,\ \ k=2,\ldots,j-1,j+1,\ldots,n-1,$ (8.86)

and

$Z_{(\xi_{j-1},\nu)}(\Psi_{\alpha}(x,\nu))=Z_{(\nu,\xi_{j})}(\Psi_{\alpha}(x,\nu))=1.$
(8.87)

Therefore,

$Z_{I_{1}}(\Psi_{\alpha}(x,\nu))=Z_{I_{n}}(\Psi_{\alpha}(x,\nu))=0.$ (8.88)

This provides the first statement of Lemma 8.1 in the case when
$\nu\neq\xi_{j},\ j=1,2,\ldots,n-1.$

To investigate the case of $\nu=\xi_{j},\ j=1,2,\ldots,n-1,$ we will use the
fourth statement of Lemma 7.1. According to this statement

$\Psi_{\alpha}(x,\xi_{j})=(x-\xi_{j})h(x),$ (8.89)

where

$h(\xi_{j})=(\alpha+1)p^{\prime\prime}(\xi_{j}).$ (8.90)

By the first statement of Lemma 5.2

$(-1)^{n-j-1}h(\xi_{j})=(-1)^{n-j-1}(\alpha+1)p^{\prime\prime}(\xi_{j})>0,\ \
j=1,2,\ldots,n-1.$ (8.91)

It follows from (8.89) and (8.91) that there exists a positive number
$\varepsilon$ such that

$(-1)^{n-j-1}\Psi_{\alpha}(\xi_{j}+\varepsilon,\xi_{j})=\varepsilon(-1)^{n-j-1}h(\xi_{j})>0,$
(8.92)

and

$(-1)^{n-j-1}\Psi_{\alpha}(\xi_{j}-\varepsilon,\xi_{j})=-\varepsilon(-1)^{n-j-1}h(\xi_{j})<0,$
(8.93)

while

$\Psi_{\alpha}(\xi_{j},\xi_{j})=0.$ (8.94)

Applying the first statement of Lemma 7.1 we obtain

$(-1)^{n-k}\Psi_{\alpha}(\xi_{k},\xi_{j})<0,\ \ \mbox{when}\ \ k>j,$ (8.95)

and

$(-1)^{n-k}\Psi_{\alpha}(\xi_{k},\xi_{j})>0,\ \ \mbox{when}\ \ k<j.$ (8.96)

In particular,

$(-1)^{n-j-1}\Psi_{\alpha}(\xi_{j+1},\xi_{j})<0,$ (8.97)

and

$(-1)^{n-j-1}\Psi_{\alpha}(\xi_{j-1},\xi_{j})>0.$ (8.98)

Based on (8.92)–(8.98) and (8.83) we conclude that

$Z_{[\xi_{j},\xi_{j}]}(\Psi_{\alpha}(x,\xi_{j}))=Z_{I_{k}}(\Psi_{\alpha}(x,\xi_{j}))=1\
\ \mbox{for any}\ \ k=2,3,\ldots,n-1,$ (8.99)

and

$Z_{I_{1}}(\Psi_{\alpha}(x,\xi_{j}))=Z_{I_{n}}(\Psi_{\alpha}(x,\xi_{j}))=0.$
(8.100)

Thus, the first statement of Lemma 8.1 is also valid for $\nu=\xi_{j},\
j=1,2,\ldots,n-1.$

Now assume that $\alpha<-(n-1).$ Let $\nu\in I_{j}=(\xi_{j-1},\xi_{j}),\
j=2,3,\ldots,n-1.$ Applying the first statement of Lemma 7.1 in the case of
finite intervals $I_{k},$ and additionally the third statement of Lemma 7.1 in
the case of infinite intervals $I_{1}$ or $I_{n},$ taking into account (8.83)
we obtain

$Z_{I_{k}}(\Psi_{\alpha}(x,\nu))=1,\ \ k\neq j,$ (8.101)

while

$Z_{I_{j}}(\Psi_{\alpha}(x,\nu))=0.$ (8.102)

Using the same idea, in the case of $\nu\geq\xi_{n-1}$ we have

$Z_{I_{k}}(\Psi_{\alpha}(x,\nu))=1,\ \ k=1,2,\ldots,n-1,$ (8.103)

while

$Z_{I_{n}}(\Psi_{\alpha}(x,\nu))=0;$ (8.104)

and in the case of $\nu\leq\xi_{1}$

$Z_{I_{k}}(\Psi_{\alpha}(x,\nu))=1,\ \ k=2,3,\ldots,n,$ (8.105)

while

$Z_{I_{1}}(\Psi_{\alpha}(x,\nu))=0.$ (8.106)

In the case of $\nu=\xi_{j},j=2,3,\ldots,n-2,$ we use again the first and
third statements of Lemma 7.1 and (8.83). We have

$Z_{[\xi_{j},\xi_{j}]}(\Psi_{\alpha}(x,\xi_{j}))=Z_{I_{k}}(\Psi_{\alpha}(x,\xi_{j}))=1,\
\ k=1,2,\ldots,j-1,j+1,\dots,n,$ (8.107)

while

$Z_{I_{j}}(\Psi_{\alpha}(x,\nu))=Z_{I_{j+1}}(\Psi_{\alpha}(x,\nu))=0.$ (8.108)

Lemma 8.1 is proved. $\square$

##  9\. Proof of Theorem 2.2 and Beyond

As we noticed in the Section 5 (see (5.47)) there is an intimate connection
between the functions $\varphi^{\prime}_{\lambda}$ and $Q_{\varkappa}[p].$ The
following lemma gives fundamental estimations for the proof of Theorem 4.4.

######  Lemma 9.1.

The following statements are true.

  1. (1)

If $\lambda>-\frac{1}{n},$ then

     * •

For any interval $I_{k}$ of the second type

$Z_{I_{k}}(\varphi_{\lambda}^{\prime}(x))=1.$ (9.109)

     * •

For any interval $I_{k}$ of the first type

$Z_{I_{k}}(\varphi_{\lambda}^{\prime}(x))=0.$ (9.110)

  2. (2)

If $\lambda\leq-1,$ then

     * •

For any finite interval of the second type $I_{k}$

$Z_{I_{k}}(\varphi_{\lambda}^{\prime}(x))=1.$ (9.111)

     * •

For any infinite interval of the first type $I_{k}$

$Z_{I_{k}}(\varphi_{\lambda}^{\prime}(x))=1.$ (9.112)

     * •

For any infinite interval of the second type $I_{k}$

$Z_{I_{k}}(\varphi_{\lambda}^{\prime}(x))=0.$ (9.113)

Proof of Lemma 9.1. By virtue of (7.71) and (7.76) the statement of Lemma 8.1
can be rewritten as follows.

######  Lemma 9.2.

The following statements are true.

  1. (1)

If $\lambda>0$ or $\lambda\leq-1,$ then for all real values of the parameter
$\mu$ we have

     * •

$Z_{I_{k}}(\Phi_{\lambda}^{\prime}(x,\mu))\leq 2$ when $k=2,3,\ldots,n-1.$

     * •

$Z_{I_{1}}(\Phi_{\lambda}^{\prime}(x,\mu))\leq 1$ and
$Z_{I_{n}}(\Phi_{\lambda}^{\prime}(x,\mu))\leq 1.$

  2. (2)

If $-\frac{1}{n}<\lambda<0,$ then for all real values of the parameter $\mu$
and all $k=1,2,\ldots,n$ we have

$Z_{I_{k}}(\Phi_{\lambda}^{\prime}(x,\mu))\leq 1.$

Let us apply Rolle’s theorem to the above statements. If $\lambda>0$ or
$\lambda\leq-1,$ then for all real values of the parameter $\mu$ we have

$Z_{I_{k}}(\Phi_{\lambda}(x,\mu))\leq 3,\ \ k=2,3,\ldots,n-1,$ (9.114)

and

$Z_{I_{1}}(\Phi_{\lambda}(x,\mu))\leq 2,\ \
Z_{I_{n}}(\Phi_{\lambda}(x,\mu))\leq 2.$ (9.115)

Assume that $-\frac{1}{n}<\lambda<0.$ Then for all real values of the
parameter $\mu$ we have

$Z_{I_{k}}(\Phi_{\lambda}(x,\mu))\leq 2,\ \ k=1,2,\ldots,n.$ (9.116)

Formulas (9.114), (9.115), (9.116) together with Lemma 5.1 and (5.53) provide
the following facts.

  1. (1)

If $\lambda>0,$ and $\mu$ is any real number, then

     * •

For any finite interval $I_{k}$ of the second type

$Z_{I_{k}}(\Phi_{\lambda}(x,\mu))=Z_{I_{k}}(\varphi_{\lambda}(x)-\mu)=0\
\mbox{or}\ 2.$ (9.117)

     * •

For any infinite interval $I_{k}$ of the first type

$Z_{I_{k}}(\Phi_{\lambda}(x,\mu))=Z_{I_{k}}(\varphi_{\lambda}(x)-\mu)=1.$
(9.118)

     * •

For any infinite interval $I_{k}$ of the second type

$Z_{I_{k}}(\Phi_{\lambda}(x,\mu))=Z_{I_{k}}(\varphi_{\lambda}(x)-\mu)=0\
\mbox{or}\ 2.$ (9.119)

  2. (2)

If $-\frac{1}{n}<\lambda<0,$ and $\mu$ is any real number, then

     * •

For any interval $I_{k},\ \ k=1,\ldots,n,$ of the first type

$Z_{I_{k}}(\Phi_{\lambda}(x,\mu))=Z_{I_{k}}(\varphi_{\lambda}(x)-\mu)=1.$
(9.120)

     * •

For any interval $I_{k},\ \ k=1,\ldots,n,$ of the second type

$Z_{I_{k}}(\Phi_{\lambda}(x,\mu))=Z_{I_{k}}(\varphi_{\lambda}(x)-\mu)=0\
\mbox{or}\ 2.$ (9.121)

  3. (3)

If $\lambda\leq-1,$ and $\mu$ is any real number, then

     * •

For any finite interval $I_{k}$ of the second type

$Z_{I_{k}}(\Phi_{\lambda}(x,\mu))=Z_{I_{k}}(\varphi_{\lambda}(x)-\mu)=0\
\mbox{or}\ 2.$ (9.122)

     * •

For any infinite interval $I_{k}$ of the first type

$Z_{I_{k}}(\Phi_{\lambda}(x,\mu))=Z_{I_{k}}(\varphi_{\lambda}(x)-\mu)=0\
\mbox{or}\ 2.$ (9.123)

     * •

For any infinite interval $I_{k}$ of the second type

$Z_{I_{k}}(\Phi_{\lambda}(x,\mu))=Z_{I_{k}}(\varphi_{\lambda}(x)-\mu)=1.$
(9.124)

Taking into account (5.49), (5.50), (5.51), and applying the second statement
of Lemma 6.1 in the case of $\lambda=0,$ we come to the conclusion.

  1. (1)

If $\lambda\geq 0,$

     * •

For any finite interval $I_{k}$ of the second type

$Z_{I_{k}}(\varphi_{\lambda}^{\prime}(x))=1.$ (9.125)

     * •

For any infinite interval $I_{k}$ of the first type

$Z_{I_{k}}(\varphi_{\lambda}^{\prime}(x))=0.$ (9.126)

     * •

For any infinite interval $I_{k}$ of the second type

$Z_{I_{k}}(\varphi_{\lambda}^{\prime}(x))=1.$ (9.127)

  2. (2)

If $-\frac{1}{n}<\lambda<0,$

     * •

For any interval $I_{k},\ \ k=1,\ldots,n,$ of the first type

$Z_{I_{k}}(\varphi_{\lambda}^{\prime}(x))=0.$ (9.128)

     * •

For any interval $I_{k},\ \ k=1,\ldots,n,$ of the second type

$Z_{I_{k}}(\varphi_{\lambda}^{\prime}(x))=1.$ (9.129)

  3. (3)

If $\lambda\leq-1,$

     * •

For any finite interval $I_{k}$ of the second type

$Z_{I_{k}}(\varphi_{\lambda}^{\prime}(x))=1.$ (9.130)

     * •

For any infinite interval $I_{k}$ of the first type

$Z_{I_{k}}(\varphi_{\lambda}^{\prime}(x))=1.$ (9.131)

     * •

For any infinite interval $I_{k}$ of the second type

$Z_{I_{k}}(\varphi_{\lambda}^{\prime}(x))=0.$ (9.132)

In order to complete the proof of Lemma 9.1 we only have to check (9.110) in
the case of $\lambda>0.$ It follows from the first statement of Lemma 6.1 that
if $\lambda>0,$ the function

$\varphi_{\lambda}(x))=\frac{p(x)}{p^{\prime}(x)}+\lambda x$

is increasing on any interval $I_{k},\ k=1,2,\ldots,n,$ of the first type.
Hence, on any interval $I_{k}$ of the first type

$Z_{I_{k}}(\varphi_{\lambda}^{\prime}(x))=0.$ (9.133)

Lemma 9.1 is proved. $\square$

Let us prove Theorem 2.2. We have $\varkappa>\frac{n-1}{n}.$ According to
(5.47)

$Z_{I_{k}}(\varphi_{\lambda}^{\prime}(x))=Z_{I_{k}}\left(Q_{\varkappa}[p](x)\right),$
(9.134)

where by (5.45)

$\lambda=\varkappa-1>-\frac{1}{n}.$

Now we can rewrite the first statement of Lemma 9.1 as follows. Let
$\varkappa>\frac{n-1}{n}$. Then

$Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=1,\quad\mbox{\it if}\ I_{k}\ \mbox{\it
is any interval of the second type}.$ (9.135)
$Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=0\quad\mbox{\it if}\ I_{k}\ \mbox{\it
is any interval of the first type}.$ (9.136)

Thus, $Z_{\mathbb{R}}\left(Q_{\varkappa}[p]\right)$ equals to the number of
all intervals of the second type, that is

$Z_{\mathbb{R}}\left(H_{\varkappa}[p](x)\right)=Z_{\mathbb{R}}\left(Q_{\varkappa}[p](x)\right)=Z_{\mathbb{R}}(\varphi_{\lambda}^{\prime}(x))=n-Z_{\mathbb{R}}(p)=Z_{\mathbb{C}}(p).$

Theorem 2.2 is proved. $\square$

Lemma 9.1 allows to make important conclusions in the case of $\varkappa\leq
0.$ If $\varkappa\leq 0,$ then $\lambda=\varkappa-1\leq-1.$ The following
lemma follows from (9.130), (9.131) and (9.132).

######  Lemma 9.3.

If $\varkappa\leq 0,$ then

  * •

For any finite interval $I_{k}$ of the second type

$Z_{I_{k}}\left(H_{\varkappa}[p](x)\right)=Z_{I_{k}}\left(Q_{\varkappa}[p](x)\right)=1.$
(9.137)

  * •

For any infinite interval $I_{k}$ of the first type

$Z_{I_{k}}\left(H_{\varkappa}[p](x)\right)=Z_{I_{k}}\left(Q_{\varkappa}[p](x)\right)=1.$
(9.138)

  * •

For any infinite interval $I_{k}$ of the second type

$Z_{I_{k}}\left(H_{\varkappa}[p](x)\right)=Z_{I_{k}}\left(Q_{\varkappa}[p](x)\right)=0.$
(9.139)

######  Remark 9.4.

Note that in order to prove Theorem 2.3 we just have to show that for any
finite interval $I_{k}$ of the first type

$Z_{I_{k}}\left(H_{\varkappa}[p]\right)=Z_{I_{k}}(\varkappa(p^{\prime})^{2}-p^{\prime\prime}p)=2.$
(9.140)

Indeed, if we use the standard notation $\\#S$ for the number of elements in
any finite set $S,$ we obtain

$Z_{\mathbb{R}}\left(H_{\varkappa}[p](x)\right)=Z_{\mathbb{R}}\left(Q_{\varkappa}[p](x)\right)=$
$\\#\\{I_{k}|\ I_{k}\ \mbox{is a finite interval of the second type}\\}+$
$+2\cdot\\#\\{I_{k}|\ I_{k}\ \mbox{is a finite interval of the first
type}\\}+$ $\\#\\{I_{k}|\ I_{k}\ \mbox{is an infinite interval of the first
type}\\}$ $=\\#\\{I_{k}|\ I_{k}\ \mbox{is a finite interval}\\}+\\#\\{I_{k}|\
I_{k}\ \mbox{is an interval of the first type}\\}=$ $n-2+Z_{\mathbb{R}}(p).$

##  10\.  Relations Between Zeros of $H_{\varkappa}[p](x)$ and Zeros of
$H_{2-\frac{1}{\varkappa}}[p^{\prime}](x).$ Proof of Theorem 2.3

Using definition (1.2) of the function $H_{\varkappa}[p]$ we obtain the
following formula for the derivative $H_{\varkappa}^{\prime}[p]$

$H_{\varkappa}^{\prime}[p]=(2\varkappa-1)p^{\prime}p^{\prime\prime}-p^{\prime\prime\prime}p,$
(10.141)

and if $\varkappa\neq 0,$ then

$H_{2-\frac{1}{\varkappa}}[p^{\prime}]=\left(2-\frac{1}{\varkappa}\right)(p^{\prime\prime})^{2}-p^{\prime\prime\prime}p^{\prime}.$
(10.142)

The lemma below can be easily verified.

######  Lemma 10.1.

For $\varkappa\neq 0,$

$H_{\varkappa}^{\prime}[p]\cdot
p^{\prime}-H_{2-\frac{1}{\varkappa}}[p^{\prime}]\cdot
p=\frac{2\varkappa-1}{\varkappa}H_{\varkappa}[p]\cdot p^{\prime\prime},$
(10.143) $H_{\varkappa}^{\prime}[p]\cdot p^{\prime\prime}-\varkappa
H_{2-\frac{1}{\varkappa}}[p^{\prime}]\cdot p^{\prime}=H_{\varkappa}[p]\cdot
p^{\prime\prime\prime}.$ (10.144)

The statement below can be considered as an analogue of Rolle’s Theorem, which
provides the relation between zeros of $H_{\varkappa}[p]$ and zeros of
$H_{\left(2-\frac{1}{\varkappa}\right)}[p^{\prime}].$

######  Lemma 10.2.

Assume that $\varkappa\neq 0,$ and

  1. (1)

$H_{\varkappa}[p](x)\neq 0$ for all $x\in(a,b).$

  2. (2)

$H_{\varkappa}[p](a)=H_{\varkappa}[p](b)=0.$

Then

  1. (1)

Let $p^{\prime}(x)\neq 0$ and $p(x)\neq 0$ for all $x\in[a,b].$ There exists
$y\in(a,b)$ such that

$H_{2-\frac{1}{\varkappa}}[p^{\prime}](y)=0.$

  2. (2)

Let $p^{\prime}(x)\neq 0$ and $p^{\prime\prime}(x)\neq 0$ for all $x\in[a,b].$
There exists $y\in(a,b)$ such that

$H_{2-\frac{1}{\varkappa}}[p^{\prime}](y)=0.$

Proof of Lemma 10.2 Let us prove the first statement of Lemma 10.2. Assume
that $p(x)\neq 0$ and $p^{\prime}(x)\neq 0$ for all $x\in(a,b).$ Let $a$ be a
root of multiplicity $m$ and $b$ be a root of multiplicity $k$ of the function
$H_{\varkappa}[p].$ It means that there are polynomials $q_{1}$ and $q_{2}$ as
well as constants $c\neq 0$ and $d\neq 0$ such that

$H_{\varkappa}[p](x)=(x-a)^{m}(c+(x-a)q_{1}(x)),\quad\mbox{and}$
$H_{\varkappa}[p](x)=(x-b)^{k}(d+(x-b)q_{2}(x)).$ (10.145)

Because $H_{\varkappa}[p](x)\neq 0$ in $(a,b),$ we have

$(-1)^{k}c\cdot d>0.$ (10.146)

Since $p(x)\neq 0$ and $p^{\prime}(x)\neq 0$ it follows from (10.143),
(10.144) that $a$ is a root of multiplicity at least $m-1$ and $b$ is a root
of multiplicity at least $k-1$ of the function
$H_{\left(2-\frac{1}{\varkappa}\right)}[p^{\prime}].$ So, there are
polynomials $T_{1}$ and $T_{2}$ such that

$H_{2-\frac{1}{\varkappa}}[p^{\prime}](x)=(x-a)^{m-1}T_{1}(x)\quad\mbox{and}\quad
H_{2-\frac{1}{\varkappa}}[p^{\prime}](x)=(x-b)^{k-1}T_{2}(x).$ (10.147)

Using (10.143), (10.145) and (10.147) we obtain

$T_{1}(a)=\frac{p^{\prime}(a)}{p(a)}mc,$ (10.148)
$T_{2}(b)=\frac{p^{\prime}(b)}{p(b)}kd.$ (10.149)

Because $p^{\prime}(x)\neq 0$ and $p(x)\neq 0$ on the interval $[a,b],$ the
following is true

$\frac{p^{\prime}(a)}{p(a)}\cdot\frac{p^{\prime}(b)}{p(b)}>0.$ (10.150)

Therefore, by (10.146)

$(-1)^{k}T_{1}(a)T_{2}(b)>0.$

Since the functions $T_{1}(x)$ and $T_{2}(x)$ are continuous, there exists
$\varepsilon>0$ such that

$(-1)^{k}T_{1}(a+\varepsilon)T_{2}(b-\varepsilon)>0,$ (10.151)

and by (10.147)

$H_{2-\frac{1}{\varkappa}}[p^{\prime}](a+\varepsilon)\cdot
H_{2-\frac{1}{\varkappa}}[p^{\prime}](b-\varepsilon)=(\varepsilon)^{m+k-2}(-1)^{k-1}T_{1}(a+\varepsilon)T_{2}(b-\varepsilon)<0.$
(10.152)

We obtain the first statement of Lemma 10.2.

The second statement of Lemma 10.2 can be derived from (10.144) in the similar
way. Lemma 10.2 is proved. $\square$

The following estimation for the number
$Z_{(a,b)}\left(H_{\varkappa}[p]\right)$ through
$Z_{(a,b)}\left(H_{2-\frac{1}{\varkappa}}[p^{\prime}]\right)$ is derived from
Lemma 10.2 in the same way as similar facts about relations between the number
of zeros of a function $f$ and the number of zeros of its derivative
$f^{\prime}$ are usually obtained from the original Rolle’s theorem.

######  Corollary 10.3.

Assume that $\varkappa\neq 0.$ Let either $p^{\prime}(x)\neq 0$ and $p(x)\neq
0$ for all $x\in[a,b],$ or $p^{\prime\prime}(x)\neq 0$ and $p^{\prime}(x)\neq
0$ for all $x\in[a,b].$ Then

$Z_{(a,b)}\left(H_{\varkappa}[p]\right)\ \leq\
Z_{(a,b)}\left(H_{2-\frac{1}{\varkappa}}[p^{\prime}]\right)+1.$ (10.153)

Denote by $\gamma_{1}<\gamma_{2}<\ldots<\gamma_{n-2}$ the roots of
$p^{\prime\prime}.$ Since the roots $\xi_{1}<\xi_{2}<\ldots<\xi_{n-1}$ of the
polynomial $p^{\prime}$ are real and simple, we have

$-\infty<\xi_{1}<\gamma_{1}<\xi_{2}<\gamma_{2}<\xi_{3}<\ldots<\gamma_{n-2}<\xi_{n-1}<\infty.$
(10.154)

We will consider the intervals

$J_{2k-1}=(\xi_{k},\gamma_{k}),\ J_{2k}=(\gamma_{k},\xi_{k+1}),\ \
k=1,2,\ldots,n-2;$ $J_{0}=I_{1}=(-\infty,\xi_{1}),\ \
J_{2n-3}=(\xi_{n-1},\infty).$ (10.155)

######  Remark 10.4.

Both $p^{\prime}$ and $p^{\prime\prime}$ do not vanish on each of the
intervals $J_{k},\ k=0,1,\ldots,2n-3.$

The following fact about the number of zeros of the function
$H_{\varkappa}[p]$ on the intervals $J_{k}$ is true.

######  Lemma 10.5.

Let $\varkappa<0.$ Then

$Z_{J_{m}}(H_{\varkappa}[p])\leq 1$ (10.156)

for any $m=0,1,\ldots,2n-3.$

Proof of Lemma 10.5 Given $\varkappa<0,$ we have $2-\frac{1}{\varkappa}>2.$
Since all roots of $p^{\prime}$ are real and simple, all roots of its
derivative $p^{\prime\prime}$ are also real and simple. Therefore, we can
apply Theorem 2.2 to the polynomial $p^{\prime}.$ We have

$Z_{\mathbb{R}}\left(H_{2-\frac{1}{\varkappa}}[p^{\prime}]\right)=Z_{\mathbb{C}}(p^{\prime})=0.$
(10.157)

Let us fix any interval $J_{m},\ m=0,1,\ldots,2n-3.$ Because by Remark 10.4
both $p^{\prime}$ and $p^{\prime\prime}$ do not vanish on $J_{m},$ we can
apply Corollary 10.3 and obtain the statement of Lemma10.5. Lemma10.5 is
proved. $\square$

The following lemma is the key to the proof of Theorem 2.3.

######  Lemma 10.6.

Let $\varkappa\leq 0.$ Then

$Z_{I_{k}}(H_{\varkappa}[p])\leq 2$ (10.158)

for any $k=2,3,\ldots,n-1,$ and

$Z_{I_{1}}(H_{\varkappa}[p])\leq 1,\ \ Z_{I_{n}}(H_{\varkappa}[p])\leq 1.$
(10.159)

Proof of Lemma 10.6. By (10.154) and (10.155)

$I_{k}=(\xi_{k-1},\xi_{k})=(\xi_{k-1},\gamma_{k-1})\cup\\{\gamma_{k-1}\\}\cup(\gamma_{k-1},\xi_{k})=J_{2k-3}\cup\\{\gamma_{k-1}\\}\cup
J_{2k-2}.$

Note that

$H_{\varkappa}[p](\gamma_{k})\neq 0,\ \ k=1,2,\ldots,n-2.$ (10.160)

Indeed, suppose that $H_{\varkappa}[p](\gamma_{k})=0.$ Since $\gamma_{k}$ is a
root of $p^{\prime\prime}$, it follows from the definition (1.2) of the
function $H_{\varkappa}[p]$ that in this case $p^{\prime}(\gamma_{k})=0$. This
fact contradicts to the assumption of Theorem 2.3 that $p^{\prime}$ doesn’t
have multiple roots.

If $\varkappa<0$ the statement of lemma follows immediately from (10.156) and
(10.160). Note that for $\varkappa=0$ we have

$H_{\varkappa}[p](x)=-p(x)p^{\prime\prime}(x).$

Since all roots of $p^{\prime}$ are real and simple, there is exactly one root
of $p^{\prime\prime}$ inside each finite interval $I_{k}.$ If $I_{k}$ is an
interval of the first type, it also contains one root of $p,$ while if it is
of the second type it doesn’t have roots of $p.$ So,

$Z_{I_{k}}(H_{\varkappa}[p])=2,\quad\mbox{\it if}\ I_{k}\ \mbox{\it is a
finite interval of the first type,}$ (10.161)

and

$Z_{I_{k}}(H_{\varkappa}[p])=1,\quad\mbox{\it if}\ I_{k}\ \mbox{\it is a
finite interval of the second type.}$ (10.162)

If $I_{k}$ is an infinite interval, there is no roots of $p^{\prime\prime}$ in
$I_{k}.$ Thus,

$Z_{I_{k}}(H_{\varkappa}[p])=1,\ \ \mbox{\it if}\ I_{k}\ \mbox{\it is an
infinite interval of the first type,}$ (10.163)

and

$Z_{I_{k}}(H_{\varkappa}[p])=0,\ \ \mbox{\it if}\ I_{k}\ \mbox{\it is an
infinite interval of the second type.}$ (10.164)

Lemma10.6 is proved. $\square$

Since $p$ doesn’t have real multiple roots, and all roots of $p^{\prime}$ are
real and simple,

$H_{\varkappa}[p](x)=0\ \Leftrightarrow\ Q_{\varkappa}[p](x)=0.$ (10.165)

So, for each $k=1,2,\ldots,n$ and any real $\varkappa$ we have

$Z_{I_{k}}\left(H_{\varkappa}[p]\right)=Z_{I_{k}}\left(Q_{\varkappa}[p]\right).$
(10.166)

Now for each $\varkappa\leq 0$ formula (9.140) follows from Lemma 10.6 and
Lemma 4.8, that is

$Z_{I_{k}}(H_{\varkappa}[p])=2,\quad\mbox{\it if}\ I_{k}\ \mbox{\it is a
finite interval of the first type}.$ (10.167)

By virtue of Remark 9.4 this completes the proof of Theorem 2.3.

##  11\.  Some properties of $M[p]$

Assume that $I_{k}$ is a finite interval of the first type. According to
(10.167) for each $\varkappa\leq 0$ there are exactly two roots of
$Q_{\varkappa}[p](x)$ inside $I_{k}.$ The statement below gives an idea about
the zero distribution of the function $Q_{\varkappa}[p](x)$ inside a finite
interval $I_{k}$ of the first type for $\varkappa<0.$

######  Lemma 11.1.

Let $\varkappa<0.$ Suppose that $I_{k}=(\xi_{k-1},\xi_{k})$ is a finite
interval of the first type, and $\alpha\in I_{k}$ is the only zero of $p$
inside the interval $I_{k}.$ Then

$Z_{(\xi_{k-1},\alpha)}\left(Q_{\varkappa}[p]\right)=Z_{(\alpha,\xi_{k})}\left(Q_{\varkappa}[p]\right)=1.$
(11.168)

Proof of Lemma 11.1 Note that by the definition (4.25) of the function $M[p]$
we have

$M[p](\alpha)=0,$

and by (4.26) for any $\varkappa<0$ we have

$Q_{\varkappa}[p](\alpha)=\varkappa\ -M[p](\alpha)<0.$ (11.169)

Using (4.32) and (4.33) we obtain

$\lim_{x\to\xi_{k-1}^{+}}Q_{\varkappa}[p](x)=\varkappa\
-\lim_{x\to\xi_{k-1}^{+}}M[p])(x)>0,$ (11.170)

and

$\lim_{x\to\xi_{k}^{-}}Q_{\varkappa}[p](x)=\varkappa-\lim_{x\to\xi_{k}^{-}}M[p](x)>0.$
(11.171)

It follows from (11.170), (11.171) and (11.169) that both numbers
$Z_{(\xi_{k-1},\alpha)}\left(Q_{\varkappa}[p]\right)$ and
$Z_{(\alpha,\xi_{k})}\left(Q_{\varkappa}[p]\right)$ are odd. By virtue of
(10.167) it is possible only if the statement of Lemma 11.1 is true. $\square$

######  Remark 11.2.

Let $I_{k}=(\xi_{k-1},\xi_{k})$ be a finite interval of the first type, and
$\gamma_{k-1}\in I_{k}$ be the only zero of $p^{\prime\prime}$ inside the
interval $I_{k}.$ In the same way as above, using (11.170), (11.171),
(10.167), (10.156), and (10.160) we can prove that

$Z_{(\xi_{k-1},\gamma)}\left(Q_{\varkappa}[p]\right)=Z_{(\gamma,\
\xi_{k})}\left(Q_{\varkappa}[p]\right)=1.$ (11.172)

######  Remark 11.3.

The relations (11.168) and (11.172) give an idea how the graph of the function
$M[p](x)$ should look on finite intervals of the first type. It follows from
(4.25) that the function $M[p]$ has exactly two roots in the interval $I_{k}$:
a root $\alpha$ of $p$ and a root $\gamma$ of $p^{\prime\prime}.$ Let us
denote these roots of $M[p]$ by $z_{1}$ and $z_{2},$ where $z_{1}\leq z_{2}.$
By virtue of (4.26) the relations (11.168) and (11.172) mean that for each
$\varkappa\leq 0$ the equation (4.27)

$M[p](x)=\varkappa$

has the unique solution in the interval $(\xi_{k-1},z_{1})$ and the unique
solution in the interval $(z_{2},\xi_{k}).$ Taking into account (4.32) and
(4.33) we can conclude that

$M[p](x)<0\quad\mbox{\it and is increasing on the interval}\
(\xi_{k-1},z_{1}).$ (11.173) $M[p](x)<0\quad\mbox{\it and is decreasing on the
interval}\ (z_{2},\xi_{k}).$ (11.174)

If additionally we use (9.136), we can state that

$M[p](x)>0,\quad x\in(z_{1},z_{2}),$

and

$\max_{x\in[z_{1},z_{2}]}M[p](x)<\frac{n-1}{n}.$

##  12\.  The case of $0<\varkappa<\frac{1}{2}$

In this section we will prove the following statement. Theorem 2.4 is a direct
corollary from this result.

######  Theorem 12.1.

Let $0\leq\varkappa<\frac{1}{2}.$ Then

  1. (1)

$Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=Z_{I_{k}}\left(H_{\varkappa}[p]\right)=1,$
if $I_{k}$ is a finite interval of the second type.

  2. (2)

$Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=Z_{I_{k}}\left(H_{\varkappa}[p]\right)=0\
\mbox{or}\ 2,$ if $I_{k}$ is a finite interval of the first type.

  3. (3)

$Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=Z_{I_{k}}\left(H_{\varkappa}[p]\right)=0,$
if $I_{k}$ is an infinite interval of the second type.

  4. (4)

$Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=Z_{I_{k}}\left(H_{\varkappa}[p]\right)=1,$
if $I_{k}$ is an infinite interval of the first type.

Proof of Theorem 12.1 If $\varkappa=0,$ then $Q_{\varkappa}[p]=M[p]$ and by
definition (4.25) of the function $M[p]$ all statements of theorem are
obvious.

Now assume that $\varkappa\neq 0$. Let $I_{k}$ be a finite interval of the
first type. Note that one of the numbers $z_{1}$ and $z_{2},$ described in
Remark 11.174, is a root $\gamma_{k-1}$ of the polynomial $p^{\prime\prime}.$
Hence,

$(z_{1},z_{2})\subset(\xi_{k-1},\gamma_{k-1})=J_{2k-3}\quad\mbox{or}\quad(z_{1},z_{2})\subset(\gamma_{k-1},\xi_{k})=J_{2k-2}.$
(12.175)

It follows from Remark 11.174 that the equation (4.27) doesn’t have solutions
in the intervals $(\xi_{k-1},z_{1})$ and $(z_{2},\xi_{k}).$ Let us choose the
interval from $J_{2k-3}$ and $J_{2k-2}$ that contains $(z_{1},z_{2}),$ and
denote it by $J.$ We have

$Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=Z_{(z_{1},z_{2})}\left(Q_{\varkappa}[p]\right)=Z_{J}\left(Q_{\varkappa}[p]\right),$
(12.176)

if $I_{k}$ is a finite interval of the first type.

If $I_{k}=(\xi_{k-1},\xi_{k})$ is a finite interval of the second type, it
contains only one root of the function $M[p]$: the root $\gamma_{k-1}$ of the
polynomial $p^{\prime\prime}.$ Let us choose the interval from
$(\xi_{k-1},\gamma_{k-1})$ and $(\gamma_{k-1},\xi_{k}),$ where $M[p](x)>0.$
Now denote this interval by $J.$ If $0<\varkappa<\frac{1}{2},$ and the
equation $M[p](x)=\varkappa$ has a solution in $I_{k},$ it must belong to $J.$
So,

$Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=Z_{J}\left(Q_{\varkappa}[p]\right),$
(12.177)

if $I_{k}$ is a finite interval of the second type.

If $0<\varkappa<\frac{1}{2},$ then $-\infty<2-\frac{1}{\varkappa}<0.$ Since
all roots $(\xi_{k})_{k=1}^{n-1}$ of $p^{\prime}$ are real and simple, all
roots $(\gamma_{k})_{k=1}^{n-2}$ of $p^{\prime\prime}$ are also real and
simple. So, $p^{\prime}$ satisfies the hypotheses of Theorem 2.3. The
intervals

$(-\infty,\gamma_{1}),(\gamma_{1},\gamma_{2}),\ldots,(\gamma_{n-3},\gamma_{n-2}),(\gamma_{n-2},\infty)$

are intervals (4.30) for $p^{\prime}.$ All of them are of the first type.
Because we denote by $\xi$ with indices the roots of $p^{\prime}$ and
$\gamma_{k-1}$ is the root of $p^{\prime\prime},$ the interval $J$ in both
(12.176) and (12.177) plays exactly the same role for $p^{\prime}$ as the
intervals $(\xi_{k-1},\alpha)$ and $(\alpha,\xi_{k})$ for $p$ in Lemma 11.1.
Applying Lemma 11.1 to $p^{\prime}$ on the interval $J$ we obtain

$Z_{J}\left(Q_{2-\frac{1}{\varkappa}}[p^{\prime}]\right)=Z_{J}\left(H_{2-\frac{1}{\varkappa}}[p^{\prime}]\right)=1.$
(12.178)

Note that $p^{\prime}(x)\neq 0$ and $p^{\prime\prime}(x)\neq 0$ for all $x\in
J.$ So, we can apply Corollary 10.3 from Lemma 10.2 and obtain

$Z_{J}\left(H_{\varkappa}[p]\right)=Z_{J}\left(Q_{\varkappa}[p]\right)\leq 2.$
(12.179)

Therefore, by (12.176) and (12.177) for any finite interval $I_{k}$ we have

$Z_{I_{k}}\left(H_{\varkappa}[p]\right)=Z_{I_{k}}\left(Q_{\varkappa}[p]\right)\leq
2.$ (12.180)

According to Lemma 4.5 the number $Z_{I_{k}}\left(Q_{\varkappa}[p]\right)$ is
even if $I_{k}$ is a finite interval of the first type, and it is odd if
$I_{k}$ is a finite interval of the second type. We obtain first two
statements of Theorem 12.1.

Now consider the infinite intervals $I_{1}=(-\infty,\xi_{1})$ and
$I_{n}=(\xi_{n-1},\infty).$ First, assume that an infinite interval $I$ is an
interval of the first type. Because all root of $p^{\prime}$ are real and
simple,

$I_{1}\subset(-\infty,\gamma_{1}),\quad\mbox{and}\quad
I_{n-1}\subset(\gamma_{n-2},\infty).$

Since $-\infty<2-\frac{1}{\varkappa}<0$, using the above remark by (10.159) we
obtain

$Z_{I}\left(H_{2-\frac{1}{\varkappa}}[p^{\prime}]\right)\leq 1.$

The infinite intervals $I$ contain neither roots of $p^{\prime}$ nor roots of
$p^{\prime\prime}.$ We apply again Corollary 10.3 from Lemma 10.2 and obtain

$Z_{I}\left(H_{\varkappa}[p]\right)=Z_{I}\left(Q_{\varkappa}[p]\right)\leq 2.$

Since $\varkappa<\frac{1}{2},$ by virtue of Lemma 4.6 if $I$ is an infinite
interval of the first type, the number $Z_{I}\left(Q_{\varkappa}[p]\right)$ is
odd. We conclude that

$Z_{I}\left(H_{\varkappa}[p]\right)=Z_{I}\left(Q_{\varkappa}[p]\right)=1.$
(12.181)

Let $I$ be an infinite interval of the second type. According to (4.25)

$M[p^{\prime}](x)=\frac{p^{\prime\prime\prime}(x)p^{\prime}(x)}{(p^{\prime\prime}(x))^{2}}.$
(12.182)

Because all roots of $p^{\prime}$ are real and simple, it is clear that
$\xi_{n-1}$ is the only root of $M[p^{\prime}]$ on $[\xi_{n-1},+\infty)$, and
$\xi_{1}$ is its only root on the interval $(-\infty,\xi_{1}].$ Since

$\lim_{x\to\pm\infty}M[p^{\prime}](x)=\frac{n-2}{n-1}>0,$

we see that

$M[p^{\prime}](x)>0,\quad\mbox{when}\
x\in(-\infty,\xi_{1})\cup(\xi_{n-1},+\infty).$ (12.183)

Since $2-\frac{1}{\varkappa}<0,$ the equation

$M[p^{\prime}](x)=2-\frac{1}{\varkappa}$

doesn’t have solutions on infinite intervals of the second type. Thus, if $I$
is an infinite interval of the second type, then

$Z_{I}\left(Q_{2-\frac{1}{\varkappa}}[p^{\prime}]\right)=Z_{I}\left(H_{2-\frac{1}{\varkappa}}(p^{\prime})\right)=0.$
(12.184)

Let us apply again Corollary 10.3 from Lemma 10.2. We have

$Z_{I}\left(Q_{\varkappa}[p^{\prime}]\right)=Z_{I}\left(H_{\varkappa}(p^{\prime})\right)\leq
1.$ (12.185)

According to Lemma 4.6 if $I$ is an infinite interval of the second type, the
number $Z_{I}\left(Q_{\varkappa}[p]\right)$ is even. Therefore,

$Z_{I}\left(H_{\varkappa}[p]\right)=Z_{I}\left(Q_{\varkappa}[p]\right)=0.$
(12.186)

Theorem 12.1 is proved. $\square$

##  13\.  Some remarks about the intervals $I_{k}$ of the first type

Because all roots of $p^{\prime}$ are real and simple, the case of intervals
of the first type is of the great interest for us. The following more general
result can be obtained in the same way as the second statement of Theorem
12.1.

######  Theorem 13.1.

Let $0\leq\varkappa<\frac{n-1}{n}.$ Then for any finite interval of the first
type $I_{k}$ the following is true

$Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=Z_{I_{k}}\left(H_{\varkappa}[p]\right)=0\
\mbox{or}\ 2.$ (13.187)

This statement is equivalent to the following fact.

######  Lemma 13.2.

For every $j=1,2,\ldots,n-1,$ and

$\frac{j-1}{j}\leq\varkappa<\frac{j}{j+1},$

on any finite interval $I_{k}$ of the first type the following estimation
holds

$Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=Z_{I_{k}}\left(H_{\varkappa}[p]\right)=0\
\mbox{or}\ 2.$ (13.188)

Proof of Lemma 13.2. In order to prove Lemma 13.2 we use induction on
$j=1,2,\ldots,n-1.$

The second statement of Theorem 12.1 provides the base of induction for $j=1.$

$\mbox{If}\quad\frac{j-1}{j}\leq\varkappa<\frac{j}{j+1},\quad\mbox{then}\quad\frac{j-2}{j-1}\leq
2-\frac{1}{\varkappa}<\frac{j-1}{j}.$

Since all roots $(\xi_{k})_{k=1}^{n-1}$ of $p^{\prime}$ are real and simple,
all roots $(\gamma_{k})_{k=1}^{n-2}$ of $p^{\prime\prime}$ are also real and
simple, and all the intervals $(\gamma_{k-1},\gamma_{k}),\ k=2,3\ldots,n-2,$
are intervals of the first type for the polynomial $p^{\prime}.$ Thus, by the
induction hypothesis

$Z_{(\gamma_{k-1},\gamma_{k})}\left(H_{2-\frac{1}{\varkappa}}[p^{\prime}]\right)=0\
\mbox{or}\ 2.$ (13.189)

Let us fix an interval $I_{k}=(\xi_{k-1},\xi_{k})$ of the first type. Denote
by $\alpha$ the only root of the polynomial $p$ inside $I_{k}.$ Note that
$\gamma_{k-1}$ is the only root of $p^{\prime\prime}$ inside $I_{k}.$ Without
loss of generality we assume that

$\xi_{k-1}<\gamma_{k-1}<\alpha<\xi_{k}$

(the case when $\xi_{k-1}<\alpha<\gamma_{k-1}<\xi_{k}$ can be considered in
the same way). Obviously, we have the following inclusion

$(\gamma_{k-1},\alpha)\ \subset\ (\gamma_{k-1},\xi_{k})\
\subset(\gamma_{k-1},\gamma_{k}).$ (13.190)

It follows from(13.189) that

$Z_{(\gamma_{k-1},\xi_{k})}\left(H_{2-\frac{1}{\varkappa}}[p^{\prime}]\right)\leq
2.$ (13.191)

Let us apply Corollary 10.3 from Lemma 10.2. We obtain

$Z_{(\gamma_{k-1},\xi_{k})}\left(H_{\varkappa}[p]\right)\leq 3.$ (13.192)

As we showed in Remark 11.174

$Z_{(\xi_{k-1},\xi_{k})}\left(H_{\varkappa}[p]\right)=Z_{(\gamma_{k-1},\alpha)}\left(H_{\varkappa}[p]\right)=Z_{(\gamma_{k-1},\xi_{k})}\left(H_{\varkappa}[p]\right).$
(13.193)

Therefore,

$Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=Z_{I_{k}}\left(H_{\varkappa}[p]\right)\leq
3.$ (13.194)

According to Lemma 4.5, the number $Z_{I_{k}}\left(Q_{\varkappa}[p]\right)$ is
even if $I_{k}$ is a finite interval of the first type. Thus, for
$\frac{j-1}{j}\leq\varkappa<\frac{j}{j+1}$ we obtain

$Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=Z_{I_{k}}\left(H_{\varkappa}[p]\right)\
=\ 0\ \mbox{or}\ 2.$ (13.195)

We obtain the conclusion of Lemma 13.2. $\square$

######  Remark 13.3.

Lemma 13.2 allows to refine Remark 11.174 about the behavior of the function
$M[p]$ on a finite interval of the first type $I_{k}=(\xi_{k-1},\xi_{k})$:
there exists $x_{max}\in I_{k}$ such that

$M[p](x)\quad\mbox{\it is increasing on}\ (\xi_{k-1},x_{max}],$ (13.196)
$M[p](x)\quad\mbox{\it is decreasing on}\ [x_{max},\xi_{k}),$ (13.197)

and

$M[p]\left(x_{max}\right)<\frac{n-1}{n}.$ (13.198)

The following statement gives the estimation for the number of real zeros of
the function $H_{\varkappa}[p]$ on infinite intervals $I_{k}$ of the first
type.

######  Theorem 13.4.

If $0\leq\varkappa<\frac{n-1}{n},$ then for any infinite interval $I_{k}$ of
the first type

$Z_{I_{k}}(H_{\varkappa}[p])=Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=1.$
(13.199)

Theorem 13.4 follows from the lemma below.

######  Lemma 13.5.

For each $j=1,2,\ldots,n-1,$ and for every

$\frac{j-1}{j}\leq\varkappa<\frac{j}{j+1},\ \ j=1,2,\ldots,n-1,$

on any infinite interval of the first type $I_{k}$ the following estimation is
true

$Z_{I_{k}}\left(H_{\varkappa}[p]\right)=Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=1.$
(13.200)

Proof of Lemma 13.5. We will use induction on $j.$ The fourth statement of
Theorem 12.1 gives the base of induction for $j=1.$

$\mbox{If}\quad\frac{j-1}{j}\leq\varkappa<\frac{j}{j+1},\quad\mbox{then}\quad\frac{j-2}{j-1}\leq
2-\frac{1}{\varkappa}<\frac{j-1}{j}.$

Given that $p^{\prime}$ has only real and simple zeros, the induction
hypothesis provides

$Z_{(-\infty,\gamma_{1})}\left(H_{2-\frac{1}{\varkappa}}[p^{\prime}]\right)=Z_{(\gamma_{n-2},\infty)}\left(H_{2-\frac{1}{\varkappa}}[p^{\prime}]\right)=1,$

where $\gamma_{1}<\gamma_{2}<\ldots<\gamma_{n-2}$ are zeros of
$p^{\prime\prime}.$ Since

$\xi_{1}<\gamma_{1}<\xi_{2}<\gamma_{2}<\ldots<\xi_{n-2}<\gamma_{n-2}<\xi_{n-1},$

we have

$Z_{(-\infty,\xi_{1})}\left(H_{2-\frac{1}{\varkappa}}[p^{\prime}]\right)\leq
1,\ \
Z_{(\xi_{n-1},\infty)}\left(H_{2-\frac{1}{\varkappa}}[p^{\prime}]\right)\leq
1.$ (13.201)

If we apply Corollary 10.3 from Lemma 10.2, we obtain

$Z_{(-\infty,\xi_{1})}\left(H_{\varkappa}[p]\right)\leq 2,\ \
Z_{(\xi_{n-1},\infty)}\left(H_{\varkappa}[p]\right)\leq 2.$ (13.202)

According to Lemma 4.5, the number $Z_{I_{k}}\left(Q_{\varkappa}[p]\right)$ is
odd if $I_{k}$ is an infinite interval of the first type. Thus, for
$\frac{j-1}{j}\leq\varkappa<\frac{j}{j+1}$ we obtain

$Z_{I_{1}}\left(H_{\varkappa}[p]\right)=1,\ \
Z_{I_{n}}\left(H_{\varkappa}[p]\right)=1.$ (13.203)

Lemma 13.5 is proved. $\square$

######  Remark 13.6.

It follows from (9.138), Theorem 12.1, and Lemma 13.5 that the function $M[p]$
decreases on the interval $(-\infty,\xi_{1})$ from $\frac{n-1}{n}$ to
$-\infty,$ and increases on the interval $(\xi_{n-1},\infty)$ from $-\infty$
to $\frac{n-1}{n}.$

##  14\. Accuracy of estimates in Theorem 2.4

Now we are going to discuss accuracy of estimates in Theorem 2.4. The
following fact could be derived from the first and the third statements of
Theorem 12.1 immediately. This theorem provides accuracy of the estimation
from below in Theorem 2.4.

######  Corollary 14.1.

Let $0<\varkappa<1/2.$ Assume that $p_{2n}(x)=x^{2n}+\ldots$ is a polynomial
having no real roots, and all roots of its derivative $p_{2n}^{\prime}$ are
real and simple. Then

$Z_{\mathbb{R}}(\varkappa(p_{2n}^{\prime})^{2}-p_{2n}p^{\prime\prime}_{2n})=2n-2.$

Denote by $T_{n}(x)=cos(n\arccos(x))$ the Chebyshev polynomial of the first
kind of degree $n$. The fact that the bound from above cannot be improved is a
corollary of the statement below.

######  Theorem 14.2.

Let $0<\varkappa<1/2.$ For every $n=1,2,\ldots,$ and each
$0<\varepsilon<\frac{1}{2}$ there exists a constant $C=C(\varepsilon,n),\
-1<C<0,$ such that for every $0<\varkappa\leq\frac{1}{2}-\varepsilon$ the
following is true

$Z_{\mathbb{R}}\left(\varkappa-M[T_{2n}+C]\right)=Z_{\mathbb{R}}\left(\varkappa-\frac{(T_{2n}(x)+C)T_{2n}^{\prime\prime}(x)}{(T_{2n}^{\prime}(x))^{2}}\right)=4n-2.$
(14.204)

Proof of Theorem 14.2. First, consider the case of $n=1,$ that is of
$T_{2}(x)+C=2x^{2}-1+C.$ If $|C|<1,$ both roots of the polynomial are real and
simple. Since $(T_{2}(x)+C)^{\prime}=4x,$ this polynomial generates two
intervals $I_{k}$: $I_{1}=(-\infty,\ 0)$ and $I_{2}=(0,\ +\infty).$ Both
intervals are intervals of the first type. Thus, the statement of Theorem 14.2
follows immediately from Remark 13.6.

Now, for any $n=2,3,\ldots,$ we consider the polynomial

$P_{2n}(x)=T_{2n}(x)-1=2^{2n-1}\prod_{k=0}^{2n-1}\left(x-cos\left(\frac{k\pi}{2n}\right)\right)$
$=2^{2n-1}(x^{2}-1)\prod_{k=1}^{n-1}\left(x-\cos\left(\frac{k\pi}{n}\right)\right)^{2}.$
(14.205)

By (4.29) for any $k=1,\ldots,n-1$ we have

$\lim_{x\to\cos\left(\frac{k\pi}{n}\right)}M[P_{2n}](x)=\frac{1}{2}.$

Let us fix any $0<\varepsilon<\frac{1}{2}$. There are numbers
$\delta_{k}=\delta_{k}(\varepsilon)$ such that for all $x,\
|x-\cos\left(\frac{k\pi}{n}\right)|<\delta_{k},\ k=1,2,\ldots,n-1,$ the
following is true

$\left|M[P_{2n}](x)-\frac{1}{2}\right|<\frac{\varepsilon}{4}.$ (14.206)

Consider the family of polynomials $\\{P_{2n}(x)+B,\ \ 0<B<1\\}$ on the
compact set

$K=\cup_{k=1}^{n-1}\left(\left[\cos\left(\frac{k\pi}{n}\right)-\frac{\delta_{k}}{2},\cos\left(\frac{k\pi}{n}\right)-\frac{\delta_{k}}{4}\right]\cup\left[\cos\left(\frac{k\pi}{n}\right)+\frac{\delta_{k}}{4},\cos\left(\frac{k\pi}{n}\right)+\frac{\delta_{k}}{2}\right]\right).$

Obviously, on $K$ we have

$M\left[P_{2n}+B\right](x)=\frac{(P_{2n}(x)+B)P_{2n}^{\prime\prime}(x)}{(P_{2n}(x))^{2}}\rightrightarrows
M[P_{2n}](x),\ \ B\to 0.$ (14.207)

Therefore, there exists $B_{0}=B_{0}(\varepsilon),\ 0<\ B_{0}<1/2,$ such that
for all $0<B<B_{0}$ and for all $x\in K$ the inequality is true

$\left|M\left[P_{2n}+B\right](x)-M[P_{2n}](x)\right|<\frac{\varepsilon}{4}.$
(14.208)

It follows from (14.206) and (14.208) that for all $0<B<B_{0}$ and for all
$x\in K$ we have

$\left|M\left[P_{2n}+B\right](x)-\frac{1}{2}\right|<\frac{\varepsilon}{2}.$
(14.209)

Thus, for all $0<B<B_{0}$ and for all $x\in K$ we obtain the estimation

$M\left[P_{2n}+B\right](x)=\frac{(P_{2n}(x)+B)P_{2n}^{\prime\prime}(x)}{(P_{2n}(x))^{2}}>\frac{1}{2}-\frac{\varepsilon}{2}>0.$
(14.210)

Note that

$(P_{2n}(x)+B)^{\prime}=(T_{2n}(x)-1+B)^{\prime}=T_{2n}^{\prime}(x)=$
$2n\frac{\sin(2n\arccos x)}{\sin(\arccos
x)}=2^{2n}n\prod_{k=1}^{2n-1}\left(x-\cos\left(\frac{k\pi}{2n}\right)\right).$

Since $0<B<\frac{1}{2},$ the roots of $P_{2n}(x)+B=T_{2n}(x)-1+B$ are all real
and simple. Therefore, all the intervals $I_{k},\ k=1,2,\ldots,2n,$ are
intervals of the first type, where

$I_{1}=\left(-\infty,\ \cos\left(\frac{\pi}{2n}\right)\right),\ \
I_{2n}=\left(\cos\left(\frac{(2n-1)\pi}{2n}\right),\ \infty\right),$ $\
I_{k}=\left(\cos\left(\frac{(k-1)\pi}{2n}\right),\
\cos\left(\frac{k\pi}{2n}\right)\right),\ \ k=2,\ldots,2n-1.$ (14.211)

Making if necessary the values of $\delta_{k},\ k=1,2,\ldots,n-1,$ smaller we
can assume that

$\left[\cos\left(\frac{k\pi}{n}\right)-\frac{\delta_{k}}{2},\cos\left(\frac{k\pi}{n}\right)-\frac{\delta_{k}}{4}\right]\subset
I_{2k},$
$\left[\cos\left(\frac{k\pi}{n}\right)+\frac{\delta_{k}}{4},\cos\left(\frac{k\pi}{n}\right)+\frac{\delta_{k}}{2}\right]\subset
I_{2k+1},\ \ k=1,2,\ldots,n-1.$

It follows from the second statements of Theorem 12.1and (14.210), that for
each $0<\varkappa\leq\frac{1}{2}-\varepsilon$ on any finite interval $I_{k},\
k=2,3,\ldots,2n-1,$ the following is true

$Z_{I_{k}}\left(\varkappa-\frac{(P_{2n}(x)+B)P_{2n}^{\prime\prime}(x)}{(P_{2n}(x))^{2}}\right)=2.$
(14.212)

Because all roots of $P_{2n}(x)+B$ are real and simple, by virtue of the
fourth statement of Theorem 12.1 for each $0<\varkappa<\frac{1}{2}$ on each of
the infinite intervals $I_{k}$ we have

$Z_{I_{k}}\left(\varkappa-\frac{(P_{2n}(x)+B)P_{2n}^{\prime\prime}(x)}{(P_{2n}(x))^{2}}\right)=1.$
(14.213)

Finally we obtain

$Z_{\mathbb{R}}\left(\varkappa-\frac{(P_{2n}(x)+B)P_{2n}^{\prime\prime}(x)}{(P_{2n}(x))^{2}}\right)=2(2n-2)+2=4n-2.$
(14.214)

Theorem 14.2 is proved. $\square$

##  15\. Polynomials with only real zeros

As we mentioned in the introductory part of this paper, in their paper [14] M.
Tyaglov and M.J. Atia found the upper and lower bounds for
$Z_{\mathbb{R}}\left(H_{\varkappa}\right)[p]$ for all real $\varkappa$ when
all roots of a polynomial $p$ are real. In Theorem D we described their
result. In this section, we are going to refine their result in the following
way. We discuss how the roots of the corresponding function $Q_{\varkappa}[p]$
can be distributed among the finite intervals $I_{k}.$

######  Lemma 15.1.

For all $a_{1}\geq 0,a_{2}\geq 0,\ldots,a_{m}\geq 0$ the following
inequalities are valid

$2\sum_{1\leq j<k\leq m}a_{j}a_{k}\ \leq\ (m-1)\sum_{k=1}^{m}a_{k}^{2},$
(15.215) $\left(\sum_{k=1}^{m}a_{k}\right)^{2}\ \leq\
m\sum_{k=1}^{m}a_{k}^{2},$ (15.216) $(a_{1}+\ldots+a_{s}-a_{s+1}-\ldots-
a_{m})^{2}\ \leq\ \max(s,m-s)\sum_{k=1}^{m}a_{k}^{2},\ \ 1\leq s\leq m-1,$
(15.217)

and, if all $a_{j}$ are nonzero numbers for $j=1,2,\ldots,m,$ then

$(a_{1}+\ldots+a_{s}-a_{s+1}-\ldots-a_{m})^{2}\ <\
\max(s,m-s)\sum_{k=1}^{m}a_{k}^{2},\ \ 1\leq s\leq m-1.$ (15.218)

Proof of Lemma 15.1. The inequality (15.216) is equivalent to the inequality
(15.215), because

$\left(\sum_{k=1}^{m}a_{k}\right)^{2}=\sum_{k=1}^{m}a_{k}^{2}+2\sum_{1\leq
j<k\leq m}a_{j}a_{k}.$

Let us prove (15.215)

$2\sum_{1\leq j<k\leq
m}a_{j}a_{k}=\sum_{j=1}^{m}\sum_{k=j+1}^{m}(2a_{j}a_{k})\leq\sum_{j=1}^{m}\sum_{k=j+1}^{m}(a_{j}^{2}+a_{k}^{2})=\sum_{j=1}^{m}\left((m-j)a_{j}^{2}+\sum_{k=j+1}^{m}a_{k}^{2}\right)$
$=\sum_{j=1}^{m}(m-j)a_{j}^{2}+\sum_{k=2}^{m}\sum_{j=1}^{k-1}a_{k}^{2}=\sum_{j=1}^{m}(m-j)a_{j}^{2}+\sum_{k=2}^{m}(k-1)a_{k}^{2}$
$=(m-1)a_{1}^{2}+\sum_{k=2}^{m}\left((m-k)a_{k}^{2}+(k-1)a_{k}^{2}\right)=(m-1)\sum_{k=1}^{m}a_{k}^{2}.$

The inequalities (15.215) and (15.216) are proved.

Let us fix $1\leq s\leq m-1$ and prove (15.217). We have

$\left(\sum_{k=1}^{s}a_{k}-\sum_{k=s+1}^{m}a_{k}\right)^{2}=\left(\sum_{k=1}^{s}a_{k}\right)^{2}+\left(\sum_{k=s+1}^{m}a_{k}\right)^{2}-2\left(\sum_{k=1}^{s}a_{k}\right)\cdot\left(\sum_{k=s+1}^{m}a_{k}\right)$
$\leq\left(\sum_{k=1}^{s}a_{k}\right)^{2}+\left(\sum_{k=s+1}^{m}a_{k}\right)^{2},$

where the inequality sign is strict if all numbers are nonzero. Applying
(15.216) to each of the terms in the right-hand side of the above inequality
we obtain

$\left(\sum_{k=1}^{s}a_{k}-\sum_{k=s+1}^{m}a_{k}\right)^{2}\ \leq\
s\sum_{k=1}^{s}a_{k}^{2}+(m-s)\sum_{k=s+1}^{m}a_{k}^{2}\leq\max(s,m-s)\sum_{k=1}^{m}a_{k}^{2}.$

Lemma 15.1 is proved. $\square$

The following statement about maximum values of the function $M[p]$ on a
finite interval $I_{k}$ in the case when all intervals $I_{k}$ are intervals
of the first type, will be obtained as a corollary of the above lemma.

######  Theorem 15.2.

Assume that a polynomial $p$ has only real and simple zeros. Then for all
$x\in I_{s}\cup I_{n-s+1}=(\xi_{s-1},\ \xi_{s})\cup(\xi_{n-s},\ \xi_{n-s+1}),\
\ s=2,3,\ldots,\lfloor\frac{n+1}{2}\rfloor,$ the following estimation is valid

$M[p](x)=\frac{p(x)p^{\prime\prime}(x)}{(p^{\prime}(x))^{2}}<\frac{n-s}{n-s+1}.$
(15.219)

Proof of Theorem 15.2. Suppose that $p(x)=(x-x_{1})(x-x_{2})\cdots(x-x_{n}).$
We will assume that $x\neq x_{k},\ \ k=1,2,\ldots,n.$ Since all roots of the
polynomial $p$ are simple, if $x=x_{k}$, then $M_{p}(x_{k})=0$. In this case
the estimation (15.219) is obvious.

Let $x\neq x_{k}.$ We have

$\frac{p(x)}{p^{\prime}(x)}=\left(\frac{p^{\prime}(x)}{p(x)}\right)^{-1}=\left(\sum_{k=1}^{n}\frac{1}{x-x_{k}}\right)^{-1}.$

Therefore,

$\left(\frac{p(x)}{p^{\prime}(x)}\right)^{\prime}=1-\frac{p(x)p^{\prime\prime}(x)}{(p^{\prime}(x))^{2}}=\frac{\sum_{k=1}^{n}\left(\frac{1}{x-x_{k}}\right)^{2}}{\left(\sum_{k=1}^{n}\frac{1}{x-x_{k}}\right)^{2}}.$
(15.220)

As before we will denote by $\xi$ with indices the roots of the derivative
$p^{\prime}.$ Since all roots of $p$ and $p^{\prime}$ are real and simple, we
have

$x_{1}<\xi_{1}<x_{2}<\xi_{2}<\ldots<\xi_{n-1}<x_{n}.$ (15.221)

So, if $x\in I_{s},\ s=2,3,\ldots,\lfloor\frac{n+1}{2}\rfloor,$ then

$x_{s-1}<\xi_{s-1}<x<\xi_{s}<x_{s+1}.$

Therefore,

$x-x_{k}>0,\ \ \mbox{when}\ \ k=1,2,\ldots,s-1,\ \ \mbox{and}$ $x-x_{k}<0,\ \
\mbox{when}\ \ k=s+1,s+2,\ldots,n,$ (15.222)

for all $x\in I_{s}.$ Similar if $x\in I_{n-s+1},\ \
s=2,3,\ldots,\lfloor\frac{n+1}{2}\rfloor,$ then

$x_{n-s}<\xi_{n-s}<x<\xi_{n-s+1}<x_{n-s+2}.$

Thus,

$x-x_{k}>0,\ \ \mbox{when}\ \ k=1,2,\ldots,n-s,\ \ \mbox{and}$ $x-x_{k}<0,\ \
\mbox{when}\ \ k=n-s+2,\ldots,n,$ (15.223)

for all $x\in I_{n-s+1}.$ Note that for $s<\lfloor\frac{n+1}{2}\rfloor$ we
have $s\leq n-s+1.$ Applying (15.218) to (15.220), we obtain

$1-\frac{p(x)p^{\prime\prime}(x)}{(p^{\prime}(x))^{2}}=\frac{\sum_{k=1}^{n}\left(\frac{1}{x-x_{k}}\right)^{2}}{\left(\sum_{k=1}^{n}\frac{1}{x-x_{k}}\right)^{2}}>\frac{1}{n-s+1}.$
(15.224)

The statement (15.219) of Theorem 15.2 follows from the last inequality.
Theorem 15.2 is proved. $\square$

######  Corollary 15.3.

Assume that a polynomial $p$ has only real and simple zeros. If

$\frac{n-2}{n-1}\leq\varkappa<\frac{n-1}{n},$

then for all finite intervals $I_{k}$ the following is true

$Z_{I_{k}}(H_{\varkappa}[p])=Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=0.$
(15.225)

The more general statement below contains Corollary 15.3 as a partial case.

######  Corollary 15.4.

Assume that a polynomial $p$ has only real and simple zeros. Let for some
$j=\lfloor\frac{n}{2}\rfloor+1,\ldots,n-1,$ the inequality

$\frac{j-1}{j}\leq\varkappa<\frac{j}{j+1}$

holds. Then for any interval $I_{k}$, $k=n-j+1,\ldots,j,$ the following is
true

$Z_{I_{k}}(H_{\varkappa}[p])=Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=0.$
(15.226)

The theorem below follows from Theorem 13.1 and Corollary 15.4.

######  Theorem 15.5.

Assume that a polynomial $p$ has only real and simple zeros. Let for some
$j=1,2,\ldots,n-1$ the inequality

$\frac{j-1}{j}\leq\varkappa<\frac{j}{j+1}$

holds. Then

  1. (1)

If $j=\lfloor\frac{n}{2}\rfloor+1,\ldots,n-1,$ then

     * •

For any finite interval $I_{k}$, where $k=n-j+1,\ldots,j,$ the estimation is
valid

$Z_{I_{k}}(H_{\varkappa}[p])=Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=0.$

     * •

For any finite interval $I_{k}$, where $k=2,\ldots,n-j,$ or
$k=j+1,\ldots,n-1,$ the estimation is valid

$Z_{I_{k}}(H_{\varkappa}[p])=Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=0\
\mbox{or}\ 2.$

  2. (2)

If $j=1,2,\ldots,\lfloor\frac{n-1}{2}\rfloor,$ then for any finite interval
$I_{k}$ the estimation is valid

$Z_{I_{k}}(H_{\varkappa}[p])=Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=0\
\mbox{or}\ 2.$

The following result shows that Theorem 15.5 cannot be improved.

######  Theorem 15.6.

For each $\varepsilon>0$ there exists a polynomial

$p_{\varepsilon}(x)=(x-x_{1}(\varepsilon))(x-x_{2}(\varepsilon))\cdot\ldots\cdot(x-x_{n}(\varepsilon)),$
(15.227)

and a system of $n-2$ points $y_{k}(\varepsilon)\in I_{k},\ \
k=2,3,\ldots,n-1,$ such that

$p_{\varepsilon}(y_{k}(\varepsilon))>\frac{k-1}{k}-\varepsilon.$ (15.228)

Proof of Theorem 15.6. We will construct the polynomial $p_{\varepsilon}$
using inductive process. We start with the polynomial

$p_{0}(x)=x^{n-1}(x-1).$ (15.229)

Its derivative has the form

$p^{\prime}_{0}(x)=nx^{n-2}\left(x-\frac{n-1}{n}\right).$ (15.230)

By virtue of (4.29) and definition (4.25) of the function $M[p]$ we have

$\lim_{x\to 0}M[p_{0}](x)=\frac{n-2}{n-1}.$ (15.231)

Let us fix any (small enough) $\varepsilon>0.$ It follows from (15.231) that
there is $0<\delta_{1}<1$ such that

$M[p_{0}](x)>\frac{n-2}{n-1}-\frac{\varepsilon}{n},\ \ \mbox{whenever}\
|x|<\delta_{1}.$ (15.232)

Denote by

$p_{1}(x)=x^{n-2}(x-x_{n-1})(x-1),$ (15.233)

where $0<x_{n-1}<1$ will be chosen later. The derivative of the above
polynomial will have the form

$p_{1}^{\prime}(x)=x^{n-3}(x-\xi_{n-2}^{1})(x-\xi_{n-1}^{1}),$ (15.234)

where $0<\xi_{n-2}^{1}<x_{1}<\xi_{n-1}^{1}<1$. Note that on the compact set
$\frac{\delta_{1}}{4}\leq x\leq\frac{\delta_{1}}{2}$ the following is true

$M[p_{1}](x)\rightrightarrows M[p_{0}](x),\ \ \mbox{when}\ \
x_{n-1}\rightarrow 0,$ (15.235)

and

$\xi_{n-2}^{1}\rightarrow 0\ \ \mbox{and}\ \
\xi_{n-1}^{1}\rightarrow\frac{n-1}{n},\ \ \mbox{when}\ \ x_{n-1}\rightarrow
0.$ (15.236)

Thus, we can find a number $x_{n-1}=x_{n-1}(\varepsilon)$ such that

$\xi_{n-2}^{1}<\frac{\delta_{1}}{4}<\frac{\delta_{1}}{2}<\xi_{n-1}^{1},$
(15.237)

and for all $x\in\left[\frac{\delta_{1}}{4},\frac{\delta_{1}}{2}\right]$ we
have

$M[p_{1}](x)>M[p_{0}](x)-\frac{\varepsilon}{n},$ (15.238)

where

$p_{1}(x)=x^{n-2}(x-x_{n-1}(\varepsilon))(x-1).$ (15.239)

Let us fix in (15.239) $x_{n-1}=x_{n-1}(\varepsilon)$ that provides (15.237)
and (15.238). It follows from (15.232) and (15.238) that

$M[p_{1}](x)>\frac{n-2}{n-1}-\frac{2\varepsilon}{n},$ (15.240)

for all $x\in\left[\frac{\delta_{1}}{4},\frac{\delta_{1}}{2}\right].$

For the next step we consider (15.239) as the initial polynomial. Its
derivative has the form (15.234). By virtue of (4.29)

$\lim_{x\to 0}M[p_{1}](x)=\frac{n-3}{n-2}.$ (15.241)

By (15.241) there is $0<\delta_{2}<\frac{\delta_{1}}{2}$ such that

$M[p_{1}](x)>\frac{n-3}{n-2}-\frac{\varepsilon}{n},\ \ \mbox{whenever}\
|x|<\delta_{2}.$ (15.242)

Denote by

$p_{2}(x)=x^{n-3}(x-x_{n-2})(x-x_{n-1}(\varepsilon))(x-1),$ (15.243)

where $0<x_{n-2}<x_{n-1}(\varepsilon)$ will be chosen later. The derivative of
the above polynomial will have the form

$p_{1}^{\prime}(x)=x^{n-4}(x-\xi_{n-3}^{2})(x-\xi_{n-2}^{2})(x-\xi_{n-1}^{2}),$
(15.244)

where
$0<\xi_{n-3}^{2}<x_{n-2}<\xi_{n-2}^{2}<x_{n-1}(\varepsilon)<\xi_{n-1}^{2}<1$.
Note that on the compact set
$\left[\frac{\delta_{2}}{4},\frac{\delta_{2}}{2}\right]\cup\left[\frac{\delta_{1}}{4},\frac{\delta_{1}}{2}\right]$
the following is true

$M[p_{2}](x)\rightrightarrows M[p_{1}](x),\ \ \mbox{when}\ \
x_{n-2}\rightarrow 0,$ (15.245)

and

$\xi_{n-3}^{2}\rightarrow 0,\ \ \xi_{n-2}^{2}\rightarrow\xi_{n-2}^{1},\ \
\xi_{n-1}^{2}\rightarrow\xi_{n-1}^{1},\ \ \mbox{when}\ \ x_{n-2}\rightarrow
0.$ (15.246)

Taking into account (15.237) and the fact that
$0<\delta_{2}<\frac{\delta_{1}}{2},$ we can find a number
$x_{n-2}=x_{n-2}(\varepsilon)$ such that

$\xi_{n-3}^{2}<\frac{\delta_{2}}{4}<\frac{\delta_{2}}{2}<\xi_{n-2}^{2}<\frac{\delta_{1}}{4}<\frac{\delta_{1}}{2}<\xi_{n-1}^{2},$
(15.247)

and such that for all
$x\in\left[\frac{\delta_{2}}{4},\frac{\delta_{2}}{2}\right]\cup\left[\frac{\delta_{1}}{4},\frac{\delta_{1}}{2}\right]$
we have

$M[p_{2}](x)>M[p_{1}](x)-\frac{\varepsilon}{n},$ (15.248)

where

$p_{2}(x)=x^{n-3}(x-x_{n-2}(\varepsilon))(x-x_{n-1}(\varepsilon))(x-1).$
(15.249)

It follows from (15.242) and (15.248) that

$M[p_{2}](x)>\frac{n-3}{n-2}-\frac{2\varepsilon}{n},$ (15.250)

for all $x\in\left[\frac{\delta_{2}}{4},\frac{\delta_{2}}{2}\right],$ and from
(15.240) and (15.248) that

$M[p_{2}](x)>\frac{n-2}{n-1}-\frac{3\varepsilon}{n},$ (15.251)

for all $x\in\left[\frac{\delta_{1}}{4},\frac{\delta_{1}}{2}\right].$

If we continue this process, finally we will obtain a polynomial

$p_{\varepsilon}(x)=(x-x_{1}(\varepsilon))(x-x_{2}(\varepsilon))\cdot\ldots\cdot(x-x_{n}(\varepsilon)),$

and the system of intervals

$\bigcup_{k=1}^{n-2}\left[\frac{\delta_{k}}{4},\frac{\delta_{k}}{2}\right],$

where

$\left[\frac{\delta_{k}}{4},\frac{\delta_{k}}{2}\right]\subset I_{n-k},\ \
k=1,2,\ldots,n-2,$ (15.252)

such that for each $k=1,2,\ldots,n-2,$ the inequalities are true

$M[p_{\varepsilon}](x)>\frac{n-k-1}{n-k}-\frac{(n-k+1)\varepsilon}{n},\ \
x\in\left[\frac{\delta_{k}}{4},\frac{\delta_{k}}{2}\right].$ (15.253)

The statement of Theorem 15.6 follows from (15.252) and (15.253). Theorem 15.6
is proved.

##  16\. Intervals $I_{k}$ of the second type

Return now to the general case. The statement below gives an estimation for
the number of real zeros of a polynomial $H_{\varkappa}[p]$ on infinite
intervals $I_{k}$ of the second type.

######  Lemma 16.1.

Assume that $I_{k}$ is an infinite interval of the second type for a
polynomial $p$ whose derivative $p^{\prime}$ has only real and simple zeros.
Then there exists a number

$C=\min_{x\in I_{k}}\frac{p(x)p^{\prime\prime}(x)}{(p^{\prime}(x))^{2}},\ \
\frac{1}{2}<C<\frac{n-1}{n},$

such that

  * •

For every $C\leq\varkappa<\frac{n-1}{n}$ the following estimation is valid

$Z_{I_{k}}(H_{\varkappa}[p])=Z_{I_{k}}(Q_{\varkappa}[p])=2.$ (16.254)

  * •

For every $\frac{1}{2}\leq\varkappa<C$ the following estimation is valid

$Z_{I_{k}}(H_{\varkappa}[p])=Z_{I_{k}}(Q_{\varkappa}[p])=0.$ (16.255)

Proof of Lemma 16.1. If $\frac{1}{2}\leq\varkappa<\frac{n-1}{n},$ then $0\leq
2-\frac{1}{\varkappa}<\frac{n-2}{n-1}.$ Since all roots of $p^{\prime}$ are
real and simple, the intervals $(-\infty,\gamma_{1}),$
$(\gamma_{n-2},+\infty),$ where $\gamma_{1}$ and $\gamma_{n-2}$ are the
smallest and the biggest roots of the polynomial $p^{\prime\prime},$ are
intervals of the first type for the polynomial $p^{\prime},$ and by Theorem
13.4.

$Z_{(-\infty,\gamma_{1})}\left(H_{2-\frac{1}{\varkappa}}[p^{\prime}]\right)=Z_{(\gamma_{n-2},+\infty)}\left(H_{2-\frac{1}{\varkappa}}[p^{\prime}]\right)=1.$

Since $\xi_{1}<\gamma_{1}<\ldots<\gamma_{n-2}<\xi_{n-1},$ we have

$Z_{(-\infty,\xi_{1})}\left(H_{2-\frac{1}{\varkappa}}[p^{\prime}]\right)\leq
1,\ \
Z_{(\xi_{n-1},+\infty)}\left(H_{2-\frac{1}{\varkappa}}[p^{\prime}]\right)\leq
1.$

By virtue of Corollary 10.3 from Lemma 10.2, if $I_{k}$ is an infinite
interval of the second type, then

$Z_{I_{k}}\left(H_{\varkappa}[p]\right)\leq 2.$

Taking into account the fourth statement of Lemma 4.6, finally we obtain

$Z_{I_{k}}\left(H_{\varkappa}(p)\right)\ =\ 0\ \ \mbox{or}\ 2.$ (16.256)

Since all roots of the polynomial $p^{\prime}$ are real, the polynomial $p$
satisfies the assumptions of Theorem 2.12. In the same way as in Theorem 2.12
we can show that (4.44) is true, that is there exists such a value $x_{0}$
that for $|x|>x_{0}>0$

$M[p](x)-\frac{n-1}{n}<0.$

So, the graph of the function $y=M[p](x)$ approaches its asymptote
$y=\frac{n-1}{n}$ from below.

If $I_{k}$ is an infinite interval of the second type, then its finite
endpoint is a wrong point (Remark 4.4). We will denote it just by $\xi$
without indices. By (4.34) and (4.35),

$\lim_{x\to\xi}M[p](x)\ =+\infty.$ (16.257)

Denote by

$C=\min_{x\in I_{1}\cup I_{n}}M[p](x).$ (16.258)

It follows from (16.257) and (4.44) that

$C<\frac{n-1}{n},$ (16.259)

and from the third statement of Theorem 12.1 that

$C>\frac{1}{2}.$ (16.260)

Lemma 16.1 is proved. $\square$

Let $I_{k}$ be a finite interval of the second type. As we noticed in Remark
4.4, one of the endpoints of $I_{k}$ is right while the other is wrong. It
follows from (4.32) – (4.35) that for any real $\varkappa$ the equation (4.27)

$M[p](x)=\varkappa$

has solutions in each finite interval $I_{k}$ of the second type. Putting
together (9.135), (9.137), and the first statement of Theorem 12.1, we
conclude that the equation (4.27) has the unique solution for all
$\varkappa\in(-\infty,\frac{1}{2})\cup(\frac{n-1}{n},\infty).$ It means that
on corresponding intervals inside $I_{k}$ the function $M[p]$ is monotone.

There may be an assumption that the function $M[p]$ is monotone on any finite
interval $I_{k}$ of the second type. If we use Corollary 10.3 from Lemma 10.2,
in the same way as above, we can easily obtain that for
$\frac{1}{2}\leq\varkappa<\frac{n-1}{n}$ on each interval $I_{k}$ of the
second type

$Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=Z_{I_{k}}\left(H_{\varkappa}[p]\right)\leq
3.$ (16.261)

The following example shows that the estimation (16.261) cannot be improved,
that is there is a polynomial $p$ all whose real roots are simple, and the
roots of its derivative $p^{\prime}$ are real and simple, such that
$Z_{I_{k}}\left(Q_{\varkappa}[p]\right)=3$ on some of the finite intervals
$I_{k}$ of the second type.

######  Example 16.2.

Consider a polynomial $p(x)=x^{2}(x-1)(x-2)(x+10)+0.1.$ Its derivative
$p^{\prime}(x)=5x^{4}+28x^{3}-84x^{2}+40x$ has four real roots

$\xi_{1}\approx-7.865,\ \xi_{2}=0,\ \xi_{3}\approx 0.617,\ \xi_{4}\approx
1.648.$

The equation

$M[p](x)=\frac{2}{3}$

has three solutions in the interval $(\xi_{1},0).$

Indeed, it is easy to check that

$\lim_{x\to 0^{-}}M[p](x)=+\infty,\ \lim_{x\to\xi_{1}^{+}}M[p](x)=-\infty,$
$M[p](-0.275)\approx 0.642,\ M[p](-1.35)\approx 0.684.$

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