# Zygmund-type inequalities for an operator preserving inequalities between
polynomials

Nisar. A. Rather, Suhail Gulzar and K.A. Thakur

###### Abstract.

In this paper, we present certain new $L_{p}$ inequalities for
$\mathcal{B}_{n}$-operators which include some known polynomial inequalities
as special cases.

00footnotetext: AMS Mathematics Subject Classification(2010): 26D10,
41A17.00footnotetext: Keywords: $L^{p}$ inequalities,
$\mathcal{B}_{n}$-operators, polynomials.

P.G.Department of Mathematics, Kashmir University, Hazratbal,

Srinagar-190006, India

e-mail<EMAIL_ADDRESS>
*Department of Mathematics, Degree College Ganderbal,

Ganderbal Kashmir- India

##  1\. Introduction and statement of results

Let $\mathscr{P}_{n}$ denote the space of all complex polynomials
$P(z)=\sum_{j=0}^{n}a_{j}{z}^{j}$ of degree $n$. For $P\in\mathscr{P}_{n}$,
define

$\left\|P(z)\right\|_{0}:=\exp\left\\{\frac{1}{2\pi}\int_{0}^{2\pi}\log\left|P(e^{i\theta})\right|d\theta\right\\},$
$\left\|P(z)\right\|_{p}:=\left\\{\frac{1}{2\pi}\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}\right\\}^{1/p},\,\,1\leq
p<\infty,$
$\left\|P(z)\right\|_{\infty}:=\underset{\left|z\right|=1}{\max}\left|P(z)\right|,\quad
m(P,k):=\underset{\left|z\right|=k}{\min}\left|P(z)\right|,\,k>0$

and denote for any complex function $\psi:\mathbb{C}\rightarrow\mathbb{C}$ the
composite function of $P$ and $\psi$, defined by
$\left(P\circ\psi\right)(z):=P\left(\psi(z)\right)\,\,\,\,(z\in\mathbb{C})$,
as $P\circ\psi$.

If $P\in\mathscr{P}_{n}$, then

(1.1) $\left\|P^{\prime}(z)\right\|_{p}\leq
n\left\|P(z)\right\|_{p},\,\,\,\,\,\,p\geq 1$

and

(1.2) $\left\|P(Rz)\right\|_{p}\leq
R^{n}\left\|P(z)\right\|_{p},\,\,\,R>1,\,\,\,\,p>0,$

Inequality (1.1) was found out by Zygmund [20] whereas inequality (1.2) is a
simple consequence of a result of Hardy [8]. Arestov [2] proved that (1.1)
remains true for $0<p<1$ as well. For $p=\infty$, the inequality (1.1) is due
to Bernstein (for reference, see [11, 15, 18]) whereas the case $p=\infty$ of
inequality (1.2) is a simple consequence of the maximum modulus principle (
see [11, 12, 15]). Both the inequalities (1.1) and (1.2) can be sharpened if
we restrict ourselves to the class of polynomials having no zero in
$\left|z\right|<1.$ In fact, if $P\in\mathscr{P}_{n}$ and $P(z)\neq 0$ in
$\left|z\right|<1$, then inequalities (1.1) and (1.2) can be respectively
replaced by

(1.3) $\left\|P^{\prime}(z)\right\|_{p}\leq
n\frac{\left\|P(z)\right\|_{p}}{\left\|1+z\right\|_{p}},\,\,\,\,p\geq 0$

and

(1.4)
$\left\|P(Rz)\right\|_{p}\leq\frac{\left\|R^{n}z+1\right\|_{p}}{\left\|1+z\right\|_{p}}\left\|P(z)\right\|_{p},\,\,\,R>1,\,\,\,p>0.$

Inequality (1.3) is due to De-Bruijn [7](see also [3]) for $p\geq 1$. Rahman
and Schmeisser [1] extended it for $0<p<1,$ whereas the inequality (1.4) was
proved by Boas and Rahman [6] for $p\geq 1$ and later it was extended for
$0<p<1$ by Rahman and Schmeisser [14]. For $p=\infty$, the inequality (1.3)
was conjectured by Erdös and later verified by Lax [9] whereas inequality
(1.4) was proved by Ankeny and Rivlin [1].

As a compact generalization of inequalities (1.3) and (1.5), Aziz and Rather
[5] proved that if $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in
$|z|<1,$ then for $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,$
$|\beta|\leq 1,$ $R>r\geq 1$ and $p>0$,

(1.5)
$\left\|P(Rz)+\phi_{n}\left(R,r,\alpha,\beta\right)P(rz)\right\|_{p}\leq\dfrac{C_{p}}{\|1+z\|_{p}}\left\|P(z)\right\|_{p}$

where

(1.6)
$C_{p}=\left\|\left(R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}\right)z+(1+\phi_{n}(R,r,\alpha,\beta))\right\|_{p}$

and

(1.7)
$\phi_{n}\left(R,r,\alpha,\beta\right)_{n}=\beta\left\\{\left(\dfrac{R+1}{r+1}\right)^{n}-|\alpha|\right\\}-\alpha.$

If we take $\beta=0,\,\alpha=1$ and $r=1$ in (1.5) and divide two sides of
(1.5) by $R-1$ then make $R\rightarrow 1,$ we obtain inequality (1.3). Whereas
inequality (1.4) is obtained from (1.5) by taking $\alpha=\beta=0.$

Rahman [13] (see also Rahman and Schmeisser [15, p. 538]) introduced a class
$\mathcal{B}_{n}$ of operators $B$ that maps $P\in\mathscr{P}_{n}$ into
itself. That is, the operator $B$ carries $P\in\mathscr{P}_{n}$ into

(1.8)
$B[P](z):=\lambda_{0}P(z)+\lambda_{1}\left(\frac{nz}{2}\right)\frac{P^{\prime}(z)}{1!}+\lambda_{2}\left(\frac{nz}{2}\right)^{2}\frac{P^{\prime\prime}(z)}{2!}$

where $\lambda_{0},\lambda_{1}$ and $\lambda_{2}$ are such that all the zeros
of

(1.9)
$u(z):=\lambda_{0}+C(n,1)\lambda_{1}z+C(n,2)\lambda_{2}z^{2},\,\,\,C(n,r)=n!/r!(n-r)!,$

lie in the half plane

(1.10) $|z|\leq|z-n/2|$

and proved that if $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in
$|z|<1$, then

(1.11)
$\left|B[P\circ\sigma](z)\right|\leq\frac{1}{2}\left\\{R^{n}\left|\Lambda_{n}\right|+\left|\lambda_{0}\right|\right\\}\left\|P(z)\right\|_{\infty}\,\,\,\mbox{}for\,\mbox{}\,\,\,|z|=1,$

(see [13, Inequalities (5.2) and (5.3)]) where $\sigma(z)=Rz$, $R\geq 1$ and

(1.12)
$\Lambda_{n}:=\lambda_{0}+\lambda_{1}\frac{n^{2}}{2}+\lambda_{2}\frac{n^{3}(n-1)}{8}.$

As an extension of inequality (1.11) to $L_{p}$-norm, recently W.M. Shah and
A. Liman [19] while seeking the desired extension, they [19, Theorem 2] have
made an incomplete attempt by claiming to have proved that if
$P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in $\left|z\right|<1$, then
for each $R\geq 1$ and $p\geq 1$,

(1.13)
$\left\|B[P\circ\sigma](z)\right\|_{p}\leq\frac{R^{n}|\Lambda_{n}|+|\lambda_{0}|}{\left\|1+z\right\|_{p}}\left\|P(z)\right\|_{p}.$

where $B\in B_{n}$ and $\sigma(z)=Rz$ and $\Lambda_{n}$ is defined by 1.12.

Rather and Shah [16] pointed an error in the proof of (1.13), they not only
provided a correct proof but also extended it for $0\leq p<1$ as well. They
proved:

######  Theorem A.

If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish for $|z|<1,$ then for
$0\leq p<\infty$ and $R>1,$

(1.14)
$\left\|B[P\circ\sigma](z)\right\|_{p}\leq\dfrac{\left\|R^{n}\Lambda_{n}z+\lambda_{0}\right\|_{p}}{\|1+z\|_{p}}\left\|P(z)\right\|_{p},$

$B\in\mathcal{B}_{n},$ $\sigma(z)=Rz$ and $\Lambda_{n}$ is defined by (1.12).
The result is sharp as shown by $P(z)=az^{n}+b,$ $|a|=|b|=1.$

Recently, Rather and Suhail Gulzar [16] obtained the following result which is
a generalization of Theorem A.

######  Theorem B.

If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish for $|z|<1,$ then for
$\alpha\in\mathbb{C}$ with $|\alpha|\leq 1,$ $0\leq p<\infty$ and $R>1,$

(1.15) $\displaystyle\left\|B[P\circ\sigma](z)-\alpha
B[P](z)\right\|_{p}\leq\dfrac{\left\|(R^{n}-\alpha)\Lambda_{n}z+(1-\alpha)\lambda_{0}\right\|_{p}}{\|1+z\|_{p}}\left\|P(z)\right\|_{p},$

where $B\in\mathcal{B}_{n},$ $\sigma(z)=Rz$ and $\Lambda_{n}$ is defined by
(1.12). The result is best possible and equality in (1.15) holds for
$P(z)=az^{n}+b,$ $|a|=|b|=1.$

If we take $\alpha=0$ in Theorem B, we obtain Theorem A.

In this paper, we investigating the dependence of

$\left\|B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)\right\|_{p}$

on $\left\|P(z)\right\|_{p}$ for $\alpha$, $\beta\in\mathbb{C}$ with
$|\alpha|\leq 1$, $|\beta|\leq 1$, $R>r\geq 1$, $0\leq p<\infty$,
$\sigma(z):=Rz$, $\rho(z):=rz$ and $\phi_{n}\left(R,r,\alpha,\beta\right)$ is
given by (1.7) and establish certain generalized $L_{p}$-mean extensions of
the inequality (1.11) for $0\leq p<\infty$ and also a generalization of (1.5).
In this direction, we first present the following result which is a compact
generalization of the inequalities (1.3), (1.4), (1.5) and (1.11) for $0\leq
p<1$ as well.

######  Theorem 1.1.

If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in $\left|z\right|<1$,
then for then for $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1$,
$|\beta|\leq 1$, $R>r\geq 1$ and $0\leq p<\infty$,

$\displaystyle\left\|B[P\circ\sigma](z)+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](z)\right\|_{p}$
(1.16)
$\displaystyle\qquad\qquad\qquad\leq\frac{\left\|\left(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n}\right)\Lambda_{n}z+\left(1+\phi_{n}(R,r,\alpha,\beta)\right)\lambda_{0}\right\|_{p}}{\left\|1+z\right\|_{p}}\left\|P(z)\right\|_{p}$

where $B\in\mathcal{B}_{n}$, $\sigma(z):=Rz$, $\rho(z):=rz$, $\Lambda_{n}$ and
$\phi_{n}\left(R,r,\alpha,\beta\right)$ are defined by(1.7) and (1.12)
respectively. The result is best possible and equality in (1.1) holds for
$P(z)=az^{n}+b,|a|=|b|\neq 0$

######  Remark 1.1.

If we take $\lambda_{1}=\lambda_{2}=0$ in (1.1), we obtain inequality (1.5).

For $\beta=0,$ inequality (1.1) reduces the following result.

######  Corollary 1.1.

If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in $\left|z\right|<1$,
then for every real or complex number $\alpha$ with $|\alpha|\leq 1$, $R>r\geq
1$ and $0\leq p<\infty$,

(1.17) $\left\|B[P\circ\sigma](z)-\alpha
B[P\circ\rho](z)\right\|_{p}\leq\frac{\left\|(R^{n}-\alpha
r^{n})\Lambda_{n}z+(1-\alpha)\lambda_{0}\right\|_{p}}{\left\|1+z\right\|_{p}}\left\|P(z)\right\|_{p}$

where $B\in\mathcal{B}_{n}$, $\sigma(z):=Rz$, $\rho(z):=rz$ and $\Lambda_{n}$
is defined by (1.12). The result is best possible and equality in (1.17) holds
for $P(z)=az^{n}+b,\,\,\,|a|=|b|\neq 0$.

######  Remark 1.2.

For taking $\alpha=0$ in (1.17), we obtain Theorem (A) and for $r=1$ in
(1.17), we get Theorem B.

Instead of proving Theorem 1.1, we prove the following more general result
which includes Theorem 1.1 as a special case.

######  Theorem 1.2.

If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in $\left|z\right|<1$,
then for then for $\alpha,\beta,\delta\in\mathbb{C}$ with $|\alpha|\leq 1$,
$|\beta|\leq 1,$ $|\delta|\leq 1,$ $R>r\geq 1$ and $0\leq p<\infty$,

$\displaystyle\Bigg{\|}B[P\circ\sigma](z)$
$\displaystyle+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)$
$\displaystyle+\delta$
$\displaystyle\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{\|}_{p}$
(1.18)
$\displaystyle\leq\frac{\left\|\left(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n}\right)\Lambda_{n}z+\left(1+\phi_{n}(R,r,\alpha,\beta)\right)\lambda_{0}\right\|_{p}}{\left\|1+z\right\|_{p}}\left\|P(z)\right\|_{p}$

where $B\in B_{n}$, $\sigma(z):=Rz$, $\rho(z):=rz$, $\Lambda_{n}$ and
$\phi_{n}\left(R,r,\alpha,\beta\right)$ are defined by(1.7) and (1.12)
respectively. The result is best possible and equality in (1.1) holds for
$P(z)=az^{n}+b,|a|=|b|\neq 0.$

######  Remark 1.3.

For $\delta=0$ in (1.2), we get Theorem 1.1.

Next, corollary which is a generalization of (1.5) follows by taking
$\lambda_{1}=\lambda_{2}=0$ in (1.2).

######  Corollary 1.2.

If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in $\left|z\right|<1$,
then for then for $\alpha,\beta,\delta\in\mathbb{C}$ with $|\alpha|\leq 1$,
$|\beta|\leq 1,$ $|\delta|\leq 1,$ $R>r\geq 1$ and $0\leq p<\infty$,

$\displaystyle\Bigg{\|}P(Rz)$
$\displaystyle+\phi_{n}\left(R,r,\alpha,\beta\right)P(rz)$
$\displaystyle+\delta\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||-|1+\phi_{n}\left(R,r,\alpha,\beta\right)|\Big{)}m}{2}\Bigg{\|}_{p}$
(1.19)
$\displaystyle\qquad\leq\frac{\left\|\left(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n}\right)z+\left(1+\phi_{n}(R,r,\alpha,\beta)\right)\right\|_{p}}{\left\|1+z\right\|_{p}}\left\|P(z)\right\|_{p}$

where $\phi_{n}\left(R,r,\alpha,\beta\right)$ is defined by(1.7). The result
is best possible and equality in (1.2) holds for $P(z)=az^{n}+b,|a|=|b|\neq
0.$

##  2\. Lemmas

For the proofs of these theorems, we need the following lemmas. The first
Lemma is easy to prove.

######  Lemma 2.1.

If $P\in\mathscr{P}_{n}$ and $P(z)$ has all its zeros in $\left|z\right|\leq
1$, then for every $R\geq r\geq 1$ and $\left|z\right|=1$,

$\left|P(Rz)\right|\geq\left(\frac{R+1}{r+1}\right)^{n}\left|P(rz)\right|.$

The following Lemma follows from [10, Corollary 18.3, p. 65].

######  Lemma 2.2.

If all the zeros of polynomial $P\in\mathscr{P}_{n}$ lie in
$\left|z\right|\leq 1$, then all the zeros of the polynomial $B[P](z)$ also
lie in $\left|z\right|\leq 1$.

######  Lemma 2.3.

If $F\in\mathscr{P}_{n}$ has all its zeros in $\left|z\right|\leq 1$ and
$P(z)$ is a polynomial of degree at most $n$ such that

$|P(z)|\leq|F(z)|\,\,\,\textrm{for}\,\,\,|z|=1,$

then for every $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1$, $|\beta|\leq
1$, $R\geq r\geq 1$, and $|z|\geq 1$,

(2.1)
$|B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)|\leq|B[F\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[F\circ\rho](z)|$

where $B\in\mathcal{B}_{n}$, $\sigma(z):=Rz$, $\rho(z):=rz$, $\Lambda_{n}$ and
$\phi_{n}\left(R,r,\alpha,\beta\right)$ are defined by (1.12) and (1.7)
respectively.

###### Proof.

Since the polynomial $F(z)$ of degree $n$ has all its zeros in $|z|\leq 1$ and
$P(z)$ is a polynomial of degree at most $n$ such that

(2.2) $|P(z)|\leq|F(z)|\,\,\,\,\textrm{for}\,\,\,\,|z|=1,$

therefore, if $F(z)$ has a zero of multiplicity $s$ at $z=e^{i\theta_{0}}$,
then $P(z)$ has a zero of multiplicity at least $s$ at $z=e^{i\theta_{0}}$. If
$P(z)/F(z)$ is a constant, then the inequality (2.1) is obvious. We now assume
that $P(z)/F(z)$ is not a constant, so that by the maximum modulus principle,
it follows that

$|P(z)|<|F(z)|\,\,\,\textrm{for}\,\,|z|>1\,\,.$

Suppose $F(z)$ has $m$ zeros on $|z|=1$ where $0\leq m\leq n$, so that we can
write

$F(z)=F_{1}(z)F_{2}(z)$

where $F_{1}(z)$ is a polynomial of degree $m$ whose all zeros lie on $|z|=1$
and $F_{2}(z)$ is a polynomial of degree exactly $n-m$ having all its zeros in
$|z|<1$. This implies with the help of inequality (2.2) that

$P(z)=P_{1}(z)F_{1}(z)$

where $P_{1}(z)$ is a polynomial of degree at most $n-m$. Now, from inequality
(2.2), we get

$|P_{1}(z)|\leq|F_{2}(z)|\,\,\,\textrm{for}\,\,|z|=1\,$

where $F_{2}(z)\neq 0\,\,for\,\,|z|=1$. Therefore for every
$\lambda\in\mathbb{C}$ with $|\lambda|>1$, a direct application of Rouche’s
theorem shows that the zeros of the polynomial $P_{1}(z)-\lambda F_{2}(z)$ of
degree $n-m\geq 1$ lie in $|z|<1$. Hence the polynomial

$f(z)=F_{1}(z)\left(P_{1}(z)-\lambda F_{2}(z)\right)=P(z)-\lambda F(z)$

has all its zeros in $|z|\leq 1$ with at least one zero in $|z|<1$, so that we
can write

$f(z)=(z-te^{i\delta})H(z)$

where $t<1$ and $H(z)$ is a polynomial of degree $n-1$ having all its zeros in
$|z|\leq 1$. Applying Lemma 2.1 to the polynomial $f(z)$ with $k=1$, we obtain
for every $R>r\geq 1$ and $0\leq\theta<2\pi$,

$\displaystyle|f(Re^{i\theta})|=$
$\displaystyle|Re^{i\theta}-te^{i\delta}||H(Re^{i\theta})|$
$\displaystyle\geq$
$\displaystyle|Re^{i\theta}-te^{i\delta}|\left(\frac{R+1}{r+1}\right)^{n-1}|H(re^{i\theta})|$
$\displaystyle=$
$\displaystyle\left(\frac{R+1}{r+1}\right)^{n-1}\frac{|Re^{i\theta}-te^{i\delta}|}{|re^{i\theta}-te^{i\delta}|}|(re^{i\theta}-te^{i\delta})H(re^{i\theta})|$
$\displaystyle\geq$
$\displaystyle\left(\frac{R+1}{r+1}\right)^{n-1}\left(\frac{R+t}{r+t}\right)|f(re^{i\theta})|.$

This implies for $R>r\geq 1$ and $0\leq\theta<2\pi$,

(2.3)
$\left(\frac{r+t}{R+t}\right)|f(Re^{i\theta})|\geq\left(\frac{R+1}{r+1}\right)^{n-1}|f(re^{i\theta})|.$

Since $R>r\geq 1>t$ so that $f(Re^{i\theta})\neq 0$ for $0\leq\theta<2\pi$ and
$\frac{1+r}{1+R}>\frac{r+t}{R+t}$, from inequality (2.3), we obtain $R>r\geq
1$ and $0\leq\theta<2\pi$,

(2.4) $|f(Re^{i\theta})|>\left(\frac{R+1}{r+1}\right)^{n}|f(re^{i\theta})|.$

Equivalently,

$|f(Rz)|>\left(\frac{R+1}{r+1}\right)^{n}|f(rz)|$

for $|z|=1$ and $R>r\geq 1$. Hence for every $\alpha\in\mathbb{C}$ with
$|\alpha|\leq 1$ and $R>r\geq 1,$ we have

$\displaystyle|f(Rz)-\alpha f(rz)|$ $\displaystyle\geq|f(Rz)|-|\alpha||f(rz)|$
(2.5)
$\displaystyle>\left\\{\left(\frac{R+1}{r+1}\right)^{n}-|\alpha|\right\\}|f(rz)|,\,\,\,\,\,|z|=1.$

Also, inequality (2.4) can be written in the form

(2.6) $|f(re^{i\theta})|<\left(\frac{r+1}{R+1}\right)^{n}|f(Re^{i\theta})|$

for every $R>r\geq 1$ and $0\leq\theta<2\pi.$ Since $f(Re^{i\theta})\neq 0$
and $\left(\frac{r+1}{R+1}\right)^{n}<1$, from inequality (2.6), we obtain for
$0\leq\theta<2\pi$ and $R>r\geq 1$,

$|f(re^{i\theta})|<|f(Re^{i\theta})|.$

Equivalently,

$|f(rz)|<|f(Rz)|\,\,\,\textrm{for}\,\,\,\,|z|=1.$

Since all the zeros of $f(Rz)$ lie in $|z|\leq(1/R)<1$, a direct application
of Rouche’s theorem shows that the polynomial $f(Rz)-\alpha f(rz)$ has all its
zeros in $|z|<1$ for every $\alpha\in\mathbb{C}$ with $|\alpha|\leq 1$.
Applying Rouche’s theorem again, it follows from (2.4) that for
$\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,|\beta|\leq 1$ and $R>r\geq
1$, all the zeros of the polynomial

$\displaystyle T(z)=$ $\displaystyle f(Rz)-\alpha
f(rz)+\beta\left\\{\left(\frac{R+1}{r+1}\right)^{n}-|\alpha|\right\\}f(rz)$
$\displaystyle=f(Rz)+\phi_{n}\left(R,r,\alpha,\beta\right)f(rz)$
$\displaystyle=\big{(}P(Rz)-\lambda
F(Rz)\big{)}+\phi_{n}\left(R,r,\alpha,\beta\right)\big{(}P(rz)-\lambda
F(rz)\big{)}$
$\displaystyle=\big{(}P(Rz)+\phi_{n}\left(R,r,\alpha,\beta\right)P(rz)\big{)}-\lambda\big{(}F(Rz)+\phi_{n}\left(R,r,\alpha,\beta\right)F(rz)\big{)}$

lie in $|z|<1$ for every $\lambda\in\mathbb{C}$ with $|\lambda|>1$. Using
Lemma 2.2 and the fact that $B$ is a linear operator, we conclude that all the
zeros of polynomial

$\displaystyle W(z)$ $\displaystyle=B[T](z)$
$\displaystyle=(B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z))$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-\lambda(B[F\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[F\circ\rho](z))$

also lie in $|z|<1$ for every $\lambda$ with $|\lambda|>1$. This implies

(2.7)
$|B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)|\leq|B[F\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[F\circ\rho](z)|$

for $|z|\geq 1$ and $R>r\geq 1$. If inequality (2.7) is not true, then exist a
point $z=z_{0}$ with $|z_{0}|\geq 1$ such that

$|B[P\circ\sigma](z_{0})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z_{0})|>|B[F\circ\sigma](z_{0})+\phi_{n}\left(R,r,\alpha,\beta\right)B[F\circ\rho](z_{0})|.$

But all the zeros of $F(Rz)$ lie in $|z|<1$, therefore, it follows (as in case
of $f(z)$) that all the zeros of
$F(Rz)+\phi_{n}\left(R,r,\alpha,\beta\right)F(rz)$ lie in $|z|<1$. Hence by
Lemma 2.2, all the zeros of
$B[F\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[F\circ\rho](z)$
also lie in $|z|<1$, which shows that

$B[F\circ\sigma](z_{0})+\phi_{n}\left(R,r,\alpha,\beta\right)B[F\circ\rho](z_{0})\neq
0.$

We take

$\lambda=\frac{B[P\circ\sigma](z_{0})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z_{0})}{B[F\circ\sigma](z_{0})+\phi_{n}\left(R,r,\alpha,\beta\right)B[F\circ\rho](z_{0})},$

then $\lambda$ is a well defined real or complex number with $|\lambda|>1$ and
with this choice of $\lambda$, we obtain $W(z_{0})=0$. This contradicts the
fact that all the zeros of $W(z)$ lie in $|z|<1$. Thus (2.7) holds and this
completes the proof of Lemma 2.3. ∎

######  Lemma 2.4.

If $P\in\mathscr{P}_{n}$ and $P(z)$ has all its zeros in $|z|\leq 1,$ then for
every $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,|\beta|\leq 1$ and
$|z|\geq 1,$

(2.8)
$\displaystyle\left|B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)\right|\geq|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}||z|^{n}m$

where $m={\min}_{|z|=1}|P(z)|,$ $B\in\mathcal{B}_{n},$
$\sigma(z)=Rz,\,\rho(z)=rz,$ $\Lambda_{n}$ and
$\phi_{n}\left(R,r,\alpha,\beta\right)$ are defined by (1.12) and (1.7)
respectively.

###### Proof.

By hypothesis, all the zeros of $P(z)$ lie in $|z|\leq 1$ and

$m|z|^{n}\leq|P(z)|\,\,\,\,\textnormal{for}\,\,\,\,|z|=1.$

We first show that the polynomial $g(z)=P(z)-\lambda mz^{n}$ has all its zeros
in $|z|\leq 1$ for every $\lambda\in\mathbb{C}$ with $|\lambda|<1.$ This is
obvious if $m=0,$ that is if $P(z)$ has a zero on $|z|=1.$ Henceforth, we
assume $P(z)$ has all its zeros in $|z|<1,$ then $m>0$ and it follows by
Rouche’s theorem that the polynomial $g(z)$ has all its zeros in $|z|<1$ for
every $\lambda\in\mathbb{C}$ with $|\lambda|<1.$ Proceeding similarly as in
the proof of Lemma 2.3, we obtain that for $\alpha,\beta\in\mathbb{C}$ with
$|\alpha|\leq 1,|\beta|\leq 1$ and $R>r\geq 1$, all the zeros of the
polynomial

$\displaystyle H(z)=$ $\displaystyle g(Rz)-\alpha
g(rz)+\beta\left\\{\left(\frac{R+1}{r+1}\right)^{n}-|\alpha|\right\\}g(rz)$
$\displaystyle=g(Rz)+\phi_{n}\left(R,r,\alpha,\beta\right)g(rz)$
$\displaystyle=\big{(}P(Rz)-\lambda
R^{n}z^{n}m\big{)}+\phi_{n}\left(R,r,\alpha,\beta\right)\big{(}P(rz)-\lambda
r^{n}z^{n}m\big{)}$
$\displaystyle=\big{(}P(Rz)+\phi_{n}\left(R,r,\alpha,\beta\right)P(rz)\big{)}-\lambda\big{(}R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}\big{)}mz^{n}$

lie in $|z|<1.$ Applying Lemma 2.1 to $H(z)$ and noting that $B$ is a linear
operator, it follows that all the zeros of polynomial

$\displaystyle B[H](z)=$
$\displaystyle\left\\{B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)\right\\}$
(2.9)
$\displaystyle\qquad\qquad\qquad\qquad-\lambda\big{(}R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}\big{)}mB[z^{n}]$

lie in $|z|<1.$ This gives

$\displaystyle\left|B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)\right|$
(2.10)
$\displaystyle\qquad\qquad\qquad\qquad\geq|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}||z|^{n}m\,\,\,\,\,\textnormal{for}\,\,\,\,\,|z|\geq
1.$

If (2) is not true, then there is point $w$ with $|w|\geq 1$ such that

(2.11)
$\displaystyle\left|B[P\circ\sigma](w)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](w)\right|<|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}||w|^{n}m.$

We choose

$\lambda=\dfrac{B[P\circ\sigma](w)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](w)}{R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}||w|^{n}m.},$

then clearly $|\lambda|<1$ and with this choice of $\lambda,$ from (2), we get
$B[H](w)=0$ with $|w|\geq 1.$ This is clearly a contradiction to the fact that
all the zeros of $H(z)$ lie in $|z|<1.$ Thus for every
$\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,$ $|\beta|\leq 1,$

$\left|B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)\right|\geq|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}||z|^{n}m$

for $|z|\geq 1$ and $R>r\geq 1.$ ∎

######  Lemma 2.5.

If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in $\left|z\right|<1$,
then for every $\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,|\beta|\leq
1,R>r\geq 1$ and $|z|\geq 1,$

$\displaystyle|B[P\circ\sigma](z)+$
$\displaystyle\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)|$ (2.12)
$\displaystyle\leq|B[P^{*}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{*}\circ\rho](z)|$

where $P^{*}(z):=z^{n}\overline{P(1/\overline{z})}$, $B\in\mathcal{B}_{n}$,
$\sigma(z):=Rz$, $\rho(z):=rz$, and $\phi_{n}\left(R,r,\alpha,\beta\right)$ is
defined by (1.7).

###### Proof.

By hypothesis the polynomial $P(z)$ of degree $n$ does not vanish in $|z|<1$,
therefore, all the zeros of the polynomial
$P^{*}(z)=z^{n}\overline{P(1/\overline{z})}$ of degree $n$ lie in $|z|\leq 1$.
Applying Lemma 2.3 with $F(z)$ replaced by $P^{*}(z)$, it follows that

$\displaystyle\left|B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)\right|$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\leq\left|B[P^{*}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{*}\circ\rho](z)\right|$

for $|z|\geq 1,|\alpha|\leq 1,|\beta|\leq 1$ and $R>r\geq 1$. This proves the
Lemma 2.5. ∎

######  Lemma 2.6.

If $P\in\mathscr{P}_{n}$ and $P(z)$ has no zero in $\left|z\right|<1,$ then
for every $\alpha\in\mathbb{C}$ with $|\alpha|\leq 1,$ $R>r\geq 1$ and
$|z|\geq 1$,

$\displaystyle\left|B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)\right|$
$\displaystyle\qquad\leq\left|B[P^{\star}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{\star}\circ\rho](z)\right|$
(2.13)
$\displaystyle\qquad\quad-\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m,$

where $P^{\star}(z)=z^{n}\overline{P(1/\overline{z})},$
$m={\min}_{|z|=1}|P(z)|,$ $B\in\mathcal{B}_{n},$ $\sigma(z)=Rz,$ $\rho(z)=rz,$
$\Lambda_{n}$ and $\phi_{n}\left(R,r,\alpha,\beta\right)$ are given by (1.12)
and (1.7) respectively.

###### Proof.

By hypothesis $P(z)$ has all its zeros in $|z|\geq 1$ and

(2.14) $\displaystyle m\leq|P(z)|\,\,\,\textnormal{for}\,\,\,\,|z|=1.$

We show $F(z)=P(z)+\lambda m$ does not vanish in $|z|<1$ for every
$\lambda\in\mathbb{C}$ with $|\lambda|<1.$ This is obvious if $m=0$ that is,
if $P(z)$ has a zero on $|z|=1.$ So we assume all the zeros of $P(z)$ lie in
$|z|>1,$ then $m>0$ and by the maximum modulus principle, it follows from
(2.14),

(2.15) $\displaystyle m<|P(z)|\,\,\,\textnormal{for}\,\,\,|z|<1.$

Now if $F(z)=P(z)+\lambda m=0$ for some $z_{0}$ with $|z_{0}|<1,$ then

$\displaystyle P(z_{0})+\lambda m=0$

This implies

(2.16) $\displaystyle|P(z_{0})|=|\lambda|m\leq
m,\,\,\,\textnormal{for}\,\,\,|z_{0}|<1$

which is clearly contradiction to (2.15). Thus the polynomial $F(z)$ does not
vanish in $|z|<1$ for every $\lambda$ with $|\lambda|<1.$ Applying Lemma 2.3
to the polynomial $F(z),$ we get

$\displaystyle|B[F\circ\sigma](z)+$
$\displaystyle\phi_{n}\left(R,r,\alpha,\beta\right)B[F\circ\rho](z)|$
$\displaystyle\leq|B[F^{\star}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[F^{\star}\circ\rho](z)|$

for $|z|=1$ and $R>r\geq 1.$ Replacing $F(z)$ by $P(z)+\lambda m,$ we obtain

$\displaystyle|B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)$
$\displaystyle
B[P\circ\rho](z)+\lambda(1+\phi_{n}\left(R,r,\alpha,\beta\right))\lambda_{0}m|$
$\displaystyle\leq|B[P^{\star}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{\star}\circ\rho](z)$
(2.17)
$\displaystyle\qquad\qquad\qquad+\bar{\lambda}(R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n})\Lambda_{n}z^{n}m|$

Now choosing the argument of $\lambda$ in the right hand side of (2) such that

$\displaystyle|B[P^{\star}\circ\sigma](z)+\phi_{n}$
$\displaystyle\left(R,r,\alpha,\beta\right)B[P^{\star}\circ\rho](z)+\bar{\lambda}(R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n})\Lambda_{n}z^{n}m|$
$\displaystyle=|B[P^{\star}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{\star}\circ\rho](z)|$
$\displaystyle\qquad\qquad\qquad-|\bar{\lambda}||R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}||z|^{n}m$

for $|z|=1,$which is possible by Lemma 2.4,we get

$\displaystyle|B[P\circ\sigma](z)$
$\displaystyle+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)|-|\lambda||1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|m$
$\displaystyle\leq$
$\displaystyle|B[P^{\star}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{\star}\circ\rho](z)|$
$\displaystyle\qquad\qquad\qquad-|\lambda||R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}||z|^{n}m$

Equivalently,

$\displaystyle|B[P\circ\sigma](z)$
$\displaystyle+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)|$
$\displaystyle\leq$
$\displaystyle|B[P^{\star}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{\star}\circ\rho](z)|$
(2.18)
$\displaystyle\quad-|\lambda|\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m.$

Letting $|\lambda|\rightarrow 1$ in (2) we obtain inequality (2.6) and this
completes the proof of Lemma 2.6. ∎

Next we describe a result of Arestov [2].

For
$\gamma=\left(\gamma_{0},\gamma_{1},\cdots,\gamma_{n}\right)\in\mathbb{C}^{n+1}\,\,\,\,\mbox{}\,\,\,\text{and}\,\,\,P(z)=\sum_{j=0}^{n}a_{j}z^{j}$,
we define

$C_{\gamma}P(z)=\sum_{j=0}^{n}\gamma_{j}a_{j}z^{j}.$

The operator $C_{\gamma}$ is said to be admissible if it preserves one of the
following properties:

  1. (i)

$P(z)$ has all its zeros in $\left\\{z\in\mathbb{C}:|z|\leq 1\right\\}$,

  2. (ii)

$P(z)$ has all its zeros in $\left\\{z\in\mathbb{C}:|z|\geq 1\right\\}$.

The result of Arestov may now be stated as follows.

######  Lemma 2.7.

[2, Theorem 2] Let $\phi(x)=\psi(\log x)$ where $\psi$ is a convex non-
decreasing function on $\mathbb{R}$. Then for all $P\in\mathscr{P}_{n}$ and
each admissible operator $\Lambda_{\gamma}$,

$\int_{0}^{2\pi}\phi\left(|C_{\gamma}P(e^{i\theta})|\right)d\theta\leq\int_{0}^{2\pi}\phi\left(c(\gamma,n)|P(e^{i\theta})|\right)d\theta$

where $c(\gamma,n)=\max\left(|\gamma_{0}|,|\gamma_{n}|\right)$.

In particular Lemma 2.7 applies with $\phi:x\rightarrow x^{p}$ for every
$p\in(0,\infty)$ and $\phi:x\rightarrow\log x$ as well. Therefore, we have for
$0\leq p<\infty$,

(2.19)
$\left\\{\int_{0}^{2\pi}\phi\left(|C_{\gamma}P(e^{i\theta})|^{p}\right)d\theta\right\\}^{1/p}\leq
c(\gamma,n)\left\\{\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta\right\\}^{1/p}.$

From Lemma 2.7, we deduce the following result.

######  Lemma 2.8.

If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in $\left|z\right|<1$,
then for each $p>0$, $R>1$ and $\eta$ real, $0\leq\eta<2\pi$,

$\displaystyle\int_{0}^{2\pi}|\big{(}B[P\circ\sigma](e^{i\theta})$
$\displaystyle+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](e^{i\theta})\big{)}e^{i\eta}$
$\displaystyle+\big{(}B[P^{*}\circ\sigma]^{*}(e^{i\theta})+\phi_{n}(R,r,\bar{\alpha},\bar{\beta})B[P^{*}\circ\rho]^{*}(e^{i\theta})\big{)}|^{p}d\theta$
$\displaystyle\leq|(R^{n}+\phi_{n}(R,r,$
$\displaystyle\alpha,\beta)r^{n})\Lambda_{n}e^{i\eta}+(1+\phi_{n}(R,r,\bar{\alpha},\bar{\beta}))\bar{\lambda_{0}}|^{p}\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta$

where $B\in\mathcal{B}_{n}$, $\sigma(z):=Rz$, $\rho(z):=rz$,
$B[P^{*}\circ\sigma]^{*}(z):=(B[P^{*}\circ\sigma](z))^{*}$, $\Lambda_{n}$ and
$\phi_{n}\left(R,r,\alpha,\beta\right)$ are defined by (1.12) and (1.7)
respectively.

###### Proof.

Since $P(z)$ does not vanish in $\left|z\right|<1$ and
$P^{*}(z)=z^{n}\overline{P(1/\bar{z})}$, by Lemma 2.5, we have for $R>r\geq
1$,

$\displaystyle|B[P\circ\sigma](z)+\phi_{n}$
$\displaystyle\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)|$ (2.20)
$\displaystyle\leq|B[P^{*}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{*}\circ\rho](z)|$

Also, since
$P^{*}(Rz)+\phi_{n}\left(R,r,\alpha,\beta\right)P^{*}(rz)=R^{n}z^{n}\overline{P(1/R\bar{z})}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}z^{n}\overline{P(1/r\bar{z})}$,
therefore,

$\displaystyle
B[P^{*}\circ\sigma](z)+\phi_{n}(R,r,\alpha,\beta)B[P^{*}\circ\rho](z)$
$\displaystyle=\lambda_{0}\big{(}R^{n}z^{n}\overline{P(1/R\bar{z})}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}z^{n}\overline{P(1/r\bar{z})}\big{)}+\lambda_{1}\left(\frac{nz}{2}\right)\Big{(}nR^{n}z^{n-1}\overline{P(1/R\bar{z})}$
$\displaystyle\quad-R^{n-1}z^{n-2}\overline{P^{\prime}(1/R\bar{z})}+\phi_{n}\left(R,r,\alpha,\beta\right)\big{(}nr^{n}z^{n-1}\overline{P(1/r\bar{z})}-r^{n-1}z^{n-2}\overline{P^{\prime}(1/r\bar{z})}\big{)}\Big{)}$
$\displaystyle\quad+\frac{\lambda_{2}}{2!}\left(\frac{nz}{2}\right)^{2}\Big{(}n(n-1)R^{n}z^{n-2}\overline{P(1/R\bar{z})}-2(n-1)R^{n-1}z^{n-3}\overline{P^{\prime}(1/R\bar{z})}$
$\displaystyle\quad+R^{n-2}z^{n-4}\overline{P^{\prime\prime}(1/R\bar{z})}+\phi_{n}\left(R,r,\alpha,\beta\right)\big{(}n(n-1)r^{n}z^{n-2}\overline{P(1/r\bar{z})}$
$\displaystyle\quad-2(n-1)r^{n-1}z^{n-3}\overline{P^{\prime}(1/r\bar{z})}+r^{n-2}z^{n-4}\overline{P^{\prime\prime}(1/r\bar{z})}\big{)}\Big{)}$

and hence,

$\displaystyle B[P^{*}\circ\sigma]^{*}$
$\displaystyle(z)+\phi\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{*}\circ\rho]^{*}(z)$
$\displaystyle=$
$\displaystyle\big{(}B[P^{*}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{*}\circ\rho](z)\big{)}^{*}$
$\displaystyle=$
$\displaystyle\left(\bar{\lambda_{0}}+\bar{\lambda_{1}}\frac{n^{2}}{2}+\bar{\lambda_{2}}\frac{n^{3}(n-1)}{8}\right)\left(R^{n}P(z/R)+\phi\left(R,r,\bar{\alpha},\bar{\beta}\right)r^{n}P(z/r)\right)$
$\displaystyle-\left(\bar{\lambda_{1}}\frac{n}{2}+\bar{\lambda_{2}}\frac{n^{2}(n-1)}{4}\right)\Big{(}R^{n-1}zP^{\prime}(z/R)+\phi\left(R,r,\bar{\alpha},\bar{\beta}\right)r^{n-1}zP^{\prime}(z/r)\Big{)}$
(2.21)
$\displaystyle+\bar{\lambda_{2}}\frac{n^{2}}{8}\Big{(}R^{n-2}z^{2}P^{\prime\prime}(z/R)+\phi\left(R,r,\bar{\alpha},\bar{\beta}\right)r^{n-2}z^{2}P^{\prime\prime}(z/r)\Big{)}.$

Also, for $|z|=1$

$\displaystyle|B[P^{*}\circ\sigma](z)+$
$\displaystyle\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{*}\circ\rho](z)|$
$\displaystyle=|B[P^{*}\circ\sigma]^{*}(z)+\phi\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{*}\circ\rho]^{*}(z)|.$

Using this in (2), we get for $|z|=1$ and $R>r\geq 1$,

$\displaystyle|B[P\circ\sigma](z)+$
$\displaystyle\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)|$
$\displaystyle\leq|B[P^{*}\circ\sigma]^{*}(z)+\phi\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{*}\circ\rho]^{*}(z)|.$

Since all the zeros of $P^{*}(z)$ lie in $|z|\leq 1$, as before, all the zeros
of $P^{*}(Rz)+\phi_{n}(R,r,\alpha,\beta)P^{*}(rz)$ lie in $|z|<1$ for all real
or complex numbers $\alpha,\beta$ with $|\alpha|\leq 1$, $|\beta|\leq 1$ and
$R>r\geq 1$. Hence by Lemma 2.2, all the zeros of
$B[P^{*}\circ\sigma](z)+\phi_{n}(R,r,\alpha,\beta)B[P^{*}\circ\rho](z)$ lie in
$|z|<1$, therefore, all the zeros of
$B[P^{*}\circ\sigma]^{*}(z)+\phi_{n}(R,r,\bar{\alpha},\bar{\beta})B[P^{*}\circ\rho]^{*}(z)$
lie in $|z|>1$. Hence by the maximum modulus principle,

$\displaystyle|B[P\circ\sigma](z)+$
$\displaystyle\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{*}\circ\rho](z)|$
(2.22)
$\displaystyle<|B[P^{*}\circ\sigma]^{*}(z)+\phi\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{*}\circ\rho]^{*}(z)|\quad\textrm{for}\quad|z|<1.$

A direct application of Rouche’s theorem shows that

$\displaystyle C_{\gamma}P(z)=$
$\displaystyle\big{(}B[P\circ\sigma](z)+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](z)\big{)}e^{i\eta}$
$\displaystyle+\big{(}B[P^{*}\circ\sigma]^{*}(z)+\phi_{n}(R,r,\bar{\alpha},\bar{\beta})B[P^{*}\circ\rho]^{*}(z)\big{)}$
$\displaystyle=$
$\displaystyle\left\\{(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n})\Lambda_{n}e^{i\eta}+(1+\phi_{n}(R,r,\bar{\alpha},\bar{\beta}))\bar{\lambda_{0}}\right\\}a_{n}z^{n}$
$\displaystyle+\cdots+\left\\{(R^{n}+\phi_{n}(R,r,\bar{\alpha},\bar{\beta})r^{n})\bar{\Lambda_{n}}+e^{i\eta}(1+\phi_{n}(R,r,\alpha,\beta))\lambda_{0}\right\\}a_{0}$

does not vanish in $|z|<1$. Therefore, $C_{\gamma}$ is an admissible operator.
Applying (2.19) of Lemma 2.7, the desired result follows immediately for each
$p>0$. ∎

We also need the following lemma [4].

######  Lemma 2.9.

If $A,B,C$ are non-negative real numbers such that $B+C\leq A,$ then for each
real number $\gamma,$

$|(A-C)e^{i\gamma}+(B+C)|\leq|Ae^{i\gamma}+B|.$

##  3\. Proof of the Theorems

###### Proof of Theorem 1.2.

By hypothesis $P(z)$ does not vanish in $|z|<1,$ therefore by Lemma 2.6, we
have

$\displaystyle\left|B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)\right|$
$\displaystyle\qquad\leq\left|B[P^{\star}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{\star}\circ\rho](z)\right|$
(3.1)
$\displaystyle\qquad\quad-\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m,$

for $|z|=1,$ $|\alpha|\leq 1$ and $R>r\geq 1$ where
$P^{\star}(z)=z^{n}\overline{P(1/\overline{z})}.$ Since
$B[P^{\star}\circ\sigma]^{\star}(z)+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(z)$
is the conjugate of
$B[P^{\star}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{\star}\circ\rho](z)$
and

$\displaystyle|B[P^{\star}\circ\sigma]^{\star}(z)$
$\displaystyle+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(z)|$
$\displaystyle=|B[P^{\star}\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P^{\star}\circ\rho](z)|$

Thus (3) can be written as

$\displaystyle\big{|}B$
$\displaystyle[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)\big{|}$
$\displaystyle\qquad\qquad+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}$
$\displaystyle\leq\left|B[P^{\star}\circ\sigma]^{\star}(z)+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(z)\right|$
(3.2)
$\displaystyle\qquad\qquad-\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2},$

for $|z|=1.$ Taking

$A=\left|B[P^{\star}\circ\sigma]^{\star}(z)+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(z)\right|$
$B=\big{|}B[P\circ\sigma](z)+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](z)\big{|},$

and

$C=\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}$

in Lemma 2.9 and noting by (3) that

$B+C\leq A-C\leq A,$

we get for every real $\gamma$,

$\displaystyle\Bigg{|}\Bigg{\\{}\big{|}B[P^{\star}$
$\displaystyle\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}$
$\displaystyle\qquad-\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{\\}}e^{i\gamma}$
$\displaystyle+\Bigg{\\{}\big{|}B$
$\displaystyle[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}$
$\displaystyle\qquad+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{|}$
$\displaystyle\leq$
$\displaystyle\Big{|}\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}e^{i\gamma}$
$\displaystyle\qquad\qquad\qquad\qquad\quad+\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}\Big{|}.$

This implies for each $p>0,$

$\displaystyle\int\limits_{0}^{2\pi}\Bigg{|}\Bigg{\\{}\big{|}B[P^{\star}$
$\displaystyle\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}$
$\displaystyle\qquad-\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{\\}}e^{i\gamma}$
$\displaystyle+\Bigg{\\{}\big{|}B$
$\displaystyle[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}$
$\displaystyle\qquad+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{|}^{p}d\theta$
$\displaystyle\leq\int\limits_{0}^{2\pi}$
$\displaystyle\Big{|}\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}e^{i\gamma}$
(3.3)
$\displaystyle\quad\quad\qquad\qquad\quad+\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}\Big{|}^{p}d\theta.$

Integrating both sides of (3) with respect to $\gamma$ from $0$ to $2\pi,$ we
get with the help of Lemma 2.8 for each $p>0,$

$\displaystyle\int\limits_{0}^{2\pi}\int\limits_{0}^{2\pi}\Bigg{|}\Bigg{\\{}\big{|}B[P^{\star}$
$\displaystyle\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}$
$\displaystyle-\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{\\}}e^{i\gamma}$
$\displaystyle+\Bigg{\\{}\big{|}B$
$\displaystyle[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}$
$\displaystyle+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{|}^{p}d\theta
d\gamma$
$\displaystyle\leq\int\limits_{0}^{2\pi}\int\limits_{0}^{2\pi}\Big{|}\big{|}B[$
$\displaystyle
P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}e^{i\gamma}$
$\displaystyle\quad\quad\qquad\qquad+\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}\Big{|}^{p}d\theta
d\gamma$
$\displaystyle\leq\int\limits_{0}^{2\pi}\Bigg{\\{}\int\limits_{0}^{2\pi}\Big{|}$
$\displaystyle\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}e^{i\gamma}$
$\displaystyle\quad\quad\qquad\qquad+\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}\Big{|}^{p}d\gamma\Bigg{\\}}\theta$
$\displaystyle\leq\int\limits_{0}^{2\pi}\Bigg{\\{}\int\limits_{0}^{2\pi}$
$\displaystyle\Bigg{|}\Big{(}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\Big{)}e^{i\gamma}$
$\displaystyle\quad\quad\qquad\qquad+\Big{(}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\Big{)}\Bigg{|}^{p}d\gamma\Bigg{\\}}\theta$
$\displaystyle\leq\int\limits_{0}^{2\pi}\Bigg{\\{}\int\limits_{0}^{2\pi}$
$\displaystyle\Bigg{|}\Big{(}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\Big{)}e^{i\gamma}$
$\displaystyle\quad\quad\qquad\qquad+\Big{(}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\Big{)}\Bigg{|}^{p}d\theta\Bigg{\\}}\gamma$
$\displaystyle\leq\int\limits_{0}^{2\pi}$
$\displaystyle\Big{|}(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n})\Lambda_{n}e^{i\gamma}+(1+\phi_{n}(R,r,\bar{\alpha},\bar{\beta}))\bar{\lambda_{0}}\Big{|}^{p}d\gamma$
(3.4)
$\displaystyle\qquad\qquad\qquad\qquad\times\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta$

Now it can be easily verified that for every real number $\gamma$ and $s\geq
1$,

$\left|s+e^{i\alpha}\right|\geq\left|1+e^{i\alpha}\right|.$

This implies for each $p>0$,

(3.5)
$\int_{0}^{2\pi}\left|s+e^{i\gamma}\right|^{p}d\gamma\geq\int_{0}^{2\pi}\left|1+e^{i\gamma}\right|^{p}d\gamma.$

If

$\displaystyle\big{|}B[P\circ\sigma]$
$\displaystyle(e^{i\theta})+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](e^{i\theta})\big{|}$
$\displaystyle+$
$\displaystyle\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\neq
0,$

we take

$s=\dfrac{\begin{multlined}\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}\\\
+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\end{multlined}\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}\\\
+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}}{\begin{multlined}\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](e^{i\theta})\big{|}\\\
+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\end{multlined}\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](e^{i\theta})\big{|}\\\
+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}},$

then by (3), $s\geq 1$ and we get with the help of (3.5),

$\displaystyle\int\limits_{0}^{2\pi}\Bigg{|}\Bigg{\\{}$
$\displaystyle\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}$
$\displaystyle\quad\quad-\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{\\}}e^{i\gamma}$
$\displaystyle+\Bigg{\\{}$
$\displaystyle\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}$
$\displaystyle\quad\quad+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{\\}}\Bigg{|}^{p}d\gamma$
$\displaystyle=\Bigg{|}$
$\displaystyle\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}$
$\displaystyle\quad\quad+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{|}^{p}$
(3.10) $\displaystyle\times\int\limits_{0}^{2\pi}$
$\displaystyle\left|e^{i\gamma}+\dfrac{\begin{multlined}\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}\\\
-\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\end{multlined}\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}\\\
-\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}}{\begin{multlined}\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](e^{i\theta})\big{|}\\\
+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\end{multlined}\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](e^{i\theta})\big{|}\\\
+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}}\right|^{p}d\gamma$
$\displaystyle=\Bigg{|}$
$\displaystyle\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}$
$\displaystyle\quad\quad+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{|}^{p}$
(3.15) $\displaystyle\times\int\limits_{0}^{2\pi}$
$\displaystyle\left|e^{i\gamma}+\left|\dfrac{\begin{multlined}\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}\\\
-\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\end{multlined}\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})+\phi_{n}\left(R,r,\bar{\alpha},\bar{\beta}\right)B[P^{\star}\circ\rho]^{\star}(e^{i\theta})\big{|}\\\
-\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}}{\begin{multlined}\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](e^{i\theta})\big{|}\\\
+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\end{multlined}\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](e^{i\theta})\big{|}\\\
+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}}\right|^{p}\,\right|d\gamma$
$\displaystyle\geq$
$\displaystyle\Bigg{|}\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}$
(3.16) $\displaystyle+$
$\displaystyle\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{|}^{p}\int\limits_{0}^{2\pi}|1+e^{i\gamma}|^{p}d\gamma.$

For

$\displaystyle\big{|}B[P\circ\sigma]$
$\displaystyle(e^{i\theta})+\phi_{n}(R,r,\alpha,\beta)B[P\circ\rho](e^{i\theta})\big{|}$
$\displaystyle+$
$\displaystyle\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\neq
0,$

then (3) is trivially true. Using this in (3), we conclude for every
$\alpha,\beta\in\mathbb{C}$ with $|\alpha|\leq 1,|\beta|\leq 1$ $R>r\geq 1$
and $p>0$,

$\displaystyle\int\limits_{0}^{2\pi}$
$\displaystyle\Bigg{|}\big{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})\big{|}$
$\displaystyle+\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{|}^{p}d\theta\int\limits_{0}^{2\pi}|1+e^{i\gamma}|^{p}d\gamma$
$\displaystyle\leq\int\limits_{0}^{2\pi}\Big{|}(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n})\Lambda_{n}e^{i\gamma}+(1+\phi_{n}(R,r,\bar{\alpha},\bar{\beta}))\bar{\lambda_{0}}\Big{|}^{p}d\gamma\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta$

This gives for every $\delta,\alpha,\beta$ with $|\delta|\leq 1,$
$|\alpha|\leq 1,$ $|\beta|\leq 1,$ $R>r\geq 1$ and $\gamma$ real

$\displaystyle\int\limits_{0}^{2\pi}$
$\displaystyle\Bigg{|}B[P\circ\sigma](e^{i\theta})+\phi_{n}\left(R,r,\alpha,\beta\right)B[P\circ\rho](e^{i\theta})$
$\displaystyle+\delta$
$\displaystyle\dfrac{\Big{(}|R^{n}+\phi_{n}\left(R,r,\alpha,\beta\right)r^{n}||\Lambda_{n}|-|1+\phi_{n}\left(R,r,\alpha,\beta\right)||\lambda_{0}|\Big{)}m}{2}\Bigg{|}^{p}d\theta\int\limits_{0}^{2\pi}|1+e^{i\gamma}|^{p}d\gamma$
(3.17)
$\displaystyle\leq\int\limits_{0}^{2\pi}\Big{|}(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n})\Lambda_{n}e^{i\gamma}+(1+\phi_{n}(R,r,\bar{\alpha},\bar{\beta}))\bar{\lambda_{0}}\Big{|}^{p}d\gamma\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta$

Since

$\displaystyle\int\limits_{0}^{2\pi}$
$\displaystyle\Big{|}(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n})\Lambda_{n}e^{i\gamma}+(1+\phi_{n}(R,r,\bar{\alpha},\bar{\beta}))\bar{\lambda_{0}}\Big{|}^{p}d\gamma\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta$
$\displaystyle=\int\limits_{0}^{2\pi}\Big{|}|(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n})\Lambda_{n}|e^{i\gamma}+|(1+\phi_{n}(R,r,\bar{\alpha},\bar{\beta}))\bar{\lambda_{0}}|\Big{|}^{p}d\gamma\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta$
$\displaystyle=\int\limits_{0}^{2\pi}\Big{|}|(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n})\Lambda_{n}|e^{i\gamma}+|(1+\phi_{n}(R,r,\alpha,\beta))\lambda_{0}|\Big{|}^{p}d\gamma\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta,$
(3.18)
$\displaystyle=\int\limits_{0}^{2\pi}\Big{|}(R^{n}+\phi_{n}(R,r,\alpha,\beta)r^{n})\Lambda_{n}e^{i\gamma}+(1+\phi_{n}(R,r,\alpha,\beta))\lambda_{0}\Big{|}^{p}d\gamma\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta,$

the desired result follows immediately by combining (3) and (3). This
completes the proof of Theorem 1.2 for $p>0$. To establish this result for
$p=0$, we simply let $p\rightarrow 0+$.

∎

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# Variational Autoencoder Learns Better Feature Representations for EEG-based
Obesity Classification

Yuan Yue  Department of Information Science
University of Otago)
Dunedin, New Zealand
<EMAIL_ADDRESS>Jeremiah D. Deng  Department of Information
Science
University of Otago
Dunedin, New Zealand
<EMAIL_ADDRESS>Dirk De Ridder  Department of Surgical Science
University of Otago
Dunedin, New Zealand
<EMAIL_ADDRESS>Patrick Manning  Department of Medicine
University of Otago
Dunedin, New Zealand
<EMAIL_ADDRESS>Matt Hall  Department of Surgical Science
University of Otago
Dunedin, New Zealand
<EMAIL_ADDRESS>Samantha Ross  Department of Medicine
University of Otago
Dunedin, New Zealand
<EMAIL_ADDRESS>Divya Adhia  Department of Surgical Science
University of Otago
Dunedin, New Zealand
<EMAIL_ADDRESS>
###### Abstract

Obesity is a common issue in modern societies today that can lead to various
diseases and significantly reduced quality of life. Currently, research has
been conducted to investigate resting state EEG (electroencephalogram) signals
with an aim to identify possible neurological characteristics associated with
obesity. In this study, we propose a deep learning-based framework to extract
the resting state EEG features for obese and lean subject classification.
Specifically, a novel variational autoencoder framework is employed to extract
subject-invariant features from the raw EEG signals, which are then classified
by a 1-D convolutional neural network. Comparing with conventional machine
learning and deep learning methods, we demonstrate the superiority of using
VAE for feature extraction, as reflected by the significantly improved
classification accuracies, better visualizations and reduced impurity measures
in the feature representations. Future work can be directed to gaining an in-
depth understanding regarding the spatial patterns that have been learned by
the proposed model from a neurological view, as well as improving the
interpretability of the proposed model by allowing it to uncover any temporal-
related information.

###### Index Terms:

deep learning, EEG, classification, variational autoencoder

##  I Introduction

Obesity is a worldwide health problem today that is associated to the
dysfunction of various body systems including the heart, the liver, kidneys,
joints, and the reproductive system [1, 2, 3, 4]. It is also the cause of
various diseases such as type 2 diabetes, cardiovascular diseases and cancers
[5].

While many researchers are investigating obesity-related clinical
characteristics, attention has been increasingly paid to the effect of obesity
on the neurological perspective [6, 7]. Studies have shown that obesity is
associated with cognitive impairment and altered structure of various brain
networks [8]. For example, it is suggested that the abnormal brain connection
in the somatosensory cortex and insula areas of obese individuals make them
less capable of predicting the energy need, thus, tending to overtake food
[9]. On the other hand, it is found that an altered hippocampal structure,
which highly correlates to the Alzheimer’s type dementia, is observed in obese
individuals [10].

As a non-invasive technique for brain activity recording, EEG technology has
been extensively used to study various brain activity patterns, including
obesity-related neurological patterns [11, 12, 13]. For instance, in [14], EEG
data are collected from a group of healthy participants and obese
participants. Traditional statistical analysis was then applied to investigate
the difference in brain activities between obese and healthy people. Their
results suggest that psychopathological mechanisms that are similar to
substance-related disorders and addictive disorders are observed in obese
brains. These mechanisms, which include an increased spectral power or
functional connectivity within a certain frequency range, suggest a different
activation mechanism of the cognitive and emotional process when exposing to
an environment which has food-related cues between healthy people and obese
people.

On the other hand, substantial progress has been made in the machine learning
and deep learning field in recent decades. This has significantly improved the
efficiency of analyzing any EEG-related tasks. However, to our best knowledge,
few studies have applied machine learning to study obesity-related brain
activities. A traditional machine learning approach was proposed in [9] to
identify distinctive EEG patterns of obese individuals, where source
localisation was performed to select the region of interest, and functional
connectivity features were used to train a classifier based on a support
vector machine (SVM) with radial basis function (RBF) kernels. An average
classification accuracy of 0.912 was reported.

In this paper, we explore obese brain activities by examining raw EEG data
recorded in resting state using a deep learning approach. The novelty of our
study includes:

  * •

Our study is the first one that focuses on identifying obese brain activities
by examining resting state EEG data using a deep learning-based framework;

  * •

We demonstrated the superior effectiveness of using a Variational Autoencoder
(VAE) model to extract features in resting state EEG classification tasks;

  * •

In additional to favourable visualization outcomes, we proposed a quantitative
measure based on impurity to evaluate the separability of the feature space,
where the superiority of VAE features was further confirmed. These added to
the explainability of our proposed model.

The overall process of our study is demonstrated in Fig 1. After data
collection and preprocessing, we first use an unsupervised VAE to learn
meaningful feature representations (i.e. the output of the encoder), a 1-D
convolutional neural network (CNN) is then applied to perform the
classification task using the feature representation as input.

The rest of the paper is structured as follows: In Section II we briefly
introduce the related works and terminologies that will be referred to in our
study. The proposed model and detailed experiment procedures including data
description, the implementation of the proposed model, as well as the
classification process, are discussed in Section III. Then, in Section IV we
evaluate the performance of the proposed model, and compare it with baseline
models. Furthermore, we present the outcome of visualization and impurity
measures on separability of the acquired latent features. The paper is
concluded in Section V.

Figure 1: Overall process of the proposed work.

##  II Related Work

###  II-A Variational Autoencoder (VAE)

Autoencoders are neural networks that can learn features from the input data
in an unsupervised way [15]. It consists of two main parts: an encoder and a
decoder. Both parts have a similar structure to a CNN without the final
classification layer. The encoder compresses the input data and learns useful
features through non-linear dimension reduction in the latent space, whereas
the decoder decompresses the features to reconstruct the input data. The
objective function of an autoencoder can be formally written as:

$\mathcal{L}_{\mathrm{AE}}=E(\|\mathbf{x}-\hat{\mathbf{x}}\|^{2}),$ (1)

where $\mathbf{x}$ is the input data sample and $\hat{\mathbf{x}}$ is the
corresponding reconstruction of the input data sample.

The biggest drawback of autoencoder is its strong tendency to overfit [16], as
it is solely trained to encode and decode with as little loss as possible
regardless of how the latent space is organized. To address this problem and
to turn an autoencoder into a generative model, VAE has been developed [17]
and found as effective solutions [16, 18]. The aim of VAE is to ensure that
the latent space is regular enough, therefore the training process can be
regularized to avoid overfitting. To achieve this regularization in the latent
space, VAE modified the traditional autoencoder by encoding an input data
sample as a distribution in the latent space. The decoder then performs the
decompressing process using data points that have been sampled from the latent
space. Moreover, the regularization in the latent space is implemented by
enforcing the encoded latent variables toward a Gaussian distribution. Thus,
the objective function of a VAE contains two parts: minimizing the
reconstruction error which makes the encoding-decoding process as precise as
possible, and forcing the distribution outputted by the encoder as close as
possible to a Gaussian distribution. The loss function of a VAE can be
formally written as:

$\mathcal{L}_{\mathrm{VAE}}=E(\|\mathbf{x}-\hat{\mathbf{x}}\|^{2})+D_{\mathrm{KL}}(q(\mathbf{z}|\mathbf{x})||p(\mathbf{z})),$
(2)

where $\mathbf{z}$ is the latent variable, $q(\mathbf{z}|\mathbf{x})$ is the
latent distribution generated by the encoder, $p(\mathbf{z})$ is the prior
distribution in the latent space which follows a Gaussian distribution, and
$D_{\mathrm{KL}}(.)$ denotes the Kullback-Leibler divergence.

VAE has been increasingly used in EEG classification tasks to learn robust
features [18, 11, 19]. For instance, [20] proposed a channel-wised VAE-based
convolutional neural network to classify motor imagery dataset that has 4
classes. The VAE here comprises 1 input layer, 5 hidden layers that follow a
deep-wise convolution structure, and 1 fully connected layer. By employing the
channel-wised VAE model, an average classification accuracy of 0.83 is
achieved comparing to the accuracy score of 0.69 achieved by using the basic
EEGNet.

###  II-B Convolutional Neural Network (CNN)

A CNN usually consists of multiple convolution-pooling layer pairs followed by
a fully connected layer. Layers with other functions such as the batch
normalization layer and activation layer can be added depending on the
modeling needs. The convolution layer is where the majority of computation has
been performed. Here, one or more kernels (i.e., filters) slide over the input
data passing by the previous layer. For each filter, the dot product (i.e.,
feature map) between the input data and the kernel is then computed. Next, the
learned feature maps are passed to a pooling layer. In this layer, the number
of parameters is reduced by taking the average or the maximum of each of the
$N$ parameters ($N$ is a user-defined integer greater than 0). Finally, the
pooled features are fed into a fully connected layer, at which each of the
input is connected to the activation unit and the final class of each data
sample can be predicted.

###  II-C EEGNet

Figure 2: Architecture of EEGNet.

EEGNet [21] is an extensively used deep learning model for EEG classification
tasks due to its advance in learning various EEG representations [22, 11, 23],
as well as its capability of achieving a relatively good performance while
requiring low computation cost. Many studies have performed EEG classification
tasks by employing EEGNet-based models with task-specific customization added
[11, 24, 22]. For example, in [24], neural structured learning (NSL) is
combined with EEGNet to classify binary motor imagery dataset. Their model
allows the relational information and structural similarities contained in the
input signal to be well maintained in the training process. Regarding their
experiment results, the basic EEGNet model demonstrated its efficiency by
achieving a classification accuracy of 0.722 when it is applied alone, with a
small increment of 0.039 after adding the NSL.

The architecture of EEGNet, as well as the parameter size adapted in our
study, are demonstrated in Fig. 2. To simplify the visualization and to
highlight the main function of EEGNet, we showed only the main convolution
layers contained in the model, without including batch normalization layers,
average pooling layers, and activation layers. EEGNet consists of a temporal
filtering layer at which a kernel size equal to $1\times\mathrm{(Sampling\
Rate/2)}$ is applied, followed by a deep-wise spatial filtering layer at which
a kernel size of $\mathrm{(\\#\ of\ channels)}\times 1$ used. A separable
convolution block is then added. The adapted deep-wise separable convolution
structure allows EEGNet to achieve a high computation efficiency by
significantly reducing the number of parameters. The final class of each
sample is then computed by a linear fully-connected layer.

###  II-D Cross-subject variability

Cross-subject variability refers to the variation in brain activities among
different individuals, as each person has a unique brain anatomy and
functionality [25]. In EEG analysis tasks, subject-dependent information
contained in the input signal tends to compromise model performances since the
model has been trained on task-unrelated whereas subject-specific information
[20, 26]. To address this problem, current research aims to find task-related
as well as robust brain activity features that are consistent across all
individuals [19, 20]. Many approaches have been developed for this purpose.
For examples, a semi-supervised model [19] was used to learn subject-
independent featuresm abd domain adaptation [27] was performed to transfer all
subjects’ data into the same latent space for further classification.

##  III Method

###  III-A Data description

The EEG datasets used in this study were collected from 30 obese females and
30 lean (i.e., healthy) females who are between 25 to 65 years old. Subjects
who have a body mass index (BMI) higher than 30 are defined as obese
individuals and those who have a BMI lower than 25 are defined as lean
individuals. The international 10–20 system was adopted for EEG recording. The
lab environment and EEG recording devices were controlled to be in the same
condition during the recording process. For each subject, resting state EEG
data are recorded during eye-closing state prior to any meal consumption.

The first five seconds of each EEG recording were discarded as they usually
contain a high level of noise. The EEG recordings were then resampled to
128Hz, and band-pass filtered between 0.1Hz - 45Hz. Next, each EEG signal is
segmented into 10-second consecutive epoch (10$\times$128=1280 timepoints).
Each epoch is then used as an independent sample.

###  III-B Feature Extraction

A novel VAE model is developed for feature extraction. As we mentioned above,
VAE works by letting the encoder learn a parametrized Gaussian distribution of
the input data in the latent space and letting the decoder reconstruct the
input data using the latent representation. We therefore can infer that the
majority of the information contained in the raw EEG signal can be captured in
the encoder’s output. Therefore, instead of using the raw EEG data as the
input for the classification task, we used the encoder output as the input
features.

The detailed architecture of the proposed VAE is shown in Fig. 1. This
architecture is designed based on the concept of EEGNet [21]. In the encoding
part of the proposed VAE, we first adapted a temporal convolution layer with a
kernel size of 1$\times$128/2=1$\times$64, to extract temporal features. We
then performed a spatial convolution by using kernels with a size of
19$\times$1, to extract spatial features. Each convolution layer is followed
by a batch normalization layer and a leaky ReLU layer. The decoder is then
designed by taking the inverse of the encoder.

We then trained the VAE on normalized raw EEG data of each subject, and the
final encoded outputs were used as the input features for later
classification.

###  III-C Train and test split

The dataset was partitioned based on subject-based cross-validation. Within
each fold, 6 subjects (3 lean and 3 obese) were held out for testing and 54
(27 lean and 27 obese) subjects were used for training. Within each training
fold, 6 subjects (3 obese and 3 lean) were further selected for validation.
The final testing score are obtained by averaging the scores of all folds.

###  III-D Classification

We used a 1-D CNN, a SVM with a RBF kernel, and a multilayer perceptron (MLP)
to predict the label of each data sample, respectively. The architecture of
the proposed 1-D CNN is shown in Fig. 3. In the first and second convolution
layers we used 8 filters with a size of 64, and 16 filters with a size of 32,
respectively. We then added an average pooling layer with a size of 4, and a
drop-out layer with a drop rate of 0.25 to reduce the number of parameters and
prevent the model from overfitting. The final convolution layer consists of 32
filters with a size of 16, followed by an average pooling layer with a size of
8 and a drop-out layer with a drop rate of 0.25. After each convolution, batch
normalization was performed, and an activation layer was added. In this way,
the network can learn and converge at a faster speed through regularization.
Next, since each subject has 26 samples (epochs), we then determined the final
label of each subject by counting how many samples that belong to the subject
are classified as obese and how many are classified as lean. The performance
of the three classifiers is evaluated in the next section.

Figure 3: Architecture of the proposed 1-D CNN used for final classification.

##  IV Results and Discussion

###  IV-A Classification performance comparison

The test scores (i.e., accuracies) obtained by using the 1-D CNN, the SVM and
the MLP are listed in Table I. The highest classification scores (0.951 and
0.937 for subject level classification and epoch level classification
respectively) are obtained when using the 1-D CNN. Therefore, we will only
consider its use in the following work on model comparison that involves both
the EEGNet and the proposed model using VAE.

TABLE I: Test scores (mean accuracy $\pm$ standard deviation) obtained using 1-D CNN, SVM, MLP as classifiers. Methods | Subject-level scores | Epoch-level scores
---|---|---
1-D CNN | 0.951 $\pm$ 0.104 | 0.937 $\pm$ 0.130
SVM | 0.917 $\pm$ 0.170 | 0.920 $\pm$ 0.139
MLP | 0.883 $\pm$ 0.299 | 0.898 $\pm$ 0.236
TABLE II: Test scores (mean accuracy $\pm$ standard deviation) of EEGNet and the proposed model. Methods | Subject-level scores | Epoch-level scores
---|---|---
EEGNet | 0.610 $\pm$0.162 | 0.578 $\pm$0.137
Proposed Model | 0.951 $\pm$0.110 | 0.937 $\pm$0.134

(a) Visualization of features learned by EEGNet.

(b) Visualization of features extracted by VAE.

(c) Subject-level view of features learned by EEGNet.

(d) Subject-level view of features learned by VAE.

Figure 4: Comparison of 2-D visualizations generated on the learned features
using t-SNE. Top row: 2-D projection of features learned by EEGNet (a) and
those learned by VAE (b). Obese data samples are represented in blue and lean
data samples are represented in orange. Clear separability can be observed
from the feature set extracted by the proposed VAE. Bottom row: 2-D projection
of features from 6 randomly selected subjects (3 obese subjects and 3 lean
subjects) by EEGNet (c), and by VAE (d) . All data points that belong to one
subject are in the same color. Subject-based clusters can be observed in the
VAE feature set.

As an extensively used model for EEG classification tasks, we chose EEGNet as
the baseline model to compare our model with [21]. We directly applied the
EEGNet on the raw EEG dataset and the test scores obtained are listed in Table
II. Our proposed model demonstrated its efficiency by significantly improved
the classification score from 0.578 to 0.937 and from 0.610 to 0.950 on epoch-
level classification and subject-level classification, respectively. The Mann-
Whitney U test was then conducted and reported a $p$-value of 0.0003 for the
subject-level test score and 0.0005 for the epoch-level test score. The
$p$-values suggest that there is a significant difference between the two
lists of test scores obtained using the EEGNet and our proposed VAE model.

###  IV-B Visual comparison of feature representations

To elaborate on the results, we applied a non-linear dimension reduction
technique named t-distributed stochastic neighbor embedding (t-SNE) [28] to
project the high dimensional feature sets learned by the EEGNet and the
proposed VAE into 2-D Euclidean spaces. We chose t-SNE here as the dimension
reduction method due to it can well preserve the local structure of high
dimensional data by minimizing the difference between high dimensional and low
dimensional data joint distributions [29].

The projected 2-D feature sets that have been learned by the EEGNet and the
proposed VAE are shown in Fig. 4(a) and Fig. 4(b), respectively. It can be
seen that features extracted by the proposed VAE are distributed in a well-
separated manner between the obese group and the lean group.

###  IV-C Quantitative comparison

To assess the discriminant ability of feature representations, here we
introduce an impurity based measure. Suppose a $D$-dimension, $N$-entry
feature set $\mathbf{X}$. Denote the value set of the $i$-th attribute as
$X_{i}$, $i=1,\cdots,D$. Using the idea of decision tree, we seek an optimal
threshold $\tau$ that splits the values in $X_{i}$ into two value subsets with
minimum impurity:

$\begin{array}[]{l}X_{i}^{L}=\\{x_{ij}|x_{ij}<\tau,j=1,\cdots,N\\}\\\
X_{i}^{R}=\\{x_{ij}|x_{ij}\geq\tau,j=1,\cdots,N\\}\end{array}$ (3)

As we are dealing with a two-class problem, the impurity can be easily
calculated using the Gini index. Suppose within a subset $S$, $p$ is the
probability of an instance $x$ belonging to Class 1, the Gini index is

$G(S)=p(1-p).$ (4)

Hence we define a “dichotomy impurity” for the $i$-th attribute based on the
minimal weighted average of impurities of the two subsets generated by the
best “cut”:

$DI_{i}=\min_{\tau}\left(\frac{|X_{i}^{L}|}{|X_{i}|}G(X_{i}^{L})+\frac{|X_{i}^{R}|}{|X_{i}|}G(X_{i}^{R})\right),$
(5)

where $|.|$ indicates cardinality. In other words, $DI_{i}$ indicates the
purest dichotomy we can get on attribute $i$. The overall separability of the
feature representation can be roughly indicated by the average $DI$:

$DI=\sum_{i}DI_{i}/D.$ (6)

The smaller $DI$ is, the better separability we can achieve.

Applying the $DI$ measure to the learned features, we have $DI=0.247$ for the
EEGNet features, and $DI=0.220$ for the VAE features. As it is usually the
good features that drive the performance of a classifier, we compare the first
quantile of these two feature schemes, i.e. the top 25% of features having the
lowest $DI_{i}$ values, as shown in Fig. 5. Clearly the $DI$ values of the VAE
features are well separated from those of EEGNet, hence are much more
promising.

Figure 5: Comparison on the first quantile of the dichotomy impurity scores
from both feature schemes.

###  IV-D Discussion

One potential reason that explains the good separability in the VAE extracted
feature set which leads to the much enhanced performance of our model is that
resting state EEG data is highly subject-dependent. By using the proposed VAE
to compute a probabilistic latent feature representation, subject-independent
elements contained in the raw signals can be extracted whereas subject-
dependent elements can be filtered out. This idea is illustrated in Fig. 4(c)
and Fig. 4(d). Here we visualized the t-SNE projected data from 3 randomly
selected obese subjects and 3 randomly selected lean subjects. All the data
points that belong to one subject are represented in the same color. A good
cross-subject separability can be observed in the VAE extracted feature set,
for instance, the data points that belong to the subjects in orange and blue
are well clustered. On the contrary, such a pattern is not shown in the
visualized feature set learned by EEGNet.

(a) Average spatial patterns for the lean group (left) and the obese group
(right). A higher color intensity indicates a higher channel importance.

(b) Absolute value of the average difference in each channel between the obese
group and the lean group

Figure 6: Visualization of latent features shown as average spatial patterns.

Moreover, we infer that EEGNet tends to tweak the learned features toward its
gradient results obtained from the classification stage, because it learns in
a supervised way. However, as an unsupervised model, VAE learns features with
the main goal of reconstructing the input data. Therefore it is less prone to
overfit the data in its training process and tends to learn more versatile
feature representations.

On the other hand, insights regarding distinctive spatial patterns between the
obese and lean groups can be gained by computing the average value of the
second convolution layer output in the encoder of the proposed VAE model.
Since in the second convolution layer of the VAE we used filters with a size
that equals to $\mathrm{(\\#\ of\ channels)}\times 1$, we can visualize the
filter outputs. A larger value in the output feature map potentially suggests
a higher importance of the corresponding brain region, furthermore, indicating
that the EEG data recorded in that channel contain less noise and therefore
makes it easier for the classifier to distinguish between obese and lean brain
signals.

The overall (averaged) spatial patterns learned by the proposed the VAE of the
obese group and the lean group are visualized in Fig. 6(a). Here, the channel
importance is represented by the color intensity, which means both a very dark
region and a very reddish region are considered important. An opposite color
pattern between two brain regions potentially suggests that the two regions
contribute oppositely to the VAE’s learning process. For a more intuitive
demonstration of the channel importance for the classification problem, we
computed and visualized the absolute value of the average difference in each
channel between the obese group and the lean group in Fig. 6(b), and this
allows us to see the net contribution of each channel. Here a darker color
simply indicates a higher channel importance. It can be observed that O2 and
O1 appear to be the most significant channels identified by the proposed VAE,
followed by F7 and Cz. T6, F8, Fp2, and Fp1 are considered as having only
minimal contribution toward the classification.

##  V Conclusion

In this study, we have investigated obesity-related brain activities by
examining resting state EEG data using a deep learning-based framework.
Specifically, we proposed using a VAE to compute the latent representations of
raw EEG signals. The features extracted by the VAE are then fed into a 1-D CNN
for the final classification. Distinctive spatial patterns learned by the
proposed VAE between obese brains and lean brains are also discussed.
Moreover, we demonstrated the instrumental role of VAE in resting state EEG
classification tasks by showing that using the learned latent representations
instead of what EEGNet learns from raw EEG signals, it can significantly
reduce the subject-dependency and achieves much improved classification
performance. The superiority of the VAE features is also reflected by
visualizations using t-SNE, and by a dichotomy impurity measure we introduced
as a quantitative indicator of class separability.

For future work, we intend to bring more discussion regarding the neurological
aspects of the spatial patterns that have been learned by the proposed model,
as well as improve the interpretability of our proposed model by including
insights regarding temporal and spectral information.

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