# Turán type oscillation inequalities in $L^{q}$ norm on the boundary of
convex polygonal domains

Polina Glazyrina, Szilárd Gy. Révész  Polina Yu. Glazyrina
Institute of Mathematics and Computer Sciences, Ural Federal University,
Ekaterinburg, RUSSIA
and
Institute of Mathematics and Mechanics,
Ural Branch of the Russian Academy of Sciences,
Ekaterinburg, RUSSIA<EMAIL_ADDRESS>Szilárd Gy. Révész
HUN-REN A. Rényi Institute of Mathematics
Budapest, Reáltanoda utca 13-15.
1053 HUNGARY<EMAIL_ADDRESS>
###### Abstract.

In 1939 Pál Turán and János Erőd initiated the study of lower estimations of
maximum norms of derivatives of polynomials, in terms of the maximum norms of
the polynomials themselves, on convex domains of the complex plane. As a
matter of normalization they considered the family ${\mathcal{P}}_{n}(K)$ of
degree $n$ polynomials with all zeros lying in the given convex, compact
subset $K\Subset{\mathbb{C}}$. While Turán obtained the first results for the
interval ${\mathbb{I}}:=[-1,1]$ and the disk
${\mathbb{D}}:=\\{z\in{\mathbb{C}}~{}:~{}|z|\leqslant 1\\}$, Erőd extended
investigations to other compact convex domains, too. The order of the optimal
constant was found to be $\sqrt{n}$ for ${\mathbb{I}}$ and $n$ for
${\mathbb{D}}$. It took until 2006 to clarify that all compact convex
_domains_ (with nonempty interior), follow the pattern of the disk, and admit
an order $n$ inequality.

For $L^{q}(\partial K)$ norms with any $1\leqslant q<\infty$ we obtained order
$n$ results for various classes of domains. Further, in the generality of all
convex, compact domains we could show a $cn/\log n$ lower bound together with
an $O(n)$ upper bound for the optimal constant. Also, we conjectured that all
compact convex domains admit an order $n$ Turán type inequality. Here we prove
this for all _polygonal_ convex domains and any $0<q<\infty$.

††footnotetext: P. Yu. Glazyrina was supported by the Ministry of Science and
Higher Education of the Russian Federation (Ural Federal University Program of
Development within the Priority-2030 Program). Sz. Gy. Révész was supported by
Hungarian Science Foundation Grant #’s K-132097, K-147153 and K-146387.

MSC 2020 Subject Classification. Primary 41A17. Secondary 30E10, 52A10.

Keywords and phrases. Bernstein-Markov Inequalities, Turán’s lower estimate of
derivative norm, logarithmic derivative, Chebyshev constant, transfinite
diameter, convex domains, outer angle, width of a convex domain, depth of a
convex domain.

###### Contents

  1. 1 Introduction
    1. 1.1 The oscillation of a polynomial in maximum norm
  2. 2 Technical preparations for the investigation of $L^{q}(\partial K)$ norms
  3. 3 Polygonal convex domains
    1. 3.1 On one side of the acute angle, $|p^{\prime}(z)|\gg n|p(z)|$ holds true
    2. 3.2 A crucial step
    3. 3.3 Proof of Theorem 2
  4. 4 Proof of Theorem 1
  5. 5 Concluding remarks

##  1\. Introduction

###  1.1. The oscillation of a polynomial in maximum norm

At the turn of the 19th and 20th centuries, the first estimates of the
derivative of a polynomial via the maximum of its values appeared. They were
obtain by A. Markov in 1889, for algebraic polynomials on an interval, and by
Bernstein and M. Riesz in 1914, for trigonometric polynomials on $[0,2\pi]$
and algebraic polynomials on the unit circle. In 1923, Szegő [27] obtained an
estimate for a large class of (not necessarily convex, but piecewise smooth)
domains. Namely, if $K\subset{\mathbb{C}}$ is a piecewise smooth simply
connected domain, with its boundary consisting of finitely many analytic
Jordan arcs, and if the maximum of the outer angles at the joining vertices of
these arcs is111If the domain is bounded, then for all directions it has
supporting lines, whence there are points where the outer angle is at least
$\pi$. $\beta\in[\pi,2\pi]$, then the domain admits a Markov type inequality
of the form ${\left\|{p^{\prime}}\right\|}_{K}\leqslant
c_{K}n^{\beta/\pi}{\left\|{p}\right\|}_{K}$ for any polynomial $p$ of degree
$n$. Here the norm ${\left\|{\cdot}\right\|}:={\left\|{\cdot}\right\|}_{K}$
denotes sup norm over values attained on $K$. This inequality is essentially
sharp for all such domains. In particular, this immediately implies that for
_analytically smooth_ convex domains the Markov factor is $O(n)$. For the unit
disk ${\mathbb{D}}:=\\{z\in{\mathbb{C}}~{}:~{}|z|\leqslant 1\\}$ even the
exact inequality is well-known:

${\left\|{p^{\prime}}\right\|}_{\mathbb{D}}\leqslant
n{\left\|{p}\right\|}_{\mathbb{D}}.$

This was conjectured, and almost proved, by Bernstein [3, 4]; for the first
published proof see [25]. Similarly, the precise result is also classical for
the unit interval ${\mathbb{I}}:=[-1,1]$ – then we have Markov’s Inequality
${\left\|{p^{\prime}}\right\|}_{\mathbb{I}}\leqslant
n^{2}{\left\|{p}\right\|}_{\mathbb{I}}$, which is sharp222Note that in this
case the outer angles at the break-points of the piecewise smooth boundary are
exactly $2\pi$ at each end., see [18].

In 1939 Pál Turán started to study converse inequalities of the form

${\left\|{p^{\prime}}\right\|}_{K}\geqslant
c_{K}n^{A}{\left\|{p}\right\|}_{K}.$

Clearly such a converse can only hold if further restrictions are imposed on
the occurring polynomials $p$. Turán assumed that all zeroes of the
polynomials belong to $K$. So denote the set of complex (algebraic)
polynomials of degree (exactly) $n$ as ${\mathcal{P}}_{n}$, and the subset
with all the $n$ (complex) roots in some set $K\subset{\mathbb{C}}$ by
${\mathcal{P}}_{n}(K)$.

Theorem A. (Turán, [29, p. 90]). If $p\in{\mathcal{P}}_{n}({\mathbb{D}})$,
then we have

(1)
${\left\|{p^{\prime}}\right\|}_{\mathbb{D}}\geqslant\frac{n}{2}{\left\|{p}\right\|}_{\mathbb{D}}~{}.$

Theorem B. (Turán, [29, p. 91]). If $p\in{\mathcal{P}}_{n}({\mathbb{I}})$,
then we have

${\left\|{p^{\prime}}\right\|}_{\mathbb{I}}\geqslant\frac{\sqrt{n}}{6}{\left\|{p}\right\|}_{\mathbb{I}}~{}.$

Inequality (1) of Theorem 1.1 is best possible. Regarding Theorem 1.1, Turán
pointed out by example of $(1-x^{2})^{n}$ that the $\sqrt{n}$ order is sharp.
Some slightly improved constants can be found in [2] and [17], however, the
exact value of the constants and the corresponding extremal polynomials were
already computed for all fixed $n$ by Erőd in [9].

Now we are going to describe Turán-type inequalities for general convex sets.
Denote by $\Gamma:=\partial K$ the boundary of $K$. The (normalized) quantity
under our study is the “inverse Markov factor” or “oscillation factor”

(2) $M_{n,q}(K):=\inf_{p\in{\mathcal{P}}_{n}(K)}M_{q}(p)\qquad\text{\rm
with}\qquad
M_{q}(p):=\frac{{\left\|{p^{\prime}}\right\|}_{L^{q}(\Gamma)}}{{\left\|{p}\right\|}_{L^{q}(\Gamma)}},$

where, as usual,

$\displaystyle{\left\|{p}\right\|}_{q}:$
$\displaystyle={\left\|{p}\right\|}_{L^{q}(\Gamma)}:=\left(\int_{\Gamma}|p(z)|^{q}|dz|\right)^{1/q}\quad(0<q<\infty),$
$\displaystyle{\left\|{p}\right\|}_{K}:$
$\displaystyle={\left\|{p}\right\|}_{L^{\infty}(\Gamma)}=\sup_{z\in\Gamma}|p(z)|=\sup_{z\in
K}|p(z)|.$

Drawing from the work of Turán, Erőd [9, p. 74] already addressed the
question: “For what kind of domains does the method of Turán apply?” Clearly,
by ”applies” he meant that it provides order $n$ oscillation for the
derivative. Moreover, he introduced new ideas into the investigation –
including the application of Chebyshev’s Inequality (4) below – so clearly he
did not simply pursue the effect of Turán’s original methods, but was indeed
after the right oscillation order of general domains. In particular, he showed
on p. 77 of [9] the following.

Theorem C. (Erőd). Let $K$ be any convex domain bounded by finitely many
Jordan arcs, joining at vertices with angles $<\pi$, with all the arcs being
$C^{2}$-smooth and being either straight lines of length $<\Delta(K)$, where
$\Delta(K)$ stands for the transfinite diameter of $K$, or having positive
curvature bounded away from $0$ by a fixed constant $\kappa>0$. Then there is
a constant $c(K)$, such that $M_{n}(K)\geqslant c(K)n$ for all
$n\in{\mathbb{N}}$.

Note that this latter result of Erőd incorporates regular $k$-gons $G_{k}$ for
large enough $k$, but not the square $Q=G_{4}$, because the side length $h$ of
a square is larger than the quarter of the transfinite diameter $\Delta$:
actually

$\Delta(G_{k})=\frac{\Gamma(1/k)}{\sqrt{\pi}2^{1+2/k}\Gamma(1/2+1/k)}h$

(see e.g. [19, p. 135]), so $\Delta(G_{k})>h$ iff $k\geqslant 7$. This implies
$M_{n}(G_{k})\geqslant c_{k}n$ for $k\geqslant 7$.

In [8], Erdélyi proved order $n$ oscillation for the square333Erdélyi also
proves similar results on rhombuses, under the further condition of some
symmetry of the polynomials in consideration – e.g. if the polynomials are
real, or odd. Note also that his work on the topic preceded [21] and
apparently was accomplished without being aware of details of [9]. $Q=G_{4}$,
too. A result of [20] also implied $M_{n}(G_{k})\geqslant c_{k}n$ for
$k\geqslant 4$, but still not for a triangle.

In the full generality of all compact convex sets, however, only an order
$\sqrt{n}$ Turán-Erőd type inequality was shown prior to 2006, see [17,
Theorem 3.2]. Note that this is sharp for ${\mathbb{I}}$, but later it turned
out that for all other compact convex domains the true order of magnitude is
similar to the disk case.

Clearly, assuming boundedness is natural, since all polynomials have
${\left\|{p_{n}}\right\|}_{K}=\infty$ when the set $K$ is unbounded. Also,
restricting ourselves to _closed_ bounded sets – i.e., to compact sets – does
not change the $\sup$ norm of polynomials under study, as all polynomials are
continuous.

Recall that the term convex domain stands for a compact, convex subset of
${\mathbb{C}}$ having nonempty interior. That is, assuming that $K$ is a
(bounded, closed) convex _domain_ , not just a compact convex _set_ , means
that we exclude only the case of the interval, for which already Turán
clarified that the order of oscillation is precisely $\sqrt{n}$.

So in order to clarify the order of oscillation for all compact convex sets it
remains to clarify the order of oscillation for compact convex domains. A
solution was published in 2006, see [21].

To study (2) some geometric parameters of the convex domain $K$ are involved
naturally. We write $d:=d(K):={\rm diam\,}(K)$ for the diameter of $K$, and
$w:=w(K):={0pt}(K)$ for the minimal width of $K$. That is,

$d(K):=\max_{z^{\prime},z^{\prime\prime}\in K}|z^{\prime}-z^{\prime\prime}|,$
$w(K):=\min_{\gamma\in[-\pi,\pi]}\left(\max_{z\in
K}\Re(ze^{i\gamma})-\min_{z\in K}\Re(ze^{i\gamma})\right).$

Note that a (closed) convex domain is a (closed), bounded, convex set
$K\subset{\mathbb{C}}$ with nonempty interior, hence $0<w(K)\leqslant
d(K)<\infty$.

Theorem D. (Halász–Révész). Let $K\subset{\mathbb{C}}$ be any bounded convex
domain. Then for all $p\in{\mathcal{P}}_{n}(K)$ we have

${\left\|{p^{\prime}}\right\|}_{K}\geqslant
C(K)n{\left\|{p}\right\|}_{K}~{}\qquad\text{\rm with}\qquad
C(K)=0.0003\frac{w(K)}{d^{2}(K)}.$

######  Remark 1.

This indeed provides the precise order, for an even larger than $n$ order
cannot occur, not for any particular compact set. Namely, let
$K\subset{\mathbb{C}}$ be any compact set with diameter $d:={\rm diam\,}(K)$.
Then for all $n$ there exists a polynomial $p\in{\mathcal{P}}_{n}(K)$ of
degree exactly $n$ satisfying

${\left\|{p^{\prime}}\right\|}\leqslant~{}\frac{1}{{\rm
diam\,}(K)}n~{}{{\left\|{p}\right\|}}.$

Indeed, considering a diameter $[z_{0},w_{0}]$ and the polynomial
$p(z)=(z-z_{0})^{d}$, the respective norm is $\|p\|_{\infty}=d^{n}$ while the
derivative norm becomes $\|p^{\prime}\|_{\infty}=nd^{n-1}$, both attained at
$w_{0}\in K$.

So, this settles the question of the _order_ , but not the precise _dependence
on the geometry_. However, up to an _absolute constant factor_ , even the
dependence on the geometrical features of the domain was also clarified in
[21].  Theorem E. (Révész) Let $K\subset{\mathbb{C}}$ be any compact,
connected set with diameter $d$ and minimal width $w$. Then for all
$n>n_{0}:=n_{0}(K):=2(d/16w)^{2}\log(d/16w)$ there exists a polynomial
$p\in{\mathcal{P}}_{n}(K)$ satisfying

${\left\|{p^{\prime}}\right\|}_{K}\leqslant
C^{\prime}(K)n{\left\|{p}\right\|}_{K}\qquad\text{\rm with}\qquad
C^{\prime}(K):=600~{}\frac{w(K)}{d^{2}(K)}~{}.$

Actually a more general pointwise inequality holds _at all points_ of
$\partial{\mathbb{D}}$. Namely, for $p\in{\mathcal{P}}_{n}(K)$ we have

$|p^{\prime}(z)|\geqslant\frac{n}{2}|p(z)|,\quad|z|=1,$

and as a corollary, for any $q>0$,

$\left(\int_{|z|=1}|p^{\prime}(z)|^{q}|dz|\right)^{1/q}\geqslant\frac{n}{2}\left(\int_{|z|=1}|p(z)|^{q}|dz|\right)^{1/q}.$

In other words, Theorem 1.1 extends to all integral norms on the perimeter,
including all $L^{q}(\partial{\mathbb{D}})$-norms, and we have for all
polynomials $p\in{\mathcal{P}}_{n}(K)$

${\left\|{p^{\prime}}\right\|}_{L^{q}(\partial{\mathbb{D}})}\geqslant\frac{n}{2}{\left\|{p}\right\|}_{L^{q}(\partial{\mathbb{D}})},\qquad
M_{n,q}({\mathbb{D}})\geqslant\frac{n}{2}~{}.$

The same way, for so-called $R$-circular domains444This term was introduced by
Levenberg and Poletsky [17] and it means that for any boundary point
$z\in\partial K$ there exists a disk $D_{R}$ of radius $R$ with $z\in\partial
D_{R}$ and $K\subset D_{R}$. the result of Theorem 1.1 extends as

${\left\|{p^{\prime}}\right\|}_{L^{q}(\partial
K)}\geqslant\frac{n}{2R}{\left\|{p}\right\|}_{L^{q}(\partial K)},\qquad
M_{n,q}(K)\geqslant\frac{n}{2R}~{}.$

In case we discuss maximum norms, one can assume that $|p(z)|$ is maximal, and
it suffices to obtain a lower estimation of $|p^{\prime}(z)|$ only at such a
special point – for general norms, however, this is not sufficient. The above
results work only for we have a pointwise inequality of the same strength
_everywhere_ , or almost everywhere. The situation becomes considerably more
difficult, when such a statement cannot be proved. E.g. if the domain in
question is not strictly convex, i.e. if there is a line segment on the
boundary, then the zeroes of the polynomial can be arranged so that even some
zeroes of the derivative lie on the boundary, and at such points
$p^{\prime}(z)$ – even $p^{\prime}(z)/p(z)$ – can vanish. As a result, at such
points no fixed lower estimation can be guaranteed, and lacking a uniformly
valid pointwise comparision of $p^{\prime}$ and $p$, a direct conclusion
cannot be drawn either.

This explains why the case of the interval ${\mathbb{I}}$ already proved to be
much more complicated for the integral mean norms.

In a series of papers [36, 37, 38, 39, 40], Zhou proved the inequality

$\left(\int_{-1}^{1}|p^{(k)}(x)|^{p}dx\right)^{1/p}\geqslant
C^{(k)}_{p,q}(n)\left(\int_{-1}^{1}|p(x)|^{q}dx\right)^{1/q},$

with $k=1$, $C^{(1)}_{p,q}(n)=c_{p,q}\left(\sqrt{n}\right)^{1-1/p+1/q}$ and
$0<p\leqslant q\leqslant\infty,$ ${1-1/p+1/q\geqslant 0}$.

The best possible constants $C^{(k)}_{p,q}(n)$ were found by Babenko and
Pichugov [1] for $p=q=\infty,$ $k=2$, by Bojanov [5] for $1\leqslant
p\leqslant\infty,$ $q=\infty$, $1\leqslant k\leqslant n$, and by Varma [33] in
the case of $p=q=2,$ $k=1$.

Exact Turán-type inequalities for trigonometric polynomials in different
$L^{q}$-metrics on ${\mathbb{T}}$ were proved in [1, 2, 6, 30, 31, 16].

Other inequalities on ${\mathbb{I}}$, ${\mathbb{D}}$, the positive semiaxes,
or on ${\mathbb{T}}$ in various weighted $L^{q}$-metrics can be found in [32,
34, 35, 16].

As said above, we also have a direct result for $R$-circular domains, and
$R$-circularity could be ascertained by some conditions on the curvature.
However, apart from these, for general domains, the situation was much less
clear. A fully general estimate, with almost optimal order, was published as
Theorem 1’ in [13].

Theorem F. (Glazyrina–Révész). Let $1\leqslant q<\infty$ and let
$K\subset{\mathbb{C}}$ be any compact, convex domain with diameter $d$ and
minimal width $w$. Then for all
$n>n_{0}:=n_{0}(K):=\max\left(10^{21},(d/w)^{5}\right)$ and for all polynomial
$p\in{\mathcal{P}}_{n}(K)$ we have

${\left\|{p^{\prime}}\right\|}_{L^{q}(\partial
K)}\geqslant\frac{1}{240~{}000}~{}\frac{w^{2}}{d^{3}}~{}\frac{n}{\log
n}{\left\|{p}\right\|}_{L^{q}(\partial K)}.$

This is close to optimal, as was shown in [14]. Here we only quote a corollary
of the more general result there, see Corollary 1 of [14]. An earlier version
with some less explicit and worse constant was given as Theorem 5 in [12].

Theorem G. (Glazyrina–Goryacheva–Révész). Let $K\subset{\mathbb{C}}$ be a
compact convex subset of ${\mathbb{C}}$ having width $w>0$ and diameter $d$.
Let $0<q\leqslant\infty$, and

$\displaystyle{n\geqslant 2(1+1/q)\frac{d^{2}}{w^{2}}\ln\frac{d}{w}}.$

If $\mu$ is the linear Lebesgue measure on the boundary of $K$ (arc length
measure $\ell$), then we have

${\left\|{p^{\prime}}\right\|}_{L^{q}(\partial K)}\leqslant
C_{q}\frac{w}{d^{2}}n{\left\|{p}\right\|}_{L^{q}(\partial K)},$

where

$C_{q}:=121\frac{3q+2+2\sqrt{q^{2}+3q+1}}{5q}\left(3+2q+2\sqrt{q^{2}+3q+1}\right)^{1/q}.$

Also we conjectured that the right order of Turán-Erőd type oscillation is
always $n$, for all convex compact domains and for all exponents $q>0$. This
was based, above the already mentioned general results, also on several
partial results, for various classes of compact convex domains, where we could
indeed prove ${\left\|{p^{\prime}}\right\|}_{L^{q}(\partial K)}\gg
n{\left\|{p}\right\|}_{L^{q}(\partial K)}$. These include generalised Erőd
type domains incorporating the class in Theorem 1.1, see [11], and some
others, too.

We recall here only one particular result, which will be needed in the
forthcoming proof as well. Let us introduce the notion of the local depth. The
local depth $h(\zeta,K)$ is the maximum of the length(s) of intersections of
$K$ and normal line(s) at $\zeta$, that is, maximum of chord lengths emanating
from $\zeta$ and perpendicular to some supporting lines. We also need the set
${\mathcal{G}}(p)$, which we define as

(3)
${\mathcal{G}}:={\mathcal{G}}(p):={\mathcal{G}}_{K}^{q}(p):=\left\\{\zeta\in\Gamma~{}:~{}|p(\zeta)|>\frac{\lambda}{n^{2/q}}\|p\|_{\infty}\right\\},\quad\lambda:=\frac{1}{\left(8\pi(q+1)\right)^{1/q}}.$

Theorem H. Let $n\in{\mathbb{N}}$ and $p\in{\mathcal{P}}_{n}(K).$ Then for any
$n\geqslant n_{0}(K)$ and $\zeta\in{\mathcal{G}}(p)$ it holds555To be precise,
this was proved in [12] only for $q\geqslant 1$, unfortunately. However, the
assumption that $q\geqslant 1$ was used only in settling the case of small
$n$; for $n\leqslant n_{0}:=32d^{4}/h^{4}$ Lemma 3 of [12] was used to get the
statement. Therefore, one can extend the result to $0<q<1$, too, at the
expense of assuming $n\geqslant n_{0}$.

$|p^{\prime}(\zeta)|\geqslant\frac{h^{4}(\zeta,K)}{1500d^{5}(K)}~{}n~{}|p(\zeta)|.$

This was important as a key tool to obtain a result for domains with _fixed
positive depth_ , that is, with $h_{K}:=\inf_{\zeta\in\partial
K}h(\zeta,K)>0$. When this happens to hold, the result immediately implies
that ${\left\|{p^{\prime}}\right\|}_{L^{q}(\partial K)}\gg
n{\left\|{p}\right\|}_{L^{q}(\partial K)}$. Note that for convex polygonal
domains $h_{K}>0$ is equivalent to assuming that the polygon has no acute
angles, see Proposition 1 of [12]. In particular, among regular $k$-gons
already for $k\geqslant 4$ we obtain the desired order $n$ oscillation result.
In view of this, regarding polygonal domains there remained only polygons with
acute angles to handle. This handling of acute angles is the crux of our
present work. The solution is based on one of the key ideas of [21], the so-
called ”tilted normal estimate lemma”, but essential changes are necessary for
making this idea work in our situation, too.

######  Theorem 1.

Let $K\Subset{\mathbb{C}}$ be a convex polygonal domain. Then for any
$0<q<\infty$ there exists a positive constant $c_{K}=c_{K}(q)$ and
$n_{0}:=n_{0}(K)$ such that for all $n\geqslant n_{0}$ and
$p\in{\mathcal{P}}_{n}(K)$

$\|p^{\prime}\|_{L^{q}(\partial K)}\geqslant c_{K}n\|p\|_{L^{q}(\partial K)}.$

##  2\. Technical preparations for the investigation of $L^{q}(\partial K)$
norms

We will use an idea going back to Erőd: the use of Chebyshev’s classical
theorem in the general form found by Fekete. This is formulated with the use
of the _transfinite diameter_ $\Delta(E)$ of a compact set $E$.

######  Lemma 1.

(Faber, Fekete, Szegő). Let $M\Subset{\mathbb{C}}$ be any compact set. Then
for all $k\in{\mathbb{N}}$ we have

(4) $\min_{w_{1},\dots,w_{k}\in{\mathbb{C}}}\max_{z\in
M}\left|\prod_{j=1}^{k}(z-w_{j})\right|\geqslant\Delta(M)^{k}.$

###### Proof.

Regarding the formulation in Lemma 1 cf. Theorem 5.5.4. (a) in [19] or [26,
(3.7) page 46]. Historically, it was first Fekete who proved the inequality.
Before that, Faber [10] has already proved $\max_{z\in
M}\left|\prod_{j=1}^{k}(z-w_{j})\right|\geqslant\rho(M)^{k}$ for $M$ a Jordan
domain bounded by an analytic Jordan curve and with $\rho(M)$ standing for the
conformal radius of the domain $M$. Note that for such domains the conformal
radius is equal to the transfinite diameter. Following Fekete, Szegő showed
that the condition of ${\mathbb{C}}\setminus M$ being simply connected is not
necessary, and that with the so-called Robin constant $\gamma(M)$ (equivalent
to capacity), the stated inequality holds true, moreover,
$\gamma(M)=\Delta(M)$ in general for all compacta. For a more detailed
discussion of the development of the topic see [12]. ∎

We will use disconnected, linearly Jordan mesurable sets in our arguments. For
their transfinite diameter, a classical estimate is well-known.

######  Lemma 2.

(Pólya, see [15, Ch. VII]). Let $J\subset{\mathbb{R}}$ be any compact set,
$|J|^{*}$ be its outer Jordan measure. Then $\Delta(J)\geqslant|J|^{*}/4$.

Next, we record a Nikolskii-type estimate, which is similar to the well-known
analogous inequality on the real line, see e.g. [28, 4.9.6 (36)]. For the
exact version below, see Lemma 1 of [12] .

######  Lemma 3.

For any $q>0$ and any polynomial of degree at most $n$ we have that

$\|p\|_{L^{q}(\partial
K)}\geqslant\left(\frac{d}{2(q+1)}\right)^{1/q}~{}\|p\|_{L^{\infty}(\partial
K)}~{}n^{-2/q}.$

Recall the above definition (3). It is important for us because we can
restrict ourselves to the points of ${\mathcal{G}}$ and neglect whatever
happens for points belonging to its complement $\Gamma\setminus{\mathcal{G}}$.
Indeed, $\Gamma$ is contained in a disk of radius $d$ around any point of $K$,
whence by the well-known property666A reference is [7, p. 52, Property 5]
about surface area, presented as a consequence of the Cauchy formula for
surface area. of convex curves, $|\Gamma|\leqslant 2\pi d$, and the above
Lemma 3 furnishes

$\int_{\Gamma\setminus{\mathcal{G}}}|p|^{q}\leqslant\frac{2\pi
d\lambda^{q}}{n^{2}}\|p\|^{q}_{\infty}\leqslant
4\pi(q+1)\lambda^{q}\|p\|_{q}^{q}=\frac{1}{2}\|p\|_{q}^{q},$

so that we find

$\int_{{\mathcal{G}}}|p|^{q}=\int_{\Gamma}|p|^{q}-\int_{\Gamma\setminus{\mathcal{G}}}|p|^{q}\geqslant\|p\|_{q}^{q}-\frac{1}{2}\|p\|_{q}^{q}\geqslant\frac{1}{2}\|p\|_{q}^{q}.$

Therefore we can restrict to (lower) estimations of $|p^{\prime}(\zeta)|$ on
the set ${\mathcal{G}}$ where $p$ is assumed to be relatively large (compared
to its maximum norm), so that we can assume that

$\displaystyle\log\dfrac{\|p\|_{\infty}}{|p(\zeta)|}$
$\displaystyle\leqslant\log(\lambda^{-1}n^{2/q})=\frac{\log(1+q)}{q}+\frac{\log\left(8\pi\right)}{q}+\frac{2}{q}\log
n$ $\displaystyle\leqslant\log(16\pi)+2\log n<4\log n$

for all $q\geqslant 1$ and $n\geqslant\sqrt{16\pi}$, i.e. already for
$n\geqslant 8$. In case $0<q<1$, a similar estimate with a factor $1/q$
obtains even more easily. Summing up we have

######  Lemma 4.

Let $n\in{\mathbb{N}}$ and $0<q<\infty$. Let $p\in{\mathcal{P}}_{n}$ and
${\mathcal{G}}={\mathcal{G}}(p)\subset\partial K$ be defined according to (3).
Then we have

$\int_{{\mathcal{G}}}|p|^{q}\geqslant\frac{1}{2}\|p\|^{q}_{L^{q}(\partial
K)}.$

Furthermore, for any point $\zeta\in{\mathcal{G}}$ we also have

$\log\frac{\|p\|_{\infty}}{|p(\zeta)|}\leqslant\frac{4}{\min(1,q)}\log
n~{}~{}(n\geqslant 8).$

Note that for $q\geqslant 1$ this was given exactly as here already in Lemma 2
of Section 5.3 of [12], so here we repeated the easy deduction for the
reader’s convenience only.

##  3\. Polygonal convex domains

In the following we denote
$D_{r}(z_{0}):=\\{z\in{\mathbb{C}}~{}:~{}|z-z_{0}|\leqslant r\\}$. As a key
step towards the proof of Theorem 1, we want to prove in this section the
following partial result.

######  Theorem 2.

Let $K\Subset{\mathbb{C}}$ be a convex polygonal domain, $0<q<\infty$, and
$n\in{\mathbb{N}}$. Let $U$, $V$, $W$ be consecutive (in the counter-clockwise
direction) vertices of $K$, and assume that $\alpha:=\angle(UVW)$ is an acute
angle of $K$ at $V$, i.e. $0<\alpha<\pi/2$.

Then there exists a positive constant $\mu(\alpha):=\mu(\alpha,q)>0$ with the
following property. For any $0<r\leqslant R_{V}/8$, where we put

$R_{V}:=\dfrac{1}{64}\min(|VU|,|VW|),$

it holds

(5) $\mu(\alpha)~{}n^{q}\int_{{\mathcal{G}}\cap
D_{r}(V)}|p|^{q}\leqslant\int_{\Gamma\cap D_{8r}(V)}|p^{\prime}|^{q}.$

###  3.1. On one side of the acute angle, $|p^{\prime}(z)|\gg n|p(z)|$ holds
true

######  Lemma 5.

With the notations of Theorem 2 let us define the segments

(6) $I_{+}=[V,U]\cap D_{R_{V}}(V),\quad I_{-}=[V,W]\cap D_{R_{V}}(V).$

Then we have for any $p\in{\mathcal{P}}_{n}(K)$ that either $(I_{+}\cup
I_{-})\subset\Gamma\setminus{\mathcal{G}}$, or at least for one of the two
segments $I_{+}$ or $I_{-}$ we have for all points $z\in I_{\pm}$ of the
relevant segment the inequality

(7)
$|p^{\prime}(z)|\geqslant\frac{\sin(\alpha/2)}{4}n|p(z)|\geqslant\frac{\sin\alpha}{8d}n|p(z)|\qquad(z\in
I_{\pm}).$

Note that Lemma 5 of [13] is a somewhat similar statement in a different
context.

###### Proof.

Without loss of generality we may assume that $V=0$ and that the segments
$I_{+}$ and $I_{-}$ are in conjugate directions, that is, with
$\beta:=\alpha/2$, we have

(8) $I_{\pm}:=\left[0,\,R_{V}e^{\pm\beta i}\right].$

Let us consider the homothety about $V=0$ and of ratio $1/64$. It brings $K$
to $K^{\prime}=\dfrac{1}{64}K$, whose boundary includes $I_{\pm}$. By
convexity, $K^{\prime}\subset K$, and for the diameters
$d^{\prime}:=d(K^{\prime})=\frac{1}{64}d(K)$.

Suppose first that some $p\in{\mathcal{P}}_{n}(K)$ has $m\geqslant n/2$ zeroes
in $K^{\prime}$. We have for any point $z\in K^{\prime}$ the estimate
$|p(z)|\leqslant(d^{\prime})^{m}d^{n-m}\leqslant(d^{\prime})^{n/2}d^{n/2}.$ To
estimate from the other side, we take any segment $I\subset K$ of the length
$d$ and apply Lemma 1, and Lemma 2:
$\|p\|_{K}\geqslant\|p\|_{I}\geqslant\Delta(I)^{n}\geqslant(d/4)^{n}.$ After
dividing these two estimates we get

$\dfrac{|p(z)|}{\|p\|_{K}}\leqslant\frac{(d^{\prime})^{n/2}d^{n/2}}{(d/4)^{n}}=\left(\dfrac{d^{\prime}}{d}\right)^{n/2}4^{n}=\left(\dfrac{1}{64}\right)^{n/2}4^{n}=2^{-n}.$

for all $z\in K^{\prime}$, in particular also on both segments (8). Whence for
$n\geqslant n_{0}$ ($n_{0}$ is necessary for $2^{-n}<\lambda n^{-2/q}$ with
$\lambda<1$ holds only for $n\geqslant n_{0}$) we must have
$I_{\pm}\cap{\mathcal{G}}=\emptyset$.

Denote ${\mathbb{H}}:=\\{z\in{\mathbb{C}}~{}:~{}\Re z\geqslant 0\\}$ the upper
halfplane. Let now assume that ${\mathcal{G}}\cap(I_{+}\cup
I_{-})\neq\emptyset$. It follows that there are $m<n/2$ zeroes of $p$ in
$K^{\prime}$, whence there are either $\geqslant n/4$ zeroes in
${\mathbb{H}}\cap(K\setminus K^{\prime})$, or there are $\geqslant n/4$ zeroes
in $-{\mathbb{H}}\cap(K\setminus K^{\prime})$.

Write ${\mathcal{Z}}=\\{z_{1},\ldots,z_{n}\\}$ for the set of zeroes of $p$,
all in $K$ by assumption. By symmetry, it suffices to deal with one case, say
when ${\mathcal{W}}:={\mathcal{Z}}\cap(-{\mathbb{H}}\cap(K\setminus
K^{\prime}))$ contains $\geqslant n/4$ zeroes. Fix any point $z\in I_{+}$,
with tangent direction $e^{i\beta}$, and write for all $z_{j}\in{\mathcal{Z}}$
$z_{j}=z+r_{j}e^{i\varphi_{j}}$ ($j=1,\ldots,n$). Then the usual Turán type
estimate yields777Note that, here again, for any $z_{j}\in{\mathcal{Z}}\subset
K$ the angle $\varphi_{j}\in[-\pi+\beta,\beta]$, whence
$\beta-\varphi_{j}\in[0,\pi]$ and $\sin(\beta-\varphi_{j})\geqslant 0$,
therefore any term can be estimated from below by $0$ and hence can be dropped
from the full sum.

$\left|\frac{p^{\prime}}{p}(z)\right|\geqslant\Im\left(-{e^{i\beta}\frac{p^{\prime}}{p}(z)}\right)=\sum_{z_{j}\in{\mathcal{Z}}}\Im\frac{e^{i\beta}}{z_{j}-z}=\sum_{z_{j}\in{\mathcal{Z}}}\frac{\sin(\beta-\varphi_{j})}{r_{j}}\geqslant\sum_{z_{j}\in{\mathcal{W}}}\frac{\sin(\beta-\varphi_{j})}{r_{j}}.$

For one thing, it is clear that $r_{j}\leqslant{\rm diam\,}(K)=d$. Also, by
construction, clearly $-\pi/2<\varphi_{j}<0$ for any $z_{j}\in{\mathcal{W}}$,
whence for such zeroes
$\beta-\varphi_{j}\in(\beta,\beta+\pi/2)\subset(\beta,\pi-\beta)$ and
$\sin(\beta-\varphi_{j})\geqslant\sin\beta$. These considerations thus yield
for any $z\in I_{+}$

$\left|\frac{p^{\prime}}{p}(z)\right|\geqslant\\#{\mathcal{W}}\frac{\sin\beta}{d}\geqslant\frac{\sin\beta}{4d}n=\frac{\sin(\alpha/2)}{4d}n\geqslant\frac{\sin\alpha}{8d}n.$

∎

###  3.2. A crucial step

Here we prove the following key lemma.

######  Lemma 6 (Tilted normal estimate).

Let $K\Subset{\mathbb{C}}$ be a convex polygonal domain, $U,$ $V,$ $W>0$ be
consecutive vertexes of $K$ (in the counter-clockwise direction),

$0<\alpha:=\angle(U,V,W)<\pi/2$

be the acute angle at $V,$ and put

(9)
$\theta:=\dfrac{1}{2}\arcsin\left(\dfrac{1}{8}\sin\alpha\right)\quad(which\
means\ that\ 8\sin(2\theta)=\sin\alpha).$

Suppose $\zeta\in(V,W)$, and that $|V\zeta|\leqslant|VU|/8$. Let $D=D(\zeta)$
be the point of intersection888The intersection point exists _inside the
interval $(V,U)$_, for according to the sine theorem
$f(\alpha):=|VD|:|V\zeta|=\sin(\pi/2-2\theta):\sin(\pi/2-\alpha+2\theta)=\cos(2\theta):\cos(\alpha-2\theta)$,
and therefore $f(\alpha)\leqslant 1/\cos(\alpha-2\theta)$, which is strictly
increasing with $\alpha$ and $2\theta:=\arcsin(\sin(\alpha)/8)$, and reaches
its maximum at $\alpha=\pi/2$ with a value
$1/\cos(\pi/2-\arcsin(\sin(\pi/2)/8))=1/\sin(\arcsin(\sin(\pi/2)/8))=8$. That
is why we assumed $r<R_{V}/8$ in Theorem 2. of $(V,U]$ and the ray

$\zeta+te^{i(\pi/2+\arg(VW)+2\theta)},\quad t\geqslant 0,$

and $T=T(\zeta)$ be the point of intersection of $(V,U]$ and the ray

$\zeta+te^{i(\pi/2+\arg(VW)+3\theta)},\quad t\geqslant 0.$

Let $\omega\in(0,8/e)$ be a positive parameter.

Then for any compact, linearly Jordan measurable set $J\subset[T,D]$, with its
measure

(10) $|J|\geqslant\omega|D\zeta|,$

and $p\in{\mathcal{P}}_{n}(K)$ we have

$\left|\frac{p^{\prime}}{p}(\zeta)\right|\geqslant\dfrac{\sin\theta}{7.5\log(8/\omega)}\left(n\frac{\sin\theta}{d}-\frac{2}{|T\zeta|}\log\frac{\|p\|_{J}}{|p(\zeta)|}\right).$

Consequently, if $|p(\zeta)|\geqslant\|p\|_{J}$, then

(11)
$\left|\frac{p^{\prime}}{p}(\zeta)\right|\geqslant\dfrac{\sin^{2}\theta}{7.5d\log(8/\omega)}\,n>\dfrac{\sin^{2}\alpha}{2000d\log(8/\omega)}\,n=:C(\alpha,\omega)\,n.$

###### Proof.

1. Without loss of generality we suppose that $[V,W]\subset{\mathbb{R}},$ and $\zeta=0$. Applying the sine theorem, we obtain

$b:=|D\zeta|=\frac{\sin\alpha}{\sin(\pi/2-\alpha+2\theta)}|V\zeta|=\frac{\sin\alpha}{\cos(\alpha-2\theta)}|V\zeta|\leqslant\frac{\sin\alpha}{\sin(2\theta)}|V\zeta|=8|V\zeta|,$
(12)
$a:=|T\zeta|=\frac{\sin\alpha}{\sin(\pi/2-\alpha+3\theta)}|V\zeta|=\frac{\sin\alpha}{\cos(\alpha-3\theta)}|V\zeta|\geqslant|V\zeta|\sin\alpha.$

Clearly, $a<b$ and for all $\xi\in[T,D]$ we have $a\leqslant|\xi|\leqslant b.$

Recall that we have denoted the zeroes of $p\in{\mathcal{P}}_{n}(K)$ as
${\mathcal{Z}}={\mathcal{Z}}(p)$ and we have written
${\mathcal{Z}}=\\{z_{j}=r_{j}e^{i\varphi_{j}},\ j=1,\ldots,n\\}.$ Further, let
us denote

$S(\varphi,\psi):=\\{z\in{\mathbb{C}}~{}:~{}\varphi<\arg
z<\psi\\}\qquad\text{\rm
and}\qquad{\mathcal{Z}}(\varphi,\psi):={\mathcal{Z}}\cap S(\varphi,\psi),$

and similarly for closed, or half-open, half-closed etc. intervals for the
arguments. We split the set ${\mathcal{Z}}$ into the following parts.

$\displaystyle{\mathcal{Z}}_{1}:$ $\displaystyle={\mathcal{Z}}[0,\theta]\,,$
$\displaystyle n_{1}:=\\#{\mathcal{Z}}_{1},$ $\displaystyle{\mathcal{Z}}_{2}:$
$\displaystyle={\mathcal{Z}}(\theta,\pi-\theta)\cap D_{2b}(0)\,,$
$\displaystyle n_{2}:=\\#{\mathcal{Z}}_{2},$ $\displaystyle{\mathcal{Z}}_{3}:$
$\displaystyle={\mathcal{Z}}(\theta,\pi-\theta)\setminus D_{2b}(0)\,,$
$\displaystyle n_{3}:=\\#{\mathcal{Z}}_{3},$ $\displaystyle{\mathcal{Z}}_{4}:$
$\displaystyle={\mathcal{Z}}[\pi-\theta,\pi]\,,$ $\displaystyle
n_{4}:=\\#{\mathcal{Z}}_{4}~{}.$

The partitioning of the set ${\mathcal{Z}}$ and, as a consequence, of the
polygonal domain $K$ is depicted in Fig. 1.

Figure 1. Partition of the set ${\mathcal{Z}}$

In the following, we estimate $\left|\dfrac{p(\xi)}{p(0)}\right|$ for any
$\xi\in J$ from below. It will be convenient to use the notation

$\xi=\rho e^{i(\pi/2+\phi)},\quad 2\theta\leqslant\phi\leqslant 3\theta,\quad
a\leqslant\rho\leqslant b.$

2. Let us start with the estimate of the distance of any $z=re^{i\varphi}\in{\mathcal{Z}}_{1}$ from $J$. By the cosine theorem, we obtain the estimate

$\displaystyle\left|\dfrac{z-\xi}{z}\right|^{2}$
$\displaystyle=\dfrac{r^{2}+\rho^{2}-2r\rho\cos(\pi/2+\phi-\varphi)}{r^{2}}$
$\displaystyle\geqslant 1+\dfrac{2\rho}{r}\sin(\phi-\varphi)\geqslant
1+\dfrac{2\rho}{r}\sin\theta\geqslant 1+\dfrac{2a}{d}\sin\theta,$

in view of $r\leqslant d$. As
$\dfrac{2a}{d}\sin\theta\leqslant\dfrac{2a\sin\alpha}{8d}\leqslant\dfrac{a}{4d}\leqslant\dfrac{1}{2}$,
we can use

$1+x>e^{x/2}\quad\mbox{for}\quad 0<x\leqslant 1/2$

to get

$\left|\dfrac{z-\xi}{z}\right|^{2}\geqslant\exp\left(\dfrac{a\sin\theta}{d}\right).$

Applying this estimate for all the $n_{1}$ zeroes $z_{j}\in{\mathcal{Z}}_{1}$
, we finally come to the inequality

(13)
$\prod_{z_{j}\in{\mathcal{Z}}_{1}}\left|\frac{z_{j}-\xi}{z_{j}}\right|\geqslant\exp\left(n_{1}\dfrac{a\sin\theta}{2d}\right)~{}\qquad\left(\xi\in
J\right).$

3. Now we estimate the distance of any $z=re^{i\varphi}\in{\mathcal{Z}}_{4}$ from $J$. Let $X$ be the intersection of the side $UV$ and the ray emanating from $\zeta$ with an angle $\pi-\theta$ from $\overrightarrow{VW}$. Then in the triangle $\triangle(V\zeta X)$ the sides compare as

$\dfrac{|\zeta X|}{|\zeta
V|}=\dfrac{\sin\alpha}{\sin(\alpha+\theta)}=\dfrac{\sin\alpha}{\sin\alpha\cos\theta+\cos\alpha\sin\theta}\leqslant\dfrac{1}{\cos\theta},$

hence by convexity any point is at most as far from $\zeta$ as
$\max(|V\zeta|,|\zeta X|)\leqslant|V\zeta|/\cos\theta$. As
${\mathcal{Z}}_{4}\subset\triangle(V\zeta X)$, we find
$r\leqslant|V\zeta|/\cos\theta$. By the cosine theorem again, and then using
$\cos(\varphi-\pi/2-\phi)\leqslant\sin(4\theta)$ we get

$\left|\dfrac{z-\xi}{z}\right|^{2}=\dfrac{r^{2}+\rho^{2}-2\rho
r\cos(\varphi-\pi/2-\phi)}{r^{2}}\geqslant
1+\dfrac{\rho}{r}\left(\dfrac{\rho}{r}-2\sin(4\theta)\right).$

To estimate the last quantity we use the following observations. First,
$\dfrac{\rho}{r}\geqslant\dfrac{a}{d}$. Second, by
$r\leqslant|V\zeta|/\cos\theta$ and (12) we find
$\dfrac{\rho}{r}\geqslant\dfrac{a}{|V\zeta|/\cos\theta}\geqslant\sin\alpha\cos\theta$.
Third, (9) implies

$\displaystyle\sin\alpha\cos\theta-2\sin(4\theta)$
$\displaystyle=8\sin(2\theta)\cos\theta-4\sin(2\theta)\cos(2\theta)\geqslant
4\sin(2\theta)\cos(2\theta).$

Hence,
$\cos\theta\cos(2\theta)\geqslant\cos^{2}(2\theta)=\frac{1}{2}(1+\cos(4\theta))\geqslant
1/2$ yields

$\displaystyle\left|\dfrac{z-\xi}{z}\right|^{2}$ $\displaystyle\geqslant
1+\dfrac{a}{d}\left(\sin\alpha\cos\theta-2\sin(4\theta)\right)$
$\displaystyle\geqslant
1+\dfrac{a8\sin\theta\cos\theta\cos(2\theta)}{d}\geqslant
1+\dfrac{4a\sin\theta}{d}.$

The same reasoning as in the previous case gives

(14)
$\prod_{z_{j}\in{\mathcal{Z}}_{4}}\left|\frac{z_{j}-\xi}{z_{j}}\right|\geqslant\exp\left(n_{4}\dfrac{a\sin\theta}{d}\right)\geqslant\exp\left(n_{4}\dfrac{a\sin\theta}{2d}\right)~{}\qquad\left(\xi\in
J\right).$

4. Next we estimate the contribution of zero factors belonging to ${\mathcal{Z}}_{3}$, i.e. the “far” zeroes $z=re^{i\varphi}$ for which we have $r\geqslant 2b$ and $\varphi\in(\theta,\pi-\theta).$ Applying the estimate $1-x\geqslant e^{-2x}$ ($0<x<1/2$) we see that

$\displaystyle\left|\dfrac{z-\xi}{z}\right|^{2}$
$\displaystyle=\left|1-\dfrac{\xi}{z}\right|^{2}\geqslant\left(1-\dfrac{\rho}{r}\right)^{2}\geqslant\left(1-\dfrac{b}{r}\right)^{2}\geqslant\exp\left(-4\dfrac{b}{r}\right).$

This yields

(15)
$\prod_{z_{j}\in{\mathcal{Z}}_{3}}\left|\frac{z_{j}-\xi}{z_{j}}\right|\geqslant\exp\left(-2b\sum_{z_{j}\in{\mathcal{Z}}_{3}}\dfrac{1}{r_{j}}\right)\geqslant\exp\left(-\dfrac{2b}{\sin\theta}\sum_{z_{j}\in{\mathcal{Z}}_{3}}\dfrac{\sin\varphi_{j}}{r_{j}}\right)\qquad(\xi\in
J).$

5. Finally we consider the contribution of the zeroes from ${\mathcal{Z}}_{2}$. By $\Delta(J)\geqslant|J|/4$ (in view of Lemma 1, and Lemma 2), $\dfrac{2b}{r_{j}}\dfrac{\sin\varphi_{j}}{\sin\theta}\geqslant 1$ for $z_{j}=r_{j}e^{i\varphi_{j}}\in{\mathcal{Z}}_{2}$, and (10), we find

(16) $\displaystyle\max_{\xi\in
J}\prod_{z_{j}\in{\mathcal{Z}}_{2}}\left|\frac{z_{j}-\xi}{z_{j}}\right|$
$\displaystyle\geqslant\left(\Delta(J)\right)^{n_{2}}\prod_{z_{j}\in{\mathcal{Z}}_{2}}\frac{1}{2b}\geqslant\left(\frac{|J|}{8b}\right)^{n_{2}}=\exp\left(-n_{2}\log(8/\omega)\right)$
$\displaystyle\geqslant\exp\left(-\log(8/\omega)\dfrac{2b}{\sin\theta}\sum_{z_{j}\in{\mathcal{Z}}_{2}}\dfrac{\sin\varphi_{j}}{r_{j}}\right).$

6. If we collect the estimates (13), (14), (15) and (16) we obtain for a certain point of maxima $\xi_{0}\in J$ in (16) the inequality

$\displaystyle\frac{\|p\|_{J}}{|p(0)|}$
$\displaystyle=\prod_{z_{j}\in{\mathcal{Z}}}\left|\frac{z_{j}-\xi_{0}}{z_{j}}\right|$
$\displaystyle\geqslant\exp\left\\{(n_{1}+n_{4})\dfrac{a\sin\theta}{2d}-\dfrac{2b\log(8/\omega)}{\sin\theta}\sum_{z_{j}\in{\mathcal{Z}}_{2}}\dfrac{\sin\varphi_{j}}{r_{j}}-\dfrac{2b}{\sin\theta}\sum_{z_{j}\in{\mathcal{Z}}_{3}}\dfrac{\sin\varphi_{j}}{r_{j}}\right\\}$
$\displaystyle\geqslant\exp\left\\{n\dfrac{a\sin\theta}{2d}-\left(\dfrac{2b\max(1,\log(8/\omega))}{\sin\theta}+\dfrac{a}{2}\right)\sum_{z_{j}\in{\mathcal{Z}}_{2}\cup{\mathcal{Z}}_{3}}\dfrac{\sin\varphi_{j}}{r_{j}}\right\\}$
$\displaystyle\geqslant\exp\left\\{n\dfrac{a\sin\theta}{2d}-\left(\dfrac{2b\max(1,\log(8/\omega))}{\sin\theta}+\dfrac{a}{2}\right)\sum_{z_{j}\in{\mathcal{Z}}}\dfrac{\sin\varphi_{j}}{r_{j}}\right\\},$

taking into account that $n_{1}+n_{2}+n_{3}+n_{4}=n$ and that for each zeroes
in ${\mathcal{Z}}_{2}\cup{\mathcal{Z}}_{3}$ we have
$\dfrac{\sin\varphi_{j}}{r_{j}}\geqslant\dfrac{\sin\theta}{d}$.

After taking logarithms, using $a\leqslant b$ and the assumption
$\omega\leqslant 8/e$ we arrive at

$\log\frac{\|p\|_{J}}{|p(0)|}\geqslant
n\frac{a\sin\theta}{2d}-\dfrac{2.5b\log(8/\omega)}{\sin\theta}\left|\frac{p^{\prime}}{p}(0)\right|~{},$

that is, writing in again the normalization $\zeta:=0$,

$\displaystyle\left|\frac{p^{\prime}}{p}(\zeta)\right|$
$\displaystyle>\dfrac{a\sin\theta}{5b\log(8/\omega)}\left(n\frac{\sin\theta}{d}-\frac{2}{a}\log\frac{\|p\|_{J}}{|p(0)|}\right).$

It remains to prove that $a/b\geqslant 2/3$, this clearly furnishing the
statement. Indeed, let

$f(\alpha):=\dfrac{a}{b}=\dfrac{\cos(\alpha-2\theta(\alpha))}{\cos(\alpha-3\theta(\alpha))}.$

Straightforward calculations give

$\displaystyle\cos^{2}(\alpha-3\theta)f^{\prime}(\alpha)=$
$\displaystyle-\sin(\alpha-2\theta)\cos(\alpha-3\theta)(1-2\theta^{\prime})$
$\displaystyle+\cos(\alpha-2\theta)\sin(\alpha-3\theta)(1-3\theta^{\prime})$
$\displaystyle=$
$\displaystyle\sin(-\theta)(1-2\theta^{\prime})-\cos(\alpha-2\theta)\sin(\alpha-3\theta)\theta^{\prime}<0.$

This clearly entails, in view of $\sin((3/2)x)\leqslant(3/2)\sin x$,
$x\in[0,\pi/2]$, that it holds

$\dfrac{a}{b}\geqslant
f(\pi/2)=\dfrac{\cos(\pi/2-2\theta(\pi/2))}{\cos(\pi/2-3\theta(\pi/2))}=\dfrac{\sin(\arcsin(1/8))}{\sin((3/2)\arcsin(1/8))}\geqslant\dfrac{2}{3}.$

∎

###  3.3. Proof of Theorem 2

Now we use a different positioning of our $K$ in the complex plane, namely
without loss of generality we suppose that $[V,W]\subset{\mathbb{R}}$ and
$V=0$, $W>0.$

Take any $p\in{\mathcal{P}}_{n}(K).$ First, if $(I_{+}\cup
I_{-})\subset\Gamma\setminus{\mathcal{G}}$, then the left hand side of (5)
vanishes and there remains nothing to prove. So consider the case when
$(I_{+}\cup I_{-})$ has points from ${\mathcal{G}}$. In view of Lemma 5 we may
assume that inequality (7) is valid on $[0,e^{i\alpha}R_{V}]$ (which
corresponds to $I_{+}$ from (6)), containing $J:=[0,8re^{i\alpha}]$. That is,
we have

(17) $|p^{\prime}(z)|\geqslant\frac{\sin\alpha}{8d}n|p(z)|\qquad(z\in
J:=[0,8re^{i\alpha}].$

Hence in particular we also have

(18)
$\int_{J}|p^{\prime}|^{q}\geqslant\left(\frac{\sin\alpha}{8d}\right)^{q}n^{q}\int_{J}|p|^{q}.$

Now we see to the estimation of $|p^{\prime}(s)|$ on
$I:=[0,r]\subset[0,R_{V}/8]$, which corresponds to $I_{-}$ from (6). First of
all we introduce a number $\eta$ such that

(19) $\int_{[0,\eta]}|p|^{q}=\dfrac{1}{2}\int_{I}|p|^{q},$

the parameter $\omega_{\alpha}:=2^{-5}\sin\alpha$, and the set

(20) $\begin{gathered}S=\\{x\in[\eta,r]\ \colon\
|p^{\prime}(x)|\leqslant\kappa n|p(x)|\\},\qquad\textrm{with}\\\
\kappa:=\kappa(\alpha):=C(\alpha,\omega_{\alpha})=\dfrac{\sin^{2}\alpha}{2000\,d\log(8/\omega_{\alpha})}=\dfrac{\sin^{2}\alpha}{2000\,d\log(2^{8}/\sin\alpha)},\end{gathered}$

where $C(\alpha,\omega)$ is the quantity from (11). By definition of $S$ we
immediately have

(21) $\int_{[\eta,r]\setminus
S}|p^{\prime}|^{q}\geqslant\kappa^{q}n^{q}\int_{[\eta,r]\setminus S}|p|^{q}.$

It remains to majorize $\displaystyle\int_{S}|p|^{q}.$ This will be pursued
not by seeking a comparison with $\displaystyle\int_{S}|p^{\prime}|^{q}$, but
with comparing to $\displaystyle\int_{J}|p|^{q}$. When succeeding, we will be
able to use that on the whole $J$ we already have the strong majorization (17)
and (18) of $|p(z)|$ by $|p^{\prime}(z)|$.

As $x$ is real, and $|p(x)|^{2}$ and $|p^{\prime}(x)|^{2}$ are polynomials,
the set $S$ consists of a finite number of segments, some of which may be
degenerate. We dissect each segment of $S$ to essentially disjoint segments
with lengths not exceding $\eta$. Thus we get a representation

$S=\bigcup_{\ell=1}^{m}I_{\ell},\qquad
I_{\ell}:=[u_{\ell},v_{\ell}],\qquad|I_{\ell}|=v_{\ell}-u_{\ell}\leqslant\eta,\quad\ell=1,\ldots,m,$
$u_{1}\leqslant v_{1}\leqslant u_{2}\leqslant\ldots\leqslant u_{m}\leqslant
v_{m}.$

For a point $x\in S$ we introduce the notations, corresponding to that of
Lemma 6:

$D(x):=J\cap\\{x+te^{i(\pi/2+2\theta)}t~{}:~{}t>0\\},\qquad
T(x):=J\cap\\{x+te^{i(\pi/2+3\theta)}t~{}:~{}t>0\\},$
$b(x):=|D(x)-x|=\frac{\sin\alpha}{\sin(\pi/2-\alpha+2\theta)}x=\frac{\sin\alpha}{\cos(\alpha-2\theta)}x,$
$h(x):=|T(x)-D(x)|=\frac{\sin\theta}{\cos(\alpha-3\theta)}b(x)=\frac{\sin\alpha\sin\theta}{\cos(\alpha-2\theta)\cos(\alpha-3\theta)}x\geqslant
2^{-4}(\sin\alpha)^{2}x.$

(Recall that the given rays emanating from $x\in S$ intersect $J$, not only
its straight line, as was said in Footnote 8 when formulating Lemma 6.)

Let $s_{\ell}=\arg\max\\{|p(x)|\colon x\in I_{\ell}\\}.$ We define the points
$D_{\ell}:=D(s_{\ell})$, $T_{\ell}:=T(s_{\ell})$ and also introduce
$b_{\ell}:=b(s_{\ell})$, $h_{\ell}:=h(s_{\ell})$, and two subsets of the
segment $[T_{\ell},D_{\ell}]\subset J$

$J_{\ell}:=\\{z\in[T_{\ell},D_{\ell}]\colon|p(z)|<|p(s_{\ell})|\\},\quad
J_{\ell}^{c}=[T_{\ell},D_{\ell}]\setminus J_{\ell}.$

We claim that $|J_{\ell}|<h_{\ell}/2$ (and hence $|J_{\ell}^{c}|\geqslant
h_{\ell}/2$). Indeed, if $|J_{\ell}|\geqslant h_{\ell}/2$ then we would have
by an application of Lemma 6 with the set $J_{\ell}$ and the parameter

$\omega:=\dfrac{|J_{\ell}|}{b_{\ell}}\geqslant\dfrac{h_{\ell}}{2b_{\ell}}=\frac{\sin\theta}{2\cos(\alpha-3\theta)}\geqslant\dfrac{\sin\theta}{2}>\dfrac{\sin
2\theta}{4}=2^{-5}\sin\alpha=\omega_{\alpha}$

the inequality $\kappa
n|p(s_{\ell})|=C(\alpha,\omega_{\alpha})n|p(s_{\ell})|<C(\alpha,\omega)n|p(s_{\ell})|<|p^{\prime}(s_{\ell})|$,
which contradicts to (20).

Our goal is to construct sets $Q_{\ell}\subset J_{\ell}^{c}$ of linear Jordan
measure $|Q_{\ell}|=2^{-6}(\sin\alpha)^{2}|I_{\ell}|$. A further criterion for
$Q_{\ell}$ is that it must be disjoint from all previous $Q_{i}$s. This can be
ascertained because

$\left|\bigcup_{i=1}^{\ell-1}Q_{i}\right|=\sum_{i=1}^{\ell-1}|Q_{i}|=2^{-6}(\sin\alpha)^{2}\sum_{i=1}^{\ell-1}|I_{i}|\leqslant
2^{-6}(\sin\alpha)^{2}v_{\ell-1}\leqslant
2^{-6}(\sin\alpha)^{2}s_{\ell}\leqslant h_{\ell}/4,$

while

$\displaystyle\left|J_{\ell}^{c}\setminus\bigcup_{i=1}^{\ell-1}Q_{i}\right|$
$\displaystyle\geqslant|J_{\ell}^{c}|-\sum_{i=1}^{\ell-1}|Q_{i}|\geqslant
h_{\ell}/4\geqslant 2^{-6}(\sin\alpha)^{2}s_{\ell}\geqslant
2^{-6}(\sin\alpha)^{2}\eta\geqslant 2^{-6}(\sin\alpha)^{2}|I_{\ell}|.$

Therefore, –taking into account that for all points $z\in Q_{\ell}\subset
J_{\ell}^{c}$ we have by construction $|p(s_{\ell})|\leqslant|p(z)|$– we are
led to

$\int_{I_{\ell}}|p|^{q}\leqslant|p(s_{\ell})||I_{\ell}|\leqslant
2^{6}(\sin\alpha)^{-2}\int_{Q_{\ell}}|p|^{q},\quad\ell=1,\ldots,m,$

and

(22) $\int_{S}|p|^{q}\leqslant 2^{6}(\sin\alpha)^{-2}\int_{J}|p|^{q}$

Finally, collecting (17), (19), (21), and (22), we are led to

$\displaystyle\int_{I\cup J}|p|^{q}$ $\displaystyle=2\int_{[\eta,r]\setminus
S}|p|^{q}+2\int_{S}|p|^{q}+\int_{J}|p|^{q}\leqslant 2\int_{[\eta,r]\setminus
S}|p|^{q}+(1+2^{7}(\sin\alpha)^{-2})\int_{J}|p|^{q}$
$\displaystyle\leqslant\left\\{2\kappa^{-q}+(1+2^{7}(\sin\alpha)^{-2})\left(\frac{8d}{\sin\alpha}\right)^{q}\right\\}n^{-q}\int_{I\cup
J}|p^{\prime}|^{q}.$

##  4\. Proof of Theorem 1

Having Theorem 2 proven, finally in this section we derive the main result of
the paper.

Observe that if $K$ is a convex polygon, and

$\delta_{0}>0$

is chosen to be a small enough constant (depending on $K$, of course), then we
can fix some other (small) positive constant parameter

$h_{0}>0$

such that a point $\zeta\in\partial K$ is either $\delta_{0}$-close to some
vertex $V$ where $K$ has an acute angle (and $\zeta$ lies on one side of the
polygon ending in the vertex $V$), or $\zeta$ has a local depth
$h(\zeta,K)\geqslant h_{0}$.

This can be made explicit in dependence of a few further geometric parameters
of the polygonal convex domain $K$. Indeed, consider any side $[A,B]$ of $K$,
any point $Z\in[A,B]$ on this side, and denote the angles at $A$ and $B$ by
$\alpha$ and $\beta$, respectively. Let us draw the normal chords
$[A,A^{\prime}]$ and $[B,B^{\prime}]$ of $K$ at $A$ and $B$, which are
perpendicular to $[A,B]$. There are three cases corresponding to the number of
acute angles among $\alpha,\beta$ being 0, 1 or 2.

If both angles are at least $\pi/2$, then both normal chords are of positive
length. Drawing the normal chord of $Z$, perpendicular to the side $[A,B]$
thus remains at least as long as the normal chord of the orthogonal trapezium
$A^{\prime}ABB^{\prime}\subset K$. This is at least as long as the minimum of
the distances $|AA^{\prime}|$ and $|BB^{\prime}|$.

If only one angle is an acute angle, say $\alpha$, then $|BB^{\prime}|$ is
still positive and the orthogonal triangle $\triangle(ABB^{\prime})$ is
included in $K$ by convexity. Therefore by simple similarity
$h(Z,K)\geqslant\frac{|AZ|}{|AB|}|BB^{\prime}|\geqslant\delta_{0}\frac{|BB^{\prime}|}{|AB|}$
if $|AZ|\geqslant\delta_{0}$.

Finally, assume that both angles $\alpha$ and $\beta$ are acute angles.
Drawing the other side lines of $K$ with angles $\alpha$ and $\beta$ at $A$
resp. $B$ to $[A,B]$, we obtain an intersection point $C$ and a triangle
$\triangle(ABC)$, which contains $K$. Given that $w:=w(K)>0$, there must exist
a point $Q$ of $K\subset\triangle(ABC)$ at least of distance $w$ from the line
of $[A,B]$. In view of $Q\in K\subset\triangle(ABC)$, the ray emanating from
$Q$ perpendicularly to $[A,B]$ intersects $[A,B]$ in a point $M$ strictly in
$(A,B)$. Note that by convexity $\triangle(ABQ)\subset K$.

Now if $Z$ is any point of $(A,B)$ – say belonging to $(A,M]$ – then the
normal chord at $Z$ is parallel to $MQ$, and as $\triangle(ABQ)\subset K$, by
similarity we find
$h(Z,K)\geqslant\frac{|AZ|}{|AM|}|MQ|\geqslant|AZ|\frac{w}{|AB|}\geqslant\delta_{0}\frac{w}{|AB|}$
if $|AZ|\geqslant\delta_{0}$.

Summing up, we find that in all cases $h(Z,K)\geqslant h_{0}$ unless $Z$ is
$\delta_{0}$-close to some vertex with an acute angle of $K$ at this vertex.

Now the full set $\Gamma:=\partial K$ splits into two parts: the set of points
$\Gamma_{0}$ which are $\delta_{0}$-close to some acute angle vertex, and the
remaining set $\Gamma\setminus\Gamma_{0}$, with local depth of its points
exceeding the preset positive value $h_{0}$.

Generally speaking, for the latter we can directly involve the local depth
Theorem 1 to derive $|p^{\prime}(\zeta)|\gg n|p(\zeta)|$ whenever
$\zeta\in{\mathcal{G}}$ and $n\geqslant n_{0}$, with the implied constant
depending only on the values of $d$ and $h_{0}$, while for $\Gamma_{0}$–so
where $|\zeta-V|<\delta_{0}$ with some vertex $V$ with acute angle–we can
invoke Theorem 2.

Let us execute this program. To make our life easier, assume that the
neighborhoods $D_{V}(8\delta_{0})$ of acute angle vertices remain disjoint –
this is certainly achieved when $\delta_{0}\leqslant\min\\{R_{V}\ :\
V~{}\textrm{is an acute angle vertex of}~{}K\\}$.

So we choose $\delta_{0}:=\frac{1}{8}\min_{V}R_{V}$, pick the corresponding
$h_{0}=h_{0}(\delta_{0})>0$, and denote
$c_{0}:=c_{0}(K,\delta_{0}):=\frac{h_{0}^{4}}{1500d^{5}}$.

Then we can write

(23) $\int_{\partial
K}|p^{\prime}|^{q}\geqslant\int_{\Gamma\setminus\Gamma_{0}}|p^{\prime}|^{q}\geqslant
c_{0}^{q}n^{q}\int_{\Gamma\setminus\Gamma_{0}}|p|^{q}.$

Next, let $V$ be any vertex with the corresponding acute angle $\alpha$. Then
an application of Theorem 2 with $r:=\delta_{0}$ furnishes the estimate

$\int_{\Gamma\cap
D_{8\delta_{0}}(V)}|p^{\prime}|^{q}\geqslant\mu(\alpha)n^{q}\int_{{\mathcal{G}}\cap
D_{\delta_{0}}(V)}|p|^{q}.$

Summing this inequality over all vertices $V$ with acute angles, the sum of
the parts on the left hand side does not exceed the full integral over
$\Gamma$, because by construction for different acute angle vertices these
neighborhoods are disjoint. (Otherwise, we could have referred to the fact
that there are at most three acute angle vertices, yielding a constant 3
here.) On the right hand side, however, the full sum is just the integral over
the totality of $\Gamma_{0}\cap{\mathcal{G}}$. So putting
$\mu:=\min_{K}\mu(\alpha)$ (with the minimum understood over all vertices $V$
of $K$ with acute angles and $\alpha$ denoting the corresponding acute angles)
yields

$\int_{\Gamma}|p^{\prime}|^{q}\geqslant\mu
n^{q}\int_{\Gamma_{0}\cap{\mathcal{G}}}|p|^{q}.$

From here, taking into account (23), we are led to

$\displaystyle 2\int_{\partial K}|p^{\prime}|^{q}$ $\displaystyle\geqslant
c_{0}^{q}n^{q}\int_{\Gamma\setminus\Gamma_{0}}|p|^{q}+\mu
n^{q}\int_{{\mathcal{G}}\cap\Gamma_{0}}|p|^{q}$
$\displaystyle\geqslant\min(c_{0}^{q},\mu)~{}n^{q}\int_{{\mathcal{G}}}|p|^{q}\geqslant\frac{1}{2}\min(c_{0}^{q},\mu)~{}n^{q}\int_{\Gamma}|p|^{q}.$

This concludes the proof of Theorem 1 with the constant
$c(K):=4^{-1/q}\min(\mu^{1/q},c_{0})$.

##  5\. Concluding remarks

As is said above in Theorem 1.1, similarly to the case of the maximum norm,
also for the $L^{q}(\partial K)$ norm any compact convex domain $K$ admits
polynomials $p\in{\mathcal{P}}_{n}(K)$ with oscillation not exceeding $O(n)$.
On the other hand we have shown for several classes of convex domains–convex
domains with fixed depth (including e.g. all smooth domains), all convex
polygons, generalized Erőd-type domains–that the order of oscillation indeed
reaches $c_{K}n$.

A natural question–quite resembling to the question posed by Erőd back in 1939
for the maximum norm case–is to identify those domains which indeed admit
order $n$ oscillation even in $L^{q}(\partial K)$ norm.

It has been clarified that, like in case of the maximum norm, also for $L^{q}$
norms the interval ${\mathbb{I}}$ behaves differently in the sense that there
the order of oscillation may be as low as $\sqrt{n}$. Therefore, it is
certainly necessary that _some_ conditions are assumed for an order $n$
oscillation. The great question is if apart from having a nonempty interior,
is there need for any additional assumption? We think that probably not.

######  Conjecture 1.

For all $0<q<\infty$ and for all compact convex domains $K\Subset{\mathbb{C}}$
there exist $n_{0}:=n_{0}(q,K)$ and $c_{K}=C(K,q)>0$ such that for all
$n\geqslant n_{0}$ and for any $p\in{\mathcal{P}}_{n}(K)$ we have
$\|p^{\prime}\|_{L^{q}(\partial K)}\geqslant c_{K}n\|p\|_{L^{q}(\partial K)}$.

We are not that far from this conjecture, c.f. Theorem 1.1. Nevertheless, in
analysis that mere $\log n$ can be the crux of the situation. Still,
encouraged by the maximum norm case and by our results for the $L^{q}$ norm so
far, we formulated an even more precise conjecture already in [11, Section 7,
Conjecture 2] (and repeated in [14], too).

######  Conjecture 2.

There exists an absolute constant $c>0$ and an $n_{0}=n_{0}(q,w,d)$ such that
for all $0<q<\infty$, for all compact convex domains $K\Subset{\mathbb{C}}$,
for all $n\geqslant n_{0}$ and for any $p\in{\mathcal{P}}_{n}(K)$ we have
$\|p^{\prime}\|_{L^{q}(\partial K)}\geqslant
c\dfrac{w}{d^{2}}n\|p\|_{L^{q}(\partial K)}$.

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# Relations between average clustering coefficient and another centralities in
graphs

Mikhail Tuzhilin Affiliation: Moscow State University, Electronic address:
<EMAIL_ADDRESS>
###### Abstract

Relations between average clustering coefficient and global clustering
coefficient, local efficiency, radiality, closeness, betweenness and stress
centralities were obtained for simple graphs.

###### keywords:

Networks, centralities, local and global properties of graphs, Watts-Strogatz
clustering coefficient, global clustering coefficient.

##  1 Introduction.

The centrality measure was introduced by Bonacich in [1]. Centrality is a
local (with relation to a vertex) or global (with relation to a whole graph)
measures in networks. There are many centrality measures (or shortly
centralities) such as local efficiency, radiality, closeness, betweenness,
stress centralities, etc. Calculation of centralities is very useful for
finding intrinsic properties of “real” networks (which can be found in
applications) [2]\- [4]. One of most important centrality measure is a
clustering coefficient, that differentiate “real” graphs (or small-world
networks) and random generated graphs [5].

There are two definitions of clustering coefficient: the average clustering
coefficient introduced by Watts-Strogatz [5] and the global clustering
coefficient. It was shown in [6] that for windmill graphs the average
clustering coefficient and the global clustering coefficient asymptotically
different. More precisely, the average clustering coefficient tends to 1 and
the global clustering coefficient tends to 0 if the number of vertices
increasing. In this paper, author provides two large class of graphs for which
the average clustering coefficient is less or equal than the global cluster
coefficient and vise versa.

Nowadays, there are also many articles where centrality measures are used for
calculations and predictions of certain network characteristics, but a very
few with theoretical basis. In the article [7] relations between different
centralities were obtained, also an estimation of the local efficiency was
obtained in terms of the average clustering coefficient. In this article
relations between the average clustering coefficient and another centralities
are proved for simple undirected graphs, in particular, it is proved that the
estimation of local efficiency in terms of the average cluster coefficient is
in fact an equality.

##  2 Main definitions.

All subsequent definitions are given for a simple undirected graph $G$ without
pendant vertices. It also can be defined to a simple graph with pendant
vertices if every function where {vertex degree $-1$} is in the denominator
are defined to be equal to $0$ for all cases where vertices degrees equal to
$1$, but this will be omitted in this article for the sake of brevity.

Let’s give necessary denotations. Let’s denote by

  * •

$V(G)$ the set of vertices, $E(G)$ the set of edges and $A=\\{a_{ij}\\}$
adjacency matrix of graph $G$.

  * •

Neighbourhood $N(v)$ — the set of vertices which adjacent to the vertex $v$,

  * •

$N^{\prime}(v)=N(v)\bigcup{v}$ subgraph in $G$ on these vertices,

  * •

$\bar{f}(x_{1},x_{2},...,x_{k})$, where $f$ is any function $V\times
V\times...\times V\rightarrow\mathbb{R}$, the restriction of this function on
$N^{\prime}(v)$ (for example $\bar{L}(x,y)$ will be the average shortest path
between $x$ and $y$ with restriction to subgraph $N^{\prime}(v)$),

  * •

$d_{i}=deg(v_{i})$,

  * •

$n=\|V(G)\|,\;\;m=\|E(G)\|$,

  * •

$X(i)=X(v_{i})$ for any $X$ — set or function corresponding to vertex $v_{i}$,

Let’s give definitions of centralities:

  1. 1.

Diameter $diam(G)=\max_{s,t\in V(G)}dist(s,t)$.

  2. 2.

Density $D(G)=\frac{\text{number of edges in }G}{\text{maximum possible number
of edges in }G}=\frac{2m}{n(n-1)}$.

  3. 3.

Global efficiency $E_{glob}(G)=\frac{1}{n(n-1)}\sum\limits_{s\neq
t}\frac{1}{dist(s,t)}$.

  4. 4.

Average shortest path length $L(G)=\frac{1}{n(n-1)}\sum\limits_{s\neq
t}dist(s,t)$.

  5. 5.

Local cluster coefficient

$c_{i}=c(i)=\frac{\text{number of edges in }N(i)}{\text{maximum possible
number of edges in }N(i)}=\frac{2\|E(N(i)))\|}{d_{i}(d_{i}-1)}$.

  6. 6.

Average clustering coefficient

$C_{WS}(G)=\frac{1}{n}\sum\limits_{i\in
V(G)}c_{i}=\frac{1}{n}\sum\limits_{i\in
V(G)}\frac{2\|E(N(i)))\|}{d_{i}(d_{i}-1)}=\frac{1}{n}\sum\limits_{i\in
V(G)}\frac{\sum\limits_{j,k\in V(G)}a_{ij}a_{jk}a_{ki}}{d_{i}(d_{i}-1)}$.

  7. 7.

Global clustering coefficient

$C(G)=\frac{\text{number of closed triplets in $G$}}{\text{number of all
triplets in $G$}}=\frac{\sum\limits_{i,j,k\in
V(G)}a_{ij}a_{jk}a_{ki}}{\sum\limits_{i\in V(G)}d_{i}(d_{i}-1)}$.

  8. 8.

Betweenness centrality $BC(i)=\sum\limits_{s,t\in V(G),\;s\neq t\neq
i}\frac{\sigma_{st}(i)}{\sigma_{st}}$, where $\sigma_{st}$ is the total number
of shortest paths from $s$ to $t$ and $\sigma_{st}(i)$ is the total number of
shortest paths which contains vertex $i$.

  9. 9.

Closeness centrality $Clo(v)=\frac{n-1}{\sum\limits_{t\in V(G)}dist(v,t)}$.

  10. 10.

Local efficiency $E_{loc}(G)=\frac{1}{n}\sum\limits_{v\in
V(G)}E_{glob}(N(v)).$

  11. 11.

Radiality $Rad(v)=\frac{\sum\limits_{t\in V(G),t\neq
v}(diam(G)+1-dist(v,t))}{n-1}$.

  12. 12.

Stress $Str(i)=\sum\limits_{s,t\in V(G),\;s\neq t\neq i}\sigma_{st}(i)$, where
$\sigma_{st}(i)$ is the total number of shortest paths from $s$ to $t$ which
contains vertex $i$.

Note that all centralities are non-negative and
$D(G),E_{glob},E_{loc},c_{i},C_{WS},C(G)$ are less or equal 1.

##  3 Main results.

All subsequent lemmas and theorems are given for a simple undirected graph $G$
without pendant vertices. It also can be defined to a simple graph with
pendant vertices if every function where $d_{i}-1$ is in the denominator are
defined to be equal to $0$ for all cases $d_{i}=1$.

First let’s prove a lemma about a relation between average shortest path
length between vertices in the neighbourhood of $i$ and local clustering
coefficient of this vertex.

######  Lemma 1.

$L(N(i))=2-c_{i}.$

[1em]0em

###### Proof.

$L(N(i))=\frac{1}{d_{i}(d_{i}-1)}\sum\limits_{s,t\in N(i),s\neq
t}dist(s,t)=\frac{1}{d_{i}(d_{i}-1)}\sum\limits_{(s,t)\in
E(N(i))}dist(s,t)+\sum\limits_{s,t\in N(i),(s,t)\notin E(N(i))}dist(s,t)=$
$=\frac{1}{d_{i}(d_{i}-1)}(2\|E(N(i))\|+\sum\limits_{(s,i),(i,t)\in
E(G),(s,t)\notin E(G)}dist(s,t))=$
$=\frac{1}{d_{i}(d_{i}-1)}(2\|E(N(i))\|+2(d_{i}(d_{i}-1)-2\|E(N(i))\|))=2-c_{i}.$

Note that shortest paths for vertices in $N(i)$ are defined corresponding to
whole graph $G$. ∎

Let’s prove theorem about a connection between local efficiency and average
clustering coefficient of a graph.

######  Theorem 1.

$E_{loc}(G)=\frac{1}{2}(1+C_{WS}(G)).$

[1em]0em

###### Proof.

Let’s give two proofs of this fact:

  1. 1.

Note that by definition $D(N(i))=c_{i}$. In the article [7] it was proved that

$3-L(N(i))\leq 2E_{glob}(N(i))\leq 1+D(N(i)).$

Using lemma 1

$3-(2-c_{i})\leq 2E_{glob}(N(i))\leq 1+c_{i}.$

Note that shortest paths for vertices in $N(i)$ are defined corresponding to
whole graph $G$. Averaging by $i$ ends the proof.

  2. 2.

Let’s rewrite the local clustering coefficient formula:

$c_{i}=\frac{\sum\limits_{(s,t)\in E(N(i))}1}{d_{i}(d_{i}-1)},$
$\frac{1}{2}(1+c_{i})=\frac{1}{2}\frac{\sum\limits_{(s,t)\in
E(N(i))}1+\sum\limits_{(s,t)\in E(N(i))}1+\sum\limits_{s,t\in
V(N(i)),(s,t)\notin E(N(i))}1}{d_{i}(d_{i}-1)}=$ $=\frac{\sum\limits_{(s,t)\in
E(N(i))}1+\sum\limits_{s,t\in V(N(i)),(s,t)\notin
E(N(i))}\frac{1}{2}}{d_{i}(d_{i}-1)}=\frac{\sum\limits_{s,t\in
V(N(i))}\frac{1}{dist(s,t)}}{d_{i}(d_{i}-1)}=E_{glob}(N(i)).$

Averaging by $i$ ends the proof.

∎

Let’s prove theorem about a connection between average clustering coefficient
and stress centrality.

######  Theorem 2.

$C_{WS}(G)\geq\frac{1}{n}\sum\limits_{i\in
V(G)}(1-\frac{Str(i)}{d_{i}(d_{i}-1)}).$

[1em]0em

###### Proof.

Note that $\forall j,k\in N(i):(j,k)\notin E(N(i))$ the shortest path between
$j$ and $k$ is $j\rightarrow i\rightarrow k$. Therefore,

$Str(i)\geq 2(\frac{d_{i}(d_{i}-1)}{2}-\|E(N(i))\|),$
$\frac{1}{d_{i}(d_{i}-1)}Str(i)\geq 1-c_{i},$

Averaging by $i$

$C_{WS}(G)\geq\frac{1}{n}\sum\limits_{i\in
V(G)}(1-\frac{Str(i)}{d_{i}(d_{i}-1)}).$

Note that for $diam(G)=2$ holds an equality. ∎

Let’s prove theorem about a relation between average clustering coefficient
and betweenness centrality.

######  Theorem 3.

Let’s $BC(i,N(i)):=\sum\limits_{j,k\in N(i),\;j\neq k\neq
i}\frac{\sigma_{jk}(i)}{\sigma_{jk}},$ then

$C_{WS}(G)\leq\frac{1}{n}\sum\limits_{i\in
V(G)}(1-\frac{BC(i,N(i))}{d_{i}(d_{i}-1)}).$

[1em]0em

###### Proof.

Let’s note that

$BC(i,N(i))=\sum\limits_{j,k\in N(i),\;(j,k)\notin
E(N(i))}\frac{1}{\sigma_{jk}}\leq\sum\limits_{j,k\in N(i),\;(j,k)\notin
E(N(i))}1=d_{i}(d_{i}-1)-2\|E(N(i))\|,$ $\frac{BC(i,N(i))}{d_{i}(d_{i}-1)}\leq
1-c_{i}.$

Averaging by $i$

$C_{WS}(G)\leq\frac{1}{n}\sum\limits_{i\in
V(G)}(1-\frac{BC(i,N(i))}{d_{i}(d_{i}-1)}).$

Note that for $diam(G)=2$ holds an equality. ∎

By using theorems 2 and 3 an estimation of average shortest path in the
neighborhood of $i$ is obtained.

######  Corollary 1.

$\frac{BC(i,N(i))}{d_{i}(d_{i}-1)}\leq
L(N(i))-1\leq\frac{Str(i)}{d_{i}(d_{i}-1)}.$

Note that shortest paths for vertices in $N(i)$ are defined corresponding to
whole graph $G$.

Let’s prove lemma about a relation between average closeness centrality and
average shortest path length in graph.

######  Lemma 2.

$\frac{1}{n}\sum\limits_{v\in V(G)}Clo(v)\geq\frac{1}{L(G)}.$

[1em]0em

###### Proof.

By the inequality of harmonic mean and arithmetic mean

$\frac{1}{n}\sum\limits_{v\in V(G)}Clo(v)=\frac{1}{n}\sum\limits_{v\in
V(G)}\frac{n-1}{\sum\limits_{t\in
V(G)}dist(v,t)}\geq\frac{n(n-1)}{\sum\limits_{v,t\in
V(G)}dist(v,t)}=\frac{1}{L(G)}.$

Note that an equality holds when all average shortest path lengths from any
vertex to all remaining vertices are equal. ∎

Now let’s prove theorem about a relation between average clustering
coefficient and closeness centrality.

######  Theorem 4.

$\frac{1}{2-C_{WS}(G)}\leq\frac{1}{n}\sum\limits_{i\in
V(G)}\frac{\sum\limits_{v\in N(i)}\overline{Clo}(v)}{d_{i}}.$

[1em]0em

###### Proof.

By lemma 2

$\frac{1}{d_{i}}\sum\limits_{v\in
N(i)}\overline{Clo}(v)\geq\frac{1}{L(N(i))}=\frac{1}{2-c_{i}}.$

By the inequality of harmonic mean and arithmetic mean (since $\forall i\in
V(G),\;0\leq c_{i}\leq 1$):

$\frac{1}{n}\sum\limits_{i\in V(G)}\frac{\sum\limits_{v\in
N(i)}\overline{Clo}(v)}{d_{i}}\geq\frac{1}{n}\sum\limits_{i\in
V(G)}\frac{1}{2-c_{i}}\geq\frac{n}{\sum\limits_{i\in
V(G)}(2-c_{i})}=\frac{1}{2-C_{WS}(G)}.$

∎

Let’s prove lemma about a relation between average shortest path length and
average radiality.

######  Lemma 3.

$\frac{1}{n}\sum\limits_{v\in V(G)}{Rad}(v)=diam(G)+1-L(G).$

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###### Proof.

The proof holds from definition

$\frac{1}{n}\sum\limits_{v\in V(G)}{Rad}(v)=\frac{1}{n}\sum\limits_{v\in
V(G)}\frac{(n-1)(diam(G)+1)-\sum\limits_{t\in V(G),\;t\neq
v}dist(v,t))}{n-1}=diam(G)+1-L(G).$

∎

Now let’s prove theorem about a relation between average clustering
coefficient and radiality.

######  Theorem 5.

$C_{WS}(G)=\frac{1}{n}\sum\limits_{i\in V(G)}(\frac{1}{d_{i}}\sum\limits_{v\in
N(i)}\overline{Rad}(v)-1)+\frac{\\#\\{N(i)\text{ which are complete
graphs}\\}}{n}.$

[1em]0em

###### Proof.

By lemma 3

$\frac{1}{d_{i}}\sum\limits_{v\in
N(i)}\overline{Rad}(v)=diam(N^{\prime}(i))+1-L(N(i))=diam(N^{\prime}(i))-1+c_{i}=c_{i}+1-\chi_{K_{d_{i}}}(N^{\prime}(i)),$

where $\chi_{K_{d_{i}}}(N^{\prime}(i))=\begin{cases}1&\text{if
$N^{\prime}(i)=K_{d_{i}}$}\\\ 0&\text{otherwise}\end{cases}$. Averaging by $i$
ends the proof. ∎

Let’s prove two theorems about a relation between average clustering
coefficient and global clustering coefficient.

######  Theorem 6.

Let’s $\forall i,j\in V(G),\;i\leq j$ hold $d_{i}\leq d_{j}\Rightarrow
c_{i}\leq c_{j}$, then

$C_{WS}(G)\leq C(G).$

[1em]0em

###### Proof.

Let’s re-numerate vertices such that $\forall i\leq j:d_{i}\leq d_{j}$. Note
that

$c_{i}=\frac{\sum\limits_{j,k\in
V(G)}a_{ij}a_{jk}a_{ki}}{d_{i}(d_{i}-1)},\;\;C(G)=\frac{\sum\limits_{i,j,k\in
V(G)}a_{ij}a_{jk}a_{ki}}{\sum\limits_{i\in V(G)}d_{i}(d_{i}-1)}.$

Indeed,

$a_{ij}a_{jk}a_{ki}=\begin{cases}1&\text{if there exists edge between vertices
$j$ and $k$ which adjacent to vertex $i$}\\\ 0&\text{otherwise}\end{cases}$

. Therefore,

$C_{WS}(G)=\frac{1}{n}\sum\limits_{i\in V(G)}\frac{\sum\limits_{j,k\in
V(G)}a_{ij}a_{jk}a_{ki}}{d_{i}(d_{i}-1)}.$

Let’s denote by $x_{i}=d_{i}(d_{i}-1)$. Since
$\|E(N(i))\|=\frac{1}{2}\sum\limits_{j,k\in V(G)}a_{ij}a_{jk}a_{ki}$ and the
maximum number of edges in subgraph $N(i)$ equals to
$\frac{d_{i}(d_{i}-1)}{2}$, then $x_{i}\geq 2,\;0\leq c_{i}\leq 1.$ Hence,
using Chebyshev’s sum inequality ($d_{i}\leq d_{j}\Rightarrow x_{i}\leq
x_{j}\text{ and }c_{i}\leq c_{j}$):

$\frac{1}{n}\sum\limits_{i\in
V(G)}x_{i}\;C_{WS}(G)=(\frac{1}{n}\sum\limits_{i\in
V(G)}x_{i})(\frac{1}{n}\sum\limits_{i\in
V(G)}c_{i})\leq\frac{1}{n}\sum\limits_{i\in
V(G)}x_{i}c_{i}=\frac{1}{n}\sum\limits_{i,j,k\in V(G)}a_{ij}a_{jk}a_{ki}.$

Therefore,

$C_{WS}(G)\leq\frac{\sum\limits_{i,j,k\in
V(G)}a_{ij}a_{jk}a_{ki}}{\sum\limits_{i\in V(G)}d_{i}(d_{i}-1)}=C(G).$

The equality holds when $\forall i,j\in V(G):d_{i}=d_{j}$, that is for a
graphs in which all vertices degrees are equal. ∎

######  Corollary 2.

Let’s $\forall i,j\in V(G),\;i\leq j$ hold $d_{i}\leq d_{j}\Rightarrow
c_{i}\geq c_{j}$, then

$C_{WS}(G)\geq C(G).$

The proof is the same as in theorem 6.

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