On the necessary condition for entire function with the increasing second quotients of Taylor coefficients to belong to the Laguerre-P\'olya class

TL;DR

This paper establishes a necessary condition involving the increasing second quotients of Taylor coefficients for an entire function with positive coefficients to belong to the Laguerre-Pólya class, identifying a critical limit constant.

Contribution

It introduces a new necessary condition based on the behavior of second quotients of Taylor coefficients for classifying entire functions within the Laguerre-Pólya class.

Findings

Functions with increasing second quotients do not belong to the Laguerre-Pólya class if the limit is below a specific constant.

Identifies a critical limit constant approximately equal to 3.2336.

Provides a criterion involving the asymptotic behavior of Taylor coefficient quotients.

Abstract

For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k, a_k>0,$ we show that $f$ does not belong to the Laguerre-P\'olya class if the quotients $\frac{a_{n-1}^2}{a_{n-2}a_n}$ are increasing in $n$, and $c:= \lim\limits_{n\to \infty} \frac{a_{n-1}^2}{a_{n-2}a_n}$ is smaller than an absolute constant $q_\infty$ $(q_\infty\approx 3{.}2336) .$

Equations

$$f(x)=c{x}^n{e}^{-\alpha x^2+\beta x}\prod _{k=1}^{\infty }\left(1-\frac{x}{x_k}\right)e^{x{x}_k^{-1}},$$

$$p_n=p_n(f):=\frac{a_{n-1}}{a_n},n\ge 1;$$

$$q_n=q_n(f):=\frac{p_n}{p_{n-1}}=\frac{a_{n-1}^2}{a_{n-2}{a}_n},n\ge 2.$$

$$a_n=\frac{a_0}{p_1{p}_2\dots p_n},n\ge 1;$$

$$a_n=\frac{a_1}{q_2^{n-1}{q}_3^{n-2}\dots q_{n-1}^2{q}_n}{\left(\frac{a_1}{a_0}\right)}^{n-1},n\ge 2.$$

$$Z_c\left(\sum _{k=0}^n{\gamma }_k{a}_k{z}^k\right)\le Z_c\left(\sum _{k=0}^n{a}_k{z}^k\right),$$

$${\left(a^{-k^2}\right)}_{k=0}^{\infty }\in \mathcal{CZ}\mathcal{D}\mathcal{S},a\ge 1,{\left(\frac{1}{k!}\right)}_{k=0}^{\infty }\in \mathcal{CZ}\mathcal{D}\mathcal{S}.$$

$$0\le \sum _{k=1}^{\infty }\frac{1}{z_k^2}={\left(\sum _{k=1}^{\infty }\frac{1}{z_k}\right)}^2-2\sum _{1\le i<j<\infty }\frac{1}{z_i{z}_j}={\left(\frac{a_1}{a_0}\right)}^2-2\frac{a_2}{a_0},$$

$$(\frac{1}{z_1}+\frac{1}{z_2}+...)(\frac{1}{z_1^3}+\frac{1}{z_2^3}+\dots )\ge (\frac{1}{z_1^2}+\frac{1}{z_2^2}+\dots {)}^2.$$

$${\sigma }_1({\sigma }_1^3-3{\sigma }_1{\sigma }_2+3{\sigma }_3)\ge ({\sigma }_1^2-2{\sigma }_2{)}^2,$$

$$\frac{a_1^2{a}_2}{a_0^3}+3\frac{a_1{a}_3}{a_0^2}-4\frac{a_2^2}{a_0^2}\ge 0.$$

$$q_3(q_2-4)+3\ge 0.$$

$$q_2(q_2-4)+3\ge 0.$$

$$1\ge x>\frac{x^2}{q_2}>\frac{x^3}{q_2^2{q}_3}>\frac{x^4}{q_2^3{q}_3^2{q}_4}>\cdots ,$$

$$\phi (x)>0\text{mbox}forallx\in [0;1].$$

$$1<x\ge \frac{x^2}{q_2}>\frac{x^3}{q_2^2{q}_3}>\cdots >\frac{x^k}{q_2^{k-1}{q}_3^{k-2}\dots q_{k-1}^2{q}_k}>\cdots$$

$$\phi (x)=\left(1-x+\sum _{k=2}^{2m+1}\frac{(-1{)}^k{x}^k}{q_2^{k-1}{q}_3^{k-2}\dots q_{k-1}^2{q}_k}\right)$$

$$+\sum _{k=2m+2}^{\infty }\frac{(-1{)}^k{x}^k}{q_2^{k-1}{q}_3^{k-2}\dots q_{k-1}^2{q}_k}=:S_{2m+1}(x,\phi )+R_{2m+2}(x,\phi ).$$

$$\phi (x)>S_{2m+1}(x,\phi )\text{mbox}forallx\in (1;q_2],m\in \mathbb{N}.$$

$$S_{2m+1}(x,\phi )=(1-x)+\left(\frac{x^2}{q_2}-\frac{x^3}{q_2^2{q}_3}\right)+\left(\frac{x^4}{q_2^3{q}_3^2{q}_4}-\frac{x^5}{q_2^4{q}_3^3{q}_4^2{q}_5}\right)+\dots$$

$$+\left(\frac{x^{2m}}{q_2^{2m-1}{q}_3^{2m-2}\cdot \dots \cdot q_{2m-1}^2{q}_{2m}}-\frac{x^{2m+1}}{q_2^{2m}{q}_3^{2m-1}\cdot \dots \cdot q_{2m}^2{q}_{2m+1}}\right).$$

$$\frac{x^{2k}}{q_2^{2k-1}{q}_3^{2k-2}\cdot \dots \cdot q_{2k}}-\frac{x^{2k+1}}{q_2^{2k}{q}_3^{2k-1}\cdot \dots \cdot q_{2k}^2{q}_{2k+1}}\ge$$

$$\frac{x^{2k}}{c^{2k-1}\cdot c^{2k-2}\cdot \dots \cdot c}-\frac{x^{2k+1}}{c^{2k}\cdot c^{2k-1}\cdot \dots \cdot c^2\cdot c}.$$

$$F(q_2,q_3,\dots ,q_{2k},q_{2k+1}):=\frac{x^{2k}}{q_2^{2k-1}{q}_3^{2k-2}\cdot \dots \cdot q_{2k}}-\frac{x^{2k+1}}{q_2^{2k}{q}_3^{2k-1}\cdot \dots \cdot q_{2k}^2{q}_{2k+1}}.$$

$$\frac{\partial F(q_2,q_3,\dots ,q_{2k},q_{2k+1})}{\partial q_2}=-\frac{(2k-1)x^{2k}}{q_2^{2k}{q}_3^{2k-2}\dots q_{2k}}+\frac{2k{x}^{2k+1}}{q_2^{2k+1}{q}_3^{2k-1}\dots q_{2k}^2{q}_{2k+1}}<0$$

$$\Leftrightarrow x<\Big{(}1-\frac{1}{2k}\Big{)}q_{2}q_{3}\ldots q_{2k}q_{2k+1}.$$

$$F(q_2,q_3,q_4,\dots ,q_{2k},q_{2k+1})\ge F(q_3,q_3,q_4,\dots ,q_{2k},q_{2k+1})=$$

$$\frac{x^{2k}}{q_3^{4k-3}{q}_4^{2k-3}\dots q_{2k}}-\frac{x^{2k+1}}{q_3^{4k-1}{q}_4^{2k-2}\dots q_{2k+1}}.$$

$$\frac{\partial F(q_3,q_3,q_4,\dots ,q_{2k},q_{2k+1})}{\partial q_3}=-\frac{(4k-3)x^{2k}}{q_3^{4k-2}{q}_4^{2k-3}\dots q_{2k}}+\frac{(4k-1)x^{2k+1}}{q_3^{4k}{q}_4^{2k-2}\dots q_{2k+1}}<0$$

$$\iff x<\frac{4k-3}{4k-1}{q}_3^2{q}_4\dots q_{2k+1}.$$

$$F(q_3,q_3,q_4\dots ,q_{2k},q_{2k+1})\ge F(q_4,q_4,q_4,q_5,\dots ,q_{2k},q_{2k+1}).$$

$$F(q_2,q_3,q_4,\dots ,q_{2k},q_{2k+1})\ge F(q_3,q_3,q_4,\dots ,q_{2k},q_{2k+1})\ge$$

Full text

1. Introduction

The zero distribution of entire functions, its sections and tails have been studied by many authors, see, for example, the remarkable survey of the topic in [20]. In this paper we investigate a new necessary condition under which some special entire functions have only real zeros. First, we need the definition of the famous Laguerre-Pólya class.

Definition 1. A real entire function $f$ is said to be in the Laguerre-Pólya class, written $f\in\mathcal{L-P}$, if it can be expressed in the form

[TABLE]

where $c,\alpha,\beta,x_{k}\in\mathbb{R}$, $x_{k}\neq 0$, $\alpha\geq 0$, $n$ is a nonnegative integer and $\sum_{k=1}^{\infty}x_{k}^{-2}<\infty$. As usual, the product on the right-hand side can be finite or empty (in the latter case the product equals 1).

This class is essential in the theory of entire functions due to the fact that the polynomials with only real zeros converge locally uniformly to these and only these functions. The following prominent theorem states an even stronger fact.

Theorem A (E.Laguerre and G.Pólya, see, for example, [4, p. 42-46]).

(i) Let $(P_{n})_{n=1}^{\infty},\ P_{n}(0)=1,$ be a sequence of complex polynomials having only real zeros which converges uniformly in the circle $|z|\leq A,A>0.$ Then this sequence converges locally uniformly to an entire function from the $\mathcal{L-P}$ class.

(ii) For any $f\in\mathcal{L-P}$ there is a sequence of complex polynomials with only real zeros which converges locally uniformly to $f$.

For various properties and characterizations of the Laguerre-Pólya class see [22, p. 100], [23] or [19, Kapitel II].

Note that for a real entire function (not identically zero) of order less than $2$ having only real zeros is equivalent to belonging to the Laguerre-Pólya class. The situation is different when an entire function is of order $2$. For example, the function $f_{1}(x)=e^{-x^{2}}$ belongs to the Laguerre-Pólya class, but the function $f_{2}(x)=e^{x^{2}}$ does not.

Let $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ be an entire function with positive coefficients. We define the quotients $p_{n}$ and $q_{n}$:

[TABLE]

The following formulas can be verified by straight forward calculations.

[TABLE]

Deciding whether a given entire function has only real zeros is a rather subtle problem. In 1926, J. I. Hutchinson found the following sufficient condition for an entire function with positive coefficients to have only real zeros.

Theorem B (J. I. Hutchinson, [5]). *Let $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$, $a_{k}>0$ for all $k$. Then $q_{n}(f)\geq 4$, for all $n\geq 2,$ if and only if the following two conditions are fulfilled:

(i) The zeros of $f(z)$ are all real, simple and negative, and

(ii) the zeros of any polynomial $\sum_{k=m}^{n}a_{k}z^{k}$, $m<n,$ formed by taking any number of consecutive terms of $f(z)$, are all real and non-positive.*

For some extensions of Hutchinson’s results see, for example, [3, §4].

We will use the well-known notion of a complex zero decreasing sequence. For a real polynomial $P$ we denote by $Z_{c}(P)$ the number of nonreal zeros of $P$ counting multiplicities.

Definition 2. A sequence $(\gamma_{k})_{k=0}^{\infty}$ of real numbers is said to be a complex zero decreasing sequence (we write $(\gamma_{k})_{k=0}^{\infty}\in\mathcal{CZDS}$), if

[TABLE]

for any real polynomial $\sum_{k=0}^{n}a_{k}z^{k}.$

The existence of nontrivial $\mathcal{CZDS}$ sequences is a consequence of the following remarkable theorem proved by Laguerre and extended by Pólya.

Theorem C (G. Pólya, see [21] or [22, pp. 314-321]). Let $f$ be an entire function from the Laguerre-Pólya class having only negative zeros. Then $\left(f(k)\right)_{k=0}^{\infty}\in\mathcal{CZDS}$.

As it follows from the theorem above,

[TABLE]

The entire function $g_{a}(z)=\sum_{j=0}^{\infty}z^{j}a^{-j^{2}}$, $a>1,$ a so called partial theta-function, was investigated in the paper [6]. Simple calculations show that $q_{n}(g_{a})=a^{2}$ for all $n.$ Since \left(a^{-k^{2}}\right)_{k=0}^{\infty}\in\mathcal{CZDS}\ , for $a\geq 1,$ we conclude that for every $n\geq 2$ there exists a constant $c_{n}>1$ such that $S_{n}(z,g_{a}):=\sum_{j=0}^{n}z^{j}a^{-j^{2}}\in\mathcal{L-P}$ $\ \Leftrightarrow\ a^{2}\geq c_{n}$.

The survey [25] by S.O. Warnaar contains the history of investigation of the partial theta-function and its interesting properties.

Theorem D (O. Katkova, T. Lobova, A. Vishnyakova, [6]). There exists a constant $q_{\infty}$ $(q_{\infty}\approx 3{.}23363666)$ such that:

1. (1)

$g_{a}(z)\in\mathcal{L-P}\Leftrightarrow\ a^{2}\geq q_{\infty};$** 2. (2)

g_{a}(z)\in\mathcal{L-P}\Leftrightarrow\* there exists $x_{0}\in(-a^{3},-a)$ such that $\ g_{a}(x_{0})\leq 0;$* 3. (3)

for a given $n\geq 2$ we have $S_{n}(z,g_{a})\in\mathcal{L-P}$ \ \Leftrightarrow\ there exists $x_{n}\in(-a^{3},-a)$ such that $\ S_{n}(x_{n},g_{a})\leq 0;$ 4. (4)

$4=c_{2}>c_{4}>c_{6}>\ldots$* and $\lim_{n\to\infty}c_{2n}=q_{\infty};$* 5. (5)

$3=c_{3}<c_{5}<c_{7}<\ldots$* and $\lim_{n\to\infty}c_{2n+1}=q_{\infty}.$*

There is a series of works by V.P. Kostov dedicated to the interesting properties of zeros of the partial theta-function and its derivative (see [8], [9], [10], [11], [12], [13], [14], [15] and [16]). For example, in [9], V.P. Kostov studied the so-called spectrum of the partial theta function, i.e. the set of values of $a>1$ for which the function $g_{a}$ has a multiple real zero.

Theorem E (V.P. Kostov, [9]).

1. (1)

The spectrum $\Gamma$ of the partial theta-function consists of countably many values of $a$ denoted by $\widetilde{a}_{1}>\widetilde{a}_{2}>\ldots>\widetilde{a}_{k}>\ldots>1,$ $\lim_{j\to\infty}\widetilde{a}_{j}=1.$ 2. (2)

For $\widetilde{a}_{k}\in\Gamma$ the function $g_{\widetilde{a}_{k}}$ has exactly one multiple real zero which is of multiplicity 2 and is the rightmost of its real zeros. 3. (3)

For $a\in(\widetilde{a}_{k+1},\widetilde{a}_{k})$ the fucntion $g_{a}$ has exactly $k$ complex conjugate pairs of zeros (counted with multiplicities).

A wonderful paper [17] among the other results explains the role of the constant $q_{\infty}$ in the study of the set of entire functions with positive coefficients having all Taylor truncations with only real zeros.

Theorem F (V.P. Kostov, B. Shapiro, [17]). *Let $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ be an entire function with positive coefficients and $S_{n}(z)=\sum_{j=0}^{n}a_{j}z^{j}$ be its sections. Suppose that there exists $N\in{\mathbb{N}},$ such that for all $n\geq N$ the sections $S_{n}(z)=\sum_{j=0}^{n}a_{j}z^{j}$ belong to the Laguerre-Pólya class. Then $\lim\inf_{n\to\infty}q_{n}(f)\geq q_{\infty}$. *

In [7], some entire functions with a convergent sequence of second quotients of coefficients are investigated. The main question of [7] is whether a function and its Taylor sections belong to the Laguerre-Pólya class. In [2] and [1], some important special functions with increasing sequence of second quotients of Taylor coefficients are studied.

In the previous paper [18], we have studied the entire functions with positive Taylor coefficients such that $q_{n}(f)$ are decreasing in $n$.

Theorem G (T. H. Nguyen, A. Vishnyakova, [18]). Let $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$, $a_{k}>0$ for all $k$, be an entire function. Suppose that $q_{n}(f)$ are decreasing in $n$, i.e. $q_{2}\geq q_{3}\geq q_{4}\geq\ldots,$ and $\lim\limits_{n\to\infty}q_{n}(f)=b\geq q_{\infty}$. Then all the zeros of $f$ are real and negative, in other words $f\in\mathcal{L-P}$.

It is easy to see that, if only the estimation of $q_{n}(f)$ from below is given and the assumption of monotonicity is omitted, then the constant $4$ in $q_{n}(f)\geq 4$ is the smallest possible to conclude that $f\in\mathcal{L-P}$.

In this paper, we study the case when $q_{n}(f)$ are increasing in $n$ and have obtained the following theorem.

Theorem 1.1**.**

Let $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$, $a_{k}>0$ for all $k,$ be an entire function. Suppose that the quotients $q_{n}(f)$ are increasing in $n$, and $\lim\limits_{n\to\infty}q_{n}(f)=c<q_{\infty}$. Then the function $f$ does not belong to the Laguerre-Pólya class.

2. Proof of Theorem 1.1

Without loss of generality, we can assume that $a_{0}=a_{1}=1,$ since we can consider a function $g(z)=a_{0}^{-1}f(a_{0}a_{1}^{-1}z)$ instead of $f(z),$ due to the fact that such rescaling of $f$ preserves its property of having real zeros and preserves the second quotients: $q_{n}(g)=q_{n}(f)$ for all $n.$ During the proof we use the notations $p_{n}$ and $q_{n}$ instead of $p_{n}(f)$ and $q_{n}(f).$ So we can write $f(x)=1+x+\sum_{k=2}^{\infty}\frac{x^{k}}{q_{2}^{k-1}q_{3}^{k-2}\ldots q_{k-1}^{2}q_{k}}.$ We will also consider a function $\varphi(x)=f(-x)=1-x+\sum_{k=2}^{\infty}\frac{(-1)^{k}x^{k}}{q_{2}^{k-1}q_{3}^{k-2}\ldots q_{k-1}^{2}q_{k}}$ instead of $f.$

Since the quotients $q_{n}$ are increasing in $n$, and $\lim\limits_{n\to\infty}q_{n}=c<q_{\infty},$ we conclude that $q_{2}\leq q_{\infty}<4.$ The following lemma shows that for $q_{2}<3$ we have $\varphi\notin\mathcal{L-P}.$

Lemma 2.1**.**

Let $\varphi(z)=\sum_{k=0}^{\infty}(-1)^{k}a_{k}z^{k}$ be an entire function, $a_{k}>0$ for all $k,$ $a_{0}=a_{1}=1$, and $q_{n}=q_{n}(\varphi)$ are increasing in $n$, i.e. $q_{2}\leq q_{3}\leq q_{4}\leq\ldots.$ If $\varphi\in\mathcal{L-P},$ then $q_{2}(f)\geq 3$.

Proof.

Denote by $0<z_{1}\leq z_{2}\leq z_{3}\leq\ldots$ the real roots of $\varphi$. We observe that

[TABLE]

whence $q_{2}\geq 2.$

According to the Cauchy-Bunyakovsky-Schwarz inequality, we obtain

[TABLE]

By Vieta’s formulas, we have $\sigma_{1}:=\sum_{k=1}^{\infty}\frac{1}{z_{k}}=\frac{a_{1}}{a_{0}},$ $\sigma_{2}=\sum_{1<i<j<\infty}\frac{1}{z_{i}z_{j}}=\frac{a_{2}}{a_{0}},$ and $\sigma_{3}=\sum_{1<i<j<k<\infty}\frac{1}{z_{i}z_{j}z_{k}}=\frac{a_{3}}{a_{0}}.$ Further, we need the following identities: $\sum_{k=1}^{\infty}\frac{1}{z_{1}^{2}}=\sigma_{1}^{2}-2\sigma_{2},$ and $\sum_{k=1}^{\infty}\frac{1}{z_{1}^{3}}=\sigma_{1}^{3}-3\sigma_{1}\sigma_{2}+3\sigma_{3}.$ Consequently, we have

[TABLE]

or

[TABLE]

Since $a_{0}=a_{1}=1$ and $a_{2}=\frac{1}{q_{2}},a_{3}=\frac{1}{q_{2}^{2}q_{3}}$, we get:

[TABLE]

Since we have the conditions that $q_{2}<4$ and $q_{2}\leq q_{3},$ we conclude that

[TABLE]

Thus, we get that $q_{2}\geq 3.$ ∎

Further, we assume that $3\leq q_{2}<q_{\infty}.$

In order to prove Theorem 1.1, we need some more Lemmas.

Lemma 2.2**.**

Let $\varphi(x)=1-x+\sum_{k=2}^{\infty}\frac{(-1)^{k}x^{k}}{q_{2}^{k-1}q_{3}^{k-2}\ldots q_{k-1}^{2}q_{k}}$ be an entire function. Suppose that $q_{2}\geq 2,$ $q_{k}$ are increasing in $k$, i.e. $q_{2}\leq q_{3}\leq q_{4}\ldots$, and $\lim\limits_{n\to\infty}q_{n}=c<q_{\infty}$. Then for any $x\in[0,q_{2}]$ we have $\varphi(x)>0$, i.e. there are no real roots of $\varphi$ in the segment $[0;q_{2}]$.

Proof.

For $x\in[0;1]$ we have

[TABLE]

whence

[TABLE]

Suppose that $x\in(1;q_{2}].$ Then we obtain

[TABLE]

For an arbitrary $m\in{\mathbb{N}}$ we have

[TABLE]

[TABLE]

By (7) we obtain $R_{2m+2}(x,\varphi)>0$ for all $x\in(1;q_{2}],$ or

[TABLE]

It remains to prove that there exists $m\in{\mathbb{N}}$ such that $S_{2m+1}(x,\varphi)>0$ for all $x\in(1;q_{2}].$ We have

[TABLE]

Under our assumptions, $q_{k}$ are increasing in $k$, and $\lim\limits_{n\to\infty}q_{n}=c.$ We prove that for any fixed $k=1,2,\ldots,m$ and $x\in(1;q_{2}]$ the following inequality holds:

[TABLE]

[TABLE]

For $x\in(1;q_{2}]$ and $k=1,2,\ldots,m,$ we define the following function

[TABLE]

We can observe that

[TABLE]

[TABLE]

Thus, under our assumptions, the function $F(q_{2},q_{3},\ldots,q_{2k},q_{2k+1})$ is decreasing in $q_{2}$. Since $q_{2}\leq q_{3},$

[TABLE]

[TABLE]

Further we have

[TABLE]

[TABLE]

Thus, under our assumptions, $F(q_{3},q_{3},q_{4},\ldots,q_{2k},q_{2k+1})$ is decreasing in $q_{3}$, and, since, $q_{3}\leq q_{4}$ we obtain

[TABLE]

Analogously, we obtain the following chain of inequalities

[TABLE]

[TABLE]

Further, we have

[TABLE]

[TABLE]

Thus, $F(q_{2k+1},q_{2k+1},\ldots,q_{2k+1},q_{2k+1})$ is decreasing in $q_{2k+1}$, and since $q_{k}$ are increasing in $k$, and $\lim\limits_{n\to\infty}q_{n}=c,$ we conclude that

[TABLE]

[TABLE]

Substituting the last inequality in (9) for every $x\in(1;q_{2}]$ and $k=1,2,\ldots,m,$ , we get

[TABLE]

where $g_{a}$ is the partial theta-function and $S_{2m+1}(y,g_{a})$ are its $(2m+1)$-th section at the point $y$. Since by our assumptions $(\sqrt{c})^{2}<q_{\infty},$ using the statement (5) of Theorem D we obtain that there exists $m\in{\mathbb{N}}$ such that $S_{2m+1}(y,g_{\sqrt{c}})\notin\mathcal{L-P}.$ Let us choose and fix such $m.$ By the statement (3) of Theorem D we obtain that for every $x$ such that $\sqrt{c}<\sqrt{c}x<(\sqrt{c})^{3}$ we have $S_{2m+1}(\sqrt{c}x,g_{\sqrt{c}})>0.$ It means that for every $x:1<x<c$ we have $S_{2m+1}(\sqrt{c}x,g_{\sqrt{c}})>0,$ and, using (10) and (8),

[TABLE]

It remains to prove that $\varphi(q_{2})>0.$ We have

[TABLE]

[TABLE]

by our assumptions on $q_{j}.$ ∎

Lemma 2.3**.**

Let $P(z)=1-z+\frac{z^{2}}{a}-\frac{z^{3}}{a^{2}b}+\frac{z^{4}}{a^{3}b^{2}c}$ be a polynomial, $3\leq a<4,$ and $a\leq b\leq c.$ Then

[TABLE]

Proof.

By the direct calculation we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Denote by $t:=\cos{\theta},t\in[-1;1].$ Since $\cos{2\theta}=2t^{2}-1,$ $\cos{3\theta}=4t^{3}-3t,$ and $\cos{4\theta}=8t^{4}-8t^{2}+1,$ we get

[TABLE]

[TABLE]

[TABLE]

We want to show that $\min_{0\leq\theta\leq 2\pi}|P(ae^{i\theta})|^{2}\geq\frac{a^{2}}{b^{4}c^{2}}.$ Using the last expression we see that the inequality we want to get is equivalent to the following: for all $t\in[-1;1]$ the next inequality holds

[TABLE]

[TABLE]

Let $y:=2t,$ $y\in[-2;2].$ We rewrite the last inequality in the form

[TABLE]

[TABLE]

[TABLE]

We note that the coefficient of $y^{4}$ is positive, and the coefficient of $y^{3}$ is negative. It is easy to show that the other coefficients are also sign-changing. For $y^{2}$: $1-\frac{4}{b^{2}c}>0$ since $b^{2}c>4$, thus, $1+\frac{a}{b}+\frac{a}{b^{2}c}-\frac{4}{b^{2}c}=(1-\frac{4}{b^{2}c})+\frac{a}{b}+\frac{a}{b^{2}c}>0.$ For $y$: $1+a+\frac{a}{b}+\frac{a}{b^{3}c}-\frac{3}{b}-\frac{3a}{b^{2}c}=(1+a-\frac{3}{b})+$ $(\frac{a}{b}-\frac{3a}{b^{2}c})+\frac{a}{b^{3}c}>0.$ Finally, $1+2a^{2}+\frac{a^{2}}{b^{2}}-2a-2\frac{a^{2}}{b}-2\frac{a^{2}}{b^{2}c}+2\frac{a}{b^{2}c}=$ $(1+a^{2}-2a)+(a^{2}-2\frac{a^{2}}{b})+(\frac{a^{2}}{b^{2}}-2\frac{a^{2}}{b^{2}c})+2\frac{a}{b^{2}c}>0$ since $1-2a+a^{2}\geq 0;$ $a^{2}-2\frac{a^{2}}{b}>0$ and $\frac{a^{2}}{b^{2}}-2\frac{a^{2}}{b^{2}c}>0$ by our assumptions.

Consequently, the inequality we need holds for any $y\in[-2;0]$, and we have to prove it for $y\in[0;2]$. Multiplying our inequality by $\frac{b^{2}c}{a}$, we get

[TABLE]

[TABLE]

and we want to prove that $\psi(y)\geq 0$ for all $y\in[0;2].$

Let $\chi(y):=\psi(y)-\frac{1}{b}(b-a)y,$ whence $\chi(y)\leq\psi(y)$ for all $y\in[0;2]$. It is sufficient to prove that $\chi(y)\geq 0$ for all $y\in[0;2].$ We have $\chi(0)=\psi(0)=\frac{b^{2}c}{a}+2ab^{2}c+ac-2b^{2}c-2abc-2a+2\geq 0,$ as it was previously shown. We also have $\chi(2)=\psi(2)-\frac{2}{c}(b-a)\geq 0$ since

[TABLE]

[TABLE]

[TABLE]

Now we consider the following function: $\nu(y):=\frac{\partial^{2}\chi(y)}{\partial y^{2}}=\frac{\partial^{2}\psi(y)}{\partial y^{2}}=12y^{2}-6(bc+a)y+2(b^{2}c+abc+a-4).$ The vertex point of this parabola is $y_{v}=\frac{bc+a}{4}\geq 3.$ Thus, we can observe that $\nu(y)$ decreases for $y\in[0;2].$ We have $\nu(0)=2(b^{2}c+abc+a-4)>0,$ and $\nu(2)=2abc+2b^{2}c-12bc-10a+40.$ We want to show that $\nu(2)$ is positive. We have

[TABLE]

[TABLE]

due to our assumptions. We conclude that $\nu(y)$ is nonnegative for $y\in[0;2]$, and it follows that $\chi^{\prime}(y)$ increases for $y\in[0;2].$

We want to show that $\chi^{\prime}(y)\leq 0$ for $y\in[0;2]$, and it is sufficient to show that $\chi^{\prime}(2)\leq 0.$ We have

[TABLE]

[TABLE]

Thus, $\chi(y)$ decreases, $\chi(2)\geq 0,$ so it is positive for $y\in[0;2]$. Since $\chi(y)\leq\psi(y)$, it follows that $\psi(y)$ is positive for $y\in[0;2].$ ∎

The function $\varphi$ can be presented in the following form:

[TABLE]

[TABLE]

By Lemma 2.3 we have

[TABLE]

Now we need the estimation on $|R_{5}(q_{2}e^{i\theta},\varphi)|$ from above.

Lemma 2.4**.**

Let $R_{5}(z,\varphi):=\sum_{k=5}^{\infty}\frac{(-1)^{k}z^{k}}{q_{2}^{k-1}q_{3}^{k-2}\ldots q_{k}}$, $q_{n}$ are increasing in $n$, and $\lim\limits_{n\to\infty}q_{n}(f)=c<q_{\infty}$. Then

[TABLE]

Proof.

We have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

∎

Let us check that $\frac{q_{2}}{q_{3}^{2}q_{4}}>\frac{q_{2}}{q_{3}^{3}q_{4}^{3}-q_{3}^{2}},$ which is equivalent to $q_{4}<q_{3}q_{4}^{3}-1.$ The last inequality obviously holds under our assumptions. Consequently, according to Rouché’s theorem, the functions $S_{4}(z,\varphi)$ and $\varphi(z)$ have the same number of zeros inside the circle $\{z:|z|<q_{2}\}$ counting multiplicities.

It remains to prove that $S_{4}(z,\varphi)$ has zeros in the circle $\{z:|z|<q_{2}\}$. To do this we need the notion of apolar polynomials and the famous theorem by J.H. Grace.

Definition 2.5**.**

(see, for example [24, Chapter 2, $3, p. 59]). Two complex polynomials$P(z)=\sum_{k=0}^{n}{n\choose k}a_{k}z^{k}$and$Q(z)=\sum_{k=0}^{n}{n\choose k}b_{k}z^{k}$of degree$n$ are called apolar if

[TABLE]

The following famous theorem due to J.H. Grace states that the complex zeros of two apolar polynomials cannot be separated by a straight line or by a circle.

Theorem H (J.H. Grace, see for example [24, Chapter 2, $3, Problem 145]). Suppose$P$and$Q$are two apolar polynomials of degree$n\geq 1.$If all zeros of$P$lie in a circular region$C,$then$Q$has at least one zero in$C.$ (A circular region is a closed or open half-plane, disk or exterior of a disk).

Lemma 2.6**.**

Let $S_{4}(z,\varphi)=1-z+\frac{1}{q_{2}}z^{2}-\frac{1}{q_{2}^{2}q_{3}}z^{3}+\frac{1}{q_{2}^{3}q_{3}^{2}q_{4}}z^{4}$ be a polynomial and $q_{2}\geq 3$. Then $S_{4}(z,\varphi)$ has at least one root in the circle $\{z:|z|\leq q_{2}\}$.

Proof.

We have $S_{4}(z,\varphi)={4\choose 0}+{4\choose 1}(-\frac{1}{4})z+{4\choose 2}\frac{1}{6q_{2}}z^{2}+{4\choose 3}(-\frac{1}{4q_{2}^{2}q_{3}})z^{3}+{4\choose 4}\frac{1}{q_{2}^{3}q_{3}^{2}q_{4}}z^{4}$. Let $Q(z)={4\choose 2}b_{2}z^{2}+{4\choose 3}b_{3}z^{3}+{4\choose 4}z^{4}.$ Then the condition for $S_{4}(z,\varphi)$ and $Q(z)$ to be apolar is the following

[TABLE]

We have $1-b_{3}+\frac{b_{2}}{q_{2}}=0.$ Further we choose $b_{3}=\frac{q_{2}-6}{2}$, and, by the apolarity condition, $b_{2}=-q_{2}(1+\frac{q_{2}-6}{2}).$ So we have

[TABLE]

[TABLE]

As we can see, the zeros of $Q$ are $z_{1}=0,z_{2}=0,z_{3}=q_{2},z_{4}=-3(q_{2}-4).$ To show that $z_{4}$ lies in the circle of radius $q_{2},$ we solve the inequality $|-3(q_{2}-4)|\leq q_{2}$. Thus, we obtain that if $q_{2}\geq 3,$ then all zeros of $Q$ are in the circle $\{z:|z|\leq q_{2}\}.$ Since all the zeros of $Q$ are in the circle $\{z:|z|\leq q_{2}\},$ we obtain by the Grace theorem that $S_{4}(z,\varphi)$ has at least one zero in the circle $\{z:|z|\leq q_{2}\}.$ ∎

Thus, $S_{4}(z,\varphi)$ has at least one zero in the circle $\{z:|z|\leq q_{2}\},$ and, applying Lemma 2.3 to the $S_{4}(z,\varphi)$, we conclude that $S_{4}(z,\varphi)$ does not have zeros on $\{z:|z|=q_{2}\}.$ So, the polynomial $S_{4}(z,\varphi)$ has at least one zero in the open circle $\{z:|z|<q_{2}\}.$ By the Rouché’s theorem, the functions $S_{4}(z,\varphi)$ and $\varphi(z)$ have the same number of zeros inside the circle $\{z:|z|<q_{2}\},$ whence $\varphi$ has at least one zero in the open circle $\{z:|z|<q_{2}\}.$ If $\varphi$ is in the Laguerre-Pólya class, this zero must be real, and, since coefficients of $\varphi$ are sign-changing, this zero belongs to $(0;q_{2}).$ However, by Lemma 2.2, we have $\varphi(x)>0$ for all $x\in[0;q_{2}].$ This contradiction shows that $\varphi\notin\mathcal{L-P}.$

Theorem 1.1 is proved.