1 Introduction

We study zero localization of real univariate polynomials and entire functions \(f(z) = a_0 + a_1 z + a_2 z^2 + \cdots \) with positive coefficients. In 1923, Hutchinson proved that, if the inequalities \(a_{k-1}^2 \geqslant 4 a_{k-2} a_k\) are valid for all \(k \geqslant 2\), then the function f belongs to the Laguerre–Pólya class. In this short note, the main object is to extend the sufficient conditions for a polynomial or an entire function to belong to the Laguerre–Pólya class obtained by Hutchinson, or, more precisely, to find intervals \([\alpha , \beta (\alpha )]\) which are not subsets of \([4, + \infty )\).

1.1 The Laguerre–Pólya class

We begin with the definitions of hyperbolic polynomials, the Laguerre–Pólya class and the Laguerre–Pólya class of type I.

Definition 1.1

A real polynomial P is said to be hyperbolic, written , if all its zeros are real.

Definition 1.2

A real entire function f is said to be in the Laguerre–Pólya class, written , if it can be expressed in the form

$$\begin{aligned} f(z) = c z^n e^{-\alpha z^2+\beta z}\prod _{k=1}^\infty \hspace{1.111pt}\biggl (1-\frac{z}{x_k} \biggr )\hspace{1.111pt}e^{zx_k^{-1}}, \end{aligned}$$

where \(c, \alpha , \beta , x_k \in \mathbb {R}\), \(x_k\ne 0\), \(\alpha \geqslant 0\), n is a nonnegative integer and \(\sum _{k=1}^\infty x_k^{-2} < \infty \).

Definition 1.3

A real entire function f is said to be in the Laguerre–Pólya class of type I, written , if it can be expressed in the following form:

$$\begin{aligned} f(z) = c z^n e^{\beta z}\prod _{k=1}^\infty \hspace{1.111pt}\biggl (1+\frac{z}{x_k} \biggr ), \end{aligned}$$

where \(c \in \mathbb {R}\), \( \beta \geqslant 0\), \(x_k >0 \), n is a nonnegative integer, and \(\sum _{k=1}^\infty x_k^{-1} < \infty \).

Note that the product on the right-hand sides in both definitions can be finite or empty (in the latter case, the product equals 1).

Various important properties and characterizations of the Laguerre–Pólya class and the Laguerre–Pólya class of type I can be found in works by Hirshman and Widder [8], Levin [19], Pólya and Szegő [26], Pólya and Schur [25], monograph by Obreschkov [23, Chapter II] and many other works. These classes are essential in the theory of entire functions since it appears that the polynomials with only real zeros (or only real and nonpositive zeros) converge locally uniformly to these and only these functions. The following prominent theorem provides an even stronger result.

Theorem A

(Laguerre and Pólya, see, for example, [8, pp. 42–46] and [19, Chapter VIII, Section 3])

(i) Let \((P_n)_{n=1}^{\infty }\), \(P_n(0)=1\), be a sequence of hyperbolic polynomials which converges uniformly on the disk \(|z|\leqslant A\), \(A > 0\). Then this sequence converges locally uniformly in \(\mathbb {C}\) to an entire function from the class. (ii) For any , there exists a sequence of hyperbolic polynomials which converges locally uniformly to f. (iii) Let \((P_n)_{n=1}^{\infty }\), \( P_n(0)=1\), be a sequence of hyperbolic polynomials having only negative zeros which converges uniformly on the disk \(|z| \leqslant A\), \(A > 0\). Then this sequence converges locally uniformly in \(\mathbb {C}\) to an entire function from the class . (iv) For any , there is a sequence of hyperbolic polynomials with only negative zeros which converges locally uniformly to f.

For a real entire function (not identically zero) of the order less than 2, the property of having only real zeros is equivalent to belonging to the Laguerre–Pólya class. Similarly, for a real entire function with positive coefficients of the order less than 1, having only real nonpositive zeros is equivalent to belonging to the Laguerre–Pólya class of type I. Strikingly, the situation changes for the functions of order 2 in the case of the Laguerre–Pólya class, and for the functions of order 1 in the case of the Laguerre–Pólya class of type I. For instance, the entire function \(f(x) = e^{-x^2}\) belongs to the class while the entire function \(g(x) = e^{x^2}\) does not.

Now we explain the connection between the functions from the Laguerre–Pólya class of type I and the generating functions of totally positive sequences.

Definition 1.4

A sequence of nonnegative numbers \((a_k)_{k=0}^\infty \) is called a totally positive sequence if all minors of the infinite matrix

$$\begin{aligned} \left\| \begin{array}{ccccc} a_0 &{} \quad a_1 &{} \quad a_2 &{} \quad a_3 &{} \quad \cdots \\ 0 &{} \quad a_0 &{} \quad a_1 &{} \quad a_2 &{} \quad \cdots \\ 0 &{} \quad 0 &{} \quad a_0 &{} \quad a_1 &{} \quad \cdots \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad a_0 &{} \quad \cdots \\ \vdots &{} \quad \vdots &{} \quad \vdots &{} \quad \vdots &{} \quad \ddots \end{array} \right\| \end{aligned}$$

are nonnegative.

The class of totally positive sequences is denoted by \(\textrm{PF}_\infty \). The corresponding class of generating functions \(f(z)=\sum _{k=0}^\infty a_kz^k\) is denoted by \(\widetilde{\textrm{PF}_\infty }\).

The class \(\widetilde{\textrm{PF}_\infty }\) was completely described in the classical theorem by Aissen, Schoenberg, Whitney and Edrei.

Theorem B

(Aissen, Schoenberg, Whitney and Edrei, [1, 2], see also [10, p. 412]) A function \(f(z)=\sum _{k=0}^\infty a_kz^k\) belongs to the class \(\widetilde{\textrm{PF}_\infty }\) if and only if

$$\begin{aligned} f(z)=C z^n e^{\gamma z}\prod _{k=1}^\infty \frac{1+\alpha _kz}{1-\beta _kz}\hspace{0.55542pt}, \end{aligned}$$

where \(C\geqslant 0\), \(n\in \mathbb {Z}\), \(\gamma \geqslant 0\), \(\alpha _k\geqslant 0\), \(\beta _k\geqslant 0\), \(\sum \hspace{0.55542pt}(\alpha _k+\beta _k) <\infty \).

A simple corollary of Theorem B is the following statement: a polynomial with nonnegative coefficients \(P(z) =\sum _{k=0}^n a_k z^k \) has only real zeros if and only if the sequence \((a_0, a_1, \ldots , a_n, 0, 0, \ldots )\) belongs to the class \(\textrm{PF}_\infty \). Also, by Theorem B, an entire function with nonnegative coefficients \(f(z)=\sum _{k=0}^\infty a_kz^k\) of order less than 1 has only real zeros if and only if the sequence \((a_k)_{k=0}^\infty \) belongs to the class \(\textrm{PF}_\infty \).

1.2 Hutchinson’s constant

The problem of understanding whether a given polynomial or entire function has only real zeros is considered subtle and complicated. A simply verified description of this class, in terms of the coefficients of a series, is impossible since it is determined by an infinite number of discriminant inequalities. In 1923, Hutchinson found a simple sufficient condition in terms of coefficients for an entire function with positive coefficients to have only real zeros, which was a generalization of the results by Petrovitch [24] and Hardy [6], or [7, pp. 95–100].

To formulate his result, let us define the second quotients of Taylor coefficients of f. Let \(f(z) = \sum _{k=0}^\infty a_k z^k\) be an entire function with real nonzero coefficients, then

In addition, it follows straightforwardly from this definition that

$$\begin{aligned}&a_n = a_1\biggl (\frac{a_1}{a_0} \biggr )^{n-1} \frac{1}{q_2^{n-1}q_3^{n-2} \cdots q_{n-1}^2 q_n}. \end{aligned}$$
(1.1)

Theorem C

(Hutchinson [9]) Let \(f(z)= \sum _{k=0}^\infty a_k z^k\), \( a_k > 0\) for all k, be an entire function. Then \(q_k(f)\geqslant 4\), for all \(k\geqslant 2\), if and only if the following two conditions are fulfilled:

(i) The zeros of f are all real, simple and negative. (ii) The zeros of any polynomial \(\sum _{k=m}^n a_kz^k\), \( m < n\), formed by taking any number of consecutive terms of f, are all real and nonpositive.

For some extensions of Hutchinson’s result see, for example, the paper by Craven and Csordas [4, Section 4]. From Hutchinson’s theorem (Theorem B) we see that f has only real zeros when \(q_k(f) \in [4, + \infty )\).

1.3 Some results related to Hutchinson’s constant

Strikingly, there are many results which are stated in the following style: there exists a constant \(c > 1\) such that if a polynomial or an entire function f with nonzero coefficients satisfies the conditions \(|q_k(f)| \geqslant c\) for all k, then we can formulate a statement about the localization of the zeros of f. For example, in [4] the authors obtained an analogue of Hutchinson’s theorem for polynomials decomposed in the Pochhammer basis. In [5], it was proved that, if for some constant \(c>0\) a polynomial P with positive coefficients satisfies the conditions \(q_k(P) > c\) for all k, then all the zeros of P lie in a special sector depending on c. In [13], the smallest possible constant \(c>0\) was found such that if a polynomial P with positive coefficients satisfies the conditions \(q_k(P) > c\) for all k, then P is stable (all the zeros of P lie in the left half-plane). In [11], the smallest possible constant \(c>0\) was found such that if a polynomial P with positive coefficients satisfies the conditions \(q_k(P) > c\) for all k, then P is a sign-independently hyperbolic polynomial. In [3], the smallest possible constant \(c>0\) was found such that if a polynomial P with complex coefficients satisfies the conditions \(|q_k(P)| > c\) for all k, then P has only simple zeros.

The following special function:

$$\begin{aligned} g_a(z) =\sum _{k=0}^{\infty } z^k a^{-k^2}, \quad a > 1, \end{aligned}$$

which is called the partial theta function, plays a significant role in the mentioned list of problems. Strikingly, \( q_k(g_a) = a^2 \) for all \( k \geqslant 2\). One of the interesting questions is for which values of a this function belongs to the Laguerre–Pólya class. The paper [12] by Katkova, Lobova, and Vishnyakova gives an exhaustive answer to this question. In particular, it is proved that there exists a constant \( q_\infty \approx 3{.}23363666\) such that if and only if \(a^2 \geqslant q_\infty \). Moreover, the authors studied analogous questions for the Taylor sections of the function \(g_a\). For more details on the partial theta function, see a series of works by Kostov dedicated to its various properties [14,15,16,17], his joint work with Shapiro [18], and a fascinating historical review by Warnaar [27].

It is easy to show that, if the estimation of \(q_k(f)\) is given only from below, then the constant 4 in \(q_k (f) \geqslant 4\) is the smallest possible to conclude that (that is, Theorem B remains valid when dropping (ii)). However, if we have only the estimation of \(q_k\) from below and require monotonicity, then the constant 4 in the condition \(q_k \geqslant 4\) can be reduced to conclude that . As an example, in [20], it was proved that if the entire functions have the decreasing \(q_k\) such that \(\lim _{\,n \rightarrow \infty } q_k = c \geqslant q_\infty \), then the function belongs to the Laguerre–Pólya class.

In this work, we show that if the estimations on \(q_k(f)\) from below and from above are given, then the constant 4 can be decreased. We would like to investigate problems where the assumption \(q_k(f) \geqslant c\) for all k is replaced by \(q_k(f) \in [\alpha , \beta ]\) for all k for some given segment \([\alpha , \beta ]\). As far as we know, the first result of such kind was obtained in [12] where the following theorem was proved.

Theorem D

(Katkova, Lobova, and Vishnyakova [12]) Let \(f(z)=\sum _{k=0}^\infty a_k z^k\), \(a_k>0\), be an entire function and \(\alpha \in [3{.}43; 4]\). Then \(q_k(f) \in [\alpha , {0{.}95}/({2\sqrt{\alpha }-\alpha }) ]\) for all \(k \geqslant 2\) implies .

2 Hutchinson’s intervals

We present our main result.

Theorem 2.1

Let \(P(x) = \sum _{k=0}^n a_k x^k\), \(a_k > 0\), be a polynomial, and \(n \geqslant 4\). Suppose that there exists \(\alpha \), \( 1 + \sqrt{5} \leqslant \alpha < 4\), such that \(q_k(P) \in [\hspace{1.111pt}\alpha , {8}/({\alpha (4 - \alpha )}) ]\hspace{1.111pt}\) for all \(k =2, 3, \ldots , n\). Then .

The following statement is a simple corollary from the above result.

Corollary 2.2

Let \(f(x) = \sum _{k=0}^\infty a_k x^k\), \(a_k > 0\), be an entire function. Suppose that there exists \(\alpha \), \( 1 + \sqrt{5} \leqslant \alpha < 4\), such that \(q_k(f) \in [\hspace{1.111pt}\alpha , {8}/({\alpha (4 - \alpha )})]\hspace{1.111pt}\) for all \(k =2, 3, \ldots \) Then .

Remark 2.3

It follows from a result about the partial theta function [12] that the constant \(\alpha \) in the statement of Theorem 2.1 cannot be less than \( q_\infty \approx 3{.}23363666\). We observe that \(1 + \sqrt{5} \approx 3.23606797\).

Another simple corollary from Theorem 2.1 is the following statement.

Corollary 2.4

Let \((a_k)_{k=0}^\infty \) be a sequence of nonnegative numbers. Suppose that there exists \(\alpha \), \( 1 + \sqrt{5} \leqslant \alpha < 4\), such that \({a_{k-1}^2}/({a_{k-2}a_k}) \in [\hspace{1.111pt}\alpha , {8}/({\alpha (4 - \alpha )})]\) for all \(k =2, 3, \ldots \) Then \((a_k)_{k=0}^\infty \in \textrm{PF}_\infty \).

2.1 Proof of Theorem 2.1

For a polynomial \(P(x) = \sum _{k=0}^n a_k x^k\) with positive coefficients, without loss of generality, we can assume that \(a_0=a_1=1\), since we can consider a polynomial \(T(x) = a_0^{-1} P (a_0 a_1^{-1}x) \) instead of P(x), due to the fact that such rescalling of P preserves its property of having real zeros and preserves the second quotients: \(q_k(T) =q_k(P)\) for all k. For the sake of brevity, we further use the notation \(q_k\) instead of \(q_k(P)\). Thereafter, we consider a polynomial

$$\begin{aligned} Q(x) = T(-x) = 1 - x + \sum _{k=2}^n\, \frac{ (-1)^k x^k}{q_2^{k-1} q_3^{k-2} \cdots q_{k-1}^2 q_k} \end{aligned}$$

instead of P (see (1.1) for the formulas for coefficients).

For further convenience, let us introduce the following notation for a set of polynomials that we are going to consider:

Our proof is based on the following lemma.

Lemma 2.5

Let \([\alpha , \beta ]\), \(0< \alpha < \beta \), be a given segment. Then the following two statements are equivalent:

  1. (a)

    For every polynomial , there exists a point \(x_0 \in (1, \alpha )\) such that \(S_{q_2, q_3, q_4}(x_0) \leqslant 0\).

  2. (b)

    The following inequalities are valid: \(\alpha \geqslant 1 + \sqrt{5}\), and, if \(\alpha < 4\), then\(\beta \leqslant {8}/({\alpha (4 - \alpha )})\).

Proof

The proof will be organized as follows. First, we will prove that (a) implies the inequality \(\alpha \geqslant 1 + \sqrt{5}\). Next, we will prove that (a) and the inequality \(\alpha \geqslant 1 + \sqrt{5}\) together are equivalent to (b). Thus, we will prove that (a) is equivalent to (b).

First stage. Suppose that (a) holds. Then, in particular, for the polynomial , there exists a point \(x_0 \in (1, \alpha )\) such that \(S_{\alpha , \alpha , \alpha }(x_0) \leqslant 0\). It is easy to check the following identity:

(2.1)

If \({1}/{4} + {2}/{\alpha ^3} - {1}/{\alpha } <0\), then for every \(x \in \mathbb {R}\) we have \(S_{\alpha , \alpha , \alpha }(x) > 0\). Thus,

$$\begin{aligned} \frac{1}{4} + \frac{2}{\alpha ^3} - \frac{1}{\alpha } = \frac{1}{4 \alpha ^3}\,\bigl (\alpha -1 + \sqrt{5}\bigr )(\alpha -2) \bigl (\alpha -1 - \sqrt{5}\bigr ) \geqslant 0, \end{aligned}$$

whence \( \alpha \in (0, 2] \cup [1 + \sqrt{5}, \infty )\). First, we consider the case \( \alpha \in (0, 2]\). By (2.1), we have

$$\begin{aligned} S_{\alpha , \alpha , \alpha }(x)&=\biggl (\frac{x^2}{\alpha ^3} - \biggl (\frac{1}{2} + \sqrt{\frac{1}{4} + \frac{2}{\alpha ^3} - \frac{1}{\alpha }} \biggr )\hspace{1.111pt}x +1\biggr ) \\&\qquad \qquad \times \biggl (\frac{x^2}{\alpha ^3} - \biggl (\frac{1}{2} - \sqrt{\frac{1}{4} + \frac{2}{\alpha ^3} - \frac{1}{\alpha }} \biggr )\hspace{1.111pt}x +1\biggr ). \end{aligned}$$

We have two quadratic polynomials in parentheses with the following discriminants:

If and , then for every \(x \in \mathbb {R}\) we have \(S_{\alpha , \alpha , \alpha }(x) > 0\). Thus, at least one of these two discriminants is nonnegative, whence , and we obtain

$$\begin{aligned} \sqrt{\frac{1}{4} + \frac{2}{\alpha ^3} - \frac{1}{\alpha }} \geqslant \frac{4 - \alpha ^{3/2}}{2\hspace{1.111pt}\alpha ^{3/2}}\hspace{0.55542pt}. \end{aligned}$$
(2.2)

We observe that \(4 - \alpha ^{3/2} \geqslant 0\) for \(\alpha \in (0, 2]\). Thus, inequality (2.2) implies

The derivative of \(\psi (\alpha )\) has a unique positive root \(\alpha _0 = {9}/{4}\), and the maximal value of \(\psi \) for \(\alpha >0\) is \(\psi ({9}/{4}) = - {5}/{16} < 0\). Thus, if (a) holds, then \(\alpha \geqslant 1 + \sqrt{5}\).

Second stage. Let us analyze the meaning of the fact that (a) and the inequality \(\alpha \geqslant 1 + \sqrt{5}\) hold true. Let \(S_{q_2, q_3, q_4}(x)\) be an arbitrary polynomial from the set . We want to investigate whether there exists \(x_0 \in (1, \alpha )\) such that \(S_{q_2, q_3, q_4}(x_0) \leqslant 0\). We observe that for all \(x>0\),

Thus, for every polynomial there exists a point \(x_0 \in (1, \alpha )\) such that \(S_{q_2, q_3, q_4}(x_0) \leqslant 0\) if and only if for every polynomial there exists a point \(x_0 \in (1, \alpha )\) such that \(S_{q_2, q_3, \alpha }(x_0) \leqslant 0\).

Next, we compute the derivative of \(S_{q_2, q_3, \alpha }(x)\) with respect to \(q_3\). We get

$$\begin{aligned} \frac{\partial }{\partial q_3}\,S_{q_2, q_3, \alpha } (x) = \frac{x^3}{q_2^2 q_3^2} - \frac{2x^4}{q_2^3 q_3^3 \alpha }\hspace{0.55542pt}. \end{aligned}$$

We observe that

$$\begin{aligned} \frac{x^3}{q_2^2 q_3^2} - \frac{2x^4}{q_2^3 q_3^3 \alpha } >0 \;\;\Longleftrightarrow \;\; x< \frac{q_2 q_3 \alpha }{2}\hspace{0.55542pt}, \end{aligned}$$

so for all \(x \in (1, \alpha )\), we get that \(S_{q_2, q_3, \alpha }(x)\) is increasing in \(q_3\). Whence, we have

$$\begin{aligned} S_{q_2, q_3, \alpha }(x) \leqslant S_{q_2, \beta , \alpha }(x) = 1 - x + \frac{x^2}{q_2} - \frac{x^3}{q_2^2 \beta } + \frac{x^4}{q_2^3 \beta ^2 \alpha }\hspace{0.55542pt}. \end{aligned}$$

Analogously, we consider the derivative of with respect to \(q_2\) to understand the monotonicity and we get

$$\begin{aligned} \frac{\partial }{\partial q_2}\,S_{q_2, \beta , \alpha } (x) ={}-\frac{x^2}{q_2^2} + \frac{2x^3}{q_2^3 \beta } - \frac{3x^4}{q_2^4 \beta ^2 \alpha }\hspace{0.55542pt}. \end{aligned}$$

We show that \({\partial } S_{q_2, \beta , \alpha } (x)/\partial q_2 < 0\) for \(x \in (1, \alpha )\), or, equivalently,

$$\begin{aligned} 3\hspace{1.111pt}x^2 -2\hspace{1.111pt}q_2 \alpha \beta x + q_2^2 \alpha \beta ^2>0. \end{aligned}$$
(2.3)

Under our assumption that \(\alpha \geqslant 1 + \sqrt{5}\), we compute the discriminant of the left-hand side of (2.3) and observe that \( {D}/{4} = q_2^2\beta ^2 \alpha (\alpha -3) > 0\), so the quadratic expression has two positive roots

$$\begin{aligned} x_{\pm } = \frac{q_2 \alpha \beta \pm q_2 \beta \sqrt{\alpha (\alpha -3)}}{3}\hspace{0.55542pt}. \end{aligned}$$

To prove (2.3), it is sufficient to check that \(\alpha < x_{-}\), or \(q_2 \beta \sqrt{\alpha (\alpha -3)} < q_2 \alpha \beta -3 \alpha \). The last inequality is equivalent to \(q_2^2 \beta ^2 +3\alpha - 2 q_2 \alpha \beta = q_2\beta (q_2\beta -2\alpha ) +3\alpha >0\), which holds under our assumptions since \(q_2 \geqslant \alpha \), and \(\beta> \alpha >2\). Thus, we have proved that for all \(x \in (1, \alpha )\):

$$\begin{aligned} S_{q_2, \beta , \alpha } (x) \leqslant S_{\alpha , \beta , \alpha } (x) = 1 - x + \frac{x^2}{\alpha } - \frac{x^3}{\alpha ^2\beta } + \frac{x^4}{\alpha ^4\beta ^2}\hspace{0.55542pt}. \end{aligned}$$

Consequently, for all polynomials there exists a point \(x_0 \in (1, \alpha )\) such that \(S_{q_2, q_3, q_4}(x_0) \leqslant 0\) if and only if for the polynomial there exists a point \(x_0 \in (1, \alpha )\) such that \(S_{\alpha , \beta , \alpha }(x_0) \leqslant 0\).

Now we consider the polynomial \(S_{\alpha , \beta , \alpha } (x)\) for \(x \in (1, \alpha )\). We set \(x =\alpha \sqrt{\beta } y\). Since \(x \in (1, \alpha )\), we have . We note that \( ( {1}/({\alpha \sqrt{\beta }}), {1}/\!{\sqrt{\beta }} ) \subset (0, 1)\). Hence, after a change of variables, we get a self-reciprocal polynomial

We set . We observe that the function \(w(y) = y + {1}/{y}\) is monotonically decreasing on the interval (0, 1), therefore, the interval is bijectively mapped to the interval \(({1}/{\beta } + \beta , {1}/({\alpha \sqrt{\beta }}) + \alpha \sqrt{\beta } )\). We want to investigate whether there exists a point such that

We consider the vertex of the parabola \(w_{v} = {\alpha \sqrt{\beta }}/{2}\), and check whether it lies in . Obviously, \(w_{v} = {\alpha \sqrt{\beta }}/{2} < \alpha \sqrt{\beta } + {1}/({\alpha \sqrt{\beta }})\). We show that the following inequality is fulfilled , or, equivalently, \(\beta > {2}/({\alpha -2})\). It is sufficient to prove that \(\alpha > {2}/({\alpha -2})\), and it is equivalent to \(\alpha ^2 - 2\alpha - 2 > 0\), which is fulfilled under our assumption \(\alpha \geqslant 1 + \sqrt{5}\). Since , there exists such that \(\widetilde{\widetilde{P}}(w_0) \leqslant 0\) if and only if the discriminant of this quadratic function is nonnegative:

$$\begin{aligned} D = \alpha ^2 \beta - 4 \alpha \beta + 8 = \beta \alpha ( \alpha -4) +8\geqslant 0. \end{aligned}$$

The inequality above is equivalent to the following statement: if \(\alpha < 4\), then \(\beta \leqslant {8}/({\alpha (4 - \alpha )})\). We have proved that (a) and the inequality \(\alpha \geqslant 1 + \sqrt{5}\) together are equivalent to (b). \(\square \)

Remark 2.6

Lemma 2.5 is an analog of [21, Theorem 1.5]. This theorem states that if \(f(x) = 1 + x + \sum _{k=2}^\infty a_k x^k\) is an entire function with positive coefficients, and \(3 \leqslant q_2(f) < 4, \) \(q_4(f) \geqslant 3\) and \(2 \leqslant q_3(f) \leqslant {8}/({d(4-d)})\), where \(d = \min \hspace{0.55542pt}(q_2(f), q_4(f))\), then there exists \(x_0 \in [-q_2(f), 0]\) such that \(f(z_0) \leqslant 0\).

Now we can prove Theorem 2.1. Let

$$\begin{aligned} Q(x) = 1 - x + \sum _{k=2}^n\, \frac{ (-1)^k x^k}{q_2^{k-1} q_3^{k-2} \cdots q_{k-1}^2 q_k} \end{aligned}$$

be a polynomial, \(n \geqslant 4\), and there exist \(\alpha \in [1 + \sqrt{5}, 4)\) such that

$$\begin{aligned} q_k \in \biggl [\alpha , \frac{8}{\alpha (4 - \alpha )} \biggr ] \end{aligned}$$

for all \(k =2, 3, \ldots , n\). Let us fix an arbitrary j such that \(1 \leqslant j \leqslant \lfloor {n}/{2}\rfloor \), and suppose that \(x \in (q_2 q_3 \cdots q_j, q_2 q_3 \cdots q_j q_{j+1})\) (for \(j=1\) we assume that \(1< x < q_2\)). Then we observe that

$$\begin{aligned} 1< x< \frac{x^2}{q_2}< \frac{x^3}{q_2^2 q_3}< \cdots < \frac{x^j}{q_2^{j-1} q_3^{j-2} \cdots q_j}\hspace{0.55542pt}, \end{aligned}$$
(2.4)

and

$$\begin{aligned} \begin{aligned} \frac{x^j}{q_2^{j-1} q_3^{j-2} \cdots q_j}&> \frac{x^{j+1}}{q_2^{j} q_3^{j-1} \cdots q_j^2 q_{j+1}}\\ {}&> \frac{x^{j+2}}{q_2^{j+1} q_3^{j} \cdots q_j^3 q_{j+1}^2 q_{j+2}}> \cdots > \frac{x^{n}}{q_2^{n-1} q_3^{n-2} \cdots q_{n-1}^2 q_{n}}\hspace{0.55542pt}. \end{aligned} \end{aligned}$$
(2.5)

We have the following representation:

We note that, for some j, the sum \(\Sigma _{2, j}(x)\) can be empty (and equal to zero), but for \(n\geqslant 5\), we have \(j +3 \leqslant \lfloor {n}/{2}\rfloor +3 \leqslant n\), so all five summands in \(g_j(x)\) are nonzero. We later consider the case \(n=4\).

For \(x \in (q_2 q_3 \cdots q_j, q_2 q_3 \cdots q_j q_{j+1})\), we observe that the terms in \(\Sigma _{1, j}(x)\) are alternating in sign and and their moduli are increasing, while the summands in \(\Sigma _{2, j}(x)\) are alternating in sign and their moduli are decreasing. Hence, \(\Sigma _{1, j}(x) < 0\) and \(\Sigma _{2, j}(x) < 0\) for all \(x \in (q_2 q_3 \cdots q_j, q_2 q_3 \cdots q_j q_{j+1})\), whence we get

$$\begin{aligned} (-1)^{j-1} Q(x) < g_j(x) \quad \text {for all}\;\; x \in (q_2 q_3 \cdots q_j, q_2 q_3 \cdots q_j q_{j+1}). \end{aligned}$$
(2.6)

We have

$$\begin{aligned} g_j(x)&= \frac{ x^{j-1}}{q_2^{j-2} q_3^{j-3} \cdots q_{j-2}^2 q_{j -1}}\, \biggl (1 - \frac{x}{q_2 q_3 \cdots q_{j-1}q_j} + \frac{x^2}{q_2^2 q_3^2 \cdots q_{j-1}^2q_j^2 q_{j+1}} \\&\qquad \qquad - \frac{x^3}{q_2^3 q_3^3 \cdots q_{j-1}^3 q_j^3 q_{j+1}^2 q_{j+2}} +\frac{x^4}{q_2^4 q_3^4 \cdots q_{j-1}^4 q_j^4 q_{j+1}^3 q_{j+2}^2 q_{j+3}} \biggr ). \end{aligned}$$

We set \(y =\frac{x}{q_2 q_3 \cdots q_{j-1}q_j}\), and for \(x \in (q_2 q_3 \cdots q_j, q_2 q_3 \cdots q_j q_{j+1})\) we have \(y \in (1, q_{j+1})\). Therefore, we obtain

Since for some \(\alpha \), where \(1 + \sqrt{5} \leqslant \alpha < 4\), the polynomial in parentheses satisfies the assumptions of Lemma 2.5. We recall that \((1, \alpha ) \subset (1, q_{j+1} )\). Thus, there exists \(y_j \in (1, q_{j+1})\) such that \(h_j(y_j) \leqslant 0\). Hence, there exists \(x_j = q_2 q_3 \cdots q_{j-1}q_j y_j \in (q_2 q_3 \cdots q_j, q_2 q_3 \cdots q_j q_{j+1})\) such that \( g_j(x_j)\leqslant 0\). Taking into account (2.6), for every \(n \geqslant 5\) we obtain

$$\begin{aligned} \forall j, \; 1 \leqslant j \leqslant \biggl \lfloor \frac{n}{2}\biggr \rfloor ,\; \exists \, x_j \in (q_2 \cdots q_j, q_2 \cdots q_j q_{j+1}): (-1)^{j-1} Q(x_j) <0. \end{aligned}$$
(2.7)

The only problem for \(n=4\) is when \(j=2\). In the latter case, we have

$$\begin{aligned} {} - Q(x)&= {}-1 + \biggl ( x - \frac{x^2}{q_2} + \frac{x^3}{q_2^2 q_3} - \frac{x^4}{q_2^3 q_3^2 q_4}\biggr )\\ {}&= {}-1 + x\biggl ( 1 - \frac{x}{q_2} + \frac{x^2}{q_2^2 q_3} - \frac{x^3}{q_2^3 q_3^2 q_4}\biggr ). \end{aligned}$$

We highlight that the polynomial in parentheses is of degree 3, however, we can estimate it from above with a polynomial of degree 4 as follows:

$$\begin{aligned} {} - Q(x)&< x\biggl ( 1 - \frac{x}{q_2} + \frac{x^2}{q_2^2 q_3} - \frac{x^3}{q_2^3 q_3^2 q_4}\biggr ) \\ {}&< x\biggl ( 1 - \frac{x}{q_2} + \frac{x^2}{q_2^2 q_3} - \frac{x^3}{q_2^3 q_3^2 q_4} + \frac{x^4}{q_2^4 q_3^3 q_4^2 q_4}\biggr ). \end{aligned}$$

Therefore, we can further reason in the same way as before. Thus, we conclude that (2.7) is valid for all \(n \geqslant 4\).

Now let us fix an arbitrary j, \( \lfloor {n}/{2}\rfloor +1 \leqslant j \leqslant n-1\), and suppose that \(x \in (q_2 q_3 \cdots q_j, q_2 q_3 \cdots q_j q_{j+1})\). Then both (2.4) and (2.5) are valid, and we have the following representation:

We note that for some j, the sum \(\Sigma _{1,j}(x)\) can be empty (and equal to zero), although for \(n\geqslant 4\), we have \(j -3 \geqslant \lfloor {n}/{2}\rfloor -2 \geqslant 0\), so all five summands in \(g_j(x)\) are nonzero.

For \(x \in (q_2 q_3 \cdots q_j, q_2 q_3 \cdots q_j q_{j+1})\), we observe that the terms in \(\Sigma _{1,j}(x)\) are alternating in sign and and their moduli are increasing, while the summands in \(\Sigma _{2,j}(x)\) are alternating in sign and their moduli are decreasing. Hence, \(\Sigma _{1,j}(x) < 0\) and \(\Sigma _{2,j}(x) < 0\) for all \(x \in (q_2 q_3 \cdots q_j, q_2 q_3 \cdots q_j q_{j+1})\), whence we get

$$\begin{aligned} (-1)^{j-1} Q(x) < g_j(x) \quad \text {for all}\;\; x \in (q_2 q_3 \cdots q_j, q_2 q_3 \cdots q_j q_{j+1}). \end{aligned}$$

We have

$$\begin{aligned} g_j(x)&= \frac{ x^{j+1}}{q_2^{j} q_3^{j-1} \cdots q_{j}^2 q_{j +1}} \, \biggl (1 - \frac{q_2 q_3 \cdots q_{j-1}q_j q_{j+1}}{x} + \frac{q_2^2 q_3^2 \cdots q_{j-1}^2q_j^2 q_{j+1}}{x^2}\\&\qquad \quad \qquad - \frac{q_2^3 q_3^3 \cdots q_{j-2}^3q_{j-1}^3 q_j^2 q_{j+1}}{x^3} +\frac{q_2^4 q_3^4 \cdots q_{j-3}^4 q_{j-2}^4q_{j-1}^3 q_j^2 q_{j+1}}{x^4} \biggr ). \end{aligned}$$

We set \(y =\frac{q_2 q_3 \cdots q_{j-1}q_j q_{j+1}}{x}\), and we observe that we have \(y \in (1, q_{j+1})\) for \(x \in (q_2 q_3 \cdots q_j, q_2 q_3 \cdots q_j q_{j+1})\). Thus, we obtain

Since \(q_{j+1}, q_{j}, q_{j-1} \in [\alpha , {8}/({\alpha (4 - \alpha )})] \) for some \(\alpha \in [1 + \sqrt{5}, 4)\), the polynomial in parentheses satisfies the assumptions of Lemma 2.5. Thus, there exists \(y_j \in (1, q_{j+1})\) such that \(h_j(y_j) \leqslant 0\). Whence, there exists \(x_j = \frac{ q_2 q_3 \cdots q_{j-1}q_j q_{j+1}}{y_j} \in (q_2 q_3 \cdots q_j, q_2 q_3 \cdots q_j q_{j+1})\) such that \( g_j(x_j)\leqslant 0\). Taking into account (2.6), for every \(n \geqslant 4\) we obtain

$$\begin{aligned} \forall j,\; \biggl \lfloor \frac{n}{2}\biggr \rfloor +1 \leqslant j \leqslant n-1, \; \exists \, x_j \in (q_2 \cdots q_j, q_2 \cdots q_{j+1}) : (-1)^{j-1} Q(x_j) <0. \end{aligned}$$
(2.8)

Since \(q_j >1\) for all \(j =2, 3, \ldots , n\), we get \(1< q_2< q_2q_3< q_2q_3q_4< \cdots < q_2q_3 q_4 \cdots q_n, \) whence \(x_1< x_2< \cdots < x_{n-1}\). By (2.7) and (2.8) we have

$$\begin{aligned}&Q(0)>0,\quad - Q(x_1)>0,\quad Q(x_2)>0,\quad -Q(x_3)>0,\quad \ldots ,\\&(-1)^{n-1} Q(x_{n-1})>0,\quad (-1)^n Q(+\infty ) >0. \end{aligned}$$

Thus, we have proved that all the zeros of Q are real. Theorem 2.1 is proved.

Remark 2.7

The assumptions on \(q_j\) in Theorem 2.1 could be slightly weakened for entire functions and polynomials of higher degrees if we obtain an analogue of Lemma 2.5 for polynomials of even degrees that are greater than 4. As it is shown in the proof of Lemma 2.5, an important role in such considerations is played by certain special polynomials which have the following property: \(\alpha =q_2 =q_4= q_6 =\cdots \), and \(\beta =q_3 =q_5= q_7 =\cdots \), when \(\alpha < \beta \). The paper [22] by Nguyen and Vishnyakova studies the entire functions with alternating second quotients of Taylor coefficients. Let

$$\begin{aligned} f_{a, b}(x)= 1 - x +\sum _{k=2}^\infty \, \frac{ (-1)^k x^k}{q_2^{k-1} q_3^{k-2} \cdots q_k} \end{aligned}$$

be an entire function such that \(q_2 = q_4 = q_6 = \cdots = \alpha \), \(q_3 = q_5 = q_7 = \cdots = \beta \), and \(1< \alpha < \beta \). In [22], it is proved that the function \(f_{a, b}\) belongs to the Laguerre–Pólya class if and only if there exists \(x_0 \in [0, q_2]\) such that \(f_{a,b}(x_0) \leqslant 0\). In addition, it is proved that if the function \(f_{a, b}\) belongs to the Laguerre–Pólya class, then \( \alpha \geqslant q_\infty \).