On the entire functions from the Laguerre–Pólya class having the decreasing second quotients of Taylor coefficients

Accepted Manuscript On the entire functions from the Laguerre-Pólya class having the decreasing second quotients of Taylor coefficients

Thu Hien Nguyen, Anna Vishnyakova

PII: DOI: Reference:

S0022-247X(18)30411-6 https://doi.org/10.1016/j.jmaa.2018.05.018 YJMAA 22250

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

25 October 2017

Please cite this article in press as: T.H. Nguyen, A. Vishnyakova, On the entire functions from the Laguerre-Pólya class having the decreasing second quotients of Taylor coefficients, J. Math. Anal. Appl. (2018), https://doi.org/10.1016/j.jmaa.2018.05.018

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ON THE ENTIRE FUNCTIONS FROM THE ´ LAGUERRE-POLYA CLASS HAVING THE DECREASING SECOND QUOTIENTS OF TAYLOR COEFFICIENTS THU HIEN NGUYEN AND ANNA VISHNYAKOVA ∞ Abstract. For an entire function f (z) = k=0 ak z k , ak > 0, we show that f belongs to the Laguerre-P´olya class if the quotients a2n−1 /an−2 an are decreasing in n, and b := lim a2n−1 /an−2 an such n→∞

that b ≥ q∞ (q∞ ≈ 3.2336). This estimation is sharp.

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 [6]. In this paper we investigate sufficient conditions under which some special entire functions have only real zeros. First, we need the definition of the famous Laguerre-P´olya class. Definition 1. A real entire function f is said to be in the LaguerreP´ olya class, written f ∈ L − P, if it can be expressed in the form  ∞   −1 x n −αx2 +βx 1− exxk , (1) f (x) = cx e xk k=1 where c, α, β, xk ∈ R, xk = 0, α ≥ 0, n is a nonnegative integer and  ∞ −2 k=1 xk < ∞. 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.P´olya, see, for example, [4, p. 4246]). (i) Let (Pn )∞ n=1 , Pn (0) = 1, be a sequence of complex polynomials having only real zeros which converges uniformly in the circle |z| ≤ 1991 Mathematics Subject Classification. 30C15; 30D15; 30D35; 26C10. Key words and phrases. Laguerre-P´olya class; entire functions of order zero; real-rooted polynomials; multiplier sequences; complex zero decreasing sequences. 1

2

T. H. NGUYEN AND A. VISHNYAKOVA

A, A > 0. Then this sequence converges locally uniformly to an entire function from the L − P class. (ii) For any f ∈ 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-P´olya class see [14, p. 100], [15] or [12, 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-P´olya class. The situation is different when an entire function 2 is of order 2. For example, the function f1 (x) = e−x belongs to the 2 Laguerre-P´olya but the function f2 (x) = ex does not. class, ∞ Let f (z) = k=0 ak z k be an entire function with positive coefficients. We define the quotients pn and qn : (2) pn = pn (f ) :=

an−1 , n ≥ 1; an

qn = qn (f ) :=

a2 pn = n−1 , n ≥ 2. pn−1 an−2 an

The following formulas can be verified by straight forward calculations. (3) a0 an = , n≥1; p 1 p 2 . . . pn

a1 an = n−1 n−2 2 q2 q3 . . . qn−1 qn



a1 a0

n−1 , n ≥ 2.

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.  k Theorem B (J. I. Hutchinson, [5]). Let f (z) = ∞ k=0 ak z , ak > 0 for all k. Then qn (f ) ≥ 4, for all n ≥ 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 nk=m ak 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 introduce the notion of a complex zero decreasing sequence. For a real polynomial P we denote by Zc (P ) the number of nonreal zeros of P counting multiplicities.

´ ENTIRE FUNCTIONS OF THE LAGUERRE-POLYA CLASS

3

Definition 2. A sequence (γk )∞ k=0 of real numbers is said to be a complex zero decreasing sequence (we write γk ∈ CZDS), if  n   n    k k (4) Zc γ k ak z ak z , ≤ Zc k=0

n

k=0

for any real polynomial k=0 ak z k . The existence of nontrivial CZDS sequences is a consequence of the following remarkable theorem proved by Laguerre and extended by P´olya. Theorem C (G. P´olya, see [13] or [14, pp. 314-321]). Let f be an entire function from the Laguerre-P´olya class having only negative zeros. Then (f (k))∞ k=0 ∈ CZDS. As it follows from the theorem above,  ∞ 2 ∞ 1 −k ∈ CZDS, a ≥ 1, ∈ CZDS. (5) a k! k=0 k=0  j −j 2 The entire function ga (z) = ∞ , a > 1, a so called partial j=0 z a theta-function, was investigated in the paper

[7]. Simple calculations 2

∞

show that qn (ga ) = a2 for all n. Since a−k ∈ CZDS , for a ≥ 1, k=0 we conclude that for every n ≥ 2 there exists a constant cn > 1 such 2 that Sn (z, ga ) := nj=0 z j a−j ∈ L − P ⇔ a2 ≥ cn . Theorem D (O. Katkova, T. Lobova, A. Vishnyakova, [7]). There exists a constant q∞ (q∞ ≈ 3.23363666) such that:

(1) ga (z) ∈ L − P ⇔ a2 ≥ q∞ ; (2) ga (z) ∈ L − P ⇔ there exists x0 ∈ (−a3 , −a) such that ga (x0 ) ≤ 0; (3) for given n ≥ 2 we have Sn (z, ga ) ∈ L − P ⇔ there exists xn ∈ (−a3 , −a) such that Sn (xn , ga ) ≤ 0; (4) 4 = c2 > c4 > c6 > . . . and limn→∞ c2n = q∞ ; (5) 3 = c3 < c5 < c7 < . . . and limn→∞ c2n+1 = q∞ . A wonderful paper [11] among the other results explains the role of the constant q∞ in the study of the set of entire functions with positive coefficients having all Taylor truncations with only real zeros. As for interesting properties of zeros of the partial theta-function see [9] and [10]. In [8] some entire functions with a convergent sequence of second quotients of coefficients are investigated. The main question of [8] is whether a function and its Taylor sections belong to the Laguerre-P´olya class. In [2] and [1], some important special functions with increasing sequence of second quotients of Taylor coefficients are studied.

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T. H. NGUYEN AND A. VISHNYAKOVA

In this paper, we study entire functions with positive Taylor coefficients such that qn (f ) are decreasing in n. We present the following theorem.  k Theorem 1.1. Let f (z) = ∞ k=0 ak z , ak > 0 for all k, be an entire function. Suppose that qn (f ) are decreasing in n, i.e. q2 ≥ q3 ≥ q4 ≥ . . . , and lim qn (f ) = b ≥ q∞ . Then all the zeros of f are real and n→∞ negative, in other words f ∈ L − P. It is easy to see that, if only the estimation of qn (f ) from below is given and the assumption of monotonisity is omitted, then the constant 4 in qn (f ) ≥ 4 is the smallest possible to conclude that f ∈ L − P. 2. Proof of Theorem 1.1 We need two Lemmas to prove Theorem 1.1.  k k Lemma 2.1. Let ϕ(z) = ∞ k=0 (−1) ak z , ak > 0 for all k, be an entire function, Sn (z, ϕ) be its partial sums up to degree n, and pk (ϕ), qk (ϕ) are defined as in (2). Suppose that qk (ϕ) are decreasing in k, i.e. q2 (ϕ) ≥ q3 (ϕ) ≥ q4 (ϕ) ≥ . . . , and lim qk (ϕ) = b > q∞ . Then there k→∞

exists a positive integer m0 and an infinite sequence of positive numbers (yj ), pj (ϕ) < yj < pj+1 (ϕ), such that for every n > 2m0 the following inequalities hold: (−1)j Sn (yj , ϕ) ≥ 0 for all j = 1, 2, 3, . . . , n − 2m0 .

(6)

Using Sn (0, ϕ) > 0 and (6), we obtain that every truncation of ϕ has at most 2m0 nonreal zeros. Proof. Without loss of generality, we can assume that a0 = a1 = 1, −1 since we can consider a function ψ(x) = a−1 0 ϕ(a0 a1 x) instead of ϕ(x). due to the fact that rescalling of ϕ preserves its property of having real zeros. Moreover, qn (ψ) = qn (ϕ) for all n. During the proof we use pn and qn instead of pn (ϕ) and qn (ϕ). Let us fix an arbitrary j ∈ N and suppose that x ∈ (pj , pj+1 ) = (q2 q3 · . . . · qj , q2 q3 · . . . · qj qj+1 ). Then we obtain 1 < x < x2 q2−1 < x3 q2−2 q3−1 < · · · < xj q21−j q32−j · . . . · qj−1 and xj q21−j q32−j · . . . · qj−1 > −1 −2 −1 xj+1 q2−j q31−j · . . . · qj−2 qj+1 > xj+2 q2−j−1 · . . . · qj−3 qj+1 qj+2 > · · · . For an arbitrary m ∈ N and n > j + 2m we have  j−2  (−1)j+k xk xj−1 j + − + (−1) Sn (x, ϕ) = j−2 j−3 k−1 k−2 q q · . . . · q q q · . . . · q k j−1 2 3 2 3 k=0 xj xj+1 xj+2m−2 − j j−1 +. . .+ j+2m−3 j+2m−4 · . . . · qj q2 q3 · . . . · qj+1 q2 q3 · . . . · qj+2m−2

q2j−1 q3j−2

´ ENTIRE FUNCTIONS OF THE LAGUERRE-POLYA CLASS

xj+2m−1 − j+2m−2 j+2m−3 q2 q3 · . . . · qj+2m−1



+

5

n 

(−1)j+k xk =: q k−1 q3k−2 · . . . · qk k=j+2m 2

=: Σ1 (x) + gm (x) + Σ2 (x). We note that the terms in Σ1 (x) are alternating in sign and increasing in moduli, whence Σ1 (x) ≥ 0. Analogously, the summands in Σ2 (x) are alternating in sign and their moduli are decreasing, whence Σ2 (x) ≥ 0. Thus, (−1)j Sn (x, ϕ) ≥ gm (x) for all x ∈ (pj ; pj+1 ).

(7) We have

 xj−1 x gm (x) = j−2 j−3 · −1 + q2 q3 · . . . · qj−1 qj q2 q3 · . . . · qj−1 −

x2 x3 + 2 3 2 q22 q32 · . . . · qj−1 qj2 qj+1 q23 q33 · . . . · qj−1 qj3 qj+1 qj+2

x2m−1 2m−2 2m−3 2 q22m−1 q32m−1 · . . . · qj2m−1 qj+1 qj+2 · . . . · qj+2m−3 qj+2m−2  x2m . − 2m 2m 2m−1 2m−2 2 q2 q3 · . . . · qj2m qj+1 qj+2 · . . . · qj+2m−2 qj+2m−1

−... +

−1 −1 We define: y = y(j) = xq2−1 q3−1 · . . . · qj−1 qj . We have the condition q2 q3 · . . . · qj < x < q2 q3 · . . . · qj qj+1 , whence 1 < y < qj+1 . In addition, let us use the following denotations for qk :

qj+1 → q2 ; qj+2 → q3 ; .. . qj+2m → q2m+1 ; qj+2m+1 → q2m+2 . Our hypothesis for qk implies the following inequalities for qk : q2 ≥ q3 ≥ q4 ≥ · · · ≥ b. Now we rewrite the expression for gm (x) in the new notations. (8)

j−2 j−1 qj · −1 + y − gm (x) = y j−1 · q2 q32 · . . . · qj−1 y4

q23 q32 q4

+ ... +

y 2m−1

q22m−2 q32m−3 ·...· q2m−1

−

y 2m

y2 q2

+

y3 − q22 q3

2 q22m−1 q32m−2 ·...· q2m−1 q2m

j−2 j−1 =: y j−1 · q2 q32 · . . . · qj−1 qj μm (y).



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T. H. NGUYEN AND A. VISHNYAKOVA

We can observe that sgn gm (x) = sgn μm (y). We choose y ∈ (1, q2 ) such that μm (y) ≥ 0. We have 2

3 4 5 (9) μm (y) = (−1 + y) + − yq2 + qy2 q3 + (− q3yq2 q4 + q4 qy3 q2 q5 ) + . . . 2

y 2m−2 +(− q2m−3 q2m−4 ·...· q2m−2 2 3

+

2 3

y 2m−1 ) q22m−2 q32m−3 ·...· q2m−1

−

2 3 4

y 2m . 2 q22m−1 q32m−2 ·...· q2m−1 q2m

We prove that for any fixed k = 1, 2, . . . , m − 1 the following inequality holds (10)

2k

y + − q2k−1 q2k−2 ·...· q 2

2k

3

2k

y 2k+1 2 q q22k q32k−1 ·...· q2k 2k+1

− b2k−1 ·by2k−2 ·...·b +

≥

y 2k+1 b2k ·b2k−1 ·...·b2 ·b

Firstly, we consider (10) for k = 1. Since q2 ≥ q3 , we have −

y3 y2 y3 y2 + 2 ≥ − + 3. q2 q2 q3 q2 q2

We observe that  2  ∂ y y 2 3y 3 y3 q2 − + 3 = 2 − 4 ≥ 0 ⇔ y ≤ 2. ∂ q2 q2 q2 q2 q2 3

2 Since y < q2 and q32 ≥ 3b > 1, we obtain that the function − yq2 + is increasing in q2 , whence (11)

−

y3 q23



y2 y3 y2 y3 y2 y3 + 2 ≥ − + 3 ≥ − + 3. q2 q2 q3 q2 q2 b b

We apply analogous arguments to prove (10) for every k = 1, 2, . . . , m− 1. Let us define the following function F ( q2 , q3 , . . . , q2k , q2k+1 ) := −

y 2k y 2k+1 + 2 q22k−1 q32k−2 · . . . · q2k q22k q32k−1 · . . . · q2k q2k+1

for q2 ≥ q3 ≥ . . . ≥ q2k+1 ≥ b. Obviously, q2 , q3 , . . . , q2k , q2k ). F ( q2 , q3 , . . . , q2k , q2k+1 ) ≥ F ( We have ∂F ( q2 , q3 , . . . , q2k , q2k ) y 2k 3y 2k+1 = 2k−1 2k−2 − . 2 2 3 4 ∂ q2k q2 q3 · . . . · q2k−1 q2k q22k q32k−1 · . . . · q2k−1 q2k Thus, 2 ∂F ( q2 , q3 , . . . , q2k , q2k ) q2 q3 · . . . · q2k−1 q2k . ≥0⇔y≤ ∂ q2k 3

´ ENTIRE FUNCTIONS OF THE LAGUERRE-POLYA CLASS

7

Since y < q2 , we obtain that the function F ( q2 , q3 , . . . , q2k , q2k ) is increasing in q2k , whence q2 , q3 , . . . , q2k , q2k ) ≥ F ( q2 , q3 , . . . , q2k−1 , b, b). F ( q2 , q3 , . . . , q2k , q2k+1 ) ≥ F ( Further we have ∂F ( q2 , q3 , . . . , q2k−1 , b, b) 2y 2k 3y 2k+1 = 2k−1 2k−2 − . 3 4 ∂ q2k−1 q2 q3 · . . . · q2k−1 b2 q22k q32k−1 · . . . · q2k−1 b3 Hence, ∂F ( q2 , q3 , . . . , q2k−1 , b, b) 2 q2 q3 · . . . · q2k−1 q2k−1 b . ≥0⇔y≤ ∂ q2k−1 3 q2 , q3 , . . . , q2k−1 , b, b) is Since y < q2 , we obtain that the function F ( increasing in q2k−1 , whence q2 , q3 , . . . , q2k−2 , b, b, b). F ( q2 , q3 , . . . , q2k−1 , b, b) ≥ F ( Applying similar arguments we get the following chain of inequalities. q2 , q3 , . . . , q2k−1 , b, b) ≥ F ( q2 , q3 , . . . , q2k , q2k+1 ) ≥ F ( F ( q2 , q3 , . . . , q2k−2 , b, b, b) ≥ . . . ≥ F (b, b, . . . , b, b). Thus, we have proved (10). Besides, −

y 2m y 2m . ≥ − 2 b2m−1 · b2m−2 · . . . · b2 · b q22m−1 q32m−2 · . . . · q2m−1 q2m

We substitute the last inequality and (10) into (9) to get the following (12)

μm (y) ≥ −

2m  (−1)k y k k=0

b

k(k−1) 2

= −S2m (y, gb ),

√ where gb (y) = g√b (− by), and ga is a partial theta function. Since, √ by assumption ( b)2 > q∞ , we have √ 2from Theorem D (4) that there exists such m0 ∈ N that c2m0 < ( b) . First, we choose and fix such − P, and, from Theorem D (3), there exists m0 . Thus, S2m0 (y, gb )√∈ L√ x0 = x0 (m0 ), x0 ∈ ( b; ( b)3 ) : S2m0 (−x0 , g√b ) ≤ 0. We put y0 = √ −1 x0 b . We have y0 ∈ (1, b) ⊂ (1, q2 ) = (1, qj+1 ), and, by (12), we obtain μm0 (y0 ) ≥ 0. We define (13)

yj := y0 · q2 q3 · . . . · qj−1 qj ,

j ∈ N.

We have the condition q2 q3 · . . . · qj < yj < q2 q3 · . . . · qj qj+1 . We recall that m0 ∈ N depends only on b, then we define a sequence yj by (13), such that pj < yj < pj+1 for all j. We fix an arbitrary j ∈ N, and suppose that n is an integer such that n > j + 2m0 . Thereafter,

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T. H. NGUYEN AND A. VISHNYAKOVA

we put x = yj in (7). Hence, using (8), (9), (12) and the fact that  μm0 (y0 ) ≥ 0, we obtain (6). ∞ Lemma 2.2. Let ϕ(z) = k=0 (−1)k ak z k , ak > 0 for all k, be an entire function, and pk (ϕ), qk (ϕ) are defined as in (2). Suppose √ that qk (ϕ) are decreasing in k, lim qk (ϕ) = b, and q∞ < b < 1 + 5. Then there k→∞

exists a sequence of positive numbers (ρj )∞ j=1 , pj (ϕ) < ρj < pj+1 (ϕ), such that for every j being large enough the function ϕ has exactly j zeros (counting multiplicities) in the circle {z : |z| < ρj }. Proof. As in the proof of Lemma 2.1, we assume that a0 = a1 = 1, and we use the denotations of pn and qn instead of pn (ϕ) and qn (ϕ). For every j ∈ N we define ρj as follows: √ (14) ρj := q2 q3 · . . . · qj qj+1 , whence q2 q3 . . . qj < ρj < q2 q3 . . . qj qj+1 . We observe that (15)

ϕ(z) =

j−3

(−1)k z k k=0 q k−1 q k−2 ·...·qk 2 3

(−1)j+2 z j+2 j+1 j 5 4 2 q2 q3 ·...·qj−2 qj−1 qj4 qj+1

+ 



+ 

(−1)j+2 z j+2 5 4 2 q2j+1 q3j ·...·qj−2 qj−1 qj4 qj+1



j+1 (−1)k z k k=j−2 q k−1 q k−2 ·...·qk + 2 3 (−1)j+2 z j+2

5 4 2 q2j+1 q3j ·...·qj−2 qj−1 qj3 qj+1 qj+2

+

−

∞

(−1)k z k k=j+3 q k−1 q k−2 ·...·qk 2 3

=: Σ1 (z) + gj (z) + ξj (z) + Σ2 (z). We show that for every j being large enough the following inequality holds: min0≤θ≤2π |gj (ρj eiθ )| > max0≤θ≤2π |ϕ(ρj eiθ ) − gj (ρj eiθ )|, so that the number of zeros of ϕ in the circle {z : |z| < ρj } is equal to the number of zeros of gj in the same circle. We have j−2  2 q2j−2 q3j−2 · . . . · qjj−2 qj+1 √ iθ i(j−2)θ ·e · 1 − qj qj+1 eiθ gj (ρj e ) = j−3 j−4 2 q2 q3 · . . . · qj−3 qj−2  j−2 √ j−3 j−2 j−2 2 2iθ 3iθ 4iθ = q2 q32 · . . . · qj−2 +qj q+1 e − qj qj+1 e + e qj−1 qj qj+1 · eijθ ·   √ √ −2iθ −iθ iθ 2iθ e = − qj qj+1 e + qj qj+1 − qj qj+1 e + e   j−2 √ j−3 j−2 j−2 2 2 ijθ q2 q3 · . . . · qj−2 qj−1 qj qj+1 · e · 2 cos 2θ − 2qj qj+1 cos θ + qj qj+1 . We would like to find min0≤θ≤2π |gj (ρj eiθ )|. Let t := cos θ ∈ [−1, 1]. We consider the expression in the big round brackets: √ gj (t) := 4t2 − 2qj qj+1 t + (qj qj+1 − 2).

´ ENTIRE FUNCTIONS OF THE LAGUERRE-POLYA CLASS

9

√ The vertex of the parabola is tj = qj qj+1 /4, and, by our assumptions, tj > 1. Further, we estimate the discriminant: limj→∞ (4(qj2 qj+1 − √ √ 4qj qj+1 + 8)) = 4(b3√ − 4b2 + 8) = 4(b − 2) · (b − (1 − 5)) · (b − (1 + 5)). Since q∞ < b < 1+ 5, we obtain that Dj < 0 for j being large enough. So we get for all sufficiently large j: √ min gj (t) = gj (1) = 2 − 2qj qj+1 + qj qj+1 . −1≤t≤1

Consequently, we have obtained the estimation from below for j being large enough   j−2 √ j−2 j−2 2 iθ 2 (16) |gj (ρj e )| ≥ q2 q3 · . . . · qj−1 qj qj+1 qj qj+1 − 2qj qj+1 + 2 . We also want to obtain the estimation of |Σ2 (ρj eiθ )| from above. k

iθ

|Σ2 (ρj e )| ≤

∞ 2  q2k q3k · . . . · qjk qj+1 k=j+3

q2k−1 q3k−2 · . . . · qk

=

j−3  2 q2 q32 · . . . · qjj−1 qj+1 1 · 1+ √ 2 qj+2 qj+3 qj+1 qj+2 qj+3 qj+4  1 + √ + ... . 2 2 2 ( qj+1 )2 qj+2 qj+3 qj+4 qj+5 Obviously, the last sum can be estimated from above by the sum of the geometric progression, so we obtain j−3

(17)

2 q2 q32 · . . . · qjj−1 qj+1 · |Σ2 (ρj eiθ )| ≤ 2 qj+2 qj+3 1−

1 √

1 qj+1 qj+2 qj+3 qj+4

.

Further, we wish to obtain the estimation of |Σ1 (ρj eiθ )| from above. k

|Σ1 (ρj eiθ )| ≤

j−3 k k 2  q2 q3 · . . . · qjk qj+1

= (we rewrite the sum from right q2k−1 q3k−2 · . . . · qk j−3 j−4 j−3 j−3 j−3 2 to left) = q2 q32 · . . . · qj−3 qj−2 qj−1 qj qj+1 + k=0

j−4

j−5 j−4 j−4 j−4 j−4 2 q2 q32 · . . . · qj−4 qj−3 qj−2 qj−1 qj qj+1 +

j−5 j−6 j−5 j−5 j−5 j−5 j−5 2 q2 q32 · . . . · qj−5 qj−4 qj−3 qj−2 qj−1 qj qj+1 + · · · =  j−3 1 j−4 j−3 j−3 j−3 2 2 q2 q3 · . . . · qj−3 qj−2 qj−1 qj qj+1 · 1 + √ qj−2 qj−1 qj qj+1

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T. H. NGUYEN AND A. VISHNYAKOVA

+



1

+ ... √ 2 2 qj−3 qj−2 qj−1 qj2 ( qj+1 )2

≤

j−3

j−4 j−3 j−3 j−3 2 qj−2 qj−1 qj qj+1 · q2 q32 · . . . · qj−3

1 1−

1 √ qj−2 qj−1 qj qj+1

(we estimate the finite sum from above by the sum of the infinite geometric progression). Finally, we obtain j−3 1 j−4 j−3 j−3 j−3 2 qj−2 qj−1 qj qj+1 · . (18) |Σ1 (ρj eiθ )| ≤ q2 q32 · . . . · qj−3 1 − qj−2 qj−11qj √qj+1 We also note that

(19) |ξj (ρj e )| = (q2 q3 · . . . · qj qj+1 ) qj+1 qj ·...·q5 q14 q3 q2 q − 2 3 j−2 j−1 j j+1 j+2

j−2 j−2 j−1 2 1 1 1 2 q · . . . · q q q − = q . j+1 j 2 3 j−1 j j+1 qj+2 qj q q ·...·q 5 q 4 q 4 q 2 √

iθ

2

3

j+2

j−2 j−1 j j+1

By (16), (17), (18) and (19), the inequality min0≤θ≤2π |gj (ρj eiθ )| > max0≤θ≤2π |ϕ(ρj eiθ ) − gj (ρj eiθ )| follows from   j−2 √ j−2 j−2 2 2 q2 q3 · . . . · qj−1 qj qj+1 qj qj+1 − 2qj qj+1 + 2 > j−3

2 q2 q32 · . . . · qjj−1 qj+1 · 2 qj+2 qj+3 1−

1 √

1 qj+1 qj+2 qj+3 qj+4

j−3

j−4 j−3 j−3 j−3 2 q2 q32 · . . . · qj−3 qj−2 qj−1 qj qj+1 ·

q2 q32

· ... ·

j−2

j−2 j−1 2 qj−1 qj qj+1



+

1 1− 1 qj+2

1 √ qj−2 qj−1 qj qj+1

1 − qj

+



,

or, equivalently,   √ qj qj+1 − 2qj qj+1 + 2 > √

√



−1 −1 qj−1 qj

qj+1 1 −

qj 1 · + 2 qj+1 qj+2 qj+3 1 − √qj+1 qj+21 qj+3 qj+4   qj qj

+ . − 1 qj+2 qj √

qj−2 qj−1 qj qj+1

Since lim qk (ϕ) = b, the last inequality (for all sufficiently large j) k→∞

derives from

√ b2 − 2b b + 2 >

b2

1 1 1 √ · √ 1√ + b 1 − b3 b b2 b 1 −

1√

b3

, b

´ ENTIRE FUNCTIONS OF THE LAGUERRE-POLYA CLASS

or, equivalently,

Denoting by c =

√ b2 − 2b b + 2 > √

b3

11

2b √ . b−1

b, we obtain after elementary transformations

c11 − 2c10 + 2c7 − c4 + 2c3 − 2c2 − 2 > 0.

(20)

We function p(c) = c11 −2c10 +2c7 −c4 +2c3 −2c2 −2 for c ∈ √ consider

the √

√ √   q∞ , 1 + 5 ⊂ ( 3, 2). We have p(c) = c11 − 2c10 + (6 3 − 9)c7 + √   (11 − 6 3)c7 − c4 + (2c3 − 2c2 − 2). Obviously, 2c3 − 2c2 − 2 > √ √ 7 √ ·4−2√ > 0. Further, 11 − 6 3 > 0, and (11 − 6 3)c − c4 > 2 · 3 3 − 2√  c4 (11 − 6 3) · 3 3 − 1 > 0. It remains to prove that c11 − 2c10 + √ √ (6 3 − 9)c7 ≥ 0, or γ(c) := c4 − 2c3 + (6√ 3 − 9) ≥ 0. We have γ  (c) = 4c3 − 6c2 = 2c2 (2c − 3) > 0, and γ( 3) = 0. Thus, we have proved (20) under our assumptions on c. This yields that for all j being large enough min0≤θ≤2π |gj (ρj eiθ )| > max0≤θ≤2π |ϕ(ρj eiθ ) − gj (ρj eiθ )|, so the number of zeros of ϕ in the circle {z : |z| < ρj } is equal to the number of zeros of gj in this circle. It remains to find the number of zeros of gj in the circle {z : |z| < ρj }. We have gj (z) =

j+1 

(−1)k z k (−1)j+2 z j+2 + . j+1 j k−1 k−2 5 4 4 2 q q · . . . · q q q · . . . · q q q q k 2 3 2 3 j−2 j−1 j j+1 k=j−2

We use the denotation w = zρ−1 , so that |w| < 1. This yields j−2

j−3 j−2 j−2 2 gj (ρj w) = (−1)j+2 wj−2 q2 q32 · . . . · qj−2 qj−1 qj qj+1 ·

  √ √ 1 − qj qj+1 w + qj qj+1 w2 − qj qj+1 w3 + w4 . It follows from (16) that gj does not have zeros on the circumference {z : |z| = ρj }, whence gj (ρj w) does not have zeros on the circumference √ √ {w : |w| = 1}. Since Pj (w) = 1−qj qj+1 w +qj qj+1 w2 −qj qj+1 w3 +w4 is the self-reciprocal polynomial on w we get that Pj has exactly two zeros in the circle {w : |w| < 1}. Hence, gj (z) has exactly j zeros in the circle {z : |z| < ρj }, and we have proved the statement of Lemma 2.2.  Now we are ready to prove Theorem 1.1. Suppose that Theorem 1.1 is proved for all entire functions g with positive coefficients such that k qn (g) are decreasing in n and lim qn (g) = q∞ . Let f (z) = ∞ k=0 ak z n→∞

12

T. H. NGUYEN AND A. VISHNYAKOVA

be an entire function with positive coefficients such that qn (f ) are decreasing in n and lim qn (f ) = b > q∞ . Let us consider a new enn→∞ 2 ∞ −1 k /2 k tire function with positive coefficients f˜(z) = z . k=0 ak (bq∞ ) Straight forward calculations show that qn (f˜) = qn (f ) · q∞ b−1 , so that qn (f˜) are decreasing in n and lim qn (f˜) = q∞ . Thus, by our assumpn→∞ ˜ has only real zeros. Since bq −1 > 1, by tion, the entire function f ∞ 

−k2 ∞

−1 bq∞ ∈ CZDS. Hence, our entire function (5) we have k=0 2  k2 /2 k −1 k /2 f (z) = ∞ (q∞ b−1 ) z has all real zeros. k=0 ak (bq∞ ) Consequently, it is sufficient to prove Theorem 1.1 for the case b = q∞ . We suppose that lim qn (f ) = q∞ , and a0 = a1 = 1. n→∞

We consider a function ϕ(z) := f (−z) = 1−z+a2 z 2 −a3 z 3 + . . . , and note that q2 (ϕ) ≥ q3 (ϕ) ≥ q4 (ϕ) ≥ . . . , and lim qn (ϕ) = q∞ . For an arn→∞  k −k2 ak z k . bitrary a > 1 we denote by ϕa the function ϕa (z) = ∞ k=0 (−1) a 2 We have qn (ϕa ) = a qn (ϕ), so we obtain q2 (ϕa ) ≥ q3 (ϕa ) ≥ q4 (ϕa ) ≥ . . . , and lim qn (ϕa ) = a2 q∞ . Let us choose and fix an arbitrary a, 1 < n→∞ √ a2 < (1+ 5)/q∞ . By Lemma 2.2, there exists a positive integer j0 such that for all l ≥ j0 the function ϕa has exactly l zeros (counting multiplicities) in the circle {z : |z| < ρl }, and pl (ϕa ) < ρl < pl+1 (ϕa ). Let us fix an arbitrary index l > j0 . Using Lemma 2.1, we obtain that there exists a positive integer m0 , and an infinite sequence of positive numbers (yj ) , pj (ϕa ) < yj < pj+1 (ϕa ), such that for every n > 2m0 the following inequalities hold: (−1)j Sn (yj , ϕa ) ≥ 0 for all j = 1, 2, 3, . . . , n − 2m0 . We mention that all the points 0 < y1 < y2 < . . . < yl−1 belong to the circle {z : |z| < ρl }. Thus, for an arbitrary n ∈ N, n > l + 2m0 , we have Sn (0, ϕa ) > 0, and (−1)j Sn (yj , ϕa ) ≥ 0 for all j = 1, 2, 3, . . . , l − 1. Tending n → ∞, we obtain ϕa (0) > 0, and (−1)j ϕa (yj ) ≥ 0 for all j = 1, 2, 3, . . . , l − 1. Hence, the function ϕa has at least l − 1 real zeros in the circle {z : |z| < ρl }. Since the number of nonreal zeros of a real function ϕa in the circle {z : |z| < ρl } is even, we conclude that all the zeros of ϕa in the circle {z : |z| < ρl } are real, and this fact is true for every l > j0 . Correspondingly, all the√zeros of ϕa are real. We have proved that for every a, 1 < a2 < (1 + 5)/q∞ , the function ϕa has only real zeros. Tending a → 1, we obtain the statement of Theorem 1.1. Theorem 1.1 is proved. 2 Acknowledgment. The authors are deeply grateful to the referee for valuable remarks.

´ ENTIRE FUNCTIONS OF THE LAGUERRE-POLYA CLASS

13

References [1] A. Bohdanov, Determining Bounds on the Values of Parameters for a Function ∞ k ϕa (z, m) = k=0 azk2 (k!)m , m ∈ (0, 1), to Belong to the Laguerre-P´olya Class, Comput. Methods Funct. Theory, (2017), doi 10.1007/s40315-017-0210-6 [2] A. Bohdanov and A. Vishnyakova, On the conditions for entire functions related to the partial theta function to belong to the Laguerre-P´olya class, J. Math. Anal. Appl., 434 , No. 2, (2016), 1740–1752, doi:10.1016/j.jmaa.2015.09.084 [3] T. Craven and G. Csordas, Complex zero decreasing sequences, Methods Appl. Anal., 2 (1995), 420–441. [4] I. I. Hirschman and D.V.Widder, The Convolution Transform, Princeton University Press, Princeton, New Jersey, 1955. [5] J. I. Hutchinson, On a remarkable class of entire functions, Trans. Amer. Math. Soc., 25 (1923), 325–332. [6] I. V. Ostrovskii, On Zero Distribution of Sections and Tails of Power Series, Israel Math. Conference Proceedings, 15 (2001), 297–310. [7] O. Katkova, T. Lobova and A. Vishnyakova, On power series having sections with only real zeros, Comput. Methods Funct. Theory, 3, No 2, (2003), 425– 441. [8] O. Katkova, T. Lobova and A. Vishnyakova, On entire functions having Taylor sections with only real zeros, J. Math. Phys., Anal., Geom., 11, No. 4, (2004), 449–469. [9] V.P.Kostov, On the zeros of a partial theta function, Bull. Sci. Math., 137, No. 8 (2013), 1018–1030. [10] V.P.Kostov, On the spectrum of a partial theta function, Proc. Roy. Soc. Edinburgh Sect. A, 144, No. 05 (2014), 925–933. [11] V.P.Kostov, B.Shapiro, Hardy-Petrovitch-Hutchinson’s problem and partial theta function, Duke Math. J., 162, No. 5 (2013), 825–861. [12] N. Obreschkov , Verteilung und Berechnung der Nullstellen reeller Polynome, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963. ¨ [13] G. P´olya, Uber einen Satz von Laguerre, Jber. Deutsch. Math.-Verein., 38(1929), pp. 161–168. [14] G. P´olya, Collected Papers, Vol. II Location of Zeros, (R.P.Boas ed.) MIT Press, Cambridge, MA, 1974. ¨ [15] G. P´olya and J.Schur, Uber zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, J. Reine Angew. Math., 144 (1914), pp. 89–113. Department of Mathematics & Computer Sciences, V. N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv, 61022, Ukraine E-mail address: [email protected] Department of Mathematics & Computer Sciences, V. N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv, 61022, Ukraine E-mail address: [email protected]