On the number of real zeros of real entire functions with a non-decreasing sequence of the second quotients of Taylor coefficients
aa r X i v : . [ m a t h . C V ] J a n ON THE NUMBER OF REAL ZEROS OF REALENTIRE FUNCTIONS WITH A NON-DECREASINGSEQUENCE OF THE SECOND QUOTIENTS OFTAYLOR COEFFICIENTS
THU HIEN NGUYEN AND ANNA VISHNYAKOVA
Abstract.
For an entire function f ( z ) = P ∞ k =0 a k z k , a k > , we define the sequence of the second quotients of Taylor coeffi-cients Q := (cid:16) a k a k − a k +1 (cid:17) ∞ k =1 . We find new necessary conditionsfor a function with a non-decreasing sequence Q to belong to theLaguerre–P´olya class of type I. We also estimate the possible num-ber of nonreal zeros for a function with a non-decreasing sequence Q. Introduction
The topic of zero distribution of entire functions has been the subjectof study and discussion of mathematicians for many years (see, forexample, [19]). In the present paper, we consider a class of entirefunctions with positive Taylor coefficients and investigate the conditionfor them to belong to the Laguerre–P´olya class of type I. We give thedefinitions of the Laguerre–P´olya class and the Laguerre–P´olya classof type I.
Definition 1 . A real entire function f is said to be in the Laguerre–P´olya class, written f ∈ L − P , if it can be expressed in the form (1) f ( z ) = cz n e − αz + βz ∞ Y k =1 (cid:18) − zx k (cid:19) e zx − k , where c, α, β, x k ∈ R , x k = 0 , α ≥ , n is a nonnegative integer and P ∞ k =1 x − k < ∞ . Definition 2 . A real entire function f is said to be in the Laguerre–P´olya class of type I, written f ∈ L − P I , if it can be expressed in the Mathematics Subject Classification.
Key words and phrases.
Laguerre–P´olya class; Laguerre–P´olya class of type I;entire functions of order zero; real-rooted polynomials; multiplier sequences; com-plex zero decreasing sequences. following form (2) f ( z ) = cz n e βz ∞ Y k =1 (cid:18) zx k (cid:19) , where c ∈ R , β ≥ , x k > , n is a nonnegative integer, and P ∞ k =1 x − k < ∞ . As usual, the product on the right-hand sides in both definitions canbe finite or empty (in the latter case the product equals 1).These classes are important for the theory of entire functions sincethe hyperbolic polynomials (i.e. real polynomials with only real ze-ros), or hyperbolic polynomials with nonnegative coefficients convergelocally uniformly to these and only these functions. The followingprominent theorem states even a stronger fact.
Theorem A (E. Laguerre and G. P´olya, see, for example, [5, p. 42–46]) and [12, chapter VIII, § (i) Let ( P n ) ∞ n =1 , P n (0) = 1 , be a sequence of real polynomials havingonly real zeros which converges uniformly on the disc | z | ≤ A, A > . Then this sequence converges locally uniformly in C to an entirefunction from the L − P class.(ii) For any f ∈ L − P there exists a sequence of real polynomialswith only real zeros which converges locally uniformly to f .(iii) Let ( P n ) ∞ n =1 , P n (0) = 1 , be a sequence of real polynomials havingonly real negative zeros which converges uniformly on the disc | z | ≤ A, A > . Then this sequence converges locally uniformly in C to anentire function from the class L − P I. (iv) For any f ∈ L − P I there is a sequence of real polynomials withonly real nonpositive zeros which converges locally uniformly to f . Numerous properties and features of the Laguerre–P´olya class andthe Laguerre–P´olya class of type I can be found in the works [20, p.100], [21] and [18, Kapitel II] (also see the survey [19] on the zerodistribution of entire functions, its sections and tails). Note that fora real entire function (not identically zero) of the order less than 2the property of having only real zeros is equivalent to belonging tothe Laguerre–P´olya class. Also, for a real entire function with positivecoefficients of the order less than 1 having only real zeros is equivalentto belonging to the Laguerre–P´olya class of type I.
NTIRE FUNCTIONS OF THE LAGUERRE–P ´OLYA I CLASS 3
Let f ( z ) = P ∞ k =0 a k z k be an entire function with real nonzero coef-ficients. We define the quotients p n and q n : p n = p n ( f ) := a n − a n , n ≥ ,q n = q n ( f ) := p n p n − = a n − a n − a n , n ≥ . From these definitions it follows straightforwardly that a n = a p p · · · p n , n ≥ ,a n = a (cid:16) a a (cid:17) n − q n − q n − · . . . · q n − q n , n ≥ . It is rather a complicated problem to understand whether a givenentire function has only real zeros. However, in 1926, J. I. Hutchinsonfound quite a simple sufficient condition for an entire function withpositive coefficients to have only real zeros.
Theorem B (J. I. Hutchinson, [6]).
Let f ( z ) = P ∞ k =0 a k z k , a k > for all k . Then q n ( f ) ≥ , for all n ≥ , if and only if the followingtwo conditions are fulfilled:(i) The zeros of f are all real, simple and negative, and(ii) the zeros of any polynomial P nk = m a k z k , m < n, formed by takingany number of consecutive terms of f , are all real and non-positive. For some extensions of Hutchinson’s results see, for example, [3, § g a ( z ) = P ∞ k =0 z k a − k , a > , known as a partial theta-function (the classical Jacobi theta-function is defined bythe series θ ( z ) := P ∞ k = −∞ z k a − k ), was investigated by many mathe-maticians and has an important role. Note that q n ( g a ) = a for all n. The survey [23] by S. O. Warnaar contains the history of investigationof the partial theta-function and some of its main properties.In particular, in the paper [7] it was explained that for every n ≥ c n > S n ( z, g a ) := P nj =0 z j a − j ∈L − P ⇔ a ≥ c n . Theorem C (O. Katkova, T. Lobova, A. Vishnyakova, [7]).
Thereexists a constant q ∞ ( q ∞ ≈ . . . . ) such that: (1) g a ( z ) ∈ L − P ⇔ a ≥ q ∞ ;(2) g a ( z ) ∈ L − P ⇔ there exists z ∈ ( − a , − a ) such that g a ( z ) ≤ if there exists z ∈ ( − a , − a ) such that g a ( z ) < , then a >q ∞ ;(4) for a given n ≥ we have S n ( z, g a ) ∈ L − P ⇔ there exists z n ∈ ( − a , − a ) such that S n ( z n , g a ) ≤ T. H. NGUYEN AND A. VISHNYAKOVA (5) if there exists z n ∈ ( − a , − a ) such that S n ( z n , g a ) < , then a > c n ;(6) 4 = c > c > c > · · · and lim n →∞ c n = q ∞ ;(7) 3 = c < c < c < · · · and lim n →∞ c n +1 = q ∞ . Partial theta function is of interest to many areas such as statisticalphysics and combinatorics [22], Ramanujan type q -series [24], asymp-totic analysis and the theory of (mock) modular forms, etc. There is aseries of works by V.P. Kostov dedicated to various properties of zerosof the partial theta-function and its derivative (see [9, 10] and the ref-erences therein). The paper [11] among the other results explains therole of the constant q ∞ in the study of the set of entire functions withpositive coefficients having all Taylor truncations with only real zeros.In [8], the following questions are investigated: whether the Taylorsections of the function ∞ Q k =1 (cid:0) za k (cid:1) , a > , and P ∞ k =0 z k k ! a k , a ≥ , belong to the Laguerre–P´olya class of type I. In [2] and [1], some im-portant special functions with non-decreasing sequence of the secondquotients of Taylor coefficients are studied.The first author studied a special function related to the partialtheta-function and the Euler function, f a ( z ) = P ∞ k =0 z k ( a k +1)( a k − +1) · ... · ( a +1) ,a > , which is also known as the q -Kummer function φ ( q ; − q ; q, − z ),where q = 1 /a (see [4], formula (1.2.22)). Note that it has increasingsecond quotients of Taylor coefficients. In [17], the conditions werefound for this function to belong to the Laguerre–P´olya class.It turns out that for many important entire functions with positivecoefficients f ( z ) = P ∞ k =0 a k z k (for example, partial theta-function from[7], functions from [2] and [1], the q -Kummer function φ ( q ; − q ; q, − z )and others) the following two conditions are equivalent:(i) f belongs to the Laguerre–P´olya class of type I,and(ii) there exists x ∈ [ − a a ,
0] such that f ( x ) ≤ . In our previous work we proved the following necessary condition fora function to belong to the Laguerre–P´olya class.
Theorem D (T. H. Nguyen, A. Vishnyakova, [15]).
Let f ( z ) = P ∞ k =0 a k z k , a k > for all k, be an entire function. Suppose that q ( f ) ≤ q ( f ) . If the function f belongs to the Laguerre-P´olya class, then thereexists x ∈ [ − a a , such that f ( x ) ≤ . In [16] we have obtained a criterion for belonging to the Laguerre–P´olya class of type I for real entire functions with the regularly non-decreasing sequence of second quotients of Taylor coefficients in termsof the existence of a point x as in Theorem D. NTIRE FUNCTIONS OF THE LAGUERRE–P ´OLYA I CLASS 5
It was previously shown in [14] that if f ( z ) = P ∞ k =0 a k z k , a k > k, is an entire function such that q ≤ q ≤ q ≤ · · · , andlim n →∞ q n ( f ) = c ≥ q ∞ , then the function f belongs to the Laguerre–P´olya class, where q ∞ is a constant from Theorem C.In the present paper we prove that the following conditions on thesecond quotients q k are necessary for the function to belong to theLaguerre–P´olya I class: Theorem 1.1.
Let f ( z ) = P ∞ k =0 a k z k , a k > , k = 0 , , , . . . , be anentire function such that q ( f ) ≤ q ( f ) ≤ q ( f ) ≤ . . . . If f ∈ L − P ,then for any k = 1 , , , . . . the following inequality holds: q n +1 > c k +1 ( c k +1 defined as in Theorem C). Calculations show that c = 1 + √ ≈ . , c ≈ . c ≈ . , c ≈ . . In [16] we also proved the following result.
Theorem E (T. H. Nguyen, A. Vishnyakova, [16]).
Let f ( z ) = P ∞ k =0 a k z k , a k > , k = 0 , , , . . . , be an entire function such that √ ≤ q ( f ) ≤ q ( f ) ≤ q ( f ) ≤ . . . Then all but a finite number ofzeros of f are real and simple. Our next theorem estimates the possible number of nonreal zeros forsuch functions.
Theorem 1.2.
Let f ( z ) = P ∞ k =0 a k z k , a k > , k = 0 , , , . . . , be anentire function such that √ ≤ q ( f ) ≤ q ( f ) ≤ q ( f ) ≤ . . . . If thereexist j = 2 , , , . . . and m ∈ N , such that q j ≥ c m , then the numberof nonreal zeros of f does not exceed j + 2 m − ( c k defined as inTheorem C). Proof of Theorem 1.1
Without loss of generality, we can assume that a = a = 1 , sincewe can consider a function g ( x ) = a − f ( a a − x ) instead of f ( x ) , dueto the fact that such rescaling of f preserves its property of havingreal zeros as well as the second quotients: q n ( g ) = q n ( f ) for all n ∈ N . During the proof we use notation p n and q n instead of p n ( f ) and q n ( f ) . We consider a function ϕ ( x ) = f ( − x ) = 1 − x + ∞ X k =2 ( − k x k q k − q k − . . . q k − q k instead of f. Theorem D states that if ϕ belongs to the Laguerre–P´olya class thenthere exists a point x ∈ [0 , a a ] = [0 , q ] such that ϕ ( x ) ≤ . T. H. NGUYEN AND A. VISHNYAKOVA
Let us introduce some more notations. For an entire function ϕ ,by S n ( x, ϕ ) and R n ( x, ϕ ) we denote the n th partial sum and the n thremainder of the series, i.e. S n ( x, ϕ ) = n X k =0 ( − k x k q k − q k − . . . q k − q k , and R n ( x, ϕ ) = ∞ X k = n ( − k x k q k − q k − . . . q k − q k . First, we need the following Lemma.
Lemma 2.1.
Let ϕ ( x ) = 1 − x + P ∞ k =2 ( − k x k q k − q k − ...q k − q k be an entirefunction. Suppose that q k are non-decreasing in k : q ≤ q ≤ q ≤ . . . . If there exists x ∈ [0 , q ] such that ϕ ( x ) ≤ , then x ∈ (1 , q ] . Proof.
For x ∈ [0 ,
1] we have:1 ≥ x > x q > x q q > x q q q > · · · , whence(3) ϕ ( x ) > x ∈ [0 , . (cid:3) Lemma 2.2.
Let ϕ ( x ) = 1 − x + P ∞ k =2 ( − k x k q k − q k − ...q k − q k be an entirefunction. Suppose that q k are non-decreasing in k : q ≤ q ≤ q ≤ . . . . If there exists x ∈ (1 , q ] such that ϕ ( x ) ≤ , then for any n ∈ N S n +1 ( x ) < . Proof.
Suppose that x ∈ (1 , q ] . Then we obtain(4) 1 < x ≥ x q > x q q > · · · > x k q k − q k − . . . q k − q k > · · · For an arbitrary n ∈ N we have: ϕ ( x ) = S n +1 ( x, ϕ ) + R n +2 ( x, ϕ ) . By (4) and the Leibniz criterion for alternating series, we concludethat R n +2 ( x, ϕ ) > x ∈ (1 , q ] , or(5) ϕ ( x ) > S n +1 ( x, ϕ ) for all x ∈ (1 , q ] , n ∈ N . Consequently, if there exists a point x ∈ (1 , q ] such that ϕ ( x ) ≤ , then for any n ∈ N we have S n +1 ( x ) < . (cid:3) NTIRE FUNCTIONS OF THE LAGUERRE–P ´OLYA I CLASS 7
Thus, we proved that if ϕ ∈ L − P , then there exists x ∈ (1 , q ]such that the inequalities S n +1 ( x ) < n ∈ N . In [14] it was proved that if an entire function ϕ ( x ) = 1 − x + P ∞ k =2 ( − k x k q k − q k − ...q k − q k belongs to the Laguerre–P´olya class, where 0 Let ϕ ( x ) = 1 − x + P ∞ k =2 ( − k x k q k − q k − ...q k − q k be an entirefunction. Suppose that ≤ q ≤ q ≤ q . . . . Then the inequality S n +1 ( x, ϕ ) ≥ S n +1 ( √ q k +1 x, g √ q k +1 ) holds for any n ∈ N and any x ∈ (1 , q ] (here g a is the partial theta-function and S n +1 ( y, g a ) is its (2 n + 1) -th section at the point y ).Proof. We have S n +1 ( x, ϕ ) = (1 − x ) + (cid:16) x q − x q q (cid:17) + (cid:16) x q q q − (6) x q q q q (cid:17) + . . . + (cid:16) x n q n − q n − · ... · q n − q n − x n +1 q n q n − · ... · q n q n +1 (cid:17) . Under our assumptions, q k are non-decreasing in k. We prove thatfor any fixed k = 1 , , . . . , n and x ∈ (1 , q ] the following inequalityholds: x k q k − q k − · . . . · q k − x k +1 q k q k − · . . . · q k q k +1 ≥ x k q k − k +1 · q k − k +1 · . . . · q k +1 − x k +1 q k k +1 · q k − k +1 · . . . · q k +1 · q k +1 . For x ∈ (1 , q ] and k = 1 , , . . . , n, we define the following function: F ( q , q , . . . , q k , q k +1 ) := x k q k − q k − · . . . · q k − x k +1 q k q k − · . . . · q k q k +1 . We can observe that ∂F ( q , q , . . . , q k , q k +1 ) ∂q = − (2 k − x k q k q k − . . . q k + 2 kx k +1 q k +12 q k − . . . q k q k +1 < ⇔ x < (cid:16) − k (cid:17) q q . . . q k q k +1 . Therefore, since (cid:16) − k (cid:17) q q . . . q k q k +1 ≥ q q > q (under ourassumptions q ≥ q ≥ F ( q , q , . . . , q k , q k +1 ) is de-creasing in q on the interval (1 , q ]. Since q ≤ q , we have for k = 1 F ( q , q ) = x q − x q q ≥ x q − x q q , T. H. NGUYEN AND A. VISHNYAKOVA and the desired inequality is proved for k = 1 . For k ≥ F ( q , q , q , . . . , q k , q k +1 ) ≥ F ( q , q , q , . . . , q k , q k +1 ) = x k q k − q k − . . . q k − x k +1 q k − q k − . . . q k +1 . Further, we have: ∂F ( q , q , q , . . . , q k , q k +1 ) ∂q = − (4 k − x k q k − q k − . . . q k + (4 k − x k +1 q k q k − . . . q k +1 < ⇔ x < k − k − q q . . . q k +1 . Since, under our assumptions, k − k − q q . . . q k +1 ≥ q q q > q , weget that F ( q , q , q , . . . , q k , q k +1 ) is decreasing in q on the interval(1 , q ] and, since q ≤ q , we obtain: F ( q , q , q . . . , q k , q k +1 ) ≥ F ( q , q , q , q , . . . , q k , q k +1 ) . Analogously, by the same computation, we obtain the following chainof inequalities: F ( q , q , q , . . . , q k , q k +1 ) ≥ F ( q , q , q , . . . , q k , q k +1 ) ≥ F ( q , q , q , q , . . . , q k , q k +1 ) ≥ . . . ≥ F ( q k , q k , . . . , q k , q k +1 ) ≥ F ( q k +1 , q k +1 , . . . , q k +1 , q k +1 ) . Thus, F ( q , q , q , . . . , q k , q k +1 ) ≥ F ( q k +1 , q k +1 , . . . , q k +1 , q k +1 )= x k q k (2 k − k +1 − x k +1 q k (2 k +1)2 k +1 . At the end we note that, under our assumptions, the expression x k q k (2 k − k +1 − x k +1 q k (2 k +1)2 k +1 is decreasing in q k +1 on the interval (1 , q ] , so weobtain F ( q , q , q , . . . , q k , q k +1 ) ≥ x k q k (2 k − k +1 − x k +1 q k (2 k +1)2 k +1 ≥ x k q k (2 k − n +1 − x k +1 q k (2 k +1)2 n +1 . NTIRE FUNCTIONS OF THE LAGUERRE–P ´OLYA I CLASS 9 Substituting the last inequality in (6) for every x ∈ (1 , q ] and k =1 , , . . . , n, we get: S n +1 ( x, ϕ ) ≥ (1 − x ) + (cid:18) x q n +1 − x q n +1 (cid:19) + (cid:18) x q n +1 − x q n +1 (cid:19) +(7) . . . + x n q n (2 n − n +1 − x n +1 q n (2 n +1)2 n +1 ! = n +1 X k =0 ( − k x k √ q n +1 k ( k − = S n +1 ( −√ q n +1 x, g √ q n +1 ) , where g a is the partial theta-function and S n +1 ( y, g a ) is its (2 n + 1)-thsection at the point y . (cid:3) Since we have S n +1 ( x, ϕ ) ≥ S n +1 ( −√ q n +1 x, g √ q k +1 ) for any n ∈ N , if there exists such a point x ∈ (1 , q ] that S n +1 ( x , ϕ ) ≤ , then S n +1 ( −√ q n +1 x , g √ q k +1 ) < . Therefore for y = √ q n +1 x , we have √ q k +1 ≤ y ≤ √ q k +1 q ≤ ( √ q k +1 ) . Using the statement (5) ofTheorem C, we obtain that q n +1 > c n +1 , which completes the proofof Theorem 1.1. Corollary 2.4. Let f ( x ) = P ∞ k =0 a k x k , a k > , k = 0 , , , . . . , be anentire function such that q ( f ) ≤ q ( f ) ≤ q ( f ) ≤ . . . . If f ∈ L − P ,then q ( f ) > . Proof. As we have proved in the previous theorem, if f ∈ L − P ,then q ( f ) > . In [15] it is proved, that, under the assumptions of theCorollary, if q ( f ) < , then q ( f ) ≤ − q ( f )(2 q ( f ) − 9) + 2( q ( f ) − p q ( f )( q ( f ) − q ( f )(4 − q ( f ))(see [15, Theorem 1.4]).We have mentioned that if f ∈ L − P , then q ( f ) ≥ . If q ( f ) = 3 , then the above inequality states q ( f ) ≤ . This contradiction provesthe Corollary. ✷ Proof of Theorem 1.2 As in the proof of Theorem 1.1 we assume that a = a = 1 , andwe consider the function ϕ ( x ) = f ( − x ) = 1 − x + P ∞ k =2 ( − k x k q k − q k − ...q k − q k instead of f. We need the following Lemma. Lemma 3.1. Let ϕ ( x ) = 1 − x + P ∞ k =2 ( − k x k q k − q k − ...q k − q k be an entirefunction. Suppose that q ≤ q ≤ q ≤ . . . . If there exist j = 2 , , , . . . and m ∈ N , such that q j ≥ c m , then for all j ≥ j + 2 m − there exists x j ∈ ( q q · · · q j , q q · · · q j q j +1 ) such that the following inequalityholds ( − j ϕ ( x j ) ≥ . The proof of this Lemma is similar to the one of [13, Lemma 2.1] Proof. Choose an arbitrary j ≥ j + 2 m − j. For every x ∈ ( q q · · · q j , q q · · · q j q j +1 ) we have1 < x < x q < x q q < . . . < x j q j − q j − . . . q j − q j and x j q j − q j − . . . q j − q j > x j +1 q j q j − . . . q j − q j q j +1 > x j +2 q j +12 q j . . . q j − q j q j +1 q j +2 > . . . We observe that( − j ϕ ( x ) = j − m X k =0 ( − k + j x k q k − q k − . . . q k − q k + j +1 X k = j − m +1 ( − k + j x k q k − q k − . . . q k − q k + ∞ X k = j +2 ( − k + j x k q k − q k − . . . q k − q k =: Σ ( x ) + h ( x ) + Σ ( x ) . Summands in Σ ( x ) are increasing in modulus and the sign of the last(biggest) summand is positive. So, for all x ∈ ( q q · · · q j , q q · · · q j q j +1 )we have Σ ( x ) > . Summands in Σ ( x ) are decreasing in modulusand the sign of the first (biggest) summand is positive. So, for all x ∈ ( q q · · · q j , q q · · · q j q j +1 ) we have Σ ( x ) > . Thus, we obtain( − j ϕ ( x ) > h ( x ) = P j +1 k = j − m +1 ( − k + j x k q k − q k − ...q k − q k (8) = − x j +1 q j q j − ...q j q j +1 + x j q j − q j − ...q j − q j − x j − q j − q j − ...q j − q j − + . . . + x j − m q j − m q j − m ...q j − m q j − m − x j − m q j − m q j − m − ...q j − m q j − m NTIRE FUNCTIONS OF THE LAGUERRE–P ´OLYA I CLASS 11 (we rewrite the sum from the end to the beginning). After factoringout the term x j +1 q j q j − ...q j q j +1 , we get( − j ϕ ( x ) > h ( x ) = x j +1 q j q j − ...q j q j +1 ( − q q ...q j q j +1 x (9) − ( q q ...q j q j +1 ) x q j +1 + ( q q ...q j q j +1 ) x q j +1 q j − . . . + ( q q ...q j q j +1 ) m − x m − q m − j +1 q m − j ...q j − m q j − m − ( q q ...q j q j +1 ) m x m q m − j +1 q m − j ...q j − m q j − m q j − m (cid:19) . Now we introduce some more notations. Set y := q q ...q j q j +1 x , and weobserve that x ∈ ( q q · · · q j , q q · · · q j q j +1 ) ⇔ y ∈ (1 , q j +1 ) . Further wechange the numeration of the second quotients: s := q j +1 , s := q j , s := q j − , . . . , s m − := q j − m +4 , s m := q j − m +3 . By our assumptions q ≤ q ≤ q ≤ . . . we get s ≥ s ≥ s ≥ . . . ≥ s m . In new notations we have(10) h ( x ) = x j +1 q j q j − . . . q j q j +1 − y − m X k =2 ( − k y k s k − s k − . . . s k − s k ! . We want to prove that there exists a point y j ∈ (1 , q j +1 ) = (1 , s )such that h ( y j ) ≥ . To do this we compare the expression in bracketswith the corresponding partial sum of the partial theta-function. Wehave ψ ( y ) := − y − P m k =2 ( − k y k s k − s k − ...s k − s k = ( − y ) +(11) (cid:16) − y s + y s s (cid:17) + (cid:16) − y s s s + y s s s s (cid:17) + . . . + (cid:18) − y m − s m − s m − ...s m − s m − + y m − s m − s m − ...s m − s m − (cid:19) − y m s m − s m − ...s m − s m − s m . We provide estimations similar to those in the proof of Lemma 2.3.Firstly, under our assumptions, one can see that(12) − y m s m − s m − . . . s m − s m ≥ − y m s m − m s m − m . . . s m s m . We prove that for any fixed k = 1 , , . . . , m − − y k s k − s k − · ... · s k + y k +1 s k s k − · ... · s k s k +1 ≥ (13) − y k s k − m · s k − m · ... · s m + y k +1 s k m · s k − m · ... · s m · s m . Firstly, we consider (13) for k = 1 . Since s ≥ s , we have − y s + y s s ≥ − y s + y s . We observe that ∂∂s (cid:18) − y s + y s (cid:19) = y s − y s > ⇔ y < s . Since y < s , from j ≥ j + 2 m − s ≥ s m = q j − m +3 ≥ q j ≥ c m > , we obtain that the function (cid:16) − y s + y s (cid:17) is increasingin s , whence(14) − y s + y s s ≥ − y s + y s ≥ − y s m + y s m . We apply analogous arguments to prove (13) for every k = 1 , , . . . , m − 1. Let us define the following function: H ( s , s , . . . , s k , s k +1 ) := − y k s k − s k − · . . . · s k + y k +1 s k s k − · . . . · s k s k +1 for s ≥ s ≥ . . . ≥ s k +1 . Obviously, H ( s , s , . . . , s k , s k +1 ) ≥ H ( s , s , . . . , s k , s k ) . We have ∂H ( s , s , . . . , s k , s k ) ∂s k = y k s k − s k − · . . . · s k − s k − y k +1 s k s k − · . . . · s k − s k . Thus, ∂H ( s , s , . . . , s k , s k ) ∂s k > ⇔ y < s s · . . . · s k − s k . Since y < s , we obtain that the function H ( s , s , . . . , s k , s k ) is in-creasing in s k , whence H ( s , s , . . . , s k , s k +1 ) ≥ H ( s , s , . . . , s k , s k ) ≥ H ( s , s , . . . , s k − , s m , s m ) . NTIRE FUNCTIONS OF THE LAGUERRE–P ´OLYA I CLASS 13 Further we have: ∂H ( s , s , . . . , s k − , s m , s m ) ∂s k − = 2 y k s k − s k − · . . . · s k − s m − y k +1 s k s k − · . . . · s k − s m . Hence, ∂H ( s , s , . . . , s k − , s m , s m ) ∂s k − > ⇔ y < s s · . . . · s k − s k − s m . Since y < s , we obtain that the function H ( s , s , . . . , s k − , s m , s m )is increasing in s k − , whence H ( s , s , . . . , s k − , s m , s m ) ≥ H ( s , s , . . . , s k − , s m , s m , s m ) . Applying similar arguments we get the following chain of inequalities. H ( s , s , . . . , s k , s k +1 ) ≥ H ( s , s , . . . , s k − , s m , s m ) ≥ H ( s , s , . . . , s k − , s m , s m , s m ) ≥ . . . ≥ H ( s m , s m , . . . , s m , s m ) . Thus, we have proved (13).We substitute the inequality (12) and (13) into (11) to get the fol-lowing(15) ψ ( y ) ≥ − m X k =0 ( − k y k s k ( k − m = − S m ( −√ s m y, g √ s m ) , where g a is a partial theta function and S n ( x, g a ) := P nj =0 x j a − j isits partial sum. By our assumption ( √ s m ) = s m = q j − m +3 and j ≥ j +2 m − , so s m = q j − m +3 ≥ q j ≥ c m , and we conclude that S m ( x, g s m ) ∈ L − P (see Theorem C). Whence, by Theorem C (4)there exists x ∈ ( − ( √ s m ) , −√ s m ) such that S m ( x , g s m ) ≤ . We put −√ s m y := x , i.e. y := − x √ s m ∈ (1 , s m ) ⊂ (1 , s ) , andwe have S m ( −√ s m y , g √ s m ) ≤ . Substituting the last inequality in (15) we obtain:(16) ψ ( y ) ≥ − S m ( −√ s m y , g √ s m ) ≥ . Substituting (16) into (15), and (15) into (9), we obtain the desiredinequality. It remains to recall that x j := q q ...q j q j +1 y , and, since y ∈ (1 , s ) = (1 , q j +1 ) , we have x j ∈ ( q q . . . q j , q q . . . q j q j +1 ) . (cid:3) Now we apply the following Lemma. Lemma ( [16, Lemma 2.1]). Let f ( z ) = P ∞ k =0 a k z k , a k > , k =0 , , , . . . , be an entire function such that √ ≤ q ( f ) ≤ q ( f ) ≤ q ( f ) ≤ . . . . For an arbitrary integer k ≥ we define ρ k ( f ) := q ( f ) q ( f ) · · · q k ( f ) p q k +1 ( f ) . Then, for all sufficiently large k , the function f has exactly k zeros onthe disk { z : | z | < ρ k ( f ) } counting multiplicities. Let us choose an arbitrary k ≥ 2, being large enough to get a state-ment of the previous Lemma, and k ≥ j + 2 m − . 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