Non-real zeros of higher derivatives of real entire functions of infinite order

Letf be a real meromorphic function of infinite order in the plane such thatf has finitely many poles. Then for eachk≥3, at least one off andf ^(k) has infinitely many non-real zeros. Together with a result of Edwards and Hellerstein, this establishes the analogue for higher derivatives of a conjecture going back to Wiman around 1911.     

Non-real zeros of derivatives of meromorphic functions

A number of results are proved concerning non-real zeros of derivatives of real and strictly non-real meromorphic functions in the plane.     

Polynomial systems with few real zeroes

We study some systems of polynomials whose support lies in the convex hull of a circuit, giving a sharp upper bound for their numbers of real solutions. This upper bound is non-trivial in that it is smaller than either the Kouchnirenko or the Khovanskii bounds for these systems. When the support is exactly a circuit whose affine span is ℤⁿ, this bound is 2n+1, while the Khovanskii bound is exponential in n². The bound 2n+1 can be attained only for non-degenerate circuits. Our methods involve a mixture of combinatorics, geometry, and arithmetic. Part of work done at MSRI was supported by NSF grant DMS-9810361. Work of Sottile is supported by the Clay Mathematical Institute. Sottile and Bihan were supported in part by NSF CAREER grant DMS-0134860. Bertrand is supported by the European research network IHP-RAAG contract HPRN-CT-2001-00271.     

On the real zeroes of the Hurwitz zeta-function and Bernoulli polynomials

The behaviour of real zeroes of the Hurwitz zeta-function

     

Sums of entire functions having only real zeros

We show that certain sums of products of Hermite-Biehler entire functions have only real zeros, extending results of Cardon. As applications of this theorem, we construct sums of exponential functions having only real zeros, we construct polynomials having zeros only on the unit circle, and we obtain the three-term recurrence relation for an arbitrary family of real orthogonal polynomials. We discuss a similarity of this result with the Lee-Yang Circle Theorem from statistical mechanics. Also, we state several open problems.

     

A property of entire functions with real taylor coefficients

Suppose thatf(z) is an entire transcendental function with real Taylor coefficients, M(r)= max¦f(z)¦on¦z¦=r, and {_n} is the sequence of sign changes of the coefficients. We will show that if (1/_n)<, then .Translated from Matematicheskie Zametki, Vol. 18, No. 3, pp. 395–402, September, 1975.The author thanks A. A. Gol'dberg for his useful comments.     

Non-real zeros of linear differential polynomials

Let f be a real entire function with finitely many non-real zeros, not of the form f = Ph with P a polynomial and h in the Laguerre-Pólya class. Lower bounds are given for the number of non-real zeros of f″ + ω f, where ω is a positive real constant.     

On universal entire functions with zero-free derivatives

We prove in this note a generalization of a theorem due to G. Herzog on zero-free universal entire functions. Specifically, it is shown that, if a nonnegative integer q and a nonconstant entire function φ of subexponential type are given, then there is a residual set in the class of entire functions with zero-free derivatives of orders q and q + 1, such that every member of that set is universal with respect to φ (D), where D is the differentiation operator. This work is supported in part by DGICYT grant PB93-0926.     

Differential operators and entire functions with simple real zeros

Let ? and f be functions in the Laguerre-Pólya class. Write ?(z)=e−αz²?₁(z) and f(z)=e−βz²f₁(z), where ?₁ and f₁ have genus 0 or 1 and α,β?0. If αβ<1/4 and ? has infinitely many zeros, then ?(D)f(z) has only simple real zeros, where D denotes differentiation.     

Uniqueness of entire functions sharing an entire function of smaller order with their derivatives

In this paper, we prove some uniqueness theorems of entire functions and their derivatives that share an entire of function whose order is less than their order. The results in this paper improve those given by G. G. Gundersen and L. Z. Yang, J. P. Wang, J. M. Chang, and Y. Z. Zhu, X. M. Li and H. X. Yi, and others. Some examples are provided to show that the results in this paper are the best possible.     

The uniqueness of an entire function sharing a small entire function with its derivatives

In this paper, we prove a theorem on the growth of a solution of a linear differential equation. From this we obtain some uniqueness theorems concerning that a nonconstant entire function and its derivatives sharing a small entire function. The results in this paper improve many known results. Some examples are provided to show that the results in this paper are the best possible.     

Migration of zeros for successive derivatives of entire functions

It is shown that if is an entire function of order less than one, all of whose zeros are real, then the minimal root of is an increasing function of which accelerates as increases.

     

On uniqueness of an entire function and its derivatives

In this paper, we study the growth of solutions of a first order linear differential equation. From this we verify that a conjecture given by Brück is true under the restriction of the hyper order less than 1/2, and obtain some uniqueness theorems of a nonconstant entire function and its first derivative sharing a nonzero constant CM. The results in this paper also improve some known results. Some examples show that the results in this paper are best possible. Project supported by the NSFC (NO. A0324617) and the RFDP (NO. 20060422049). Received: 20 April 2006 Revised: 4 February 2007