Zeros of Entire Functions

Abstract

Introduction.This paper is the outgrowth of the author's Ph.D. thesis written under the supervision of Professor A. Peyerimhoff, whose paper on zeros of power series [4] has been the starting point.1 contains some lemmas due to A. Peyerimhoff, which are needed in 2.In all theorems (with the exception of Theorem 4 only) of the remaining part of this paper, real entire functions flz) of finite order with infinitely many real and finitely many complex^) zeros are taken as starting point.Then in 2 functions of the form g(z) = A(z)f(z) + B(z)f'(z) are considered where A(z) and B(z) are real polynomials.In Theorem 1 upper bounds for the number of zeros of g(z) in the complex plane with the exception of certain real intervals are obtained.In certain instances the exact number of zeros is obtained.At the end of 2 examples are given which show that the results of Theorem 1 are best possible.Questions of this kind have been considered by Laguerre and Borel especially [1],[2] for the particular case g(z) =f'(z).One of the results of [4] deals with functions g(z) = af(z) + zf'(z) where a is real and/(z) of order < 1.The method of proof of this result has been generalized in the present paper.In 3 a couple of theorems (2, 3, 5, 6) which are derived from Theorem 1 are partly slight generalizations of known theorems, especially of Laguerre [2], [3],[5], [6].These theorems deal with questions of the following kind.If one has some information on the zeros of the entire function flz) = 2 anzn, one wants to gain information on the zeros of F(z) = 2 anG(n)zn, where G(z) is an entire function of a certain type.For Theorem 4, which is known already, a short proof is given here which is independent of Theorem 1. From this theorem follows immediately the wellknown fact that the Besselfunctions of real order > -1 have real zeros only.From Theorem 6 Hurwitz's Theorem on the complex zeros of the Besselfunctions of real order < -1 follows as a special case.In Theorem 9 functions g(z) = af(z) + zlf'(z) are considered, where a is real, / odd, and flz) is a real canonical product of finite genus.Then the exact number of complex zeros of g(z) is obtained.Especially we find that for a>0 and / odd the functions a sin z-l-z' cos z and a cos z-z' sin z have exactly I-1 complex zeros.1.In the proof of Lemma 1 we will need some elementary properties of difference quotients.In the following we suppose that flz) is a holomorphic function.