Non-real zeros of derivatives of meromorphic functions

Abstract

A number of results are proved concerning non-real zeros of derivatives of real and strictly non-real meromorphic functions in the plane. MSC 2000: 30D35.

Non-real zeros of derivatives

“…Furthermore, Theorem 1.2 and [24] together show that there are no functions f satisfying the hypotheses of Conjecture 1.1 such that either of the following holds: f has infinite order and the zeros or poles of f have finite exponent of convergence; f has finite order and infinitely many poles but finitely many zeros. The conjecture was also proved in [28,Theorem 1.4] for real transcendental meromorphic functions in the plane which map the open upper half-plane H into itself. All zeros and poles of such functions are automatically real and simple and interlaced [29]: that is, between any two consecutive poles of f there is a zero, and between consecutive zeros of f lies a pole (this follows from a consideration of residues for f and 1/f ).…”

“…Theorem 1.7 will be deduced from [28,Theorem 1.4] and the following result involving real meromorphic functions with real zeros and poles such that, with finitely many exceptions, all poles are simple and adjacent poles are separated by at least one zero.…”

“…If X is neither bounded above nor bounded below, using a translation makes it possible to assume that the poles x k and zeros y k satisfy x k < y k < x k+1 and x k /y k > 0 for each k. Hence f = ψe h where ψ is defined as in (9) and maps H into itself, while h is an entire function. If h is constant then (6) follows from [28,Theorem 1.4], as does (5) if f ′′ has only real zeros. Furthermore, if h is non-constant then a contradiction arises via part (ii) or (iii) of Theorem 1.8.…”

Necessary and Sufficient Condition for zeros of Derivative of Meromorphic and Entire Functions

“…Many theorems about the zeros of derivatives of meromorphic functions, since the zeros of derivatives of meromorphic functions play an important role in the study of value distributions of meromorphic functions. The recent research about derivatives of meromorphic functions can be found from [8,9,10,11,12].…”