Zeros of derivatives of strictly nonreal meromorphic functions

J K Langley

doi:10.1215/00192082-8513350

Abstract

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

Highlights

Let f be a meromorphic function in the plane and let f (z) = f (z)
The first aim of the present paper is to prove a result in the spirit of Theorem 1.1, but with no assumption on the location of poles
In either case the set {α∗, |am1|, |am2|, |am3|} has a least positive member, which will be denoted by β

Summary

Introduction

Theorem 1.4 Let f be a strictly non-real meromorphic function in the plane, such that all but finitely many zeros and poles of f and f ′′ are real. Lemma 2.2 ([22, 24]) Let f be a non-constant meromorphic function in the plane which satisfies at least one of the following two conditions: (a) f and f ′′ have finitely many non-real zeros and poles; (b) f and f (m) have finitely many non-real zeros, for some m ≥ 3. There exists a non-constant meromorphic function H with finitely many zeros and poles such that, using (12), g = Hf, H. Lemma 3.6 Assume that either f ′/f is a rational function or f satisfies (1), and that f , f ′, f ′′ and f ′′′ have only real zeros.

Hence it suffices to take

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