Value Distribution of the &amp;lt;i&amp;gt;k&amp;lt;/i&amp;gt;th Derivatives of Meromorphic Functions

Pai Yang,Xiaojun Liu

doi:10.4236/apm.2014.41002

Abstract

In the paper, we take up a new method to prove a result of value distribution of meromorphic functions: let f be a meromorphic function in , and let , where P is a polynomial. Suppose that all zeros of f have multiplicity at least , except possibly finite many, and as . Then has infinitely many zeros.

Highlights

The value distribution theory of meromorphic functions occupies one of the central places in Complex Analysis which has been applied to complex dynanics, complex differential and functional equations, Diophantine equations and others
We study the value distribution of transcendental meromorphic functions, all but finitely many of whose zeros have multiplicity at least k +1, where k is a positive integer
( ) ( ) ( ) ( ) S ∆ an ,δ * , g = S ∆ 0,δ * , gn ≤ M2 which contradicts (1.3)

Summary

Introduction

The value distribution theory of meromorphic functions occupies one of the central places in Complex Analysis which has been applied to complex dynanics, complex differential and functional equations, Diophantine equations and others. He proves that if f(z) is a transcendental meromorphic function in the plane, either f(z) assumes every finite value infinitely often, or every derivative of f(z) assumes every finite nonzero value infinitely often. We study the value distribution of transcendental meromorphic functions, all but finitely many of whose zeros have multiplicity at least k +1, where k is a positive integer. Theorem B Let k ≥ 2 be an integer, let f ( z) be a meromorphic function of finite order ρ ( f ) in , and let= a ( z) P ( z) ez ≡/ 0 , where P is a polynomial. Suppose that all zeros of f have multiplicity at least k +1, except possibly finitely many.

Notation and Some Lemmas

Proof of Theorem