Derivatives of Meromorphic functions with multiple zeros and elliptic functions

TL;DR

This paper investigates the solutions of the equation f' = h where f is a meromorphic function with mostly multiple zeros and h is an elliptic function, showing that under certain growth conditions, solutions are infinite.

Contribution

It establishes a new result linking the zeros of meromorphic functions with elliptic functions, extending previous work on derivatives of meromorphic functions.

Findings

f' = h has infinitely many solutions under specified conditions

Most zeros of f are multiple except finitely many

Growth condition T(r,h) = o(T(r,f)) as r → ∞

Abstract

Let f be a nonconstant meromorphic function in the plane and h be a nonconstant elliptic function. We show that if all zeros of f are multiple exept finitely many and T(r,h)=o{T(r,f)} as r tends to infinity, then f'=h has infinitely many solutions (including poles).