A note on the Wiman-Valiron inequality

TL;DR

This paper revisits the Wiman-Valiron inequality, providing a new, more general version that broadens understanding of the relationship between an analytic function's maximum modulus and its Taylor coefficients, and discusses open problems.

Contribution

It introduces a novel, more general form of the Wiman-Valiron inequality that had not been previously documented in the literature.

Findings

Derived a new general version of the Wiman-Valiron inequality

Summarized existing results on the inequality

Presented an open problem for future research

Abstract

The Wiman-Valiron inequality relates the maximum modulus of an analytic function to its Taylor coefficients via the maximum term. After a short overview of the known results, we obtain a general version of this inequality that seems to have been overlooked in the literature so far. We end the paper with an open problem.