Rapidly growing entire functions with three singular values

TL;DR

This paper solves a longstanding problem by constructing entire functions with three singular values that can dominate any given growth function, advancing the understanding of complex function behavior.

Contribution

It introduces a method to explicitly construct entire functions with three singular values matching any prescribed growth, filling a gap in complex analysis.

Findings

Constructed entire functions with three singular values dominating arbitrary functions.

Established a method for controlling the growth of such functions.

Resolved a major open problem in the theory of entire functions.

Abstract

We settle the problem of finding an entire function with three singular values whose Nevanlinna characteristic dominates an arbitrarily prescribed function.