All admissible meromorphic solutions of Hayman's equation

TL;DR

This paper classifies all meromorphic solutions of Hayman's differential equation with rational coefficients, proving they have finite order and identifying conditions for non-rational solutions using Nevanlinna theory.

Contribution

It provides a complete characterization of all admissible meromorphic solutions to Hayman's equation, including non-rational solutions, extending previous results.

Findings

All solutions have finite order.

Existence of one-parameter families of non-rational solutions.

Solutions are classified based on coefficient functions.

Abstract

We find all non-rational meromorphic solutions of the equation $ww"-(w')^2=\alpha(z)w+\beta(z)w'+\gamma(z)$, where $\alpha$, $\beta$ and $\gamma$ are rational functions of $z$. In so doing we answer a question of Hayman by showing that all such solutions have finite order. Apart from special choices of the coefficient functions, the general solution is not meromorphic and contains movable branch points. For some choices for the coefficient functions the equation admits a one-parameter family of non-rational meromorphic solutions. Nevanlinna theory is used to show that all such solutions have been found and allows us to avoid issues that can arise from the fact that resonances can occur at arbitrarily high orders. We actually solve the more general problem of finding all meromorphic solutions that are admissible in the sense of Nevanlinna theory, where the coefficients $\alpha$, $\beta$ and $\gamma$ are meromorphic functions.