On the modulus of solutions of a first order differential equation

TL;DR

This paper investigates the growth behavior of solutions to a specific first order differential equation involving exponential polynomials, establishing lower bounds and connecting the results to the hyper-order and Brück's conjecture in meromorphic function theory.

Contribution

It provides new lower bounds for solutions of a class of differential equations and relates these bounds to the hyper-order, partially confirming Brück's conjecture.

Findings

Solutions have infinite logarithmic measure sets where their magnitude is significantly large.

The hyper-order of solutions is exactly equal to the degree of the polynomial in the exponential.

Extended methods describe growth orders of solutions to certain second order algebraic differential equations.

Abstract

Let $P(z)=z^{n}+a_{n-2}z^{n-2}+\cdots+a_0$ be a nonconstant polynomial and $S(z)$ be a nonzero rational function and denote $h(z)=S(z)e^{P(z)}$. Let $\theta\in(0,\pi/2n)$ be a constant and $\varepsilon>0$ be a small constant. It is shown that if $f(z)$ is a solution of the first order differential equation $f'(z)=h(z)f(z)+1$, then there is a sequence $\{r_{k}\}$ such that the set $E=\cup_{l=0}^{\infty}[r_{2l},r_{2l+1}]$ has infinite logarithmic measure and for all $r\in E$, \begin{equation}\tag{\dag} \begin{split} |f(re^{i\theta})|\geq (1-\varepsilon)\frac{\sqrt[n]{\sin n\theta}}{n}r\exp\left(e^{(1-\varepsilon)r^n\cos n\theta}\sin\varepsilon\right). \end{split} \end{equation} When $h(z)=e^{z}$, we also give a lower bound for $|f(re^{i\theta})|$ for other values of $r$. The estimate in $(\dag)$ yields that the hyper-order $\varsigma(f)$ of $f(z)$ is equal to $n$, giving a partial answer to Br\"{u}ck's conjecture in uniqueness theory of meromorphic functions. An extension of the method also yields a complete description on the order of growth of entire solutions of a second order algebraic differential equation of Hayman in the autonomous case.