On the modulus of solutions of a first order differential equation

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.