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 $\epsilon >0$ be a small constant. It is shown that if $f(z)$ is a solution of the first order differential equation $f^{\prime }(z)=h(z)f(z)+1$, 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† \begin{split} |f(re^{iθ})|\geq (1-\varepsilon)\frac{\sqrt[n]{\sin nθ}}{n}r\exp\left(e^{(1-\varepsilon)r^n\cos nθ}\sin\varepsilon\right). \end{split} \end{equation} When $h(z)=e^z$, we also give a lower bound for $|f(r{e}^{i\theta })|$ for other values of $r$. The estimate in $(†)$ yields that the hyper-order $\varsigma (f)$ of $f(z)$ is equal to $n$, giving a partial answer to Brück'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.

arXiv comment30 pages