On the modulus of meromorphic solutions of a first order differential equation (2407.00580v4)

Abstract: Let $P(z)=z^{n}+a_{n-2}z^{n-2}+\cdots+a_0$ be a polynomial of degree $n\geq 1$ and $S(z)$ be a nonzero rational function. Let $\theta\in(0,\pi/2n)$ be a constant. It is shown that if $f(z)$ is a meromorphic solution of the first order differential equation $f'(z)=S(z)e^{P(z)}f(z)+1$, then, for any small constant $\varepsilon>0$, there is a union $\mathbf{\Gamma}=\cup_{k=1}^{\infty}\Gamma_k$ of parts $\Gamma_k$ of the ray $\Gamma:z=re^{i\theta}$, $r\in[r_0,\infty)$ such that for all $z\in \mathbf{\Gamma}$, \begin{equation}\tag{\dag} |f(z)|\geq (1-\varepsilon)\left(\frac{\sqrt[n]{\sin n\theta}}{n\cos\theta}\right)x\exp\left(e^{(1-\varepsilon)\frac{\cos n\theta}{\cos^n\theta}x^n}\sin\varepsilon\right). \end{equation} In particular, the estimate in $(\dag)$ together with the Wiman--Valiron theory implies that the hyper-order $\varsigma(f)$ of $f(z)$ is equal to $n$, which provides partial answers to Br\"{u}ck's conjecture in uniqueness theory of meromorphic functions and also a problem on a second order algebraic differential equation of Hayman. We also give a lower bound for $|f(z)|$ on other parts of the ray $\Gamma$.