On the modulus of meromorphic solutions of a first order differential equation

TL;DR

The paper analyzes meromorphic solutions of the differential equation $f'(z)=h(z)f(z)+1$ with $h(z)=S(z)e^{P(z)}$, where $P$ is a nonconstant polynomial of degree $n\ge 1$ and $S$ is rational. By employing Nevanlinna theory and Bank–Langley-type asymptotics for $G(z)$, the authors construct a carefully chosen union of rays $\Psi$ along which a sharp lower bound for $|f(z)|$ can be established, culminating in a lower bound that grows like $|f(z)| \ge (1-\varepsilon)\left( \frac{(\sin n\theta)^{1/n}}{n\cos\theta}\right)x e^{e^{(1-\varepsilon)\frac{\cos n\theta}{\cos^n\theta}x^n}\sin\varepsilon}$ on $\Psi$. In the simplest case $h(z)=e^{z}$, an additional lower bound on the rest of the ray is obtained, and the obtained growth along the rays, together with Wiman–Valiron theory, yields that the hyper-order of $f$ equals $n$. These results provide partial confirmations of Brück’s conjecture in uniqueness theory and relate to Hayman’s second-order differential equations by clarifying the growth behavior of transcendental meromorphic solutions. The paper also outlines several technical steps, including the definition of integration paths, estimates for multiple integrals along these paths, and a detailed asymptotic analysis of $G(z)$ and $U(z)$.

Abstract

Let $h(z)=S(z)e^{P(z)}$, where $P(z)=z^{n}+a_{n-2}z^{n-2}+\cdots+a_0$ is a polynomial of degree $n\geq 1$ and $S(z)$ is a nonzero rational function. Let $θ\in(0,π/2n)$ be a constant. It is shown that if $f(z)$ is a meromorphic solution of the first order differential equation $f'(z)=h(z)f(z)+1$, then, for any small constant $\varepsilon>0$, there is a union $Ψ=\cup_{k=1}^{\infty}Φ_k$, where $Φ_k=Φ([\bar{x}_{k-1},\bar{x}_k])$ along the ray $Φ:[x_0,\infty)\to \mathbb{C}$, $Φ(x)=x+iy$ and $y=x\tanθ$, such that for all $z\in Ψ$, \begin{equation}\tag† |f(z)|\geq (1-\varepsilon)\left(\frac{\sqrt[n]{\sin nθ}}{n\cosθ}\right)x\exp\left(e^{(1-\varepsilon)\frac{\cos nθ}{\cos^nθ}x^n}\sin\varepsilon\right). \end{equation} In the simplest case $h(z)=e^{z}$, we also give a lower bound for $|f(z)|$ on the rest of the ray $Φ$. The estimate in $(†)$ together with the Wiman--Valiron theory implies that the hyper-order $ς(f)$ of $f(z)$ is equal to $n$, providing partial answers to Brück's conjecture in uniqueness theory of meromorphic functions and also a problem on a second order algebraic differential equation of Hayman.

Theorems & Definitions (7)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• proof
• Theorem 3.1
• proof : Proof of Theorem \ref{['maintheorem2']}
• Theorem 4.1