On entire function $e^{p(z)}\int_0^{z}\beta(t)e^{-p(t)}dt$ with applications to Tumura--Clunie equations and complex dynamics

TL;DR

This paper analyzes the growth of a specific entire function involving exponential polynomials, solves related Tumura--Clunie differential equations, and explores applications to complex dynamics and second-order differential equations.

Contribution

It provides new growth estimates for a class of entire functions and characterizes solutions to Tumura--Clunie equations with applications to complex dynamics.

Findings

Growth behavior of $H(z)$ described on the complex plane.

All solutions with few zeros of a specific second-order differential equation identified.

A theorem on first-order linear differential equations related to complex dynamics proved.

Abstract

Let $p(z)$ be a nonconstant polynomial and $\beta(z)$ be a small entire function of $e^{p(z)}$ in the sense of Nevanlinna. We first describe the growth behavior of the entire function $H(z):=e^{p(z)}\int_0^{z}\beta(t)e^{-p(t)}dt$ on the complex plane $\mathbb{C}$. As an application, we solve entire solutions of Tumura--Clunie type differential equation $f(z)^n+P(z,f)=b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}$, where $b_1(z)$ and $b_2(z)$ are nonzero polynomials, $p_1(z)$ and $p_2(z)$ are two polynomials of the same degree~$k\geq 1$ and $P(z,f)$ is a differential polynomial in $f$ of degree $\leq n-1$ with meromorphic functions of order~$<k$ as coefficients. These results allow us to determine all solutions with relatively few zeros of the second-order differential equation $f''-[b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}+b_3(z)]f=0$, where $b_3(z)$ is a polynomial. We also prove a theorem on certain first-order linear differential equation related to complex dynamics.