On Entire Function $$e^{p(z)}\int _0^{z}\beta (t)e^{-p(t)}dt$$ with Applications to Tumura–Clunie Equations and Complex Dynamics

Abstract

Let p(z) be a non-constant polynomial and \(\beta (z)\) be a small entire function of \(e^{p(z)}\) in the sense of Nevanlinna. We describe the growth behavior of the entire function \(H(z):=e^{p(z)}\int _0^{z}\beta (t)e^{-p(t)}dt\) in the complex plane \(\mathbb {C}\). As an application, we find entire solutions of the 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 non-zero polynomials, \(p_1(z)\) and \(p_2(z)\) are two polynomials of the same degree \(k\ge 1\) and P(z, f) is a differential polynomial in f of degree at most \(n-1\) with meromorphic functions of order less than 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 equations related to complex dynamics.