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

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. By using the classical Phragm\'{e}n--Lindel\"{o}f theorem, we analyze 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 $\mathbb{C}$. We then apply these results to 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~$k\geq 1$ and $P(z,f)$ is a differential polynomial in $f$ of degree $\leq n-1$ with meromorphic functions of order less than~$k$ as coefficients, and precisely characterize entire solutions of this equation. This gives an answer to a problem in the literature and allows to find all zero-free solutions of the second-order differential equation $f''-(b_1e^{p_1}+b_2e^{p_2}+b_3)f=0$, where $b_3$ is a polynomial. We also use the Phragm\'{e}n--Lindel\"{o}f theorem to prove a theorem on certain first-order non-homogeneous linear differential equation related to complex dynamics.