关于整函数$e^{p(z)}\int_0^{z}β(t)e^{-p(t)}dt$及其在Tumura-Clunie方程和复动力学中的应用
On entire function $e^{p(z)}\int_0^{z}β(t)e^{-p(t)}dt$ with applications to Tumura--Clunie equations and complex dynamics
摘要：
设$p(z)$是非常数多项式，$β(z)$是$e^{p(z)}$在Nevanlinna意义下的小整函数.我们首先描述整个函数$H(z):=e^{p(z)}\int_0^{z}β(t)e^{-p(t)}dt$在复平面$\mathbb{C}$上的增长行为。作为应用，我们求出了Tumura-Clunie型微分方程$f(z)^n+P(z,f)=b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}$的整体解，其中$b_1(z)$和$b_2(z)$是非零多项式，$p_1(z)$和$p_2(z)$是两个同次多项式$k\geq 1$，$P(z,f)$是$f$中的微分多项式，其次为$\leq n-1$，系数为$<k$级亚纯函数.这些结果使我们能够确定二阶微分方程$f''-[b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}+b_3(z)]f=0$的所有具有相对较少零点的解，其中$b_3(z)$是一个多项式。我们还证明了与复动力学有关的一阶线性微分方程的一个定理。
英文摘要：
Let $p(z)$ be a nonconstant polynomial and $β(z)$ be a small entire function of $e^{p(z)}$ in the sense of Nevanlinna. We first describe the growth behavior of the entire function $H(z):=e^{p(z)}\int_0^{z}β(t)e^{-p(t)}dt$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
