ASYMPTOTIC BEHAVIOR OF THE LOGARITHMIC DERIVATIVE OF ENTIRE FUNCTION OF IMPROVED REGULAR GROWTH IN THE METRIC OF $L^q[0,2\pi]$

Abstract

Let $f$ be an entire function with $f(0)=1$, $(\lambda_n)_{n\in\mathbb N}$ be the sequence of its zeros, $n(t)=\sum_{|\lambda_n|\le t}1$, $N(r)=\int_0^r t^{-1}n(t)\, dt$, $r>0$, $h(\varphi)$ be the indicator of $f$, and $F(z)=zf'(z)/f(z)$, $z=re^{i\varphi}$. An entire function $f$ is called a function of improved regular growth if for some $\rho\in (0,+\infty)$ and $\rho_1\in (0,\rho)$, and a $2\pi$-periodic $\rho$-trigonometrically convex function $h(\varphi)\not\equiv -\infty$ there exists a set $U\subset\mathbb C$ contained in the union of disks with finite sum of radii and such that \begin{equation*} \log |{f(z)}|=|z|^\rho h(\varphi)+o(|z|^{\rho_1}),\quad U\not\ni z=re^{i\varphi}\to\infty. \end{equation*} In this paper, we prove that an entire function $f$ of order $\rho\in (0,+\infty)$ with zeros on a finite system of rays $\{z: \arg z=\psi_{j}\}$, $j\in\{1,\ldots,m\}$, $0\le\psi_1<\psi_2<\ldots<\psi_m<2\pi$, is a function of improved regular growth if and only if for some $\rho_3\in (0,\rho)$ \begin{equation*} N(r)=c_0r^\rho+o(r^{\rho_3}),\quad r\to +\infty,\quad c_0\in [0,+\infty), \end{equation*} and for some $\rho_2\in (0,\rho)$ and any $q\in [1,+\infty)$, one has \begin{equation*} \left\{\frac{1}{2\pi}\int_0^{2\pi}\left|\frac{\Im F(re^{i\varphi})}{r^\rho}+h'(\varphi)\right|^q\, d\varphi\right\}^{1/q}=o(r^{\rho_2-\rho}),\quad r\to +\infty. \end{equation*}