A criterion for the improved regular growth of an entire function in terms of the asymptotic behavior of its logarithmic derivative in the metric of $L^q[0;2\pi

Abstract

Let $f$ be an entire function, $f(0)=1$, $F(z)=zf^{\prime }(z)/f(z)$, and $\Gamma_m=\bigcup\limits_{j=1}^ m \{z: \arg z=\psi_{j}\}$, $0\le\psi_1<\psi_2<\ldots<\psi_m<2\pi$. An entire function $f$ is called a function of improved regular growth if for some $\rho\in (0;+\infty)$ and $\rho_2\in (0;\rho)$, 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 such that \begin{equation*} \log |{f(z)}|=|z|^\rho h(\varphi)+o(|z|^{\rho_2}),\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 $\Gamma_m$ is a function of improved regular growth if and only if for some $\rho_2 \in (0;\rho)$ and every $q\in [1;+\infty)$, one has \begin{equation*} \left\{\frac{1}{2\pi}\int_0^{2\pi}\left|\frac{F(re^{i\varphi})}{r^\rho}-\rho% \widetilde {h}(\varphi)\right|^q\, d\varphi\right\}^{1/q}=o(r^{\rho_2-\rho}),\quad r\to +\infty, \end{equation*} where $\widetilde{h}(\varphi)=h(\varphi)-i{h^{\prime }(\varphi)}/{\rho% }$ and $h(\varphi)$ is the indicator of the function $f$.