Sufficient conditions for the improved regular growth of entire functions in terms of their averaging

Abstract

Let $f$ be an entire function of order $\rho\in (0,+\infty)$ with zeros on a finite system of rays $\{z: \arg z=\psi_{j}\}$, $j\in\{1,\ldots,m\}$, $0\le\psi_1<\psi_2<\ldots<\psi_m<2\pi$ and $h(\varphi)$ be its indicator. In 2011, the author of the article has been proved that if $f$ is of improved regular growth (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 $\log |{f(z)}|=|z|^\rho h(\varphi)+o(|z|^{\rho_1})$, $U\not\ni z=re^{i\varphi}\to\infty$), then for some $\rho_3\in (0,\rho)$ the relation \begin{equation*} \int_1^r {\frac{\log |{f(te^{i\varphi})}|}{t}}\, dt=\frac{r^\rho}{\rho}h(\varphi)+o(r^{\rho_3}),\quad r\to +\infty, \end{equation*} holds uniformly in $\varphi\in [0,2\pi]$. In the present paper, using the Fourier coefficients method, we establish the converse statement, that is, if for some $\rho_3\in (0,\rho)$ the last asymptotic relation holds uniformly in $\varphi\in [0,2\pi]$, then $f$ is a function of improved regular growth. It complements similar results on functions of completely regular growth due to B. Levin, A. Grishin, A. Kondratyuk, Ya. Vasyl'kiv and Yu. Lapenko.