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- W4380591746 abstract "Let $f$ be an entire function, $f(0)=1$, $F(z)=zf^{prime }(z)/f(z)$, and $Gamma_m=bigcuplimits_{j=1}^ m {z: arg z=psi_{j}}$, $0lepsi_1<psi_2<ldots<psi_m<2pi$. An entire function $f$ is called a function of improved regular growth if for some $rhoin (0;+infty)$ and $rho_2in (0;rho)$, and a $2pi$-periodic $rho$-trigonometrically convex function $h(varphi)notequiv -infty$, there exists a set $Usubsetmathbb C$ 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 Unotni z=re^{ivarphi}toinfty. end{equation*} In this paper, we prove that an entire function $f$ of order $rhoin (0;+infty)$ 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 $qin [1;+infty)$, one has begin{equation*} left{frac{1}{2pi}int_0^{2pi}left|frac{F(re^{ivarphi})}{r^rho}-rho% widetilde {h}(varphi)right|^q, dvarphiright}^{1/q}=o(r^{rho_2-rho}),quad rto +infty, end{equation*} where $widetilde{h}(varphi)=h(varphi)-i{h^{prime }(varphi)}/{rho% }$ and $h(varphi)$ is the indicator of the function $f$." @default.
- W4380591746 created "2023-06-15" @default.
- W4380591746 creator A5046822603 @default.
- W4380591746 date "2023-04-28" @default.
- W4380591746 modified "2023-09-23" @default.
- W4380591746 title "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 $L^q[0;2pi]$" @default.
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- W4380591746 doi "https://doi.org/10.37069/1810-3200-2023-20-1-3" @default.
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