Regular growth of Fourier coefficients of the logarithmic derivative of entire functions of improved regular growth

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*}

show abstract

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 $L^q[0;2\pi]$

Khats

2023

UMB

View full text Add to dashboard Cite

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 $f$.

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Regular growth of Fourier coefficients of the logarithmic derivative of entire functions of improved regular growth

Cited by 3 publications

References 7 publications

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 Lq[0; 2π]

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 Lq[0; 2π]

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

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 $L^q[0;2\pi]$

Contact Info

Product

Resources

About