scholarly journals Convergence in $L^p[0,2\pi]$-metric of logarithmic derivative and angular $\upsilon$-density for zeros of entire function of slowly growth

2015 ◽  
Vol 7 (2) ◽  
pp. 209-214
Author(s):  
M.R. Mostova ◽  
M.V. Zabolotskyj
Keyword(s):  
Entire Function ◽  
Logarithmic Derivative ◽  
Zero Order

The subclass of a zero order entire function $f$ is pointed out for which the existence of angular $\upsilon$-density for zeros of entire function of zero order is equivalent to convergence in $L^p[0,2\pi]$-metric of its  logarithmic derivative.

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

2021 ◽  
Vol 9 (1) ◽  
pp. 49-55
Author(s):  
R. Khats’
Keyword(s):  
Asymptotic Behavior ◽  
Entire Function ◽  
Convex Function ◽  
Logarithmic Derivative ◽  
Finite System ◽  
Regular Growth ◽  
Finite Sum

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*}

On a conjecture of Fuchs

2001 ◽  
Vol 131 (5) ◽  
pp. 1209-1216 ◽  
Author(s):  
Joseph Miles ◽  
John Rossi
Keyword(s):  
Entire Function ◽  
Logarithmic Derivative ◽  
Image Position ◽  
Nevanlinna Deficiency

If f is an entire function of order ρ, 0 < ρ < 2−11, it is shown that the Nevanlinna deficiency d(0, f′/f) of the logarithmic derivative of f satisfies For small positive ρ, this result strengthens an earlier estimate of Eremenko et al. concerning a conjecture of Fuchs.

scholarly journals Asymptotics of the entire functions with $\upsilon$-density of zeros along the logarithmic spirals

2019 ◽  
Vol 11 (1) ◽  
pp. 26-32
Author(s):  
M.V. Zabolotskyj ◽  
Yu.V. Basiuk
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Logarithmic Spiral ◽  
Growth Function ◽  
Zero Order ◽  
Logarithmic Spirals

Let $\upsilon$ be the growth function such that $r\upsilon'(r)/\upsilon (r) \to 0$ as $r \to +\infty$, $l_\varphi^c = \{z=te^{i(\varphi+c \ln t)}, 1 \leqslant t < +\infty\}$ be the logarithmic spiral, $f$ be the entire function of zero order. The asymptotics of $\ln f(re^{i(\theta +c \ln r)})$ along ordinary logarithmic spirals $l_\theta^c$ of the function $f$ with $\upsilon$-density of zeros along $l_\varphi^c$ outside the $C_0$-set is found. The inverse statement is true just in case zeros of $f$ are placed on the finite logarithmic spirals system $\Gamma_m = \bigcup_{j=0}^m l_{\theta_j}^c$.

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