Асимптотичне Поводження Логарифмічної Похідної Цілих Функцій Нульового Порядку

Zabolotskyj, M. V.; Mostova, M. R.

doi:10.15330/cmp.6.2.237-241

Cited by 2 publications

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References 2 publications

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“…where ln(1 − z an ) is a single-valued branch in (C \ l γ −π ) of the multivalued function Ln(1 − z an ) such that ln(1 − z an )| z=0 = 0. For z = re i(φ+β(r)) , −π < φ < π, we have (see (10), (11))…”

Section: Proof Of Main Resultsmentioning

confidence: 99%

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Binomial asymptotics for the logarithmic derivative of zero-order entire functions with zeros along curves of regular rotation

Zabolots’kyi¹,

Basiuk²,

Mostova³

2019

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Section: Proof Of Main Resultsmentioning

confidence: 99%

“…Criteria of s.r.g. for functions f ∈ H 0 in terms of quantities related to their logarithmic derivatives are found in [10,11].…”

mentioning

confidence: 99%

Binomial asymptotics for the logarithmic derivative of zero-order entire functions with zeros along curves of regular rotation

Zabolots’kyi¹,

Basiuk²,

Mostova³

2019

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Logarithmic Derivative of the Blaschke Product With Slowly Increasing Counting Function of Zeros

Zabolots’kyi

Gal

Mostova

2021

BMJ

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The Blaschke products form an important subclass of analytic functions on the unit disc with bounded Nevanlinna characteristic and also are meromorphic functions on $\mathbb{C}$ except for the accumulation points of zeros $B(z)$. Asymptotics and estimates of the logarithmic derivative of meromorphic functions play an important role in various fields of mathematics. In particular, such problems in Nevanlinna's theory of value distribution were studied by Goldberg A.A., Korenkov N.E., Hayman W.K., Miles J. and in the analytic theory of differential equations -- by Chyzhykov I.E., Strelitz Sh.I. Let $z_0=1$ be the only boundary point of zeros $(a_n)$ %=1-r_ne^{i\psi_n},$ $-\pi/2+\eta<\psi_n<\pi/2-\eta,$ $r_n\to0+$ as $n\to+\infty,$ of the Blaschke product $B(z);$ $\Gamma_m=\bigcup\limits_{j=1}^{m}\{z:|z|<1,\mathop{\text{arg}}(1-z)=-\theta_j\}=\bigcup\limits_{j=1}^{m}l_{\theta_j},$ $-\pi/2+\eta<\theta_1<\theta_2<\ldots<\theta_m<\pi/2-\eta,$ be a finite system of rays, $0<\eta<1$; $\upsilon(t)$ be continuous on $[0,1)$, $\upsilon(0)=0$, slowly increasing at the point 1 function, that is $\upsilon(t)\sim\upsilon\left({(1+t)}/2\right),$ $t\to1-;$ $n(t,\theta_j;B)$ be a number of zeros $a_n=1-r_ne^{i\theta_j}$ of the product $B(z)$ on the ray $l_{\theta_j}$ such that $1-r_n\leq t,$ $0<t<1.$ We found asymptotics of the logarithmic derivative of $B(z)$ as $z=1-re^{-i\varphi}\to1,$ $-\pi/2<\varphi<\pi/2,$ $\varphi\neq\theta_j,$ under the condition that zeros of $B(z)$ lay on $\Gamma_m$ and $n(t,\theta_j;B)\sim \Delta_j\upsilon(t),$ $t\to1-,$ for all $j=\overline{1,m},$ $0\leq\Delta_j<+\infty.$ We also considered the inverse problem for such $B(z).$

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Асимптотичне Поводження Логарифмічної Похідної Цілих Функцій Нульового Порядку

Cited by 2 publications

References 2 publications

Binomial asymptotics for the logarithmic derivative of zero-order entire functions with zeros along curves of regular rotation

Binomial asymptotics for the logarithmic derivative of zero-order entire functions with zeros along curves of regular rotation

Logarithmic Derivative of the Blaschke Product With Slowly Increasing Counting Function of Zeros

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