scholarly journals Integral mean of Green’s potentials and their conjugate

2013 ◽  
Vol 5 (1) ◽  
pp. 19-29
Author(s):  
Ya.V. Vasyl'kiv ◽  
M.Ya. Kravec
Keyword(s):  
Counting Function ◽  
Blaschke Products ◽  
Lebesgue Integral ◽  
Integral Means ◽  
Function Conjugate

The best possible estimates for Lebesgue integral means $m_q(r,F)\; (1\le q<+\infty)$ for the pair of functions $F= g+i\:\breve{g}$, here $g$ - Green's potential, $\breve{g}$ - function conjugate to $g$, was obtained. It generalizes well-known results of Ya.V. Vasyl'kiv and A.A. Kondratyuk for logarithms $\log\; B$ of Blaschke products $B$ in terms of counting function $n(r,0,B)\; (0<r<1)$ of their zeroes.

LOGARITHMIC DERIVATIVE OF THE BLASCHKE PRODUCT WITH SLOWLY INCREASING COUNTING FUNCTION OF ZEROS

2021 ◽  
Vol 9 (1) ◽  
pp. 164-170
Author(s):  
Y. Gal ◽  
M. Zabolotskyi ◽  
M. Mostova
Keyword(s):  
Blaschke Product ◽  
Meromorphic Functions ◽  
Logarithmic Derivative ◽  
Counting Function ◽  
Finite System ◽  
Blaschke Products ◽  
Nevanlinna Characteristic ◽  
Number Of Zeros ◽  
Accumulation Points ◽  
Theory Of Value

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 $B(z).$

Integral Means and Zero Distributions of Blaschke Products

1972 ◽  
Vol 24 (5) ◽  
pp. 755-760 ◽  
Author(s):  
C. N. Linden
Keyword(s):  
Half Plane ◽  
Blaschke Product ◽  
Blaschke Products ◽  
Integral Means ◽  
Elementary Theorem ◽  
Image Position ◽  
Analogous Theorem

A sequence {zn} in D = {z: |z| < 1} is a Blaschke sequence if and only ifIf 0 appears m times in {zn} thenis the Blaschke product defined by {zn}. The set of all Blaschke products will be denoted by . If B ∊ it is well-known that B is regular in D, and |B(z, {zn})| < 1 when z ∊ D.For a given pair of values p in (0, ∞) and q in [0, ∞) we denote by ℐ(p, g) the class of all Blaschke products B(z, {zn}) such thatas r → 1 — 0. In the case q ≦ max(p — 1,0) the classes of functions B and ℐ(p, q) are identical: this is a particular case of an elementary theorem for functions subharmonic in a disc, the analogous theorem for functions subharmonic in a half-plane appearing in [1],

Remark on Integral Means of Derivatives of Blaschke Products

2018 ◽  
Vol 61 (3) ◽  
pp. 640-649 ◽  
Author(s):  
Atte Reijonen
Keyword(s):  
Blaschke Product ◽  
Unit Disc ◽  
Regularity Condition ◽  
Blaschke Products ◽  
Integral Means ◽  
Area Measure ◽  
Lebesgue Area ◽  
Derivatives Of

AbstractIf B is the Blachke product with zeros {zn}, then , whereMoreover, it is a well-known fact that, for 0 < p < ∞,is bounded if and only if Mp(r, ΨB) is bounded. We find a Blaschke product B0 such that Mp(r, ) and Mp(r, ) are not comparable for any < p < ∞. In addition, it is shown that, if 0 < p < ∞, B is a Carleson–Newman Blaschke product and a weight ω satisfies a certain regularity condition, thenwhere d A(z) is the Lebesgue area measure on the unit disc.

scholarly journals INTEGRAL MEANS OF THE DERIVATIVES OF BLASCHKE PRODUCTS

2008 ◽  
Vol 50 (2) ◽  
pp. 233-249 ◽  
Author(s):  
EMMANUEL FRICAIN ◽  
JAVAD MASHREGHI
Keyword(s):  
Blaschke Product ◽  
Blaschke Products ◽  
Integral Means ◽  
Rate Of Growth ◽  
Model Subspace ◽  
Derivatives Of

AbstractWe study the rate of growth of some integral means of the derivatives of a Blaschke product and we generalize several classical results. Moreover, we obtain the rate of growth of integral means of the derivative of functions in the model subspaceKBgenerated by the Blaschke productB.

scholarly journals ON BOUNDEDNESS OF INTEGRAL MEANS OF BLASCHKE PRODUCT LOGARITHMS

2003 ◽  
Vol 8 (3) ◽  
pp. 259-265
Author(s):  
YA. V. Vasylkiv ◽  
A. A. Kondratyuk ◽  
S. I. Tarasyuk
Keyword(s):  
Fourier Series ◽  
Blaschke Product ◽  
Analytic Functions ◽  
Counting Function ◽  
Positive Function ◽  
Series Method ◽  
Integral Means ◽  
Fourier Series Method

Using the Fourier series method for the analytic functions, we obtain a result characterizing the behaviour of the integral means of Blaschke product logarithms. Namely, if the zero counting function n(r, B) of the Blaschke product B satisfies the conditionwhere l is a positive function on (0, 1) such thatthen the q‐integral mean mq (r, log B) = [] is bounded on (0,1), where log B is a branch of the logarithm of B. Šiame straipsnyje Furje eilučiu metodu gauta analitiniu funkciju Blaschke sandaugos logaritmu integraliniu reikšmiu elgsenos charakteristika. Jeigu Blaschke sandaugos B nuliu funkcija n(r, B) tenkina salyga [], čia l yra neneigiama funkcija intervale (0,1) ir [], tuomet q‐integraline reikšme [] yra aprežta intervale (0,1), kai log B yra B logaritmo šaka.

Sign in / Sign up

Export Citation Format

Share Document