scholarly journals On weights which admit the reproducing kernel of Bergman type

1992 ◽  
Vol 15 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Zbigniew Pasternak-Winiarski
Keyword(s):  
Unit Disk ◽  
Bergman Kernel ◽  
Reproducing Kernel ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Admissible Weight ◽  
Necessary And Sufficient ◽  
Weighted Bergman Kernel

In this paper we consider (1) the weights of integration for which the reproducing kernel of the Bergman type can be defined, i.e., the admissible weights, and (2) the kernels defined by such weights. It is verified that the weighted Bergman kernel has the analogous properties as the classical one. We prove several sufficient conditions and necessary and sufficient conditions for a weight to be an admissible weight. We give also an example of a weight which is not of this class. As a positive example we consider the weightμ(z)=(Imz)2defined on the unit disk inℂ.

scholarly journals QKtype spaces of analytic functions

2006 ◽  
Vol 4 (1) ◽  
pp. 73-84 ◽  
Author(s):  
Hasi Wulan ◽  
Jizhen Zhou
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Nondecreasing Function ◽  
Necessary And Sufficient Conditions ◽  
Type Space ◽  
Necessary And Sufficient ◽  
Spaces Of Analytic Functions

For a nondecreasing functionK:[0,8)?[0,8)and0<p<8,-2<q<8, we introduceQK(p,q), aQKtype space of functions analytic in the unit disk and study the characterizations ofQK(p,q). Necessary and sufficient conditions onKsuch thatQK(p,q)become some known spaces are given.

Coefficient Regions for Univalent Trinomials

1980 ◽  
Vol 32 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Q. I. Rahman ◽  
J. Waniurski
Keyword(s):  
Unit Disk ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Image Position ◽  
Necessary And Sufficient

The problem of determining necessary and sufficient conditions bearing upon the numbers a2 and a3 in order that the polynomial z + a2z2 + a3z3 be univalent in the unit disk |z| < 1 was solved by Brannan ([3], [4]) and by Cowling and Royster [6], at about the same time. For his investigation Brannan used the following result due to Dieudonné [7] and the well-known Cohn rule [9].THEOREM A (Dieudonné criterion). The polynomial1is univalent in |z| < 1 if and only if for every Θ in [0, π/2] the associated polynomial2does not vanish in |z| < 1. For Θ = 0, (2) is to be interpreted as the derivative of (1).The procedure of Cowling and Royster was based on the observation that is univalent in |z| < 1 if and only if for all α such that 0 ≧ |α| ≧ 1, α ≠ 1 the functionis regular in the unit disk.

scholarly journals Some properties of Glisson distances in the upper half-plane

2021 ◽  
Vol 2131 (3) ◽  
pp. 032039
Author(s):  
M Ovchintsev
Keyword(s):  
Unit Ball ◽  
Unit Disk ◽  
Half Plane ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ordinate Axis ◽  
Upper Half Plane ◽  
Necessary And Sufficient

Abstract The author compares the Gleason distance with the distance of Euclid in the unit disk in the upper half plane. The concept of “the Gleason distance” was formulated in the work of H.S. Bear [1] The Gleason distance is defined as follows (see [1]): d = sup |f(z2)-f(z1)|, f(Z)εB1(K) where B 1 (K) is the unit ball in the space of bounded analytic in K functions. The author of the article proves that in the circle K the distances of Gleason and Euclid are equal only when the points are opposite. He found necessary and sufficient conditions, when the distances are equal for the two given points which are symmetrical about the ordinate axis.

scholarly journals A Subclass of q-Starlike Functions Defined by Using a Symmetric q-Derivative Operator and Related with Generalized Symmetric Conic Domains

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 917
Author(s):  
Shahid Khan ◽  
Saqib Hussain ◽  
Muhammad Naeem ◽  
Maslina Darus ◽  
Akhter Rasheed
Keyword(s):  
Unit Disk ◽  
Sufficient Conditions ◽  
Starlike Functions ◽  
Necessary And Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Derivative Operator ◽  
Structural Formulas ◽  
Necessary And Sufficient

In this paper, the concepts of symmetric q-calculus and conic regions are used to define a new domain Ωk,q,α˜, which generalizes the symmetric conic domains. By using the domain Ωk,q,α˜, we define a new subclass of analytic and q-starlike functions in the open unit disk U and establish some new results for functions of this class. We also investigate a number of useful properties and characteristics of this subclass, such as coefficients estimates, structural formulas, distortion inequalities, necessary and sufficient conditions, closure and subordination results. The proposed approach is also compared with some existing methods to show the reliability and effectiveness of the proposed methods.

scholarly journals SOME NOTICES ON ZEROS AND POLES OF MEROMORPHIC FUNCTIONS IN A UNIT DISK FROM THE CLASSES DEFINED BY THE ARBITRARY GROWTH MAJORANT

2021 ◽  
Vol 9 (2) ◽  
pp. 124-130
Author(s):  
I. Sheparovych
Keyword(s):  
Meromorphic Function ◽  
Unit Disk ◽  
Fourier Coefficients ◽  
Meromorphic Functions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient ◽  
The Given ◽  
Increasing Function ◽  
Zeros And Poles

In [4] by the Fourier coefficients method there were obtained some necessary and sufficient conditions for the sequence of zeros $(\lambda_{\nu})$ of holomorphic in the unit disk $\{z:|z|<1\}$ functions $f$ from the class that determined by the majorant $\eta :[0;+\infty)\to [0;+\infty )$ that is an increasing function of arbitrary growth. Using that result in present paper it is proved that if $(\lambda_{\nu})$ is a sequence of zeros and $(\mu_ {j})$ is a sequence of poles of the meromorphic function $f$ in the unit disk, such that for some $A>0, B>0$ and for all $r\in(0;1):\ T(r;f)\leqslant A\eta\left(\frac B{1-|z|}\right)$, where $T(r;f):=m(r;f)+N(r;f);\ m(r;f)=\frac{1}{2\pi }\int\limits_0^{2\pi } \ln ^{+}|f(re^{i\varphi})|d\varphi$, then for some positive constants $A_1, A’_1, B_1, B’_1, A_2, B_2$ and for all $k \in\mathbb{N}$, $r,\ r_1$ from $(0;1)$, $r_2\in(r_1;1)$ and $\sigma\in(1;1/r_2)$ the next conditions hold $N (r,1/f) \leq A_1 \eta\left(\frac{B_1}{1-r}\right)$, $N(r,f)\leq A'_1\eta \left( \frac{B'_1}{1-r}\right) $, $$\frac1{2k}\left|\sum\limits_{r_1 <|\lambda_{\nu}|\leqslant r_{2}} \frac1{\lambda_{\nu}^k} -\sum\limits_{r_1 < |\mu_j|\leqslant r_2} \frac 1{\mu_j^{k}} \right| \leq \frac{A_{2}}{r_{1}^{k}}\eta\left(\frac{B_{2}}{1 -r_1}\right ) +\frac{A_{2}}{r_{2}^{k}}\max\left\{ 1;\frac 1{k\ln \sigma}\right\}\eta\left(\frac{B_{2}}{1 -\sigma r_{2}}\right)$$ It is also shown that if sequence $(\lambda_{\nu})$ satisfies the condition $N (r,1/f) \leq A_1 \eta\left(\frac{B_1}{1-r}\right)$ and $$\frac1{2k}\left|\sum\limits_{r_1 <|\lambda_{\nu}|\leqslant r_{2}} \frac1{\lambda_{\nu}^k} \right| \leq \frac{A_{2}}{r_{1}^{k}}\eta\left(\frac{B_{2}}{1-r_{1}}\right) +\frac{A_{2}}{r_{2}^{k}}\max\left\{ 1;\frac 1{k\ln \sigma}\right\}\eta\left(\frac{B_{2}}{1 -\sigma r_{2}}\right)$$ there is possible to construct a meromorphic function from the class $T(r;f)\leqslant \frac{A}{\sqrt{1-r}}\eta\left(\frac B{1-r}\right)$, for which the given sequence is a sequence of zeros or poles.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj&gt; 0 for eachj&gt; 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

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