Applications of the Hadamard product in geometric function theory

Zbigniew Jerzy Jakubowski; Piotr Liczberski; Łucja Żywień

Mathematica Bohemica (1991)

  • Volume: 116, Issue: 2, page 148-159
  • ISSN: 0862-7959

Abstract

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Let 𝒜 denote the set of functions F holomorphic in the unit disc, normalized clasically: F ( 0 ) = 0 , F ' ( 0 ) = 1 , whereas A ⊂ 𝒜 is an arbitrarily fixed subset. In this paper various properties of the classes A α , α ∈ C { - 1 , - 1 2 , ... } , of functions of the form f = F * k α are studied, where F ∈ . A , k α ( z ) = k ( z , α ) = z + 1 1 + α z 2 + ... + 1 1 + ( n - 1 ) α z n + ... , and F * k α denotes the Hadamard product of the functions F and k α . Some special cases of the set A were considered by other authors (see, for example, [15],[6],[3]).

How to cite

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Jakubowski, Zbigniew Jerzy, Liczberski, Piotr, and Żywień, Łucja. "Applications of the Hadamard product in geometric function theory." Mathematica Bohemica 116.2 (1991): 148-159. <http://eudml.org/doc/29295>.

@article{Jakubowski1991,
abstract = {Let $\mathcal \{A\}$ denote the set of functions $F$ holomorphic in the unit disc, normalized clasically: $F(0)=0, F^\{\prime \}(0)=1$, whereas $A\subset \mathcal \{A\}$ is an arbitrarily fixed subset. In this paper various properties of the classes $A_\alpha , \alpha \in C \lbrace -1,-\frac\{1\}\{2\},\ldots \rbrace $, of functions of the form $f=F*k_\alpha $ are studied, where $F\in .A$, $k_\alpha (z)=k(z,\alpha )=z+\frac\{1\}\{1+\alpha \}z^2+\ldots + \frac\{1\}\{1+(n-1)\alpha \}z^n+\ldots $, and $F*k_\alpha $ denotes the Hadamard product of the functions $F$ and $k_\alpha $. Some special cases of the set $A$ were considered by other authors (see, for example, [15],[6],[3]).},
author = {Jakubowski, Zbigniew Jerzy, Liczberski, Piotr, Żywień, Łucja},
journal = {Mathematica Bohemica},
keywords = {Hadamard product; typically real functions; class of type $A_\alpha $; Hadamard product},
language = {eng},
number = {2},
pages = {148-159},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Applications of the Hadamard product in geometric function theory},
url = {http://eudml.org/doc/29295},
volume = {116},
year = {1991},
}

TY - JOUR
AU - Jakubowski, Zbigniew Jerzy
AU - Liczberski, Piotr
AU - Żywień, Łucja
TI - Applications of the Hadamard product in geometric function theory
JO - Mathematica Bohemica
PY - 1991
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 116
IS - 2
SP - 148
EP - 159
AB - Let $\mathcal {A}$ denote the set of functions $F$ holomorphic in the unit disc, normalized clasically: $F(0)=0, F^{\prime }(0)=1$, whereas $A\subset \mathcal {A}$ is an arbitrarily fixed subset. In this paper various properties of the classes $A_\alpha , \alpha \in C \lbrace -1,-\frac{1}{2},\ldots \rbrace $, of functions of the form $f=F*k_\alpha $ are studied, where $F\in .A$, $k_\alpha (z)=k(z,\alpha )=z+\frac{1}{1+\alpha }z^2+\ldots + \frac{1}{1+(n-1)\alpha }z^n+\ldots $, and $F*k_\alpha $ denotes the Hadamard product of the functions $F$ and $k_\alpha $. Some special cases of the set $A$ were considered by other authors (see, for example, [15],[6],[3]).
LA - eng
KW - Hadamard product; typically real functions; class of type $A_\alpha $; Hadamard product
UR - http://eudml.org/doc/29295
ER -

References

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