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On an estimate of a functional in the class of holomorphic univalent functions

Zbigniew Jerzy Jakubowski;  Krystyna Zyskowska — 1993

Mathematica Bohemica

Let S denote the class of functions f ( x ) = z + a 2 z 2 + a 3 z 3 + ... univalent and holomorphic in the unit disc Δ = { z : z < 1 } . In the paper we obtain an estimate of the functional a 3 - c a 2 2 + c a 2 n in the class S for arbitrarily fixed x ∈ 𝐑 and n = 1 , 2 , 3 , ... . Hence, for some special values of the parameters, we obtain estimates of several interesting functionals and numerous applications. A few open problems of a similar type are also formulated.

Applications of the Hadamard product in geometric function theory

Zbigniew Jerzy Jakubowski;  Piotr Liczberski;  Łucja Żywień — 1991

Mathematica Bohemica

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]).

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