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On a generalization of close-to-convex functions

Swadesh Kumar Sahoo, Navneet Lal Sharma (2015)

Annales Polonici Mathematici

The paper of M. Ismail et al. [Complex Variables Theory Appl. 14 (1990), 77-84] motivates the study of a generalization of close-to-convex functions by means of a q-analog of the difference operator acting on analytic functions in the unit disk 𝔻 = {z ∈ ℂ:|z| < 1}. We use the term q-close-to-convex functions for the q-analog of close-to-convex functions. We obtain conditions on the coefficients of power series of functions analytic in the unit disk which ensure that they generate functions in...

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.

On an extremal problem

Krystyna Zyskowska (1995)

Mathematica Bohemica

Let S denote the class of functions f ( z ) = 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 a sharp estimate of the functional | a 3 - α a 2 2 | + α | a 2 | 2 in the class S for an arbitrary α .

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