On a class of analytic functions governed by subordination

Ravinder Krishna Raina; Janusz Sokół

doi:10.2478/ausm-2019-0012

Abstract

The purpose of this paper is to introduce a class of functions ℱ_λ, λ ∈ [0, 1], consisting of analytic functions f normalized by f(0) = f´(0) − 1 = 0 in the open unit disk U which satisfies the subordination condition that

$z f^{'} (z) / {(1 - λ) f (z) + λ z} ≺ q (z), z \in 𝕌,$ $${\rm{z}}f'\left( {\rm{z}} \right)/\left\{ {\left( {1 - \lambda } \right){\rm{f}}\left( {\rm{z}} \right) + \lambda {\rm{z}}} \right\} \prec {\rm{q}}\left( {\rm{z}} \right),\,\,\,\,\,{\rm{z}} \in {\rm{\mathbb{U},}}$$

where $q (z) = \sqrt{1 + z^{2}} + z$ ${\rm{q}}\left( {\rm{z}} \right) = \sqrt {1 + {{\rm{z}}^{\rm{2}}}} + {\rm{z}}$. Some basic properties (including the radius of convexity) are obtained for this class of functions.

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On a class of analytic functions governed by subordination

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