Sufficient conditions for Carathéodory functions and applications to univalent functions

Author:

Kwon Oh Sang,Sim Young Jae

Abstract

Abstract In this paper, the authors derive several sufficient conditions for a function to be the Carathéodory function in the unit disk 𝔻: = {z ∈ ℂ: |z| < 1}. More precisely, for given β ∈ (–π/2, π/2), γ ∈ [0, cosβ) and δ ∈ (0, π/2], we find some sufficient conditions for an analytic function p such that p(0) = 1 to satisfy Re{e−iβ p(z)} > γ or | arg {p(z)–γ} | < δ for all z ∈ 𝔻 by using the first-order differential subordination. We then apply the results obtained here in order to find some conditions for univalent functions with geometric properties such as spirallikeness and strongly starlikeness.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics

Reference24 articles.

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3. Contribution à la theorie des fonctions univalentes;Časopis Pěst. Mat.,1932

4. New conditions for starlikeness and strongly starlikeness of order alpha;Houston J. Math.,2017

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