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.
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