Sufficient conditions for Carathéodory functions and applications to univalent functions

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 givenβ∈ (–π/2,π/2),γ∈ [0, cosβ) andδ∈ (0,π/2], we find some sufficient conditions for an analytic functionpsuch thatp(0) = 1 to satisfy Re{e−iβp(z)} >γor | arg {p(z)–γ} | <δfor allz∈ 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.