Sufficient conditions for Carathéodory functions and applications to univalent functions

Year, pages: 2019, 1065 - 1076

In this paper, the authors derive several sufficient conditions for a function to be the Carathéodory function in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}: |z|<1 \}$. More precisely, for given $β \in (-π/2,π/2)$, $γ \in [0,\cosβ)$ and $δ\in(0,π/2]$, we find some sufficient conditions for an analytic function $p$ such that $p(0)=1$ to satisfy ${\re}\{{\e}^{-{\ii}β} p(z) \} > γ$ or $| \arg \{p(z)-γ\} |<δ$ for all $z\in\mathbb{D}$ 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.

Kwon, O., Sim, Y. 2019. Sufficient conditions for Carathéodory functions and applications to univalent functions. In Mathematica Slovaca, vol. 69, no.5, pp. 1065-1076. 0139-9918. DOI: https://doi.org/ 10.1515/ms-2017-0290

Kwon, O., Sim, Y. (2019). Sufficient conditions for Carathéodory functions and applications to univalent functions. Mathematica Slovaca, 69(5), 1065-1076. 0139-9918. DOI: https://doi.org/ 10.1515/ms-2017-0290

Publisher: Mathematical Institute, Slovak Academy of Sciences, Bratislava