Sharp Robertson-type coefficient estimates for radial starlike and quasi-convex of type B mappings of order α in complex Banach spaces

Abstract

In this paper, we study biholomorphic mappings $F: \mathbb{B} \to X$ of the form $F(z) = f(z)z$ for $z \in \mathbb{B}$, where $f: \mathbb{B} \to \mathbb{C}$ is a holomorphic function satisfying $f(0) = 1$. Our main contributions are as follows: We establish sharp estimates for the coefficients of starlike functions of order $α$ as well as quasi-convex functions of type $B$ and order $α$ by employing the notion of majorization. The results established here directly extend the results due to Robertson \cite{Robertson-1936} to the broader setting of complex Banach spaces.