Sharp inequalities of homogeneous expansions for quasi-convex mappings of type B and almost starlike mappings of order alpha

TL;DR

This paper establishes sharp inequalities for homogeneous expansions of specific classes of biholomorphic mappings in complex Banach spaces, providing precise estimates for their third and fourth order expansions.

Contribution

It introduces new sharp inequalities for quasi-convex and almost starlike biholomorphic mappings, advancing understanding of their homogeneous expansions.

Findings

Sharp inequalities for homogeneous expansions of quasi-convex mappings

Sharp estimates for third and fourth homogeneous expansions

Results applicable to mappings on unit ball and polydisk

Abstract

In this paper, we first obtain several sharp inequalities of homogeneous expansion for both the subclass of all normalized biholomorphic quasi-convex mappings of type B and order alpha and the subclass of all normalized biholomorphic almost starlike mappings of order alpha defined on the unit ball $B$ of a complex Banach space X. Then, with these sharp inequalities, we derive the sharp estimates of the third and fourth homogeneous expansions for the above mappings defined on the unit polydisk $D^n$ in C(n).