Biholomorphic Convex Mappings of Order alpha on the Unit Ball in Hilbert Spaces

TL;DR

This paper introduces and characterizes biholomorphic convex mappings of order alpha on the unit ball in complex Hilbert spaces, providing conditions, relationships with starlike mappings, and concrete examples.

Contribution

It defines biholomorphic convex mappings of order alpha and establishes conditions, relationships, and examples in the context of Hilbert space mappings.

Findings

Provided sufficient conditions for biholomorphic convexity of order alpha.

Established an Alexander's theorem linking convex and starlike mappings.

Constructed explicit examples using linear operators.

Abstract

In this paper, we first introduce the concept of biholomorphic convex mapping of order alpha on the unit ball B in a complex Hilbert space X. Next we provide some sufficient conditions that a locally biholomorphic mapping f is a biholomorphic convex mapping of order alpha and give an Alexander's theorem between the subclass of convex mappings and the subclass of starlike mappings on B in Hilbert space. We also obtain the order of starlikeness of biholomorphic convex mappings of order alpha on B in Hilbert spaces. Finally, we construct some concrete examples of biholomorphic convex mappings of order alpha on B in Hilbert spaces by means of a linear operator.