Boundedness, univalence and quasiconformal extension of Robertson functions

TL;DR

This paper investigates properties of mbda-Robertson functions, establishing conditions for their boundedness, quasiconformal extension, and univalence, using Lfner chains, contributing to geometric function theory.

Contribution

It provides new criteria for boundedness and quasiconformal extension of Robertson functions and offers an alternative proof of their univalence via Lfner chains.

Findings

Conditions for boundedness of Robertson functions

Criteria for quasiconformal extension

Alternative proof of univalence using Lfner chains

Abstract

This article contains several results for \lambda-Robertson functions, i.e., analytic functions $f$ defined on the unit disk $D$ satisfying $f(0) = f'(0)-1=0$ and $Re e^{-i\lambda} {1+zf"(z)/f'(z)} > 0$ in $D$, where $\lambda \epsilon (-\pi/2,\pi/2)$. We will discuss about conditions for boundedness and quasiconformal extension of Robertson functions. In the last section we provide another proof of univalence for Robertson functions by using the theory of L\"owner chains.