Prawitz's area theorem and the mixed Aharonov sequence

TL;DR

This paper introduces the mixed Aharonov sequence related to locally univalent functions, establishing a new univalence criterion and properties, including an inequality for functions with quasiconformal extensions.

Contribution

It presents a novel mixed Aharonov sequence and a new univalence criterion that generalizes previous results, along with new properties and inequalities.

Findings

Established a new univalence criterion for locally univalent functions.

Proved new properties of the mixed Aharonov sequence.

Derived an inequality for univalent functions with quasiconformal extensions.

Abstract

In this paper, motivated by the Prawitz area theorem and the work of Aharonov, we introduce the mixed Aharonov sequence associated with a locally univalent analytic function. By using the mixed Aharonov sequence, we establish a new univalence criterion for the locally univalent analytic functions in the unit disk, which generalizes some related results of Aharonov in \cite{Ah}. We also prove some new properties about the (mixed) Aharonov sequence, in particular, a new inequality for the Aharonov sequence is established for the univalent functions with a quasiconformal extension.