Linear combinations of univalent harmonic mappings convex in the direction of the imaginary axis

TL;DR

This paper introduces a family of univalent harmonic functions mapping the unit disk to domains convex in the imaginary axis direction, and establishes conditions for their linear combinations to preserve univalence and convexity.

Contribution

It provides new criteria for linear combinations of harmonic mappings to remain univalent and convex in the specified direction, expanding understanding of harmonic function combinations.

Findings

Conditions for linear combinations to be univalent

Criteria for convexity in the imaginary axis direction

Analysis of sheared harmonic mappings

Abstract

In the present paper, we introduce a family of univalent harmonic functions, which map the unit disk onto domains convex in the direction of the imaginary axis. We find conditions for the linear combinations of mappings from this family to be univalent and convex in the direction of the imaginary axis. Linear combinations of functions from this family and harmonic mappings obtained by shearing of analytic vertical strip maps are also studied.