Always-convex harmonic shears

TL;DR

This work characterizes when the shear construction applied to a convex analytic datum $\varphi$ yields convex, univalent harmonic mappings for every Schwarz dilatation $\omega$. By leveraging rotation-invariance, convex-analytic function theory, and boundary-rotation results, it proves that the resulting harmonic shear must be a rotation of a function $F=H+G$ with $H+G$ belonging to the model family $\mathcal{A}=\{\mathcal{H},\mathcal{H}_{-1},\mathcal{L}_\lambda\}$ (for $|\lambda|=1$, $\lambda\neq -1,1$). The main theorem uses a four-case analysis to show that, up to rotation, $\varphi$ must lie in $\mathcal{A}$, thereby completely classifying always-convex harmonic shears. The results connect the shear construction with half-planes and strips, and yield corollaries for broader dilatations like $\omega(z)=\lambda z^N$. This provides a precise, rotation-variant criterion for constructing convex harmonic mappings via shear.

Abstract

We determine completely the analytic functions $\varphi$ in the unit disk $\mathbb D$ such that for all (normalized) orientation-preserving harmonic mappings $f=h+\overline g$ produced by the shear construction with $h+g=\varphi$, the condition that each $f$ maps $\mathbb D$ onto a convex domain holds. As a consequence, we obtain the following more general result: for a given complex number $η$, with $|η|=1$, we characterize those holomorphic mappings $\varphi$ in $\mathbb D$ such that every harmonic function $f=h+\overline g$ as above with $h-ηg=\varphi$ maps $\mathbb D$ onto a convex domain. The resulting functions are mappings onto a half-plane and mappings onto a strip, and the shear direction, determined by the parameter $η$ above, is parallel to the linear boundaries of the half-planes and strips.