Always-convex harmonic shears

We determine completely the analytic functions $φ$ 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 $h+g=φ$, 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 $φ$ in $\mathbb D$ such that every harmonic function $f=h+\overline g$ as above with $h-ηg=φ$ 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.

💡 Research Summary

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The paper investigates a very natural but surprisingly restrictive problem in the theory of planar harmonic mappings: for which analytic functions φ defined on the unit disk 𝔻 does the shear construction always produce a convex harmonic mapping, regardless of the choice of dilatation ω?

A harmonic mapping f=h+ \overline g in 𝔻 is called orientation‑preserving when its Jacobian J_f=|h′|²−|g′|²>0, and the shear construction consists of solving the linear system
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