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.