Always-convex harmonic shears

We determine completely the analytic functions $\phi$ in the unit disk $\mathbb{D}$ such that for all (normalized) orientation-preserving harmonic mappings $f=h+\bar{g}$ produced by the shear construction with $h+g=\phi$, 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 $\eta$, with $\mathrm{\mid }\eta \mathrm{\mid }=1$, we characterize those holomorphic mappings $\phi$ in $\mathbb{D}$ such that every harmonic function $f=h+\bar{g}$ as above with $h-\eta g=\phi$ 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 $\eta$ above, is parallel to the linear boundaries of the half-planes and strips.