Always-convex Harmonic Shears

We determine completely the analytic functions φ in the unit disk D such that for all (normalized) orientation-preserving harmonic mappings f=h+g¯ produced by the shear construction with h+g=φ, the condition that each f maps 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 D such that every harmonic function f=h+g¯ as above with h-ηg=φ maps 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.