A one-parameter family of normalized harmonic mappings of the unit disk, defined by requiring Re[(1-alpha)h'(z) + alpha z h''(z)] > -M + |(1-alpha)g'(z) + alpha z g''(z)| with g'(0) = 0, for M > 0 and alpha in (0, 1]. The parameter alpha interpolates between a first-derivative condition of Ponnusamy type and, at alpha = 1, the second-derivative class of Ghosh and Allu, so results proved once for the family recover both endpoints. For this class the sharp coefficient bounds on |a_n|, |b_n|, |a_n| + |b_n| and ||a_n| - |b_n|| are established with explicit extremal functions, together with a two-sided growth estimate on |f(z)|, closure under convex combinations and under harmonic convolution, and the radii of univalence and of convexity.
A complex-valued harmonic mapping of the unit disk is a solution of the Laplace equation, and every such map splits canonically as f = h + conjugate(g) with h and g analytic; by Lewy's theorem it is …