Given a sense-preserving injective harmonic mapping F in the unit disk D and a is an element of C we consider a simple deformation C (sic) bar right arrow F-a := H + a (G) over bar of F, where H and G are holomorphic mappings in D determined by F = H + (G) over bar and G(0) = 0. We introduce a natural generalization of convexity called alpha-convexity. Then we study the bi-Lipschitz behaviour of mappings F-a under the assumption that F is a quasiconformal harmonic mapping of D onto an alpha-convex domain F(D). As an application we show that if F is a quasiconformal harmonic self-mapping of D, then H is a bi-Lipschitz mapping. Consequently, a sense-preserving harmonic self-mapping F of D is quasiconforrnal if H is Lipschitz with the Jacobian of F separated from zero by a positive constant in D.