In 1958, E. Heinz obtained a lower bound for , where is a one-to-one harmonic mapping of the unit disk onto itself keeping the origin fixed. We show various variants of Heinz's inequality in the case where is the Poisson integral of a function of bounded variation in the unit circle. In particular, we obtain such inequalities for when it is a locally injective quasiregular mapping or an injective mapping of the unit disk onto a bounded convex domain in the complex plane.