The Schwarz-Pick lemma for planar harmonic mappings

The classical Schwarz-Pick lemma for holomorphic mappings is generalized to planar harmonic mappings of the unit disk D completely. (I) For any 0 < r < 1 and 0 less than or similar to rho < 1, the author constructs a closed convex domain E(r,rho) such that F(<(Delta)over bar>(z, r)) subset of e(i alpha)E(r,rho) = {e(i alpha)z : z is an element of E(r,rho)} holds for every z is an element of D, w = rho e(i alpha) and harmonic mapping F with F(D) subset of D and F(z) = w, where Delta(z, r) is the pseudo-disk of center z and pseudo-radius r; conversely, for every z is an element of D, w = rho e(i alpha) and w' is an element of e(i alpha)E(r,rho), there exists a harmonic mapping F such that F(D) subset of D, F(z) = w and F(z') = w' for some z' is an element of partial derivative Delta(z, r). (II) The author establishes a Finsler metric H(z)(u) on the unit disk D such that H(F(z))(e(i theta)F(z)(z) + e(-i theta)F((z) over bar)(z)) <= 1/1-|z|(2) holds for any z is an element of D, 0 <= theta <= 2 pi and harmonic mapping F with F(D) subset of D; furthermore, this result is precise and the equality may be attained for any values of z, theta, F(z) and arg(e(i theta)F(z)(z) + e(-i theta)F((z) over bar)(z)).