Abstract. We give a lower estimate for the Bloch constant for planar harmonic mappings which are quasiregular and for those which are open. The latter includes the classical Bloch theorem for holomorphic functions as a special case. Also, for bounded planar harmonic mappings, we obtain results similar to a theorem of Landau on bounded holomorphic functions.

Abstract. For a normalized quasiregular pluriharmonic mapping f of the unit ball B n of C n into C n , we estimate the supremum of numbers R such that some subdomain Ω of the ball is mapped by f diffeomorphically onto some ball of radius R. Our estimates significantly improve earlier estimates, even in the case of harmonic functions in the disc.

Abstract. We give a necessary and sufficient condition for a composition operator on an α-Bloch space with α ≥ 1 to be bounded below. This extends a known result for the Bloch space due to P. Ghatage,

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 ρ < 1, the author constructs a closed convex domain Er,ρ such thatis the pseudo-disk of center z and pseudo-radius r; conversely, for every z ∈ D, w = ρe iα and w ∈ e iα Er,ρ, there exists a harmonic mapping F such that F (D) ⊂ D, F (z) = w and F (z ) = w for some z ∈ ∂Δ(z, r). (II) The author establishes a Finsler metric Hz(u) on the unit disk D such that H F (z) (e iθ Fz(z) + e −iθ F z (z)) 1 1 − |z| 2 holds for any z ∈ D, 0 θ 2π and harmonic mapping F with F (D) ⊂ D; furthermore, this result is precise and the equality may be attained for any values of z, θ, F (z) and arg(e iθ Fz(z) + e −iθ F z (z)).

Abstract. We give lower estimates for Bloch's constant for quasiregular holomorphic mappings. A holomorphic mapping of the unit ball B n into C n is K-quasiregular if it maps infinitesimal spheres to infinitesimal ellipsoids whose major axes are less than or equal to K times their minor axes. We show that if f is a K-quasiregular holomorphic mapping with the normalization det f (0) = 1, then the image f (B n ) contains a schlicht ball of radius at least 1/12K 1−1/n . This result is best possible in terms of powers of K. Also, we extend to several variables an analogous result of Landau for bounded holomorphic functions in the unit disk.