On the nitsche conjecture for harmonic mappings in R2 and R3

We give the new inequality related to the J. C. C. Nitsche conjecture (see [6]). Moreover, we consider the two- and three-dimensional case. Let A(r, 1) = {z : r < vertical bar z vertical bar < 1}. Nitsche's conjecture states that if there exists a univalent harmonic mapping from an annulus A(r, 1) to an annulus A(s, 1), then 8 is at most 2r/(r(2) + 1). Lyzzaik's result states that s < t where t is the length of the Grotzsch's ring domain associated with A(r, 1) (see [5]). Weitsman's result states that s <= 1/(1+1/2(r log r)(2)) (see [8]). Our result for two-dimensional space states that s <= 1/(1+1/2 log(2) r) which improves Weitsman's bound for all r, and Lyzzaik's bound for r close to 1. For three-dimensional space the result states that s <= 1/(r - logr).