The Modulus of the Image Annuli Under Univalent Harmonic Mappings and a Conjecture of Nitsche

The object of the paper is to show that if f is a univalent,harmonic mapping of the annulus A(r, 1) = {z : r < |z| <1} onto the annulus A(R, 1), and if s is the length of the segmentof the Grötzsch ring domain associated with A(r, 1), thenR < s. This gives the first, quantitative upper bound ofR, which relates to a question of J. C. C. Nitsche that he raisedin 1962. The question of whether this bound is sharp remainsopen.     

Uniqueness of harmonic mappings with Blaschke dilatations

Let Ω be a bounded convex domain and let ω be a finite Blaschke product of order N = 1, 2, .... It is known that the elliptic differential equation admits a one-to-one solution normalized by ƒ(0) = 0, ƒ_z(0) > 0 and maps the open unit disc onto a convex (n + 2)-gon whose vertices belong to ∂Ω. In this article it is shown that this solution is unique.     

Close-to-Convexity Criteria for Planar Harmonic Mappings

We give a criteria for planar harmonic mappings to be univalent close-to-convex which settles a conjecture of P. T. Mocanu.     

On a conjecture of D. Styer regarding univalent geometric and annular starlike functions

The aim of this paper is two-fold. First, to give a direct proof for the already established result of Styer which states that a univalent geometrically starlike function is a univalent annular starlike function if is bounded. Second, to show that the boundedness condition of is necessary, thus disproving a conjecture of Styer.