The object of this paper is to study the local properties of light harmonic mappings.

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

Abstract.The authors study open continuous functions which map the unit disc to compact Riemann surfaces and which assume each value in the range space (with a finite number of exceptions) either p or q times for some positive integers p, q . Although the questions here originated in efforts to understand mapping properties of locally univalent analytic functions, the authors remove analyticity assumptions and show that the underlying issues are topological and combinatoric in nature. The mappings are studied by embedding their image surfaces in compact covering spaces, a setting which allows the consideration of fairly general ranges and which accommodates branch and exceptional points. Known results are generalized and extended; several open questions are posed, particularly regarding the higher dimensional analogues of the results.

The object of the paper is to show that if f is a univalent, harmonic mapping of the annulus A(r, 1) = {z : r < [mid ]z[mid ] < 1} onto the annulus A(R, 1), and if s is the length of the segment of the Grötzsch ring domain associated with A(r, 1), then R < s. This gives the first, quantitative upper bound of R, which relates to a question of J. C. C. Nitsche that he raised in 1962. The question of whether this bound is sharp remains open.

Let be an open Riemann surface with finite genus and finite number of boundary components, and let be a closed Riemann surface. An open continuous function from to is termed a (p, q)-map, 0 < q < p, if it has a finite number of branch points and assumes every point in either p or q times, counting multiplicity, with possibly a finite number of exceptions. These comprise the most general class of all non-trivial functions having two valences between and .The object of this paper is to study the geometry of (p, q)-maps and establish a generalized embedding theorem which asserts that the image surfaces of (p, q)-maps embed in p-fold closed coverings possibly having branch points off the image surfaces.