Abstract. In this paper, we first prove the coefficient conjecture of Clunie and Sheil-Small for a class of univalent harmonic functions which includes functions convex in some direction. Next, we prove growth and covering theorems and some related results. Finally, we propose two conjectures. An affirmative answer to one of which would then imply for example a solution to the conjecture of Clunie and Sheil-Small.

Abstract. In this article, we determine the radius of univalence of sections of normalized univalent harmonic mappings for which the range is convex (resp. starlike, close-to-convex, convex in one direction). Our result on the radius of univalence of section s n,n (f ) is sharp especially when the corresponding mappings have convex range. In this case, each section s n,n (f ) is univalent in the disk of radius 1/4 for all n ≥ 2, which may be compared with classical result of Szegö on conformal mappings.

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In this paper, we prove necessary and sufficient conditions for a sensepreserving harmonic function to be absolutely convex in the open unit disk. We also estimate the coefficient bound and obtain growth, covering and area theorems for absolutely convex harmonic mappings. A natural generalization of the classical Bernardi type operator for harmonic functions is considered and its connection between certain classes of uniformly starlike harmonic functions and uniformly convex harmonic functions is also investigated. At the end, as applications, we present a number of results connected with hypergeometric and polylogarithm functions.