DEDICATED TO KSU LATE PROFESSOR KENNETH B. CUMMINS 7r27r1911᎐5r13r1998Complex-valued harmonic functions that are univalent and sense-preserving in the unit disk ⌬ can be written in the form f s h q g, where h and g are analytic in ⌬. We give univalence criteria and sufficient coefficient conditions for normalized harmonic functions that are starlike of order ␣, 0 F ␣ -1. These coefficient conditions are also shown to be necessary when h has negative and g has positive coefficients. These lead to extreme points and distortion bounds. ᮊ 1999 Academic Press

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A function analytic in the open unit disk D is said to be bi-univalent in D if both the function and its inverse map are univalent there. The bi-univalency condition imposed on the functions analytic in D makes the behavior of their coefficients unpredictable. Not much is known about the behavior of the higher order coefficients of classes of bi-univalent functions. We use Faber polynomial expansions of bi-univalent functions to obtain estimates for their general coefficients subject to certain gap series as well as providing bounds for early coefficients of such functions.

Some classes of uniformly starlike and convex functions are introduced. The geometrical properties of these classes and their behavior under certain integral operators are investigated

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We consider meromorphic starlike univalent functions that are also bi-starlike and find Faber polynomial coefficient estimates for these types of functions. A function is said to be bi-starlike if both the function and its inverse are starlike univalent.