A Convolution Approach on Partial Sums of Certain Harmonic Univalent Functions

A continuous complex-valued function f u iv is said to be harmonic in a simply connected domain D if both u and v are real harmonic in D. In any simply-connected domain we can write f h g, where h and g are analytic in D. We call h the analytic part and g the co-analytic part of f . A necessary and sufficient condition for f to be locally univalent and sense-preserving in D is that |h′ z | > |g ′ z |, z ∈ D, see 1 . For more basic results on harmonic functions one may refer to the following standard text book by Duren 2 . See also Ahuja 3 and Ponnusamy and Rasila 4, 5 . Denote by SH the class of functions f h g which are harmonic univalent and sensepreserving in the open unit disk U {z : |z| < 1} for which f 0 fz 0 − 1 0. Then for f h g ∈ SH we may express the analytic functions h and g as