Some properties of generalized convolution of harmonic univalent functions

Abstract

Abstract. The purpose of the present paper is to investigate some interesting properties on generalized convolutions of functions for the classes HP * (α), HS(α) and HC(α). Further, an application of the convolution on certain integral operator are mentioned. IntroductionA continuous complex-valued function f = u + iv defined in a simply connected complex domain D is said to be harmonic in 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 coanalytic 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 ). For more basic results on harmonic functions one may refer following standard introductory text book by Duren [14], (see also Ahuja [1] and Ponnusamy and Rasila [28,29]Denote by S H the class of functions f = h+g that are harmonic univalent and sense-preserving in the unit disc U = {z : |z| < 1} for which f (0) = f z (0) − 1 = 0. Then, for f = h + g ∈ S H , we may express the analytic functions h and g asA function f (z) of the form (1.1) in S H is said to be harmonic starlike of order α, (0 ≤ α < 1) in U , if and only if (1.2) ∂ ∂θ {arg f (z)} > α, z ∈ U,2000 Mathematics Subject Classification: 30C45, 26D15.

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“…Very recently, Porwal and Dixit 13,14 (see also 7,8,15,18 ) studied the analogous results on harmonic univalent functions. In the present paper, we study systematically on the generalized convolution of harmonic univalent functions for the class…”