and Applied Analysis 3 2. Preliminaries A continuous function f u iv is a complex-valued harmonic function in a domain D ⊂ if both u and v are real harmonic in D. In any simply connected domain, we can write

The purpose of the present paper is to investigate some interesting properties on generalized convolutions of functions for the classes HP ∗(α),HS(α) andHC(α). Further, an application of the convolution on certain integral operator are mentioned. AMS 2010 Mathematics Subject Classification : 30C45, 26D15.

Ruscheweyh‘and She&Small proved the P6lya-Schoenberg conjecture that the class of convex analytic functions is closed under convolution or Hadamard product. They also showed that clos&o-convexity is preserved under convolution with convex analytic functions. In this note, we investigate harmonic analogs. Beginning with convex analytic functions, we form certain harmonic functions which preserve...

A continuous function f(x+ iy) = u(x, y) + iv(x, y) defined in a domain D ⊂ C (Complex plane) is harmonic in D if u and v are real harmonic in D. Clunie and Shiel-Small [3] showed that in a simply connected domain such functions can be written in the form f = h+g, where both h and g are analytic. We call h the analytic part and g, the co-analytic part of f . Let w(z) = g ′(z) h′(z) be the dilat...