Convolution Properties of Harmonic Univalent Functions Preserved by Some Integral Operator

Abstract

A complex valued function f = u+ iv defined in a domain D ⊂ C, is harmonic in D, if u and v are real harmonic. Such functions can be represented as f(z) = h(z) + g(z), where h an g are analytic in D. In this paper we study some convolution properties preserved by the integral operator In H,λf, n ∈ N0 = N ∪ {0}, λ > 0, where the functions f are univalent harmonic and sense-preserving in the open unit disc E = {z : |z| < 1}, I H,λf(z) = I λh(z) + In λg(z), and I n hh(z) = z + ∞ ∑ k=2 ak [1 + λ(k − 1)]n z, I λg(z) = ∞ ∑ k=1 bk [1 + λ(k − 1)]n z. Relevant connections of the results presented here with those obtained in earlier works are pointed out. 2000 Mathematics Subject Classification: 30C45.

Cite this paper

@inproceedings{AlOboudi2010ConvolutionPO, title={Convolution Properties of Harmonic Univalent Functions Preserved by Some Integral Operator}, author={Fatima M. Al-Oboudi}, year={2010} }