Journal of Inequalities and Applications
Volume 2008 (2008), Article ID 263413, 10 pages
doi:10.1155/2008/263413
Research Article
On Harmonic Functions Defined by Derivative Operator
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600 Selangor D. Ehsan, Malaysia
Received 16 September 2007; Revised 20 November 2007; Accepted 26 November 2007
Academic Editor: Vijay Gupta
Copyright © 2008 K. Al-Shaqsi and M. Darus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let 𝒮ℋ denote the class of functions f=h+g¯ that are harmonic univalent and sense-preserving in the unit disk U={z:|z|<1}, where h(z)=z+∑k=2∞akzk, g(z)=∑k=1∞bkzk(|b1|<1). In this paper, we introduce the class Mℋ(n,λ,α) of functions f=h+g¯ which are harmonic in U. A sufficient coefficient of this class is determined. It is shown that this coefficient bound is also necessary for the class Mℋ¯(n,λ,α) if fn(z)=h+gn¯∈Mℋ(n,λ,α), where h(z)=z−∑k=2∞|ak|zk, gn(z)=(−1)n∑k=1∞|bk|zk and n∈ℕ0. Coefficient conditions, such as distortion bounds, convolution conditions, convex combination, extreme points, and neighborhood for the class Mℋ¯(n,λ,α), are obtained.