On Harmonic Functions Defined by Derivative Operator. (3rd December 2007)

• Record Type: Journal Article
• Authors: Al-Shaqsi, K.; Darus, M.
• Other Names: Gupta Vijay Academic Editor.
• Is Part Of: Journal of inequalities and applications. Volume 2008(2008)
• Publication Date: 2007-12-03
• Subjects: Inequalities (Mathematics) -- Periodicals; 512.97
• DOI: 10.1155/2008/263413
• Languages: English
• ISSNs: 1029-242X

Abstract

Abstract : Let𝒮 ℋ denote the class of functionsf = h + g ¯ that are harmonic univalent and sense-preserv- ing in the unit disk𝕌 = { z : | z | < 1 }, whereh ( z ) = z + ∑ k = 2 ∞ a k z k, g ( z ) = ∑ k = 1 ∞ b k z k ( | b 1 | < 1 ) . In this paper, we introduce the classM ℋ ( n, λ, α ) of functionsf = h + g ¯ which are harmonic in𝕌 . A sufficient coefficient of this class is determined. It is shown that this coefficient bound is also necessary for the classM ℋ ¯ ( n, λ, α ) iff n ( z ) = h + g n ¯ ∈ M ℋ ( n, λ, α ), whereh ( z ) = z − ∑ k = 2 ∞ | a k | z k, g n ( z ) = ( − 1 ) n ∑ k = 1 ∞ | b k | z k andn ∈ ℕ 0 . Coefficient conditions, such as distortion bounds, convolution conditions, convex combination, extreme points, and neighborhood for the classM ℋ ¯ ( n, λ, α ), are obtained.