On Harmonic Functions Defined by Derivative Operator. (3rd December 2007) - Al-Shaqsi, K.; Darus, M.
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.