On certain classes of harmonic functions defined by the fractional derivatives

2009/07/16 by M. Eshaghi Gordji, Gordji, M. Eshaghi, S. Shams +3
Mathematics · #30C45 (Primary) #30C80 (Secondary) #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:30C45 #msc:30C80

paper · pdf · doi:10.48550/arxiv.0907.2834

arxiv created 2009/07/16 · arxiv updated 2009/12/01

Abstract

In this paper we have introduced two new classes HM(β, λ, k, ν) and HM (β, λ, k, ν) of complex valued harmonic multivalent functions of the form f = h + g, satisfying the condition Re \(1 - λ) (Ω^vf)/(z) + λ(1-k) ((Ω^vf)')/(z') + λk ((Ω^vf)'')/(z'') \ > β, (z∈ D) where h and g are analytic in the unit disk D = \z : |z| < 1\. A sufficient coefficient condition for this function in the class HM(β, λ, k, ν) and a necessary and sufficient coefficient condition for the function f in the class HM(β, λ, k, ν) are determined. We investigate inclusion relations, distortion theorem, extreme points, convex combination and other interesting properties for these families of harmonic functions.

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