On certain classes of harmonic functions defined by the fractional derivatives

Abstract

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