On certain classes of harmonic functions defined by the fractional derivatives

英文摘要：

In this paper we have introduced two new classes $\mathcal{H}\mathcal{M}(β, λ, k, ν)$ and $\overline{\mathcal{H}\mathcal{M}} (β, λ, k, ν)$ 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 $\mathcal{H}\mathcal{M}(β, λ, k, ν)$ and a necessary and sufficient coefficient condition for the function $f$ in the class $\overline{\mathcal{H}\mathcal{M}}(β, λ, k, ν)$ are determined. We investigate inclusion relations, distortion theorem, extreme points, convex combination and other interesting properties for these families of harmonic functions.