On certain classes of harmonic functions defined by the fractional derivatives

TL;DR

This paper introduces new classes of harmonic functions defined via fractional derivatives, providing coefficient conditions, inclusion relations, and properties such as distortion and extreme points.

Contribution

It defines two new classes of harmonic functions with fractional derivative conditions and establishes coefficient criteria and fundamental properties for these classes.

Findings

Coefficient conditions for the classes are derived.

Inclusion relations and distortion theorems are established.

Properties like extreme points and convex combinations are analyzed.

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 \[ Re \{(1 - \lambda) \frac{\Omega^vf}{z} + \lambda(1-k) \frac{(\Omega^vf)'}{z'} + \lambda k \frac{(\Omega^vf)''}{z''} \} > \beta, (z\in \mathcal{D})\] 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.