On certain classes of harmonic functions defined by the fractional derivatives

Details On certain classes of harmonic functions defined by the fractional derivatives On certain classes of harmonic functions defined by the fractional derivatives

2009-07-16

arxiv.org/abs/0907.2834v1

Mathematics

Complex Variables

Scientific paper

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.

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