On a class of harmonic functions associated with the $\mu $th order differential subordination

Abstract

In this paper, a class $\mathcal{P}_{H}^{\mu }(\alpha ,\beta )$ of functions
$f=h+\overline{g}$ which are the harmonic shears of the analytic functions $
h+g$ is defined and studied. A sufficient coefficient condition for
functions $f=h+\overline{g}\in \mathcal{P}_{H}^{\mu }(\alpha ,\beta )$ is
obtained. It is proved that this coefficient condition is necessary and
sufficient for functions belonging to its subclass $\mathcal{NP}_{H}^{\mu
}(\alpha ,\beta ).$ Results on bounds, extreme points, convolution, convex
combinations and integral operator for functions belonging to the subclass $%
\mathcal{NP}_{H}^{\mu }(\alpha ,\beta )$ are obtained. Inequalities for
certain hypergeometric harmonic functions belonging to these classes are
also given and the results based on one special case when $\mu =4$ are
included