Sufficient Conditions for a Certain General Class of Caratheodory Functions

Abstract

In the present paper, the authors derive sufficient conditions for functions to be in a certain
general class of Carath\'{e}odory functions in the open unit disk by using
the Miller-Mocanu lemma. As an application of our main result, we deduce
sufficient
conditions for functions belonging to the class
$\mathcal{ST}\hat{\mathcal{S}}_\beta^\mu$ which is introduced here. The various results
presented here would generalize and extend many known results.