Some results for certain subclasses of functions with differential equation and subordination

Abstract

By applying the differential subordination theorem, we further investigate the subclass $H^{ngamma}_lambda [alpha,beta]$ of functions which are analytic in the unit disk. Several subordination results on a convex function and a incomplete beta function are obtained. Moreover, the function that belongs to the $H^{ngamma}_lambda [alpha,beta]$ with a Cauchy-Euler differential equation is also discussed on similar subject. Our results extend some earlier works.