We make a survey of recent results on the geometrical properties of smooth bump functions and the geometrical and topological properties of starlike bodies in Banach spaces. We try to stress the interplay between infinitedimensional topology and nonlinear functional analysis by relating questions about topological and geometrical properties of starlike bodies to other interesting problems on no...

In this paper, we consider a new class of analytic functions in the unit disk using polynomials of order alpha. We give some sufficient conditions for functions belonging to this class.