Convolution and radius problems of analytic functions associated with the tilted Carathéodory functions

Abstract

The concept of convolution is applied to investigate some subordination results for the normalized analytic functions whose first derivative belongs to the class of the tilted Carathéodory functions. The sharp radius of starlikeness of order $\alpha$ of  the product of two normalized analytic functions satisfying certain specified conditions is computed. In addition, the various sharp radii constants such as the radius of lemniscate of Bernoulli starlikeness, radius of parabolic starlikeness and several other radius constants of product of two normalized analytic functions are also determined. Relevant connections of our results with the existing results are also pointed out.