Convolution Properties of <i>q</i>-Janowski-Type Functions Associated with (<i>x</i>,<i>y</i>)-Symmetrical Functions

Abstract

The purpose of this paper is to define new classes of analytic functions by amalgamating the concepts of q-calculus, Janowski type functions and (x,y)-symmetrical functions. We use the technique of convolution and quantum calculus to investigate the convolution conditions which will be used as a supporting result for further investigation in our work, we deduce the sufficient conditions, Po´lya-Schoenberg theorem and the application. Finally motivated by definition of the neighborhood, we give analogous definition of neighborhood for the classes S˜qx,y(α,β) and K˜qx,y(α,β), and then investigate the related neighborhood results, which are also pointed out.

Keywords

• analytic functions
• Hadamard product
• (<i>x</i>,<i>y</i>)-symmetrical functions
• <i>q</i>-calculus
• (<i>ρ</i>,<i>q</i>)-neighborhood