Starlikeness and Convexity of q-polyharmonic Univalent Mappings

Abstract

The connection of polyharmonic mappings with q-calculus is not explored yet. In this paper, we attempt to construct a bridge between polyharmonic mappings and q-calculus by genaralizing the results obtained by Chen et al. (Bull Belg Math Soc Simon Stevin 21:67-82, 2014) to q-calculus. We first introduce the notion of q-polyharmonic mappings and define two new subclasses of q-polyharmonic mappings, denoted by SHp,q & lowast;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}<^>*_{{\mathcal {H}}_{p,q}}$$\end{document} and CHp,q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {C}}_{{\mathcal {H}}_{p,q}}$$\end{document}, associated with q-calculus. Then, we prove that these classes are univalent and sense-preserving in the open unit disk D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {D}}$$\end{document}. Further, we explore two characterizations in terms of convolution for q-polyharmonic mappings to be starlike and convex, respectively. We next investigate coefficient estimates and extreme points for functions belonging to these classes. Finally, we introduce radii of starlikeness and radii of convexity for these function classes.