Coefficient estimates for the class of q-starlike functions

TL;DR

This paper investigates the coefficient bounds of q-starlike functions using convolution, Carathéodory-Toeplitz, and Parseval's theorems, providing new insights into their properties for real and complex q.

Contribution

It offers a comprehensive analysis of coefficient estimates for q-starlike functions, extending previous work with new bounds and determinant evaluations for real and complex q.

Findings

Derived upper bounds for initial coefficients for real q in (0,1)

Analyzed Hankel and Toeplitz determinants of q-starlike functions

Established coefficient bounds for complex q using Parseval's theorem

Abstract

By employing the $q$-difference operator, various classes of $q$-extensions of starlike functions have emerged from many different viewpoints and perspectives. Ruscheweyh's work unified these $q$-extensions with convolution operations. Inspired by prior research, this paper delves into the class of $q$-starlike functions defined via convolution. First, we provide comprehensive analysis of its Taylor coefficients by using the Carath\'eodory-Toeplitz Theorem and its corollaries. Precise upper bounds for the initial few coefficients are obtained for real $q\in(0,1)$. Furthermore, we analyze both Hankel and Toeplitz determinants by using the foundational results. Second, we establish the upper bounds of its coefficients with the application of Parseval's theorem for $q$-starlike functions when $q$ is a complex number.