Subclasses of Analytic Functions with Negative Coefficients Involving q-Derivative Operator

Abstract

Let A denote the class of functions f which are analytic in the open unit disk U. The subclass of A consisting of univalent functions is denoted by M. In this paper, we also consider a subclass of M which is denoted by V, consisting of functions with negative coefficients. In addition, this paper also studies the q-derivative operator. By combining the ideas, this paper introduced three subclasses of A with negative coefficients involving q-derivative. Furthermore, the coefficient estimates, growth results and extreme points were obtained for all of these classes.

The Subclasses of Analytic Functions of Complex Order with Application of q-Derivative Operators

“…Further, if q → 1 − , the class S * q (𝛾) and C q (𝛾) reduces to the starlike functions of order 𝛾, S * (𝛾), and convex functions of order 𝛾, C (𝛾), respectively. Furthermore, there are several new subclasses of A involving q-derivative that have been introduced by other mathematicians (see Aldweby and Darus, 2017;Alsoboh and Darus, 2019;Altinkaya and Yalçin, 2017;Bulut, 2017;Janteng and Halim, 2009b;Karahuseyin, 2017;Lashin et al, 2021;Hern et al, 2022;Hern et al, 2020;Olatunji and Dutta, 2018;Piejko and Sokół, 2020;Ramachandran et al, 2017;Shamsan et al, 2021;Shilpa, 2022). In fact Seoudy and Aouf (2016) have introduced how classes of q-convex and q-starlike of complex order can be obtained using the principle of subordination and q-derivative.…”