Some inclusion properties for the class of $q$-calculus operators

Abstract

UDC 517.54

In recent years, the $q$-difference operator has been primarily used to examine the characteristics of analytic functions. We~develop and analyze a unified $q$-analog of a family of linear operators arising in the geometric function theory. Our~primary objective is to generate some subclasses of analytic functions by using the $q$-analog of a linear operator. We~use the idea of $q$-analog to construct certain differential and integral operators $\mathfrak{C}_q^\nu (\eta,\mu;.)$ and $\mathfrak{Z}_{q,\zeta}$, respectively, to~extend the C\v{a}ta\c{s} and Noor operators over the set of univalent functions. By means of these operators and convolution, we establish a new operator $\mathfrak{ZC}_{q,\zeta}^{\nu} (\eta,\mu;.)$. Further, we study inclusion relations that have integral preservation features.