Generalized \(q\)-Srivastava-Attiya operator on multivalent functions

Let f be analytic in open unit disc $E=\{z:|z|<1\}$ with $f\left(0\right)=0$ and $f^{{}^{\prime }}\left(0\right)=1$ . The q-derivative of f is defined by: $D_{q}f\left(z\right)=\frac{f\left(z\right)-f\left(qz\right)}{(1-q)z},q\in (0,1),z\in \mathcal{B}-\left\{0\right\},$ where $\mathcal{B}$ is a q-geometric subset of $\mathbb{C}$ . Using operator $D_q$ , q-analogue class $k-U{M}_q(\alpha ,\beta )$ , k-uniformly Mocanu functions are defined as: For $k=0$ and $q\to 1^{-}$ , $k-$ reduces to $M\left(\alpha \right)$ of Mocanu functions. Subordination is used to investigate many important properties of these functions. Several interesting results are derived as special cases.

This article investigates the applications of the q-Carlson–Shaffer operator on subclasses of q-uniformly starlike functions, introducing the class $S{T}_q(m,c,d,\beta )$. The study establishes a necessary condition for membership in this class and examines its behavior within conic domains. The article delves into properties such as coefficient bounds, the Fekete–Szegö inequality, and criteria defined via the Hadamard product, providing both necessary and sufficient conditions for these properties.

This article presents a q-generalized linear operator in Geometric Function Theory (GFT) and investigates its application to classes of analytic bounded functions of complex order $S_q(c;M)$ and $C_q(c;M)$ where $0<q<1,$ $0\ne c\in \mathbb{C}$ , and $M>\frac{1}{2}.$ Integral inclusion of the classes related to the q-Bernardi operator is also proven.

The aim of the article is to investigate the applications of $q$-Srivastava-Attiya integral operator on the subclasses of $q$-starlike, $q$-convex and $q$-close-to-convex functions. The results are established by using the subordination result and $q$-analogue of Jack's Lemma. The characteristic properties including necessary condition and its applications, inclusions and integral preservations of the defined classes in the context of $q$-Bernardi operator are established.