This article presents certain families of analytic functions regarding q-starlikeness and q-convexity of complex order $\gamma (\gamma \in \mathbb{C}\backslash \left\{0\right\})$ . This introduced a q-integral operator and certain subclasses of the newly introduced classes are defined by using this q-integral operator. Coefficient bounds for these subclasses are obtained. Furthermore, the $(\delta ,q)$-neighborhood of analytic functions are introduced and the inclusion relations between the $(\delta ,q)$-neighborhood and these subclasses of analytic functions are established. Moreover, the generalized hyper-Bessel function is defined, and application of main results are discussed.

The goal of this article is to establish several new formulas and new results related to the Marichev-Saigo-Maeda fractional integral and fractional derivative operators which are applied on the $(p,q)$-extended Bessel function. The results are expressed as the Hadamard product of the $(p,q)$-extended Gauss hypergeometric function $F_{p,q}$ and the Fox-Wright function ${}_r{\mathsf{\Psi }}_s\left(z\right)$. Some special cases of our main results are considered. Furthermore, the $(p,q)$-extended Bessel-Wright function is introduced. Finally, a variety of formulas for the Marichev-Saigo-Maeda fractional integral and derivative operators involving the $(p,q)$-extended Bessel-Wright function is established.

In the present research paper, our aim is to introduce a new subfamily of meromorphic p-valent (multivalent) functions. Moreover, we investigate sufficiency criterion for such defined family.