Properties for Close-to-Convex and Quasi-Convex Functions Using q -Linear Operator

In this work, we describe the q -analogue of a multiplier–Ruscheweyh operator of a specific family of linear operators I q , ρ s ( ν , τ ) , and we obtain findings related to geometric function theory (GFT) by utilizing approaches established through subordination and knowledge of q -calculus operators. By using this operator, we develop generalized classes of quasi-convex and close-to-convex functions in this paper. Additionally, the classes K q , ρ s ( ν , τ ) φ , Q q , ρ s ( ν , τ ) φ are introduced. The invariance of these recently formed classes under the q -Bernardi integral operator is investigated, along with a number of intriguing inclusion relationships between them. Additionally, several unique situations and the beneficial outcomes of these studies are taken into account.