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 Iq,ρs(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 Kq,ρs(ν,τ)φ, Qq,ρs(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.