On Generalizations of the Close-to-Convex Functions Associated with <semantics>q</semantics>-Srivastava–Attiya Operator - Daniel Breaz; Abdullah A. Alahmari; Luminiţa-Ioana Cotîrlă; Shujaat Ali Shah (Mathematics, 2023)
The study of the <semantics>q</semantics>-analogue of the classical results of geometric function theory is currently of great interest to scholars. In this article, we define generalized classes of close-to-convex functions and quasi-convex functions with the help of the <semantics>q</semantics>-difference operator. Moreover, by using the <semantics>q</semantics>-analogues of a certain family of linear operators, the classes <semantics><msubsup>Kq,bs</msubsup><mfenced open="(" close=")">h</mfenced></semantics>, <semantics><msubsup><mover accent="true">K˜</mover>q,sb</msubsup><mfenced open="(" close=")">h</mfenced></semantics>, <semantics><msubsup>Qq,bs</msubsup><mfenced open="(" close=")">h</mfenced></semantics>, and <semantics><msubsup><mover accent="true">Q˜</mover>q,sb</msubsup><mfenced open="(" close=")">h</mfenced></semantics> are introduced. Several interesting inclusion relationships between these newly defined classes are discussed, and the invariance of these classes under the <semantics>q</semantics>-Bernadi integral operator was examined. Furthermore, some special cases and useful consequences of these investigations were taken into consideration.