On the Biholomorphic Convex Mappings of Order Alpha on $$D_p^n$$ D p n

Abstract

In this paper, we first give the definition of biholomorphic convex mappings of order \(\alpha \) on the Reinhardt domain \(D_{p}^n\) in \(C^{n}\). Next, we provide several sufficient conditions for it. Then we introduce a subclass \(SK(D_{p}^n, \alpha )\) of biholomorphic convex mappings of order \(\alpha \) on \(D_{p}^n\), and give a necessary and sufficient condition for the subclass \(SK(D_{p}^n, \alpha )\). From these, we construct some concrete examples of biholomorphic convex mappings of order \(\alpha \) on \(D_{p}^n\). The results presented extend the related results of earlier authors.