On Weakly H-Quasiconvex Functions on the Heisenberg Group

The aim of this paper is the investigation of weakly H-quasiconvex functions in the framework of the Heisenberg group $\mathbb{H}$. The functions of this class, recently introduced by M. B. Sun and X. P. Yang ["Some properties of quasiconvex functions on the Heisenberg Group", Acta Math. Appl. Sinica 21 (2005) 571–580], are defined via the property that their sublevels are weakly H-convex subsets of $\mathbb{H}$. Following Crouzeix's approach, we prove conditions of first and second order, easily testable, for regular weakly H-quasiconvex functions, and we provide a characterization of them.