New generalization of geodesic convex function

Abstract

As a generalization of geodesic function, this paper introduces the notion of geodesic $ \varphi_{E} $-convex function. Some properties of $ \varphi_{E} $-convex function and geodesic $ \varphi_{E} $-convex function are established. The concepts of geodesic $ \varphi_{E} $-convex set and $ \varphi_{E} $-epigraph are equally given. The characterization of geodesic $ \varphi_{E} $-convex functions in terms of their $ \varphi_{E} $-epigraphs are also obtained.