New generalization of geodesic convex function

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.