New generalization of geodesic convex function

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