Geodesic $gamma$-Pre-$E$-Convex Functions on Riemannian Manifolds

In this paper, we generalize geodesic $E$-convex function and define geodesic $gamma$-pre-$E$-convex and geodesic $gamma$-$E$-convex functions on Riemannian manifolds. The sufficient condition of equivalence class of geodesic $gamma$-pre-$E$-convexity and geodesic $gamma$-$E$-convexity for differentiable function on Riemannian manifolds is studied. We discuss the sufficient condition for $E$-epigraph to be geodesic $E$-convex set. At the end, we establish some optimality results with the aid of geodesic $gamma$-pre-$E$-convex and geodesic $gamma$-$E$-convex functions and discuss the mean value inequality for geodesic $gamma$-pre-$E$-convex function.