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

中文关键词:  geodesic $E$-convex set  geodesic $\gamma$-pre-$E$-convex function  geodesic $\gamma$-$E$-convex function  optimality conditions

英文关键词:geodesic $E$-convex set  geodesic $\gamma$-pre-$E$-convex function  geodesic $\gamma$-$E$-convex function  optimality conditions

中文摘要:

      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.

英文摘要:

      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.