Nonsmooth convexity and monotonicity in terms of a bifunction on Riemannian manifolds

In this paper, we introduce geodesic h-convexity, geodesic h-pseudoconvexity and geodesic h-quasiconvexity of a real-valued function defined on a geodesic convex subset of a Riemannian manifold in terms of a bifunction h. We extend Diewart's mean value theorem for Dini directional derivatives to the Riemannian manifolds. By using this mean value theorem, we present some relations between geodesic convexity and geodesic h-convexity, geodesic pseudoconvexity and geodesic h-pseudoconvexity, and geodesic quasiconvexity and geodesic h-quasiconvexity. We also introduce monotonicity, quasimonotonicity and pseudomonotonicity for the bifunction h. We investigate the relations between Geodesic h-convexity of a real-valued function and monotonicity of h, geodesic h-pseudoconvexity of a real-valued function and pseudomonotonicity of h, and geodesic h-quasiconvexity of a real-valued function and quasimonotonicity of h. We introduce the geodesic h-pseudolinearity of a real-valued function defined on geodesic convex subset of a Riemannian manifold. We provide some characterizations of geodesic h-pseudolinearity, and give some relations between geodesic h-pseudolinearity and geodesic pseudolinearity. The pseudoaffiness of a bifunction h is introduced and some of its characterizations are also presented.