Abstract
We introduce log-preinvex and log-invex functions on a Riemannian manifold. Some properties and relationships of these functions are discussed. A characterization for the existence of a global minimum point of a mathematical programming problem is presented. Moreover, a mean value inequality under geodesic log-preinvexity is extended to Cartan-Hadamard manifolds.
Highlights
In Mathematical Sciences, convexity plays very crucial role and contributes a fundamental character in optimization theory, engineering, economics, management science, variational inequalities and Riemannian manifolds etc
In 1981, Hanson [1] gave a generalization of convex function, which was later known as an invex function
Let U ⊆ Mbe a geodesic invex set with respect to η : M × M → T Mand h : U → R be a geodesic log-preinvex function at ū ∈ U
Summary
In Mathematical Sciences, convexity plays very crucial role and contributes a fundamental character in optimization theory, engineering, economics, management science, variational inequalities and Riemannian manifolds etc. The geodesic geodesic E-convexity was proposed on a Riemannian manifold in [20] These results were further generalized in [21,22,23]. Analyzing the discussion of Pini [18], Barani and Pouryayevali [12], and Noor [7,8], we attempt an effort to introduce the geodesic log-preinvex and geodesic log-invex functions on Riemannian manifolds.
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