and Applied Analysis 3 Example 2.3. Let X be a subset in Rn defined as follows: X : { x1, x2 | 0 < x2 < x2 1, 0 < x1 < 2 } ∪ { 0, 0 }. 2.2 Consider the point u 0, 0 . Since the tangent line of the curve x2 x2 1 at point u is the line x2 0. Then, for any x ∈ X \ {u}, there exists 0 < λ0 < 1 such that u λη x, u / ∈ X, ∀λ ∈ 0, λ0 . 2.3 Therefore, there exists no vector-valued function η x, u / 0 s...

The present paper deals with the properties of geodesic η-preinvex functions and their relationships with η-invex functions and strictly geodesic η-preinvex functions. The geodesic η-pre-pseudo-invex and geodesic η-pre-quasi-invex functions on the geodesic invex set are introduced and some of their properties are discussed.

In this paper, we define and introduce some new concepts of strongly φ-preinvex (φ-invex) functions and strongly φη-monotone operators. We establish some new relationships among various concepts of φ-preinvex (φ-invex) functions. As special cases, one can obtain various new and known results from our results. Results obtained in this paper can be viewed as refinement and improvement of previous...

A class of functions called E-B-vex functions is defined as a generalization of E-convex and B-vex functions. Similarly, a class of E-B-preinvex functions, which are generalizations of E-convex and B-preinvex functions, is introduced. In addition, the concept of B-linear functions is also generalized to E-B-linear functions. Some properties of these proposed classes are studied. Furthermore, th...

The concept of a geodesic invex subset of a Riemannian manifold is introduced. Geodesic invex and preinvex functions on a geodesic invex set with respect to particular maps are defined. The relation between geodesic invexity and preinvexity of functions on manifolds is studied. Using proximal subdifferential, certain results concerning extremum points of a non smooth geodesic preinvex function ...

Abstract. In this note, we give some estimates of the generalized quadrature formula of Gauss-Jacobi type for phi-preinvex functions.