Akhlad Iqbal

On Geodesic E-Convex Sets, Geodesic E-Convex Functions and E-Epigraphs

In this paper, we introduce a new class of sets and a new class of functions called geodesic E-convex sets and geodesic E-convex functions on a Riemannian manifold. The concept of E-quasiconvex functions on Rn is extended to geodesic E-quasiconvex functions on Riemannian manifold and some of its properties are investigated. Afterwards, we generalize the notion of epigraph called E-epigraph and discuss a characterization of geodesic E-convex functions in terms of its E-epigraph. Some properties of geodesic E-convex sets are also studied.

On Properties of Geodesic -Preinvex Functions

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.

Strong Geodesic α-Preinvexity and Invariant α-Monotonicity on Riemannian Manifolds

In this article, we introduce and study a new class of generalized convex functions on Riemannian manifold, called strongly α-invex and strongly geodesic α-preinvex functions. Several kinds of invariant α-monotonicities on Riemannian manifold are introduced. We establish the relationships among the strong α-invexity, strong geodesic α-preinvexity and invariant α-monotonicities under suitable conditions. Various types of α-invexities for functions on Riemannian manifolds are introduced and relations among them are established.

Geodesic G-invex sets and semistrictly geodesic η-preinvex functions

In this article, we introduce semistrictly geodesic η-preinvex functions on Riemannian manifolds, geodesic G-invex sets and study their properties. Our results extend and improve the results of Yang and Li [X. Yang and D. Li, Semistrictly preinvex functions, J. Math. Anal. Appl. 258 (2001), pp. 287–308]. Example is constructed in support of our defnition.

Some Integral Inequalities for Log-Preinvex Functions

In this paper, a new type of Hermite–Hadamard inequalities is established for log-preinvex functions. Some natural applications to special means of real numbers are also discussed.