The present paper provides several corrected dual-Simpson-type inequalities for functions whose local fractional derivatives are generalized convex. To that end, we derive a new local fractional integral identity as an auxiliary result. Using this new identity along with generalized Hölder’s inequality and generalized power mean inequality, we establish some new variants of fractal corrected dual-Simpson-type integral inequalities. Furthermore, some applications for error estimates of quadrature formulas as well as some special means involving arithmetic and p-logarithmic mean are offered to demonstrate the efficacy of our findings.

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.

The growing dependence on optimization models in decision-making has created a demand for tools that can facilitate the formulation and resolution of a broader range of real-world processes and systems associated with human activity. These situations often involve assumptions that diverge from traditional optimization methodologies. One viable approach for addressing optimization problems in real-life scenarios with uncertainty is interval-valued optimization. Taking into account the significance of interval-valued optimization, in this paper, we derive first and second order necessary optimality conditions for a multi-objective programming problem with interval-valued objective functions defined on a Riemannian manifold. To establish these conditions, we consider the objective functions to be weakly differentiable and twice weakly differentiable for first and second order, respectively. Additionally, we assume that the constraints, both equality and inequality constraints, are differentiable and twice differentiable for first and second order conditions respectively. The first order as well as second order necessary conditions are derived under two types of constraint qualifications. Furthermore, we provide illustrative examples to demonstrate the application of the established results.