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In this paper, a new class of nonconvex optimization problem is considered, namely  $(h,\varphi)$-$(b,F,\rho)$-convexity is defined for $(h,\varphi)$-differentiable mathematical programming problem. The sufficiency of the so-called Karush-Kuhn-Tucker optimality conditions are established for the considered $(h,\varphi)$-differentiable mathematical programming problem under (Generalized) $(h,\varphi)$-$(b,F,\rho)$-convexity hypotheses. Further, the so-called  Mond-Weir $(h,\varphi)$-dual problem is defined for the considered $(h,\varphi)$-differentiable mathematical programming problem and several duality theorems in the sense of Mond-Weir are derived under appropriate (Generalized) $(h,\varphi)$-$(b,F,\rho)$-convex assumptions.

We are interested in proposing the convex fuzzy functions from R into E to describe the concepts of left and right Generalized derivatives. In this way, several properties of the concepts are discussed. As well, the notions of Generalized subdifferential for these functions are studied and calculated. Finally, the properties of the concepts are illustrated and applied through some examples.

In the present work, we prove a parametrized identity for a di erentiable function via Generalized integral operators. By applying the established identity and the new so{called Generalized m-convex function, some Generalized trapezium, Ostrowski and Simpson type integral inequalities have been discovered. Various special cases have been studied as well. Some applications of the present results to special means and new error estimates for the trapezium and midpoint quadrature formula have been investigated. It is hoped that the methods and techniques of this paper could further stimulate the research conducted in the  eld of integral inequalities.

This survey investigates some developments in the second-order characterization of Generalized convex functions using the coderivative of subdifferential mapping. More precisely, it presents the second-order characterization for quasiconvex, pseudoconvex and invex functions. Furthermore, it gives some applications of the second-order subdifferentials in optimization problems such as constrained and unconstrained nonlinear programming.

New variants of the Hermite - Hadamard inequality within the framework of Generalized fractional integrals for $(h,m,s)$-convex modified second type functions have been obtained in this article. To achieve these results, we used the Holder inequality and another form of it - power means. Some of the known results described in the literature can be considered as particular cases of the results obtained in our study.

We first construct new Hermite-Hadamard type inequalities which include Generalized fractional integrals for convex functions by using an operator which generates some significant fractional integrals such as Riemann-Liouville fractional and the Hadamard fractional integrals. Afterwards, Trapezoid and Midpoint type results involving Generalized fractional integrals for functions whose the derivatives in modulus and their certain powers are convex are established. We also recapture the previous results in the particular situations of the inequalities which are given in the earlier works.

Following a recent work of Chademan-Mirzapour (1995&1999). we generalize the classical theorems of Jensen, Bernstein-Doetsch, Ostrowski, Blumberg-Sierpinski and Mehdi on approximately midconvex functions in real vactor spaces to approximately midconvex functions on topological groups. We define also E -Wright-convexity in topological groups and prove a theorem on it