In this article, a new concept of convexity, B −1 -convexity, is defined. In this context, B −1 -convex sets and B −1 -measurable maps are discussed and examined.
A subset U of n + is -convex if for all x, y ∈ U and all ∈ [0, 1] one has x ∨ y ∈ U . These sets were introduced and studied by Briec, Horvath, Rubinov and Adilov [7,8,10]
This concept is defined and studied by Adilov, Briec, and Yesilce. In this work, -convex and−1 -convex functions are defined and some fundamental theorems about these functions are proved, additionally some important properties of -convex and −1 -convex sets are compared then the construction of sets is described with graphics.
Recently, fractional calculus has become a very popular and important area. Specially, fractional integral inequalities have been studied by different authors. In this article, we give new Hermite-Hadamard type inequalities for B-convex functions via Riemann-Liouville and Hadamard fractional integrals. Also, we show that the inequalities involve the fractional integrals of a function with respect to the function g which are the more general form of these obtained Hermite-Hadamard inequalities.
B −1 -convexity is an abstract convexity type. We obtained Hermite-Hadamard inequality for B −1 -convex functions. But now, there are new and more general integral operator types that are fractional integrals. Thus, we need to prove Hermite-Hadamard inequalities involving different fractional integral operator types with this article.
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