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In this paper we present different variants of the well-known Hermite–Hadamard inequality, in a generalized context. We consider general fractional integral operators for h-convex and r-convex functions. [ABSTRACT FROM AUTHOR]

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In a recently published paper with the same title, Debreu and Koopmans have studied conditions which imply the quasiconvexity of the function where x = ( x, x,⋯, x) and, for i = 1, 2,⋯, n, X is a finite-dimensional open convex set and f a real-valued nonconstant function on X These conditions involve the convexity indices of functions f, a concept introduced in the Debreu and Koopmans paper. First, we give a new definition of the convexity index equivalent to that of Debreu and Koopmans. Then, by means of this definition, we can simplify the proofs given by Debreu and Koopmans and extend some of their results. [ABSTRACT FROM AUTHOR]