Inequalities of Hermite-Hadamard type for \(h\)-convex functions on linear spaces (Q2812610)

In recent years, many extensions of the concept of convexity have been given by modifying the basic Jensen inequality. These various extensions were unified by \textit{S. Varošanec} [J. Math. Anal. Appl. 326, No. 1, 303--311 (2007; Zbl 1111.26015)], with the class of \(h\)-convex functions defined by the inequality: \(f \bigl(tx + (1-t)y\bigr)\leq h(t)f(x) + h(1-t)f(y), \forall t, 0<t<1\), where \(h:(0,1)\mapsto \mathbb R\) and \(f\), \(h\) non-negative. The author extends this to functions defined on convex sub-sets of a linear space and obtains an extension of the Hermite-Hadamard inequality for this general situation. Refinements of the immediate inequality are given and there is an extensive bibliography.