engleski

Integral mean value bounds for h-convex functions

Let I and J be intervals in {; ; \bf R}; ; , ( 0, 1 ) \subseteq J and let h: J \rightarrow {; ; \bf R}; ; be a non-negative function, h\not\equiv 0. We say that f:I\rightarrow {; ; \bf R}; ; is an h-convex function if f is non-negative and for all x, y\in I, \alpha \in (0, 1), we have f(\alpha x +(1-\alpha)y)\leq h(\alpha)f(x)+h(1-\alpha)f(y). The h-convex functions are a generalization of non-negative convex, s-convex, Godunova-Levin functions and P-functions. This observation leads us to the unified treatment of these several varieties of convexity. Some properties of h-convex functions are discussed. Especially, integral mean value bounds for h-convex function and related results are derived.

h-convex function ; integral mean

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