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Convex functions and the Hadamard inequality. (English) Zbl 0832.26015
Summary: The Hadamard inequality is proven without resorting to any properties of the derivative. Only the convexity of the function in a closed interval is needed. Furthermore, if the existence of the integral is assumed, then the convexity requirement is weakened to convexity in the sense of Jensen. Both the Hadamard inequality and a corresponding upper bound are generalized for integrals of the Stieltjes type.

MSC:
26D15 Inequalities for sums, series and integrals
26A51 Convexity of real functions in one variable, generalizations
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