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Hadamard-type inequalities for quasiconvex functions. (English) Zbl 1025.26013
Among others, the following theorem is offered, weakening an assumption in a result by A. M. Rubinov and J. Dutta [J. Math. Anal. Appl. 270, 80-91 (2002; Zbl 1010.26017)]. Let $$X$$ be a convex Borel subset of $$\mathbb{R}^n$$, $$\mu$$ a finite Borel measure on $$X,$$ and $$f$$ a nonnegative Borel-measurable quasiconvex function, mapping $$X$$ into $$\mathbb{R}$$ extended by $$+\infty.$$ Then $$\inf_{v\in S} \mu(X_{v,u})f(u)\leq\int_X f\: d\mu$$ for all $$u\in X.$$

##### MSC:
 26D15 Inequalities for sums, series and integrals 26B25 Convexity of real functions of several variables, generalizations 28A05 Classes of sets (Borel fields, $$\sigma$$-rings, etc.), measurable sets, Suslin sets, analytic sets 28A25 Integration with respect to measures and other set functions
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