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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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