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Further development of an open problem concerning an integral inequality. (English) Zbl 1169.26007

The main result of this paper is as follows: Let \(\alpha\) and \(\beta\) be two positive numbers and let \(f\) be a nonnegative continuous function on an interval \([a,b]\) such that \(\int_x^b f^{\min\{\beta,1\}}(t)\,dt\geq \int _x^b(t-a)^{\min\{\beta,1\}}\,dt\) for every \(x\in[a,b].\) Then \(\int_{a}^{b}f^{\alpha+\beta}(t)\,dt\geq \int_{a}^{b}(t-a)^\alpha f^\beta (t)\,dt.\) This solves an open problem raised by Q. A. NgĂ´ et al. in their “Note on an integral inequality” [JIPAM, J. Inequal. Pure Appl. Math. 7, No. 4, Paper No. 120, 5 p., electronic only (2006; Zbl 1154.26312)].

MSC:

26D15 Inequalities for sums, series and integrals

Citations:

Zbl 1154.26312
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