Liu, Wen-Jun; Cheng, Guo-Sheng; Li, Chun-Cheng Further development of an open problem concerning an integral inequality. (English) Zbl 1169.26007 JIPAM, J. Inequal. Pure Appl. Math. 9, No. 1, Paper No. 14, 5 p. (2008). 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)]. Reviewer: Constantin Niculescu (Craiova) Cited in 5 Documents MSC: 26D15 Inequalities for sums, series and integrals Keywords:integral inequality; Young’s inequality; integration by parts Citations:Zbl 1154.26312 PDFBibTeX XMLCite \textit{W.-J. Liu} et al., JIPAM, J. Inequal. Pure Appl. Math. 9, No. 1, Paper No. 14, 5 p. (2008; Zbl 1169.26007) Full Text: EuDML EMIS