A generalization of a result of A. Meir for non-decreasing sequences
\textit{A. Meir} [Rocky Mt. J. Math. 11, 577-579 (1981; Zbl 0482.26007)] has proved the following result: Let \(0\leq p_ 1\leq p_ 2\leq...\leq p_ n\) and \(0=a_ 0\leq a_ 1\leq...\leq a_ n\), satisfying \(a_ i-a_{i- 1}=p_ i\quad (i=1,2,...,n).\) If \(r\geq 1\) and \(s+1\geq 2(r+1),\) then \[ (1)\quad ((s+1)\sum^{n-1}_{i=1}a_ i^ s\frac{p_ i+p_{i+1}}{2})^{1/(s+1)}\leq ((r+1)\sum^{n-1}_{i=1}a^ r_ i\frac{p_ i+p_{i+1}}{2})^{1/(r+1)}. \] For \(p_ i=1\) \((i=1,...,n)\) we have an inequality of \textit{M. S. Klamkin} and \textit{D. J. Newman} [Am. Math. Mon. 83, 26-30 (1976; Zbl 0329.26018)]. In this paper, the authors proved an inequality which is stronger than inequality (1). They also proved a generalization of (1). \{The reviewer notes that similar integral inequalities were considered in two papers. One is the mentioned paper of Klamkin and Newman, and the other is by the reviewer [Rad. 78, Odjeljenje Prir. Mat. Nauka 24, 91-93 (1985; Zbl 0579.26005).\}
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