An extension of an inequality for nondecreasing sequences (Q1193066)

\textit{A. Meir} [Rocky Mt. J. Math. 11, 577-579 (1981; Zbl 0482.26007)] proved: Theorem A. Let \(0=a_ 0\leq a_ 1\leq\cdots\leq a_ n\). Suppose \(a_ i- a_{i-1}\leq p_ i\) \((i=1,2,\dots,n)\) and \((*)\) \(p_ 1\leq p_ 2\leq\cdots\leq p_ n\). If \(r\geq 1\) and \(s+1\geq 2(r+1)\), then \[ \left((s+1)\sum^{n-1}_{i=1} a^ s_ i{p_ i+p_{i+1}\over 2}\right)^{1/(s+1)}\leq \left((r+1)\sum^{n-1}_{i=1} a^ r_ i{p_ i+p_{i+1}\over 2}\right)^{1/(r+1)}. \] In this paper we have proved that the conclusion of Theorem A remains valid if \((*)\) is replaced by \((**)\) \(p_ i\leq p_ n\) \((i=1,2,\dots,n-1)\).