On some reverse mean inequalities (Q1815451)

\textit{L. Nanian} [J. Aust. Math. Soc., Ser. A 49, 319-326 (1990; Zbl 0715.26010)] and \textit{H. Alzer} [J. Math. Anal. Appl. 190, No. 3, 774-779 (1995; Zbl 0823.26013)] found upper bounds for the quotient of the \(p\)th and the \(q\)th (\(p>q\), \(p\) close to \(q\)) integral power means of positive decreasing functions while supposing that another integral inequality holds. The present authors weaken somewhat the monotonicity requirement in both and the integral inequality in Alzer's result.