Generalizations of Alzer's and Kuang's inequality (Q2724878)

Let \(f\) be a strictly increasing convex (or concave) function on the interval \((0,1]\), then the sequence \(\frac 1n\sum_{i=k+1}^{n+k}f(\frac{i}{n+k})\) is decreasing with natural number \(n\) and nonnegative integer \(k\) and has a lower bound \(\int_0^1f(t)dt\). If taking \(k=0\), Kuang's results can be deduced. If taking \(f(x)=x^r\) for \(r>0\), an Alzer type inequality \([({\frac {1}{n} \sum_{i=1}^{n}i^r})/({\frac {1}{n+1} \sum_{i=1}^{n+1}i^r })]^{1/r}>\frac{n+k}{n+m+k}\) can be obtained. Moreover, some inequalities involving factorials and ratios of geometric means are gained as easy consequences.NEWLINENEWLINENEWLINERecently, some new extended results are proven. Please refer to \([1]\) and \([2]\) and references therein: [1] \textit{F. Qi}, ``Monotonicity of sequences involving convex function and sequence'', RGMIA Res. Rep. Coll. 3, No. 2, Art.~14, 321-329 (2000), available online at\break \texttt{http://rgmia.vu.edu.au/v3n2.html}, [2] \textit{Ch.-P. Chen, F. Qi, P. Cerone}, and \textit{S. S. Dragomir}, ``Monotonicity of sequences involving convex and concave functions'', RGMIA Res. Rep. Coll. 5, No. 1, Art.~1 (2002), available online at \texttt{http://rgmia.vu.edu.au/v5n1.html}.