Inequalities of Rado-Popoviciu type. (Q2776287)

The first result obtained in this paper is the inequality \(\frac{n}{n+1}\leq\frac{[(n-1)!]^{1/(n-1)}}{[n!]^{1/n}}\) which is an improvement of the inequality \(\frac{n}{n+1}<\frac{[n!]^{1/n}}{[(n+1)!]^{1/(n+1)}}\) by H. Minc and L. Sathre, where \(n\) denotes any natural number. Recently, these inequalities were generalized to \(\frac{n+k+1}{n+m+k+1}<\frac{[(n+k)!/k!]^{1/n}}{[(n+m+k)!/k!]^{1/(n+m)}}< \sqrt{\frac{n+k}{n+m+k}}\) for positive integers \(m\) and \(n\) and nonnegative integer \(k\) by \textit{B.-N. Guo} and \textit{F. Qi} in [``Inequalities and monotonicity of sequences involving \({[(n+k)!/k!]^{1/n}}\)'', RGMIA Res. Rep. Coll. 2, No.~5, Art.~8, 685--692 (1999), available online at \url{http://rgmia.vu.edu.au/v2n5.html}; ``Inequalities and monotonicity for the ratio of gamma functions'', Taiwanese J. Math. 7, No.~2, 239--247 (2003; Zbl 1050.26012); ``Some inequalities involving the geometric mean of natural numbers and the ratio of gamma functions'', RGMIA Res. Rep. Coll. 4, No.~1, Art.~6, 41--48 (2001), available online at \url{http://rgmia.vu.edu.au/v4n1.html}]. More general, the function \(\frac{[{\Gamma(x+y+1)}/{\Gamma(y+1)}]^{1/x}}{x+y+1}\) is decreasing in \(x\geq1\) for fixed \(y\geq0\). Some weeks ago, the following monotonicity and convexity results were proved by Ch.-P. Chen and F. Qi: The function \(\frac{[\Gamma(x+1)]^{1/x}}{x+1}\) is strictly decreasing and strictly logarithmically convex on \((0,+\infty)\); the function \(\frac{[\Gamma(x+1)]^{1/x}}{\sqrt{x+1}}\) is strictly increasing and strictly logarithmically concave on \((0,+\infty)\). NEWLINENEWLINEThe second result means that the sequence \(\frac{\frac1n\sum_{i=1}^ni^r}{[n!]^{r/n}}\) for natural number \(n\) is increasing for any given \(r\geq1\) which is equivalent to an inequality \(\left(\frac{\frac {1}{n} \sum_{i=1}^{n}i^r}{\frac {1}{n+1} \sum_{i=1}^{n+1}i^r }\right)^{1/r} < \frac {[n!]^{1/n}}{[(n+1)!]^{1/(n+1)}}\) for \(r\geq1\) and natural number \(n\). As immediate consequences of this monotonicity and the arithmetic-geometric inequality, two Rado-Popoviciu type inequalities were established. In [\textit{T. H. Chan, P. Gao} and \textit{F. Qi}, ``On a generalization of Martins' inequality'', RGMIA Research Rep. Coll. 4, No.~1, Art.~12, 93--101 (2001), available online at \url{http://rgmia.vu.edu.au/v4n1.html}], these results were extended and generalized to \(\left(\frac{\frac {1}{n} \sum_{i=k+1}^{n+k}(ai+b)^r}{\frac {1}{n+m} \sum_{i=k+1}^{n+m+k}(ai+b)^r}\right)^{1/r} < \frac {\left[\prod_{i=k+1}^{n+k}(ai+b)\right]^{1/n}} {\left[\prod_{i=k+1}^{n+m+k}(ai+b)\right]^{1/(n+1)}}\) for \(r>0\), where \(a\) and \(b\) are positive real numbers, \(k\) is a nonnegative integer, and \(m\) and \(n\) be natural numbers.NEWLINENEWLINEMoreover, the monotonicity of the function \(\phi(r)=\left(\frac{ \frac1n\sum_{i=1}^{n}i^r}{\frac1{n-1}\sum_{i=1}^{n-1}i^r}\right)^{1/r }\) was discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0978.00017].