A matrix inequality (Q1601618)

Using Hölder's inequality the inequality below is proved for an \(m\times n\) matrix \((x_{ij})\) with non-negative entries which are not all equal to zero for \(p\geq 1\) \[ {m^{p-1}+ n^{p-1}\over (mn)^{p-1}+ \min(m^{p-1}, n^{p-1})}\leq {m^{p-1}\sum^m_{i=1} (\sum^n_{j=1} x_{ij})^p+n^{p-1}\sum^n_{j=1} (\sum^m_{i=1} x_{ij})^p\over (\sum^m_{i=1}\sum^n_{j=1} x_{ij})^p+(mn)^{p-1} \sum^m_{i=1} \sum^n_{j=1} x^p_{ij}}. \] For \(p=2\) the result reduces to the inequality by \textit{H. Alzer} [Linear Algebra Appl. 323, 195-199 (2001; Zbl 0978.15017)].