G. Bennett [Linear Algebra Appl. 82, 81–98 (1986; Zbl 0601.15014)] proved the inequality
\[ \|Ax\|_q\geq \lambda \|x\|_p, \]
where
\[ \|Ax\|_q^q=\sum_{j=1}^m\left (\sum_{k=1}^n a_{j,k}x_k \right )^q \text{ and } \lambda ^q=\min_{1\leq r\leq n}r^{-q/p}\sum_{j=1}^m\left (\sum_{k=1}^r a_{j,k}\right )^q, \]
for \(x=(x_1,\dots ,x_n)\), \(x_1\geq \cdots \geq x_n\geq 0\), \(p\geq 1\), \(0<q\leq p\), \(A=(a_{j,k})\), \(1\leq j\leq m\), \(1\leq k\leq n\) with \(a_{j,k}\geq 0\). The inequality is reversed when \(0<p\leq 1\) and \(q\geq p\) with min replaced by max in \(\lambda.\) The author gives an alternate proof of this result using the approach of J. Bergh [Math. Z. 202, No. 1, 147–149 (1989; Zbl 0661.46022)]. He then uses this inequality to prove the following theorem.
Theorem: Let \(0<p<1\), \(\alpha \geq 1\), \(\alpha p<1\), \(0\leq t\leq 1\), \(x_1\geq x_2\geq \cdots \geq 0\). Then
\[ \sum _{n=1}^\infty\left (\frac{1}{n^\alpha}\sum_{k=n}^\infty\left ( (k+t)^\alpha -(k+t-1)^\alpha \right )x_k\right )^p \leq \frac{1}{\alpha}B\left (\frac{1}{\alpha }-p,p+1\right )\sum_{n=1}^\infty x_n^p, \]
where \(B(x,y)\), \(x>0\), \(y>0\), is the beta function. Some consequences of this theorem are deduced in Section 4.