On a converse Cauchy inequality of D. Zagier (Q1205556)

The following discrete analogue of an inequality of \textit{D. Zagier} [Nederl. Akad. Wet., Proc., Ser. A 80, 349-351 (1977; Zbl 0368.26003)] is proved: If \(a_ i\) and \(b_ i\) \((i=1,\dots,n)\) are real numbers such that \(a_ 1\geq a_ 2\geq\cdots\geq a_ n>0\) and \(b_ 1\geq b_ 2\geq\cdots\geq b_ n>0\), then \[ \sum^ n_{i=1}a^ 2_ i\sum^ n_{i=1}b^ 2_ i\left/\max\left(a_ 1\sum^ n_{i=1}b_ i,\;b_ 1\sum^ n_{i=1}a_ i\right)\right.\leq\sum^ n_{i=1}a_ ib_ i, \] with equality if and only if \(a_ 1=\cdots=a_ n\) and \(b_ 1=\cdots=b_ n\).