On an inequality of Tchebyschef (Q915895)

Let \(n>k\geq 2\), \(p_ i>0\), \(a_ i>0\), \(b_ i>0\), \(i=1,...,n\), \(a_ 1\leq...\leq a_ n,\) \(b_ 1\leq...\leq b_ n,\) \(a_ 1<a_ k,\) \(b_ 1<b_ k.\) Let \(P_ n=p_ 1+...+p_ n\) and \(T_ n=\sum^{n}_{i=1}p_ ia_ ib_ i-(1/P_ n)\sum^{n}_{i=1}p_ ia_ i\sum^{n}_{i=1}p_ ib_ i.\) The author proves that \[ T_ n/T_ k>P_ k(P_ n-p_ 1)/(P_ n(P_ k-p_ 1)). \] Moreover, the right lower bound cannot be replaced by a greater value depending only on the weights \(p_ i\).