Complementary inequalities involving the Stolarsky mean (Q607971)

It is proved that the maximum of the function \[ f_{p,q}(a_1,\dots,a_n)=\frac{a_1^p+\dots+a_n^p}{n}-\left(\frac{a_1^q+\dots+a_n^q}{n}\right)^{p/q} \] (where \(p>q>0,\, b>a>0,\, a_1,\dots,a_n\in[a,b]\)) is attained if and only if \(k(n)\) of the numbers \(a_1,\dots,a_n\) are equal to \(a\) and the other \(n-k(n)\) are equal to \(b,\) while \(k(n)\) is one of the values \[ \left[n\frac{b^q-D^q_{p,q}(a,b)}{b^q-a^q}\right],\,\,\left[n\frac{b^q-D^q_{p,q}(a,b)}{b^q-a^q}\right]+1. \] Here \(D_{p,q}(a,b)\) is the Stolarsky mean of \(a\) and \(b\) of powers \(p\) and \(q.\) Also some asymptotic results for \(k(n)\) are given.