Sharp Stolarsky mean bounds for the complete elliptic integral of the second kind

In the article, we prove that the double inequality \[25/16<ε(r)/S5/2,2(1,{r})<π/2,\] holds for all \(r ∈ (0, 1)\) with the best possible constants \(25/16\) and \(π/2\), where \({r}=(1-r^2)1/2, ε(r)=∫π/2_0√1-r^2sin^2(t)dt\) , is the complete elliptic integral of the second kind and \(Sp,q(a,b)=[q(a^p-b^p)/(p(a^q-b^q))]1/(p-q)\), is the Stolarsky mean of a and b.