On comparison of Stolarsky and Gini means (Q1874549)

A Stolarski mean is defined by \[ D_{a,b}(x,y)=[(b/a)(x^a-y^a)/(x^b-y^b)]^{1/(a-b)} \] if \((a-b)ab\neq 0,\) while a Gini mean by \[ S_{a,b}(x,y)=[(x^a +y^a)/(x^b+y^b)]^{1/(a-b)} \] if \(a\neq b,\) with limiting cases when these conditions about \(a,b\) are not satisfied. Authors offer inequalities between \(D_{a,b}\) and \(S_{a,b}\), and between \(D_{a,b}\) and \(S_{a-1,b-1}\) with necessary and sufficient conditions for the former but mostly just necessary conditions for the latter, and it is stated as open problem whether the latter conditions are also sufficient.