An inequality between ratio of the extended logarithmic means and ratio of the exponential means. (Q1432836)

Let \(S_p(a,b)\) be the Stolarsky mean of \(a,b> 0\), where for \(a\neq b\), \(S_p(a,b)= ({b^p- a^p\over p(b- a)})^{1/(p-1)}\) \((p\neq 0,1)\); \(S_0(a,b)= L(a,b)\), \(S_1(a, b)= I(a, b)\), with \(L\) and \(I\) denoting the logarithmic, resp. identric mean. The authors prove that \(S_{r+1}(a, b)/S_{r+ 1}(a,b+ k)< S_1(a, b)/S_1(a, b+k)\) for all \(r\in\mathbb{R}\) and \(k> 0\).