Optimal Lehmer mean bounds for the combinations of identric and logarithmic means (Q463186)

Summary: For any \(\alpha\in(0,1)\), we answer the questions: what are the greatest values \(p\) and \(\lambda\) and the least values \(q\) and \(\mu\), such that the inequalities \(L_p(a,b)< I^\alpha(a,b)L^{1-\alpha}(a,b)<L_q(a,b)\) and \(L_\lambda(a,b)<\alpha I(a,b)+(1-\alpha)L(a,b)<L_\mu(a,b)\) hold for all \(a,b>0\) with \(a\neq b\)? Here, \(I(a,b)\), \(L(a,b)\), and \(L_p(a,b)\) denote the identric, logarithmic, and \(p\)th Lehmer means of two positive numbers \(a\) and \(b\), respectively.