Sharp power mean bounds for two Sandor-Yang means

In the article, we prove that the double inequalities Ma(a, b) < Q(a, b) eG(a, b)/U(a, b)-1 < M beta(a, b), M.(a, b) < G(a, b) eQ(a, b)/V(a, b)-1 < M mu(a, b) hold for all a, b > 0 with a = b if and only if a = 2 log2/(2 + log 2) = 0.5147 center dot center dot center dot, beta = 2/3,. = 2 log2/(2 -log 2) = 1.0607 center dot center dot center dot and mu = 4/3, where Mp(a, b) = [(a p + bp)/2] 1/p (p = 0), M0(a, b) = G(a, b) = v ab, Q(a, b) = (a2 + b2)/2, U(a, b) = (a -b)/[ v 2. arctan((a -b)/v 2ab)] and V(a, b) = (a -b)/[ v 2 sinh -1((a -b)/v 2ab)] are respectively the pth power, geometric, quadratic, first Yang and second Yang means, and sinh -1(x) is the inverse hyperbolic sine function.