A chain of mean value inequalities

Abstract

G = G(x, y) = ?xy, L = L(x,y) = x?y/log(x)?log(y)' I=I(x,y)= 1/e(xx?yy) 1/(x-y) be the geometric, logarithmic, identric, and arithmetic means of x and y. We prove that the inequalities L(G2,A2) < G(L2,I2) < A(L2,I2) < I(G2,A2) are valid for all x, y > 0 with x ? y. This refines a result of Seiffert.