A CHAIN OF EIGHT INEQUALITIES INVOLVING MEANS OF TWO ARGUMENTS

For two positive real numbers a and b, let H: = H(a, b ), G:= G(a, b), A := A(a, b ) and Q : Q(a, b) be the harmonic mean, the geometric mean, the arithmetic mean and the quadratic mean of a and b, respectively. In this short note, we prove the following interesting chain involving eight inequalities: G <= root QH <= root AG <= A+G/2 <= Q + H/2 <= root A(2) + G(2)/2 <= root Q(2) + H-2/2 <= Q + G/2 <= A, where equality holds in each of these inequalities if and only if a = b . Some remarks, in particular connected with Muirhead's inequality, and two questions related to a similar form of chain of inequalities are also given.