An Optimal Double Inequality between the harmonic root mean Square and Identric Mean

摘要

We present the largest value a and the least value β in (0, 1/2) such that the double inequality H(αα + (l-α)b,αb + (l-α)a) 0 with a ≠ b. Here, I(a,b) = l/e(b~b/a~a)~(l/(b-a)) and H(a,b) = 1/2ab/1/(a~2 + b~2) denote the identric mean and harmonic root mean square of two positive numbers a and b, respectively. Mathematics Subject Classification (2010): 26E60.