A Sharp Double Inequality between Harmonic and Identric Means

and Applied Analysis 3 Theorem 1.1. If p, q ∈ 0, 1/2 , then the double inequality H ( pa ( 1 − pb, pb 1 − pa < I a, b < H ( qa ( 1 − qb, qb 1 − qa 1.8 holds for all a, b > 0 with a/ b if and only if p ≤ 1 − √ 1 − 2/e /2 and q ≥ 6 − √6 /12. 2. Proof of Theorem 1.1 Proof of Theorem 1.1. Let λ 6 − √6 /12 and μ 1 − √ 1 − 2/e /2. Then from the monotonicity of the function f x H xa 1 − x b, xb 1 − x a in 0, 1/2 we know that to prove inequality 1.8 we only need to prove that inequalities I a, b < H λa 1 − λ b, λb 1 − λ a , 2.1 I a, b > H ( μa ( 1 − μb, μb 1 − μa, 2.2 hold for all a, b > 0 with a/ b. Without loss of generality, we assume that a > b. Let t a/b > 1 and r ∈ 0, 1/2 , then from 1.1 and 1.2 one has logH ra 1 − r b, rb 1 − r a − log I a, b log { r 1 − r t2 [ r2 1 − r 2 ] t r 1 − r } − log t 1 − t log t t − 1 1 log 2. 2.3