A double inequality for the combination of Toader mean and the arithmetic mean in terms of the contraharmonic mean

Abstract

We find the greatest value λ and the least value μ such that the double inequality C(λa + (1 - λ)b, λb + (1 - λ)a) < αA(a, b) + (1 - α)T(a, b) < C(μa + (1 - μ)b, μb + (1 - μ)a) holds for all α ε (0, 1) and a, b > 0 with a 6 ≠ b, where C(a, b), A(a, b), and T(a, b) denote respectively the contraharmonic, arithmetic, and Toader means of two positive numbers a and b.