doiSerbia

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

Jiang Wei-Dong (Weihai Vocational College, Department of Information Engineering Shandong Province, Weihai City, China)
Qi Feng (Inner Mongolia University for Nationalities Tongliao City, College of Mathematics, Inner Mongolia Autonomous Region, China + Tianjin Polytechnic University College of Science, Department of Mathematics, Tianjin City, China + Henan Polytechnic University I)

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  (0,1) and a, b > 0 with a ≠ 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.

Keywords: bound, contraharmonic mean, arithmetic mean, Toader mean, complete elliptic integrals