Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters

摘要

In the article, we present the best possible parameters λ = λ(p) and μ = μ(p) on the interval [0, 1/2] such that the double inequality Gp[λa+(1−λ)b,λb+(1−λ)a]A1−p(a,b)<E(a,b)<Gp[μa+(1−μ)b,μb+(1−μ)a]A1−p(a,b) holds for any p ∈ [1, ∞) and all a, b  0 with a ≠ b, where A(a, b) = (a + b)/2, G(a,b)=ab and E(a,b)=[2∫0π/2acos2θ+bsin2θdθ/π]2 are the arithmetic, geometric and special quasi-arithmetic means of a and b, respectively.