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

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 aligned& Gᵖ[λ a+(1-λ)b, λ b+(1-λ)a ]A¹⁻ᵖ(a,b) & < E(a,b) < Gᵖ[μ a+(1-μ)b, μ b+(1-μ)a ]A¹⁻ᵖ(a,b) aligned 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 /2√acos²θ+bsin²θ\,dθ/π]² are the arithmetic, geometric and special quasi-arithmetic means of a and b, respectively.