Sharp Bounds for Neuman Means in Terms of Geometric, Arithemtic and Quadratic Means

الأصل · EN

In this paper, we find the greatest values α₁, α₂, α₃, α₄, α₅, α₆, α₇, α₈ and the least values β₁, β₂, β₃, β₄, β₅, β₆, β₇, β₈ such that the double inequalities Aα₁(a,b)G¹⁻α₁(a,b)<NGA(a,b)<Aβ₁(a,b)G¹⁻β₁(a,b), α₂G(a,b)+1-α₂A(a,b)<1NGA(a,b)<β₂G(a,b)+1-β₂A(a,b), Aα₃(a,b)G¹⁻α₃(a,b)<NAG(a,b)<Aβ₃(a,b)G¹⁻β₃(a,b), α₄G(a,b)+1-α₄A(a,b)<1NAG(a,b)<β₄G(a,b)+1-β₄A(a,b), Qα₅(a,b)A¹⁻α₅(a,b)<NAQ(a,b)<Qβ₅(a,b)A¹⁻β₅(a,b), α₆A(a,b)+1-α₆Q(a,b)<1NAQ(a,b)<β₆A(a,b)+1-β₆Q(a,b), Qα₇(a,b)A¹⁻α₇(a,b)<NQA(a,b)<Qβ₇(a,b)A¹⁻β₇(a,b), α₈A(a,b)+1-α₈Q(a,b)<1NQA(a,b)<β₈A(a,b)+1-β₈Q(a,b) hold for all a, b>0 with a≠ b, where G, A and Q are respectively the geometric, arithmetic and quadratic means, and NGA, NAG, NAQ and NQA are the Neuman means derived from the Schwab-Borchardt mean.