Toader型平均与几种经典平均的确界

给出了最佳参数α_1,α_2,α_3,β_1,β_2,β_3∈R,使得双向不等式α_1Q(a,b)+(1-α_1)G(a,b)0且a≠b成立.其中A(a,b)=(a+b)/2,H(a,b)=2ab/(a+b),G(a,b)=(ab)~(1/2),Q(a,b)=((a~2+b~2)/2)~(1/2),C(a,b)=(a~2+b~2)/(a+b),T(a,b)=2/π∫_0~(π/2)(a~2cos~2t+b~2sin~2)~(1/2)tdt分别是两个正数a和b的算术平均,调和平均,几何平均,二次平均,反调和平均和Toader平均.

Sharp Bounds for Toader-type Mean by Other Classic Bivariate Means

In this paper,we present the best possible parameters α1,α2,α3,β1,β2,β3 ∈ R such that the double inequalitiesα1Q(a,b) + (1-α1) G(a,b) ＜ T A(a,b),Q(a,b)] ＜β1Q(a,b) + (1-β1) G(a,b),α2Q(a,b) + (1-α2) H(a,b) ＜ T A(a,b),Q(a,b)] ＜β2Q(a,b) + (1-β2) H(a,b),α3C(a,b) + (1-α3) H(a,b) ＜ T A(a,b),Q(a,b)] ＜ β3C(a,b) + (1-β3) H(a,b)hold for all a,b ＞ 0 with a ≠ b,where A(a,b) =(a + b)/2,H(a,b) =2ab/ (a + b),G(a,b) =√ab,Q(a,b)=√(a2+b2)/2,C(a,b)=(a2 +b2) /(a+b) and T(a,b)=2/π.∫on/2.√a2 cos2 t + b2 sin2 tdt are respectively the arithmetic,harmonic,geometric,quadratic,contraharmonic and Toader means of a and b,respectively.