Sharp Bounds for Toader Mean in terms of Arithmetic and Second Contraharmonic Means - Wei-Mao Qian; Ying-Qing Song; Xiao-Hui Zhang; Yu-Ming Chu (Journal of Function Spaces)
We present the best possible parameters λ1,μ1∈R and λ2,μ2∈1/2,1 such that double inequalities λ1C(a,b)+1-λ1A(a,b)<T(a,b)<μ1C(a,b)+1-μ1A(a,b), Cλ2a+1-λ2b,λ2b+1-λ2a<T(a,b)<Cμ2a+1-μ2b,μ2b+1-μ2a hold for all a,b>0 with a≠b, where A(a,b)=(a+b)/2, C(a,b)=a3+b3/a2+b2 and T(a,b)=2∫0π/2a2cos2θ+b2sin2θdθ/π are the arithmetic, second contraharmonic, and Toader means of a and b, respectively.