Sharp Bounds for Toader Mean in terms of Arithmetic and Second Contraharmonic Means

Abstract

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)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.