Bounds for Combinations of Toader Mean and Arithmetic Mean in Terms of Centroidal Mean

Abstract EN

The authors find the greatest value λ and the least value μ, such that the double inequality C¯(λa+(1-λb),λb+(1-λ)a)<αA(a,b)+(1-α)T(a,b)0 with a≠b, where C¯(a,b)=2(a2+ab+b2)/3(a+b), A(a,b)=(a+b)/2, and Ta,b=2/π∫0π/2a2cos2θ+b2sin2θdθ denote, respectively, the centroidal, arithmetic, and Toader means of the two positive numbers a and b.