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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)<C¯(μa+(1-μ)b,μb+(1-μ)a) holds for all α∈(0,1) and 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.