Sharp bounds for the Neuman mean in terms of the quadratic and second Seiffert means

In this paper, we prove that alpha = 0 and beta = root 3 pi-4log(2+root 3)/(root 2 pi-4)log(2+root 3) = 0.29758 ... are the best possible constants such that the double inequality alpha Q(a, b) + (1 - alpha)T(a, b) < S-CA(a, b) < beta Q(a, b) + (1 - beta)T(a, b) holds for all a, b > 0 with a not equal b, where Q(a, b) = root(a(2) + b(2))/2, S-CA(a, b) = (a - b)root 3(a(2) + b(2)) + 2ab/2(a + b) sinh(-1)((a-b)root 3(a(2)+ b(2))+ 2ab/(a+ b)(2)) and T(a, b) = (a - b)/[2 arctan((a - b)/(a + b))] are the quadratic, Neuman and second Seiffert means of a and b, respectively.