SHARP BOUNDS FOR NEUMAN-SANDOR MEAN IN TERMS OF THE CONVEX COMBINATION OF QUADRATIC AND FIRST SEIFFERT MEANS

In this article, we prove that the double inequality alpha P(a, b) + (1-alpha)Q(a, b) < M (a,b) < beta P(a, b) + (1-beta)Q (a,b) holds for any a, b > 0 with a not equal b if and only if alpha > 1/2 and beta <= [pi root 2(log(1 + root 2 -1)] /[root 2 pi-2) log(1 + root 2)] = 0.359..., where M(a, b), Q(a, b), and P(a, b) are the Neuman-Sandor, quadratic, and first Seiffert means of a and b, respectively.