Sharp Geometric Mean Bounds for Neuman Means

We find the best possible constants alpha(1), alpha(2), beta(1), beta(2) is an element of [0,1/2] and alpha(3), alpha(4), beta(3), beta(4) is an element of [1/2,1] such that the double inequalities G(alpha(1)a + (1 - alpha(1))b, alpha(1)b + (1 - alpha(1))a) < N-AG(a,b) < G(beta(1)a + (1 - beta(1))b, beta(1)b + (1 - beta(1))a), G(alpha(2)a + (1 - alpha(2))b, alpha(2)b + (1 - alpha(2))a) < N-GA (a,b) < G(beta(2)a + (1 - beta(2))b, beta(2)b + (1 - beta(2))a), Q(alpha(3)a + (1 - alpha(3))b, alpha(3)b + (1 - alpha(3))a) < N-QA(a,b) < Q(beta(3)a + (1 - beta(3))b, beta(3)b + (1 - beta(3))a), Q(alpha(4)a + (1 - alpha(4))b, alpha(4)b + (1 - alpha(4))a) < N-AQ(a,b) < Q(beta(4)a + (1 - beta(4))b, beta(4)b + (1 - beta(4))a) hold for all a,b > 0 with a not equal b, where G,A, and Q are, respectively, the geometric, arithmetic, and quadratic means and N-AG, N-GA, N-QA, and N-AQ are the Neuman means.