科研之友
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摘要
For fixed s >= 1 and any t(1), t(2) is an element of (0, 1/2) we prove that the double inequality G(s) (t(1)a + (1 - t(1))b, t(1)b + (1 - t(1))a)A(1-s) (a, b) < P(a, b) < G(s)(t(2)a + (1 - t(2))b, t(2)b + (1 - t(2))a)A(1-s)(a, b) holds for all a, b > 0 with a not equal b if and only if t(1) <= (1 - root 1 - (2/pi)(2/s))/2 and t(2) >= (1 - 1/root 3s)/2. Here, P(a, b), A(a, b) and G(a, b) denote the Seiffert, arithmetic, and geometric means of two positive numbers a and b, respectively.
