A sharp double inequality between Seiffert, arithmetic, and geometric means
Summary: For fixed \(s \geq 1\) and any \(t_1, t_2 \in (0, 1/2)\) 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 \(a \neq b\) if and only if \(t_1 \leq (1 - \sqrt{1 - (2/\pi)^{2/s}})/2\) and \(t_2 \geq (1 - 1/\sqrt{3s})/2\). 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.
scientific article; zbMATH DE number 6078783