An optimal double inequality between Seiffert and geometric means (Q410774)

Summary: For \(\alpha, \beta \in (0, 1/2)\) we prove that the double inequality \(G(\alpha a + (1 - \alpha)b, \alpha b + (1 - \alpha)a) < P(a, b) < G(\beta a + (1 - \beta)b, \beta b + (1 - \beta)a)\) holds for all \(a, b > 0\) with \(a \neq b\) if and only if \(\alpha \leq (1 - \sqrt{1 - 4/\pi^2})/2\) and \(\beta \geq (3 - \sqrt{3})/6\). Here, \(G(a, b)\) and \(P(a, b)\) denote the geometric and Seiffert means of two positive numbers \(a\) and \(b\), respectively.