Best possible bounds for Yang mean using generalized logarithmic mean (Q1793713)

Summary: We prove that the double inequality \(L_p(a, b) < U(a, b) < L_q(a, b)\) holds for all \(a, b > 0\) with \(a \neq b\) if and only if \(p \leq p_0\) and \(q \geq 2\) and find several sharp inequalities involving the trigonometric, hyperbolic, and inverse trigonometric functions, where \(p_0 = 0.5451 \cdots\) is the unique solution of the equation \((p + 1)^{1 / p} = \sqrt{2} \pi / 2\) on the interval \((0, \infty)\), \(U(a, b) = (a - b) / [\sqrt{2} \arctan((a - b) / \sqrt{2 a b})]\), and \(L_p(a, b) = [(a^{p + 1} - b^{p + 1}) /((p + 1)(a - b))]^{1 / p}(p \neq - 1,0)\), \(L_{- 1}(a, b) = (a - b) /(\log a - \log b)\) and \(L_0(a, b) = (a^a / b^b)^{1 /(a - b)} / e\) are the Yang, and \(p\)th generalized logarithmic means of \(a\) and \(b\), respectively.