Proof of one optimal inequality for generalized logarithmic, arithmetic, and geometric means (Q608012)

The main theorem is an answer on two open problems given by \textit{B.-Y. Long} and \textit{Y.-M. Chu} [J. Inequal. Appl. 2010, Article ID 806825 (2010; Zbl 1187.26015)]. Theorem: Let \(\alpha \in (0,1/2)\cup (1/2,1)\), \(a\not= b\), \(a>0\), \(b>0\). Let \(p(\alpha)\) be a solution of \[ \frac 1p \ln (1+p)+\ln (\frac{\alpha}{2})=0 \] in \((-1,1)\). Then, if \(\alpha \in (0,1/2)\), then \[ \alpha A(a,b)+(1-\alpha)G(a,b) <L_p(a,b) \] for \(p\geq p(\alpha) \) and \(p(\alpha)\) is the best constant, and if \(\alpha \in (1/2,1)\), then \[ \alpha A(a,b)+(1-\alpha)G(a,b) >L_p(a,b) \] for \(p\leq p(\alpha) \) and \(p(\alpha)\) is the best constant. \(A(a,b), G(a,b)\) and \(L_p(a,b)\) are notations for an arithmetic mean, a geometric mean and a generalized logarithmic mean respectively.