Best Possible Inequalities between Generalized Logarithmic Mean and Classical Means

Abstract

We answer the question: for $\alpha ,\beta ,\gamma \in \left(\mathrm{0,1}\right)$ with $\alpha +\beta +\gamma =1$, what are the greatest value $p$ and the least value $q$, such that the double inequality $L_p(a,b)<A^{\alpha }(a,b)G^{\beta }(a,b)H^{\gamma }(a,b)<L_q(a,b)$ holds for all $a,b>0$ with $a\ne b$? Here $L_p(a,b)$, $A(a,b)$, $G(a,b)$, and $H(a,b)$ denote the generalized logarithmic, arithmetic, geometric, and harmonic means of two positive numbers $a$ and $b$, respectively.