In the article, we prove that the double inequalities
$M_{\alpha }(a,b)< S_{QA}(a,b)< M_{\beta}(a,b)$
and
$M_{\lambda }(a,b)< S_{AQ}(a,b)< M_{\mu}(a,b)$
hold for all
$a, b>0$
with
$a\neq b$
if and only if
$\alpha\leq\log 2/[1+\log2-\sqrt{2}\log(1+\sqrt{2})]=1.5517\ldots$
,
$\beta\geq5/3$
,
$\lambda\leq4\log2/[4+2\log2-\pi]=1.2351\ldots$
and
$\mu\geq4/3$
, where
$S_{QA}(a,b)=A(a,b)e^{Q(a,b)/M(a,b)-1}$
and
...
