In this paper, we present the sharp bounds of the ratios
$U(a,b)/L_{4}(a,b)$
,
$P_{2}(a,b)/U(a,b)$
,
$NS(a,b)/P_{2}(a,b)$
and
$B(a,b)/NS(a,b)$
for all
$a, b>0$
with
$a\neq b$
, where
$L_{4}(a,b)=[(b^{4}-a^{4})/(4(\log b-\log a))]^{1/4}$
,
$U(a,b)=(b-a)/[\sqrt{2}\arctan((b-a)/\sqrt{2ab})]$
,
$P_{2}(a,b)=[(b^{2}-a^{2})/(2\arcsin ((b^{2}-a^{2})/(b^{2}+a^{2})))]^{1/2}$
,
$NS(a,b)=(b-a)/[2\sinh ^{-1}((b-a)/(b+a))]$
,
$B(a,b)=Q(a,b)e^{A(a...
