Let
K
ν
K_{\nu }
be the modified Bessel functions of the second kind of order
ν
\nu
. The ratio
Q
ν
(
x
)
=
x
K
ν
−
1
(
x
)
/
K
ν
(
x
)
Q_{\nu }\left ( x\right ) =xK_{\nu -1}\left ( x\right ) /K_{\nu }\left ( x\right )
appeared in physics and probability. In this paper, we collate properties of this ratio, and prove the conjecture that
(
−
1
)
n
Q
ν
(
n
)
(
x
)
>
(
>
)
0
\left ( -1\right ) ^{n}Q_{\nu }^{\left ( n\right ) }\left ( x\right ) >\left ( >\right ) 0
for
x
>
0
x>0
and
n
=
2
,
3
n=2,3
if
|
ν
|
>
(
>
)
1
/
2
\left \vert \nu \right \vert >\left ( >\right ) 1/2
holds for
n
=
2
n=2
. This yields several new consequences and improves some known results. Finally, two conjectures are proposed.