We prove the inequalities
\[
A
n
′
/
G
n
′
⩽
(
1
−
G
n
′
)
/
(
1
−
A
n
′
)
⩽
A
n
/
G
n
A_n’ /G_n’ \leqslant (1 - G_n’ )/(1 - A_n’ ) \leqslant {A_n}/{G_n}
\]
and
\[
A
n
′
/
G
n
′
⩽
(
1
−
G
n
)
/
(
1
−
A
n
)
⩽
A
n
/
G
n
,
A_n’ /G_n’ \leqslant (1 - {G_n})/(1 - {A_n}) \leqslant {A_n}/{G_n},
\]
where
A
n
{A_n}
and
G
n
{G_n}
(respectively,
A
n
′
A_n’
and
G
n
′
G_n’
) denote the unweighted arithmetic and geometric means of
x
1
,
…
,
x
n
{x_1}, \ldots ,{x_n}
(respectively,
1
−
x
1
,
…
,
1
−
x
n
1 - {x_1}, \ldots ,\;1 - {x_n}
) with
x
i
∈
(
0
,
1
2
]
(
i
=
1
,
…
,
n
;
n
⩾
2
{x_i} \in (0,\tfrac {1} {2}](i = 1, \ldots ,n;n \geqslant 2
. Further we show that the ratios
(
1
−
G
n
′
)
/
(
1
−
A
n
′
)
(1 - G_n’ )/(1 - A_n’)
and
(
1
−
G
n
)
/
(
1
−
A
n
)
(1 - {G_n})/(1 - {A_n})
can be compared if and only if
n
=
2
n = 2
.