We prove the inequality
λ
2
[
V
1
]
−
λ
1
[
V
1
]
≥
λ
2
[
V
0
]
−
λ
1
[
V
0
]
{\lambda _2}[{V_1}] - {\lambda _1}[{V_1}] \geq {\lambda _2}[{V_0}] - {\lambda _1}[{V_0}]
for the difference of the first two eigenvalues of one-dimensional Schrödinger operators
−
d
2
d
x
2
+
V
i
(
x
)
,
i
=
0
,
1
- \frac {{{d^2}}}{{d{x^2}}} + {V_i}(x),i = 0,1
, where
V
1
{V_1}
and
V
0
{V_0}
are symmetric potentials on
(
a
,
b
)
(a,b)
and on
(
a
,
(
a
+
b
)
/
2
)
(a,(a + b)/2)
, and
V
0
−
V
1
{V_0} - {V_1}
is decreasing on
(
a
,
(
3
a
+
b
)
/
4
)
(a,(3a + b)/4)
.