We consider the eigenvalue problem
u
+
λ
u
+
p
(
x
)
u
=
0
u+\lambda u +p(x)u=0
in
(
0
,
π
)
(0,\pi )
,
u
(
0
)
=
u
(
π
)
=
0
u(0)=u(\pi )=0
, where
p
∈
L
1
(
0
,
π
)
p\in L^{1}(0,\pi )
keeps a fixed sign and
‖
p
‖
L
1
>
0
\|p\|_{L^{1}}> 0
, and we obtain some lower and upper bounds for
‖
p
‖
L
1
\|p\|_{L^{1}}
in terms of its nonnegative eigenvalues
λ
\lambda
. Two typical results are: (1)
‖
p
‖
L
1
>
λ
|
sin
λ
π
|
\|p\|_{L^{1}}>\sqrt {\lambda }\,|\sin {\sqrt {\lambda }\,\pi }|
if
λ
>
1
\lambda > 1
and is not the square of a positive integer; (2)
‖
p
‖
L
1
≤
16
/
π
\|p\|_{L^{1}}\le 16/\pi
if
λ
=
0
\lambda =0
is the smallest eigenvalue.