Comparison theorems for a nonlinear eigenvalue problem as well as a Lyapunov type of inequality are derived. They are used to establish upper and lower bounds for various integral functionals associated with real solutions of the nonlinear boundary value problem
y
+
p
(
x
)
y
2
n
+
1
=
0
,
y
(
a
)
=
y
′
(
b
)
=
0
y + p\left ( x \right ){y^{2n + 1}} = 0, y\left ( a \right ) = y’\left ( b \right ) = 0
, where
a
>
b
a > b
are real,
n
n
is a positive integer and
p
p
is positive and continuous on
[
a
,
b
]
\left [ {a,b} \right ]
. Some of the results are analogues of a distance between zeros problem for the linear case of
n
=
0
n = 0
.