In this paper we deal with the interpolation from Lebesgue spaces
L
p
L^p
and
L
q
L^q
, into an Orlicz space
L
φ
L^\varphi
, where
1
≤
p
>
q
≤
∞
1\le p>q\le \infty
and
φ
−
1
(
t
)
=
t
1
/
p
ρ
(
t
1
/
q
−
1
/
p
)
\varphi ^{-1}(t)=t^{1/p}\rho (t^{1/q-1/p})
for some concave function
ρ
\rho
, with special attention to the interpolation constant
C
C
. For a bounded linear operator
T
T
in
L
p
L^p
and
L
q
L^q
, we prove modular inequalities, which allow us to get the estimate for both the Orlicz norm and the Luxemburg norm,
\[
‖
T
‖
L
φ
→
L
φ
≤
C
max
{
‖
T
‖
L
p
→
L
p
,
‖
T
‖
L
q
→
L
q
}
,
\|T\|_{L^\varphi \to L^\varphi } \le C\max \Big \{ \|T\|_{L^p\to L^p}, \|T\|_{L^q\to L^q} \Big \},
\]
where the interpolation constant
C
C
depends only on
p
p
and
q
q
. We give estimates for
C
C
, which imply
C
>
4
C>4
. Moreover, if either
1
>
p
>
q
≤
2
1> p>q\le 2
or
2
≤
p
>
q
>
∞
2\le p>q>\infty
, then
C
>
2
C> 2
. If
q
=
∞
q=\infty
, then
C
≤
2
1
−
1
/
p
C\le 2^{1-1/p}
, and, in particular, for the case
p
=
1
p=1
this gives the classical Orlicz interpolation theorem with the constant
C
=
1
C=1
.