We show that if
X
X
is a rearrangement invariant space on
[
0
,
1
]
[0, 1]
that is an interpolation space between
L
1
L_{1}
and
L
∞
L_{\infty }
and for which we have only a one-sided estimate of the Boyd index
α
(
X
)
>
1
/
p
,
1
>
p
>
∞
\alpha (X) > 1/p, 1 > p > \infty
, then
X
X
is an interpolation space between
L
1
L_{1}
and
L
p
L_{p}
. This gives a positive answer for a question posed by Semenov. We also present the one-sided interpolation theorem about operators of strong type
(
1
,
1
)
(1, 1)
and weak type
(
p
,
p
)
,
1
>
p
>
∞
(p, p), 1 > p > \infty
.