Interpolation between $L_{1}$ and $L_{p}, 1 < p < \infty$

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