On the interpolation constant for Orlicz spaces

In this paper we deal with the interpolation from Lebesgue spaces $L^p$ and $L^q$, into an Orlicz space $L^\varphi$, where $1\le p<q\le\infty$ and $\varphi^{-1}(t)=t^{1/p}\rho(t^{1/q-1/p})$for some concave function $\rho$, with special attention to the interpolation constant $C$. For a bounded linear operator $T$ in $L^p$ and $L^q$, we prove modular inequalities, which allow us to get the estimate for both the Orlicz norm and the Luxemburg norm,

where the interpolation constant $C$ depends only on $p$ and $q$. We give estimates for $C$, which imply $C<4$. Moreover, if either $1< p<q\le 2$ or $2\le p<q<\infty$, then $C< 2$. If $q=\infty$, then $C\le 2^{1-1/p}$, and, in particular, for the case $p=1$ this gives the classical Orlicz interpolation theorem with the constant $C=1$.