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-phi, where 1 less than or equal to p <q <less th an or equal to> infinity and phi (-1) (t) = t(1/p) rho (t(1/q-1/p)) for som e concave function rho, with special attention to the interpolation constan t C. For a bounded linear operator T in L-p and L-q, we prove modular inequ alities, which allow us to get the estimate for both the Orlicz norm and th e Luxemburg norm, parallel toT parallel to (L phi --> L phi) less than or equal to C max {par allel toT parallel to (Lp-->Lp), parallel toT parallel to (Lq-->Lq)}, where the interpolation constant C depends only on p and q. We give estimat es for C, which imply C <4. Moreover, if either 1 < p<q <less than or equal to> 2 or 2 less than or equal to p <q<infinity, then C <2. If q = <infinit y>, then C less than or equal to 2(1-1/p), and, in particular, for the case p = 1 this gives the classical Orlicz interpolation theorem with the const ant C = 1.