Interpolation of uniformly convex Banach spaces

If AO and A1 are a compatible couple of Banach spaces, one of which is uniformly convex, then the complex interpolation spaces [AO, Aj]0I are also uniformly convex for 0 0 and equivalent to AA near 0). Also, PA(T) > 0 for T > 0 (this can be concluded from the duality formula PA(T) = 2 SUp,,O{TE 28A.*(e)} and the fact that AA*(e) < E2) and being an Orlicz function, it turns out that PA is strictly increasing. 8A is also strictly increasing if A is uniformly convex (8A itself is strictly increasing when A is uniformly convex since A(,(e)/c is nondecreasing). If L is a Banach lattice of measurable functions on a measure space (2, 2, M) we denote by L(A) the space of Received by the editors April 13, 1981 and, in revised form, June 29, 1981. 1980 Mathematics Subject Classification. Primary 46B20. ?01982 American Mathematical Society 0002-9939/81/0000-0343/$02.25