Uniform convexity of ψ-direct sums of Banach spaces

Let X and Y be Banach spaces and psi a continuous convex function on the unit interval [0, 1] satisfying certain conditions. Let X circle pluspsi Y be the direct sum of X and Y equipped with the associated norm with psi. We show that X circle pluspsi Y is uniformly convex if and only if X, Y are uniformly convex and psi is strictly convex. As a corollary we obtain that the iota(p,q)-direct sum X circle plus (p,q) Y, 1 less than or equal to q less than or equal to p less than or equal to infinity (not p = q = 1 nor infinity), is uniformly convex if and only if X, Y are, where iota(p,q) is the Lorentz sequence space. These results extend the well-known fact for the iota(p)-sum X circle plus (p) Y, 1 < p < infinity. Some other examples are also presented. (C) 2002 Elsevier Science (USA). All rights reserved.