Uniform convexity of ψ-direct sums of Banach spaces

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