[EN] Given a Banach space (X,k · k), we study the connection between uniformly convex functions f : X ¿ R bounded above by k · kp , and the existence of norms on X with moduli of convexity of power type. In particular, we show that there exists a uniformly convex function f : X ¿ R bounded above by k · k2 if and only if X admits an equivalent norm with modulus of convexity of power type 2.

We thank Professor C. Zalinescu and an anonymous referee for several helpful comments which led to improvements in both the accuracy and presentation of this note.