Uniformly convex renormings and generalized cotypes

Abstract We are concerned about improvements of the modulus convexity by renormings a super-reflexive Banach space. Typically optimal results beyond Pisier's power functions bounds t p , with ≥ 2 and they related to notion generalized cotype. obtain an explicit upper bound for all moduli equivalent we show that if this is best possible, then space admits renorming type 2. UMD bigger, up multiplicative constant, than its also prove super-multiplicativity supremum set cotypes.