Uniformly smooth renorming of Banach spaces with modulus of convexity of power type 2

An upper bound q(c) for the best, under equivalent renorming, possible power type of the modulus of smoothness of a Banach space with modulus of convexity satisfying δX(ε)⩾cε2, is found. The estimate is asymptotically sharp and is expressed in terms of linear fractional function q(c).