Uniformly convex functions on Banach spaces

Given a Banach space ($X$,$\Vert\cdot\Vert$), we study the connection between uniformly convex functions $f:X \to \mathbb{R}$ bounded above by $\Vert\cdot\Vert^p$ 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 $f:X \to \mathbb{R}$ bounded above by $\Vert\cdot\Vert^2$ if and only if $X$ admits an equivalent norm with modulus of convexity of power type 2.