  The Loewner equation encrypts a growing simple curve in the plane into a
real-valued driving function. We show that if the driving function $\lambda$ is
in $C^{\beta}$ with $\beta>2$ (or real analytic) then the Loewner curve is in
$C^{\beta + \frac{1}{2}}$ (respectively analytic). This is a converse to a
result by Earle and Epstein and extends a result of Wong.
