U.R Freiberg and M.R. Lancia, Energy Form on a Closed Fractal Curve (2004):
The Koch snow flake is the union of three Koch curves of Hausdorff dimension $D=\ln 4/\ln 3$ and Hölder exponent $\beta=\ln 2/\ln 3$.
See Proposition 2.2, attributed to a 1999 paper which I did not find online, quite possibly their $\beta$ is the $\alpha$ in the OP.