U.R Freiberg and M.R. Lancia, <A HREF="https://pdfs.semanticscholar.org/4928/071439ea10140b21dd0a0ccd124812c84f31.pdf">Energy Form on a Closed Fractal Curve</A> (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 <A HREF="https://iris.uniroma1.it/handle/11573/243201#.XSB1Sy2B3OQ">1999 paper</A> which I did not find online, quite possibly their $\beta$ is the $\alpha$ in the OP.