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.

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 $\beta=\log 2/\log 3$.

See Proposition 2.2, attributed to a 1999 paper which I did not find online.

(For reference, Pietro Majer's argument for $\alpha=\log 3/\log 4$ is in this 2016 MO answer.)

Energy Form on a Closed Fractal Curve (2004). See Proposition 2.2, attributed to Ref. 15, which I did not find online:

The Koch snow flake is the union of three Koch curves of Hausdorff dimension $D=\ln 4/\ln 3$ and Hölder exponent $\beta=\ln 2/\ln 3$.

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.