Does there exist an entire function with $f(0)=1$ and $f(4i)=i$ and $|f(z_1) - f(z_2)| < |z_1 - z_2|^{\frac{\pi}{3}}$ in $1<|z|<3$?

Question

Does there exist an entire function with $f(0)=1$ and $f(4i)=i$ and $|f(z_1) - f(z_2)| < |z_1 - z_2|^{\frac{\pi}{3}}$ whenever $z_1,z_2 \in \{z: 1<|z|<3\}$? I think it has something to do with the Maximum-modulus principle or Liouville's theorem. I don't know where to start! If not answer, little hint will also be appreciated.