Check whether the statement true or false? There exists an analytic function $f:\mathbb C → \mathbb C$ such that $f(0) = 1, f(4i) = i$ and for all $z_j$ such that $1 < |z_j | < 3, j = 1, 2$, we have $$|f(z_1) − f(z_2)| ≤ |z_1 − z_2| ^{π /3}.$$
My attempt:- $\because \pi/3>1.$ We can write $$\frac{|f(z_1) − f(z_2)|}{|z_1 − z_2|} ≤ |z_1 − z_2| ^{π /3-1}.$$ By the definition of derivative. $f'(z)=0,1 < |z | < 3$. By Identity Theorem $f'(z)=0$. Hence, $f(z)=Constant$. Which contradict to the proposition $f(0) = 1, f(4i) = i$. Am I correct?