Analytic $f : f(z^5) = 5 \cdot f(z) $ for $0 < arg(z) < \pi / 100$

Question

Consider the complex region $ W : $ with $Re(z) > 1$ and $0 < arg(z) < \pi / 100$

Is there a nonconstant analytic function $f(z)$ such that

$$f(z^5) = 5 \cdot f(z) $$

Holds for all such $z$ and $z^5$ in $W$ and is analytic in that region $W$ ?

Notice that the natural logarithm is NOT a solution.

Answer 1

Contrary to your claim, expressed in the question and in a comment to this answer, the principal value of the log, i.e., the function $${\rm Log}(z):=\ln|z|+i\,{\rm Arg}(z)\ ,$$ is an analytical solution of your functional equation, valid for $z\ne0$ in the sector $$\bigl|{\rm Arg}(z)\bigr|<{\pi\over5}\ .$$