Can there be a holomorphic on a disc for which $f(z)=f(\alpha z)$ whereas $\alpha$ is transcendental and $\alpha z \in D$? where D is the disc. I believe it to be true as I find no reason for contradiction. But I find no example.
Edit: it seems that the constant function does hold for this question with any transcendental $|\alpha|=1$. Is there a non constant function?
Choosing an holomorphic branch of $\log(z)$ then $f(z) =\exp(2i\pi \log(z)/\log(\alpha))$ works on any disk not containing $0$.
If the disk contains $0$ then write the power series of $f(z)$ and $f(\alpha z)$ and equate the coefficients.