Maximum principles for multivalued analytic functions

The function $f(z)$ is supposed to be regular in the multiply connected closed domain $\mathfrak {D}$ and $\left|f(z)\right|$ single-valued in $\mathfrak {D}$. [$f(z)$ is not necessarily single-valued.] The absolute value $\left|f(z)\right|$ attains its maximum at a boundary point of $\mathfrak {D}$. The maximum cannot be attained at an inner point of $\mathfrak {D}$ unless $f(z)$ is a constant.

The function $f(z)$ is supposed to be regular on that piece of the Riemann surface of $\log z$ that covers the annulus $0<\left|z\right|\leq 1$. If $f(z)$ is bounded and in particular if $\left|f(z)\right|\leq 1$ for $\left|z\right|=1$ then $\left|f(z)\right|\leq 1$ in the entire domain.