Does exist an holomorphic function $f$ on $|z|>1$, non-constant, and bounded by the limit at $\infty$? I have a question in my book that asks to show that if a function $f$ is holomorphic on $\{z:|z|>1\}$, continuous on $\{z: |z|=1\}$and has limit on $\infty$ then its maximum is for some point $w$ with $|w| = 1$. I have solved the problem except for one case in which the function is bounded by its limit at $\infty$. So, that is my question, is that possible? Can the maximum be at $\infty$ instead of the unit circle? I have come across with another similar question but with no response. I have tried using Liouville's theorem but the function isn't even defined in the whole plane. Any help is appreciated.

If $f$ is holomorphic in $\{z:|z|>1\}$, continuous on $\{z: |z|=1\}$and has limit at $z =\infty$, then it is bounded by its values on $|z|=1$, i.e.
$$
 \max \{ |f(z) | : |z| > 1 \} = \max \{ |f(z) | : |z| = 1 \} \, .
$$
For a proof consider $g(z) =f(1/z)$, which is holomorphic in $\{z: 0 < |z|<1\}$, continuous on $\{z: |z|=1\}$and has a limit at $z=0$.
Now apply Riemann's theorem on removable singularities to conclude that $g$ can be extended to a holomorphic function on the entire unit disk. The maximum modulus principle then shows that $g$ is bounded by its values on $|z|=1$, so that the same is true for $f$.
In addition, if $f$ is bounded by its value at $\infty$ then $g$ is bounded by $|g(0)|$, which implies that $g$ (and consequently $f$) is constant.
