Analytic Function on a Half Plane Bounded at the Boundary Must Be Bounded

Question

I want to verify an argument that supposedly shows that any analytic function in, let's say, the right half plane that is bounded in the boundary must be itself bounded by the same bound. That is if we take $H^+=\{z\in\mathbb{C}\; |\; \text{Re}(z)\geq 0\}$ and a continuous function $f:H^+\to\mathbb{C}$ that is analytic in the interior and such that there is some $M\geq 0$ such that $|f(z)|\leq M$ for all $z\in\partial H^+$ then it must be that this inequality is true for all $z\in H^+$. The argument is as follows: If we consider the the function $f\left(\frac{z+1}{1-z}\right)$ this is an analytic function on the the unit disk so by the maximum modulus principle, this function must be bounded by the points in the unit circle which are sent to the imaginary axis by the mobius transformation even the point at infinity. From this it seems to obviously follow what I claim. However this sounds too good to be true and so I would like to ask for someone to verify or counter the argument.