complex analysis - Bounded real part on the disk implies bounded imaginary part

If the real part of a holomorphic function on the unit disk is bounded, then the Borel-Caratheodory theorem implies that the function is bounded, thereby implying the imaginary part is in fact bounded. Is there a direct way to show that the imaginary part is bounded using the standard theory of maximum modulus principle and conformal maps?

Answer

The mapping $w$, defined by $w(z)=i\log\dfrac{1+z}{1-z}$ maps the unit disk to the strip $|{\bf Re}\,w|\leq\dfrac{\pi}{2}$.