Let $f$ be holomorphic function on unit disk and it is continuous on boundary of the disk.
It is known that $f$ is constant and equal to zero if $f$ is vanishing on sub-arc of boundary (Maximum Principle + Reflection theorem).
Also, if real part $Ref$ is vanishing on whole boundary then we have $f$ is constant (by open mapping thm + harmonic function's maximum principle).
However, can we construct a "non-constant" such function $f$ satisfying real part $Ref$ is vanishing only on sub-arc of boundary?