On $\mathbb{C}$, holomorphic functions coincide with analytic functions. Wikipedia therefore intimates that (a) two holomorphic functions composed are holomorphic, and (b) the functional inverse of a holomorphic function is also holomorphic provided the function's derivative vanishes nowhere.
Without loss of generality, suppose $g$ and $f$ are two of the three holomorphic functions. If $z$ is holomorphic and z' vanishes nowhere, it then follows that $w=z^{-1}\circ g\circ f$ is holomorphic. If $w$ is holomorphic and w' vanishes nowhere, then $z=g\circ f\circ w^{-1}$ is holomorphic. However, if one of the functions on the other side, $z$ or $w$, is the third holomorphic function and has a derivative that vanishes somewhere, then it stands to reason the fourth function might not be holomorphic.