Can i get the answer to the following problem. Please point out: Is this very trivial, True or is there a trivial counterexample. Thanks in advance..

Let $D\subset \mathbb C$ be a simply connected domain, and $\gamma: [0,1]\to D$, be a smooth embedding. Given a continuous one form $\phi$ along $\gamma$ and $\epsilon >0$, Does there exists a holomorphic function $h$ on some open subset $U$ of $\gamma$, $U\subset D$ such that $|dh-\phi|<\epsilon$.