I will denote by $\mathbb{C}^*$ the punctured complex plane, $\mathbb{C} \setminus \{0\}$. Let's say I have a holomorphic map on the punctured plane, $w: \mathbb{C}^* \to \mathbb{C}$, such that the map $e^{w}: \mathbb{C}^* \to \mathbb{C}^*$ has a (holomorphic) antiderivative $F: \mathbb{C}^* \to \mathbb{C}$.
Obviously I have a homotopy between $e^{w}$ and the constant map 1 where the intermediate maps are holomorphic functions $\mathbb{C}^* \to \mathbb{C}^*$: i.e., I simply consider $e^{tw}, t \in [0,1]$. However, the intermediate maps of this homotopy do not necessarily have antiderivatives, obviously. I was wondering if this homotopy could be modified in some way to ensure that the intermediate maps $\mathbb{C}^* \to \mathbb{C}^*$ do have antiderivatives.
Is there a theorem that gives this property, or if not, does anyone have any ideas as to how I could approach this problem? (I'd appreciate any ideas for approaches - even if they are vague. Or any references that might be helpful. Thank you!)