Let $U\subseteq\Bbb{R}^2$ be a star-shaped region with respect to some $x_0\in U$. It's easy to prove that every two curves $\gamma_1,\gamma_2\subseteq U$ are homotopic (since by definition each is homotopic to constant curve $e_{x_0}$ and homotopy is an equivalence relation).
But is it true that if $\gamma_1,\gamma_2\subseteq U$ are smooth, then there exists a smooth homotopy between them? One can easily find a smooth homotopy between each of them to $e_{x_0}$ (for example $(1-t)\gamma_1(x)+tx_0$), but I wasn't able to find a smooth homotopy between them - I thought about $(1-t)\gamma_1(x)+t\gamma_2(x)$ but since $U$ is not necessarily convex, there's no reason for this to work.
Is it even true that there must exist one?
Any help would be appreciated.