Polynomial estimates over exponential curves in ${\mathbb{C}}^2$

For any complex $\alpha$ with non-zero imaginary part we show that Bernstein-Walsh type inequality holds on the piece of the curve $\{(e^z,e^{\alpha z}):z\in \mathbb{C}\}$. Our result extends a theorem of Coman-Poletsky \cite{CP10} where they considered real-valued $\alpha$.