Bernstein-Walsh inequalities and the exponential curve in $\mathbb{C}^2$

It is shown that for the pluripolar set $K=\{(z,e^z):\,\vert z\vert\leq1\}$ in ${\Bbb C}^2$ there is a global Bernstein-Walsh inequality: If $P$ is a polynomial of degree $n$ on ${\Bbb C}^2$ and $\vert P\vert\leq1$ on $K$, this inequality gives an upper bound for $\vert P(z,w)\vert$ which grows like $\exp(\frac12n^2\log n)$. The result is used to obtain sharp estimates for $\vert P(z,e^z)\vert$.