1) Find all entire functions that are uniformly continuous on $\mathbb{C}$.
2) Find all entire functions $f(z)$ such that such that for every integer $n \geq 1$,
$\oint_{\partial\mathbb{D}} f(z)\bar{z}^ndz = 0,$ where $\mathbb{D}$ is the unit disk.
I'm a bit shaky on the first one, but I think it's that an entire function has an infinite radius of convergence, so is everywhere normally convergent. So if each term in it's power series is uniformly continuous on $\mathbb{C}$, then the function will be uniformly continuous on $\mathbb{C}$. Am I on the right track?
For the second, I'm not sure how to use the Cauchy Integral Formula since $f(z)\bar{z}^n$ isn't holomorphic.