Problem #514: Let $f(z)$ be an entire function

Let $f(z)$ be an entire function. Does there exist a path $L$ so that, for every $n$,\[\lvert f(z)/z^n\rvert \to \infty\]as $z\to \infty$ along $L$?

Can the length of this path be estimated in terms of $M(r)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$? Does there exist a path along which $\lvert f(z)\rvert$ tends to $\infty$ faster than a fixed function of $M(r)$ (such that $M(r)^\epsilon$)?