Let $f(z)$ be an entire function of finite order, and let $\Gamma$ be a rectifiable path on which $f(z)\to \infty$. Let $\ell (r)$ be the length of $\Gamma$ in the disc $|z|<r$. Find a path for which $\ell (r)$ grows as slowly as possible, and estimate $\ell (r)$ in terms of $M(r)={\max }_{|z|=r}|f(z)|$.In particular, can such a path $\Gamma$ be found for which $\ell (r)\ll r$?