OPEN

Let $f(z)$ be an entire function, not a polynomial. Does there exist a path $C$ such that, for every $\lambda>0$, the integral
\[\int_C \lvert f(z)\rvert^{-\lambda} \mathrm{d}z\]
is finite?

Huber [Hu57] proved that for every $\lambda>0$ there is a path $C_\lambda$ such that this integral is finite.