A Degree-Four Lemniscate Path Theorem

We prove the degree-four case of a path problem of Erdős, Herzog, and Piranian. If $f$ is monic of degree four and all zeros of $f$, counted with multiplicity, lie in the open unit disk, then two zeros from this list can be joined inside $$\{z:\mathrm{\mid }f(z)\mathrm{\mid }<1\}$$ by a possibly degenerate polygonal path of length less than $2$.