Area and the inradius of lemniscates

We study geometric properties of filled lemniscates $E(p,c)=\{z:\, |p(z)|\le c\}$ of a complex polynomial $p(z)$ of degree $n$. In particular, we answer a question raised by H.H. Cuenya and F.E. Levis (2007) by showing that there is a constant $C(n)$ such that $\frac{\mu(E(p,c))}{\pi r^2(E(p,c))}\le C(n)$ for every lemniscate $E(p,c)$. Here $\mu(E(p,c))$ and $r(E(p,c))$ denote the area and the inradius of $E(p,c)$.