Area and the inradius of lemniscates

We study geometric properties of filled lemniscates 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 for every lemniscate E(p,c). Here μ(E(p,c)) and r(E(p,c)) denote the area and the inradius of E(p,c).