Area and the inradius of lemniscates

We study geometric properties of filled lemniscates E ( p , c ) = { z ∈ C : | p ( z ) | ⩽ 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 μ ( E ( p , c ) ) π r 2 ( E ( p , c ) ) ⩽ C ( n ) 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 ) .