Area and the inradius of lemniscates
Let \(p\) be a complex polynomial of degree \(n\), \(c\) a positive constant, \(E(p,c)\) the filled lemniscate of \(p\) of radius \(c\), i.e. \(E(p,c)=\{z\in\mathbb{C}:| p(z)|\leq c\}\) and \(L(E)\) the perimeter of \(E\). An open problem posed by Erdös et al. in 1958 is to maximize the length \(L(E(p,1))\) over all monic polynomials \(p_n\). Erdös conjectured that \(p_n(z)=z^n-1\) provides the desired maximum. In the present paper the authors study the similar problem on the maximum of the ratio of the area of \(E\) divided by \(\pi r^2\), where \(r\) denotes the inradius of \(E\), and prove that it is bounded by some constant depending only of the degree \(n\) for all \(p_n\) and all \(0<c<\infty\). This proves a conjecture posed by \textit{H. H. Cuenya} and \textit{F. E. Levis} [J. Math. Anal. Appl. 336, No. 2, 953--961 (2007; Zbl 1123.32006)].
scientific article; zbMATH DE number 5555208