Lemniscates and inequalities for the logarithmic capacities of continua (Q881051)

The lemniscate of a polynomial \(P(z)=\Pi_{j=1}^n (z-z_j),\,z_j\in {\mathbb C}, n\geq 2\), is the set \(E(P)=\{z:| P(z)| \leq 1\}\). The critical points of \(P\) are the zeros of its derivative. The author proves an inequality which provides some information on the dispersion of the zeros and the critical points of \(P\) inside \(E(P)\). More precisely, suppose that \(| P(\zeta)| \leq 1\) for all critical points \(\zeta\) of \(P\). Let \(m\) be the number of critical points of \(P\). Then for any \(n-m+1\) points in \(E(P)\), there exists a continuum \(\gamma\subset E(P)\) containing all zeros and critical points of \(P\) such that cap\(\gamma\leq 2^{-1/n}\). Here cap denotes the logarithmic capacity. As corollaries, estimates for continua of minimal capacity containing given points are obtained.