Moment inequalities for equilibrium measures in the plane (Q652913)

Let \(K\subset \mathbb{C}\) be a compact set satisfying \(\mathrm{cap}(K)=1\) and \(\int_{K}zd\mu _{K}(z)=0\), where \(\mathrm{cap}(\cdot )\;\)denotes the logarithmic capacity and \(\mu _{K}\) the equilibrium measure of \(K\). One such compact set is the interval \(L=[-2,2]\). The main results of this paper concern the extremal role played by \(L\) for the quantity \(M_{\phi}(K)=\int_{K}\phi (\text{Re}z)d\mu _{K}(z)\), where \(\phi :\mathbb{R}\rightarrow \mathbb{R}\) is any convex function. Theorem 1. If \(K\subset \mathbb{R}\), then \(M_{\phi }(K)\geq M_{\phi }(L)\), and this inequality is strict if \(\mathrm{cap}(K\backslash L)>0\) and \(\phi \left| _{L}\right. \) is not linear. Theorem 2. If \(K\) is connected, then \(M_{\phi }(K)\leq M_{\phi }(L)\), and this inequality is strict if \(K\neq L\) and \(\phi \left| _{L}\right. \) is not linear. The proofs are potential-theoretic in nature. The authors describe some interesting applications (for example, to zeros of polynomials) and connections with existing open problems (for example, an old conjecture of Pommerenke).