$L_p$ extensions of Gonchar's inequality for rational functions

A. L. Levin ; E. B. Saff

Sbornik. Mathematics, Tome 76 (1993) no. 1, pp. 199-210

Résumé

Given a condenser $~(E,\, F)$ in the complex plane, let $~C(E,\, F)$ denote its capacity and let $~\mu^*=\mu_E^*-\mu_F^*$ be the (signed) equilibrium distribution for $~(E,\, F)$. Given a finite positive measure $\mu$ on $E\cup F$, let $$ G(\mu_E')=\exp\biggl(\,\int\log(d\mu/d\mu_E^*)\,d\mu_E^*\biggr),\quad G(\mu_F')=\exp\biggr(\,\int\log(d\mu/d\mu_F^*)\,d\mu_F^*\biggr). $$ We show that for $0$ and for any rational function $r_n$ of order $n$ \begin{equation} \|r_n\|_{L_p(d\mu,E)}\|1/r_n\|_{L_q(d\mu,F)}\geqslant e^{-n/C(E,F)}G^{1/p}(\mu_E') G^{1/q}(\mu_E'), \tag{1} \end{equation} which extends a classical result due to A. A. Gonchar. For a symmetric condenser we also obtain a sharp lower bound for $\|r_n-\lambda\|_{L_p(d\mu,\,E\cup F)}$, where $\lambda=\lambda(z)$ is equal to $0$ on $E$ and $1$ on $F$. The question of exactness of (1) and the relation to certain $n$-widths are also discussed.