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$L_p$ extensions of Gonchar's inequality for rational functions
A. L. Levin,
E. B. Saff
Abstract:
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
$0<p,q<\infty$ 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.
UDC:
517.5
MSC: Primary
30A10,
30C85; Secondary
31A15 Received: 12.06.1991