This article is cited in 3 papers

On estimates for Goncharov polynomials

V. A. Oskolkov


Abstract: The following is proved in the article.
Theorem. {\it If a sequence of interpolation points satisfies the conditions $|\arg z_n|\leqslant\frac\pi2\left(1-\frac1\rho\right)$ for all sufficiently large $n$ and $\varlimsup_{n\to\infty}n^{-1/\rho}|z_n|=\varlimsup_{n\to\infty}n^{-1/\rho}S_n=1,$ where $S_n=\sum_{\nu=0}^{n-1}|z_\nu-z_{\nu+1}|,$ for $1\leqslant\rho<\infty,$ and $\arg z_n=0,$ $z_n\leqslant z_{n+1}$ $(n=0,1,\dots),$ $\lim_{n\to\infty}n^{-1/\rho}z_n=1$ for $0<\rho<1,$ then the assertions}
1) $\varlimsup_{n\to\infty}\{n^{-n/\rho}n!\max_{|z|\leqslant r}|P_n(z)|\}^{1/n}\equiv1$ for $1\leqslant\rho<\infty$,
2) $\frac1\rho\exp\left(1-\frac1\rho\right)\leqslant\varlimsup_{n\to\infty}\{n^{-n/\rho}n!\max_{|z|\leqslant r}|P_n(z)|\}^{1/n}\leqslant1$ for $0<\rho<1$
\noindentare valid for any $r<\infty$. Here $P_n(z)$ is the Goncharov polynomial of degree $n$.
Bibliography: 3 titles.

UDC: 517.535.4

MSC: 30A06, 30A80, 30A04

Received: 23.10.1972

 English version: Mathematics of the USSR-Sbornik, 1973, 21:1, 57–62

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