On estimates for Goncharov polynomials

V. A. Oskolkov

Sbornik. Mathematics, Tome 21 (1973) no. 1, pp. 57-62

Résumé

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.

Bibliographie

[1] M. A. Evgrafov, “Metod blizkikh sistem v prostranstve analiticheskikh funktsii i ego primeneniya k interpolyatsii”, Trudy Mosk. matem. ob-va, V (1956), 89–201 | MR

[2] V. L. Goncharov, “Recherches sur les dérivées successives des fonctions analytiques. Généralisation de la série d'Abel”, Ann. Ecole Norm. super, 47 (1930), 1–78

[3] I. I. Ibragimov, Metody interpolyatsii funktsii i nekotorye ikh primeneniya, izd-vo «Nauka», Moskva, 1965 | MR