Estimates of derivatives of simplest fractions and other questions

V. I. Danchenko

Sbornik. Mathematics, Tome 197 (2006) no. 4, pp. 505-524

Résumé

The approximation properties of simplest fractions (s.f.'s), that is, of the logarithmic derivatives of complex polynomials, have recently become a subject of intensive research. These properties of s.f.'s prove to have many similarities with those of polynomials. For instance, one has for them analogues of Mergelyan's and Jackson's classical results on uniform polynomial approximation. In connection with approximation by s.f.'s estimates of the Markov–Bernstein kind for derivatives of s.f.'s on various subsets of the complex plane arouse interest. Such estimates are obtained in this paper on circles, straight lines and their intervals, and some applications of these estimates are indicated. Several other questions relating to approximation properties of s.f.'s are also considered. Bibliography: 28 titles.

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