Abstract:
Applications of the geometric theory of functions to inequalities for algebraic polynomials are considered. The main attention is paid to constructing a univalent conformal mapping for a given polynomial and to applying the Lebedev and Nehari theorems to this mapping. A new sharp inequality of Bernshtein type for polynomials with restrictions on the growth on a segment or on a circle, inequalities with restrictions on the zeros of the polynomial, and other inequalities are obtained. In particular, classical inequalities by Markov, Bernshtein, and Schur are strengthened.
Citation:
V. N. Dubinin, A. V. Olesov, “Application of conformal mappings to the inequalities for polynomials”, Analytical theory of numbers and theory of functions. Part 18, Zap. Nauchn. Sem. POMI, 286, POMI, St. Petersburg, 2002, 85–102; J. Math. Sci. (N. Y.), 122:6 (2004), 3630–3640
\Bibitem{DubOle02}
\by V.~N.~Dubinin, A.~V.~Olesov
\paper Application of conformal mappings to the inequalities for polynomials
\inbook Analytical theory of numbers and theory of functions. Part~18
\serial Zap. Nauchn. Sem. POMI
\yr 2002
\vol 286
\pages 85--102
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl1569}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1937370}
\zmath{https://zbmath.org/?q=an:1074.30003}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2004
\vol 122
\issue 6
\pages 3630--3640
\crossref{https://doi.org/10.1023/B:JOTH.0000035238.79760.70}
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https://www.mathnet.ru/eng/znsl1569
https://www.mathnet.ru/eng/znsl/v286/p85
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