Abstract:
Inequalities for the derivatives with respect to φ=argz the functions ReP(z), |P(z)|2 and argP(z) are established for an algebraic polynomial P(z) at points on the circle |z|=1. These estimates depend, in particular, on the constant term and the leading coefficient of the polynomial P(z) and improve the classical Bernstein and Turan inequalities. The method of proof is based on the techniques of generalized reduced moduli.
\Bibitem{Dub00}
\by V.~N.~Dubinin
\paper Distortion theorems for polynomials on a~circle
\jour Sb. Math.
\yr 2000
\vol 191
\issue 12
\pages 1797--1807
\mathnet{http://mi.mathnet.ru/eng/sm528}
\crossref{https://doi.org/10.1070/sm2000v191n12ABEH000528}
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This publication is cited in the following 20 articles:
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Thangjam Birkramjit Singh, Robinson Soraisam, Barchand Chanam, “Sharpening of Turán type inequalities for polar derivative of a polynomial”, Complex Anal Synerg, 9:1 (2023)
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V. N. Dubinin, “Some remarks on rotation theorems for complex polynomials”, Sib. elektron. matem. izv., 18:1 (2021), 369–376
Singh T.B., Chanam B., “Generalizations and Sharpenings of Certain Bernstein and Turan Types of Inequalities For the Polar Derivative of a Polynomial”, J. Math. Inequal., 15:4 (2021), 1663–1675
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V. N. Dubinin, “Lower Bound for the Discrete Norm of a Polynomial on the Circle”, Math. Notes, 90:2 (2011), 284–287
Dubinin V.N., Karp D.B., “Generalized condensers and distortion theorems for conformal mappings of planar domains”, Interaction of Analysis and Geometry, Contemporary Mathematics Series, 424, 2007, 33–51
V. N. Dubinin, “Applications of the Schwarz lemma to inequalities for entire functions with constraints on zeros”, J. Math. Sci. (N. Y.), 143:3 (2007), 3069–3076
A. V. Olesov, “Inequalities for majorizing analytical functions”, J. Math. Sci. (N. Y.), 133:6 (2006), 1693–1703
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A. V. Olesov, “Application of Conformal Mappings to Inequalities for Trigonometric Polynomials”, Math. Notes, 76:3 (2004), 368–378
V. N. Dubinin, “Conformal mappings and inequalities for algebraic polynomials. II”, J. Math. Sci. (N. Y.), 129:3 (2005), 3823–3834
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