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Inequalities for polynomials with restricted coefficients. (English) Zbl 0655.30005

In a previous paper [C. Frappier, Q. I. Rahman and St. Ruscheweyh, Trans. Am. Math. Soc. 288, 69-99 (1985; Zbl 0567.30006)], various results pertaining to refinements of the inequalities of Bernstein and M. Riesz for polynomials have been obtained. The author continues the study further and using similar methods of proof obtains new results of the same kind. It is discussed that in a certain sense the results obtained are best possible.
Reviewer: G.D.Dikshit

MSC:

30C10 Polynomials and rational functions of one complex variable
26C05 Real polynomials: analytic properties, etc.
26D20 Other analytical inequalities
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)

Citations:

Zbl 0567.30006
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References:

[1] S. N. Bernstein,Leçons sur les propriétés extrémales et la meilleure approximation des fonctions analytiques d’une variable réelle, Gauthier-Villars, Paris, 1926.
[2] C. Frappier,Some inequalities for trigonometric polynomials. J. Austral. Math. Soc. (Series A)39 (1985), 216–226. · Zbl 0653.42001
[3] C. Frappier, Q. I. Rahman and St. Ruscheweyh,Inequalities for polynomials with two equal coefficients, J. Approximation Theory44 (1985), 73–81. · Zbl 0608.41010
[4] C. Frappier, Q. I. Rahman and St. Ruscheweyh,New inequalities for polynomials, Trans. Amer. Math. Soc.288 (1985), 69–99. · Zbl 0567.30006
[5] F. R. Gantmacher,The Theory of Matrices, Chelsea, New York, 1959. · Zbl 0085.01001
[6] M. Riesz,Ueber einen Satz des Herrn Serge Bernstein, Acta Math.40 (1916), 337–347. · JFM 46.0472.01
[7] W. Rogosinski and G. Szegö,Über die Abschnitte von Potenzreihen die in einen Kreisse beschrankt bleiben, Math. Z.28 (1928), 73–94. · JFM 54.0336.03
[8] St. Ruscheweyh,Convolutions in Geometric Function Theory Les Presses de l’Université de Montréal, 1982.
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