Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

New inequalities for polynomials


Authors: C. Frappier, Q. I. Rahman and St. Ruscheweyh
Journal: Trans. Amer. Math. Soc. 288 (1985), 69-99
MSC: Primary 26D05; Secondary 30A10, 41A17
DOI: https://doi.org/10.1090/S0002-9947-1985-0773048-1
MathSciNet review: 773048
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Using a recently developed method to determine bound-preserving convolution operators in the unit disk, we derive various refinements and generalizations of the well-known inequalities of S. Bernstein and M. Riesz for polynomials. Many of these results take into account the size of one or more of the coefficients of the polynomial in question. Other results of similar nature are obtained from a new interpolation formula.


References [Enhancements On Off] (What's this?)

  • [1] R. J. Duffin and A. C. Schaeffer, A refinement of an inequality of the brothers Markoff, Trans. Amer. Math. Soc. 50 (1941), 517-528. MR 0005942 (3:235c)
  • [2] W. L. Ferrar, Algebra, 2nd ed., Oxford Univ. Press, London, 1957.
  • [3] C. Frappier and Q. I. Rahman, On an inequality of S. Bernstein, Canad. J. Math. 34 (1982), 932-944. MR 672687 (84c:30005)
  • [4] F. R. Gantmacher, The theory of matrices, Chelsea, New York, 1959.
  • [5] A. Giroux and Q. I. Rahman, Inequalities for polynomials with a prescribed zero, Trans. Amer. Math. Soc. 193 (1974), 67-98. MR 0352427 (50:4914)
  • [6] N. K. Govil, V. K. Jain and G Labelle, Inequalities for polynomials satisfying $ P(z) \equiv {z^n}P(1/z)$, Proc. Amer. Math. Soc. 57 (1976), 238-242. MR 0414838 (54:2930)
  • [7] M. Lachance, E. B. Saff and R. S. Varga, Inequalities for polynomials with a prescribed zero, Math. Z. 168 (1979), 105-116. MR 544699 (80j:30009)
  • [8] M. A. Malik, On the derivative of a polynomial, J. London Math. Soc. 1 (1969), 57-60. MR 0249583 (40:2827)
  • [9] Z. Nehari, Conformal mapping, McGraw-Hill, New York, 1952. MR 0045823 (13:640h)
  • [10] D. J. Newman, Polynomials and rational functions, Approximation Theory and Applications (Z. Ziegler, ed.), Academic Press 1981. MR 615416 (82h:30006)
  • [11] G. Pólya and G. Szegö, Aufgaben und Lehrsatze aus der Analysis, Springer, Berliń, 1925.
  • [12] Q. I. Rahman, Applications of functional analysis to extremal problems for polynomials, Sém. Math. Supérieures, Presses Univ. Montréal, 1967. MR 0251195 (40:4426)
  • [13] Q. I. Rahman and G. Schmeisser, Some inequalities for polynomials with a prescribed zero, Trans. Amer. Math. Soc. 216 (1976), 91-103. MR 0399427 (53:3271)
  • [14] M. Riesz, Über einen Satz des Herrn Serge Bernslein, Acta Math. 40 (1916), 337-347. MR 1555142
  • [15] W. Rogosinski and G. Szegö, Über die Abschnitte von Potenzreihen die in einen Kreise beschränkt bleiben, Math. Z. 28 (1928), 73-94. MR 1544940
  • [16] W. W. Rogosinski, Extremum problems for polynomials and trigonometrical polynomials, J. London Math. Soc. 29 (1954), 259-275. MR 0062859 (16:29f)
  • [17] St. Ruscheweyh, Convolutions in geometric function theory, Sém. Math. Supérieures, Presses Univ. Montréal, 1982. MR 674296 (84a:30029)
  • [18] A. C. Schaeffer, Inequalities of A. Markoff and S. Bernstein for polynomials and related functions, Bull. Amer. Math. Soc. 47 (1941), 565-579. MR 0005163 (3:111a)
  • [19] T. Sheil-Small, On the convolution of analytic functions, J. Reine Angew. Math. 258 (1973), 137-152. MR 0320761 (47:9295)
  • [20] G. Szegö, Über einen Satz des Herrn Serge Bernslein, Schriften Königsberger Gelehrten Gesellschaft 5 (1928), 59-70.
  • [21] M. Tsuji, Potential theory in modern function theory, Maruzen, Tokyo, 1959. MR 0114894 (22:5712)
  • [22] C. Visser, A simple proof of certain inequalities concerning polynomials, Nederl. Akad. Wetensch. Proc. 48, 276-281 = Indag. Math. 7 (1945), 81-86. MR 0015568 (7:440a)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 26D05, 30A10, 41A17

Retrieve articles in all journals with MSC: 26D05, 30A10, 41A17


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0773048-1
Keywords: Polynomials, Bernstein's inequality, interpolation formula
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society