Inequalities for polynomials with a prescribed zero
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- by A. Giroux and Q. I. Rahman PDF
- Trans. Amer. Math. Soc. 193 (1974), 67-98 Request permission
Abstract:
Inequalities for the derivative and for the maximum modulus on a larger circle of a polynomial with a given zero on the unit circle are obtained in terms of its degree and maximum modulus on the unit circle; examples are given to show that these are sharp with respect to the degree (best constants are not known). Inequalities for ${L^p}$ norms, in particular ${L^2}$ norms, are also derived. Also certain functions of exponential type are considered and similar inequalities are obtained for them. Finally, the problem of estimating ${P_n}(r)$ (with $0 < r < 1$) given ${P_n}(1) = 0$ is taken up.References
- N. I. Ahiezer, Lekcii po Teorii Approksimacii, OGIZ, Moscow-Leningrad, 1947 (Russian). MR 0025598
- N. C. Ankeny and T. J. Rivlin, On a theorem of S. Bernstein, Pacific J. Math. 5 (1955), 849–852. MR 76020, DOI 10.2140/pjm.1955.5.849
- Ralph Philip Boas Jr., Entire functions, Academic Press, Inc., New York, 1954. MR 0068627
- R. P. Boas Jr., Inequalities for functions of exponential type, Math. Scand. 4 (1956), 29–32. MR 85343, DOI 10.7146/math.scand.a-10453
- R. P. Boas Jr., Inequalities for asymmetric entire functions, Illinois J. Math. 1 (1957), 94–97. MR 84577
- R. P. Boas Jr., Inequalities for polynomials with a prescribed zero, Studies in mathematical analysis and related topics, Stanford Univ. Press, Stanford, Calif., 1962, pp. 42–47. MR 0150269
- R. P. Boas Jr. and Q. I. Rahman, $L^{p}$ inequalities for polynomials and entire functions, Arch. Rational Mech. Anal. 11 (1962), 34–39. MR 158994, DOI 10.1007/BF00253927
- N. G. de Bruijn, Inequalities concerning polynomials in the complex domain, Nederl. Akad. Wetensch., Proc. 50 (1947), 1265–1272 = Indagationes Math. 9, 591–598 (1947). MR 23380
- R. J. Duffin and A. C. Schaeffer, Some inequalities concerning functions of exponential type, Bull. Amer. Math. Soc. 43 (1937), no. 8, 554–556. MR 1563585, DOI 10.1090/S0002-9904-1937-06602-X G. H. Hardy, The mean value of the modulus of an analytic function, Proc. London Math. Soc. 14 (1915), 269-277.
- Peter D. Lax, Proof of a conjecture of P. Erdös on the derivative of a polynomial, Bull. Amer. Math. Soc. 50 (1944), 509–513. MR 10731, DOI 10.1090/S0002-9904-1944-08177-9 G. Pólya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Springer, Berlin, 1925.
- Q. I. Rahman, On asymmetric entire functions. II, Math. Ann. 167 (1966), 49–52. MR 200450, DOI 10.1007/BF01361214
- Q. I. Rahman and Frank Stenger, An extremal problem for polynomials with a prescribed zero, Proc. Amer. Math. Soc. 43 (1974), 84–90. MR 333123, DOI 10.1090/S0002-9939-1974-0333123-0
- Marcel Riesz, Über Einen Sat’z des Herrn Serge Bernstein, Acta Math. 40 (1916), no. 1, 337–347 (German). MR 1555142, DOI 10.1007/BF02418550 —, Eine trigonometrische Interpolationsformel und einige Ungleichungen für Polynome, Jahresbericht der Deutschen Mathematiker Vereinigung 23 (1914), 354-368.
- W. Rogosinski and G. Szegö, Über die Abschnitte von Potenzreihen, die in einem Kreise beschränkt bleiben, Math. Z. 28 (1928), no. 1, 73–94 (German). MR 1544940, DOI 10.1007/BF01181146
- A. C. Schaeffer, Inequalities of A. Markoff and S. Bernstein for polynomials and related functions, Bull. Amer. Math. Soc. 47 (1941), 565–579. MR 5163, DOI 10.1090/S0002-9904-1941-07510-5 G. Szegö, Über einen Satz des Herrn Serge Bernstein, Schriften der Königsberger Gelehrten Gesellschaft 5 (1928), 59-70. A. Zygmund, A remark On conjugate series, Proc. London Math. Soc. (2) 34 (1932), 392-400.
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 193 (1974), 67-98
- MSC: Primary 30A06
- DOI: https://doi.org/10.1090/S0002-9947-1974-0352427-3
- MathSciNet review: 0352427