Inequalities for polynomials with a prescribed zero

Giroux, A.; Rahman, Q. I.

doi:10.1090/S0002-9947-1974-0352427-3

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

The mean value of the modulus of an analytic function

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

Aufgaben und Lehrsätze aus der Analysis

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

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

Über einen Satz des Herrn Serge Bernstein

A remark On conjugate series