Integral mean estimate for polynomials with restricted zeros
Problemy analiza, Tome 13 (2024) no. 3, pp. 101-117.

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In this paper, we present certain sharp $L^{p}$-inequalities for polynomials with restricted zeros. Our results improve and generalize some known integral inequalities for polynomials in the complex domain.
Keywords: polynomials, inequalities
Mots-clés : complex domain.
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N. A. Rather; N. Wani; A. Bhat. Integral mean estimate for polynomials with restricted zeros. Problemy analiza, Tome 13 (2024) no. 3, pp. 101-117. http://geodesic.mathdoc.fr/item/PA_2024_13_3_a6/

[1] Ankeny N. C., Rivlin T. J., “On a theorem of S. Bernstein”, Pacific J. Math., 5:6 (1955), 849–852 | DOI | MR | Zbl

[2] Arestov V. V., “On integral inequalities for trigonometric polynomials and their derivatives”, Math. USSR-Izv., 18:1 (1982), 1–17 | DOI | MR | Zbl

[3] Arestov V. V., “Integral inequalities for algebraic polynomials on the unit circle”, Math. Notes Acad. Sci. USSR, 48:4 (1990), 977–984 | DOI | MR | Zbl

[4] Arestov V. V., “Integral inequalities for algebraic polynomials with a restriction on their zeros”, Anal. Math., 17:1 (1991), 1–20 | DOI | MR | Zbl

[5] Aziz A., “Integral mean estimates for polynomials with restricted zeros”, J. Approx. Theory, 55:2 (1988), 232–239 | DOI | MR | Zbl

[6] Aziz A., Rather N. A., “$L_p$ inequalities for polynomials”, Glas. Math., 32 (1997), 39–43 | DOI | MR | Zbl

[7] Aziz A., Rather N. A., “Some new generalizations of Zygmund-type inequalities for polynomials”, Math. Inequalities Appl., 15:2 (2012), 469–486 | DOI | MR | Zbl

[8] Bernstein S., Lezons sur les propriitis extrimales et la meilleure approximation des fonctions analytiques d'une fonction rielle, v. 30, Paris, 1926

[9] Boas R. P., Rahman Q. I., “$L^{p}$ inequalities for polynomials and entire functions”, Arch. Rational Mech. Anal., 11:1 (1962), 34–39 | DOI | MR | Zbl

[10] De Bruijn N. G., Springer T. A., “On the zeros of composition-polynomials”, Proc. Sect. Sci. Kon. Ned. Akad. Wet. Amst., 50:7–8 (1947), 895–903 | MR | Zbl

[11] Govil N. K., “On the derivative of a polynomial”, Proc. Amer. Math. Soc., 41:2 (1973), 543–546 | DOI | MR | Zbl

[12] Gulzar S., Rather N. A., “On a Composition Preserving Inequalities between Polynomials”, J. Contemp. Math. Anal., 53:3 (2018), 21–26 | DOI | MR | Zbl

[13] Hardy G. H., Littlewood J. E., Pulya G., Inequalities, Cambridge University Press, 1988 | DOI | MR | Zbl

[14] Lax P. D., “Proof of a conjecture of P. Erd?s on the derivative of a polynomial”, Bull. Amer. Math. Soc., 50 (1944), 509–513 | DOI | MR | Zbl

[15] Mahler K., “On the zeros of the derivative of a polynomial”, Proc. R. Soc. Lond. A, 264:1317 (1961), 145–154 | DOI | MR | Zbl

[16] Osserman R., “A sharp Schwarz inequality on the boundary”, Proc. Amer. Math. Soc., 128:12 (2000), 3513–3517 | DOI | MR | Zbl

[17] Rahman Q. I., Schmeisser G., Analytic Theory of Polynomials, Oxford University Press, 2002 | DOI | MR | Zbl

[18] Rahman Q. I., Schmeisser G., “$L^{p}$ inequalities for polynomials”, J. Approx. Theory, 53:1 (1988), 26–32 | DOI | MR | Zbl

[19] Rather N. A., Bhat A., Shafi M., “Integral inequalities for the growth and higher derivative of polynomials”, J. Contemp. Math. Anal., 57:4 (2022), 242–251 | DOI | MR | Zbl

[20] Rather N. A., Gulzar S., Bhat A., “$L_p$ inequalities for the growth of polynomials with restricted zeros”, Arch. Math. (Brno), 58:3 (2022), 159–167 | DOI | MR | Zbl

[21] Smyth C. The, “Mahler measure of algebraic numbers: a survey”, Lond. Math. Soc. Lecture Note Ser., 352, 2008, 322–349 | DOI | MR | Zbl

[22] Turbn P., “ber die Ableitung von Polynomen”, Compositio Math., 7 (1940), 89–95 http://eudml.org/doc/88754

[23] Zygmund A., “A remark on conjugate series”, Proc. Lond. Math. Soc., s2–34:1 (1932), 392–400 | DOI | MR | Zbl