Estimates for the polar derivative of a constrained polynomial on a disk

Gradimir V. MilovanoviÄ‡; Abdullah Mir; Adil Hussain

doi:10.56754/0719-0646.2403.0541

DOI:

https://doi.org/10.56754/0719-0646.2403.0541

Abstract

This work is a part of a recent wave of studies on inequalities which relate the uniform-norm of a univariate complex coefficient polynomial to its derivative on the unit disk in the plane. When there is a limit on the zeros of a polynomial, we develop some additional inequalities that relate the uniform-norm of the polynomial to its polar derivative. The obtained results support some recently established ErdÅ‘s-Lax and Turán-type inequalities for constrained polynomials, as well as produce a number of inequalities that are sharper than those previously known in a very large literature on this subject.

Keywords

Complex domain , Constrained polynomial , Rouché‘s theorem , Zeros

Gradimir V. MilovanoviÄ‡ Serbian Academy of Sciences and Arts, 11000 Belgrade, Serbia University of Niš, Faculty of Sciences and Mathematics, P.O. Box 224, 18000 Niš, Serbia. https://orcid.org/0000-0002-3255-8127
Abdullah Mir Department of Mathematics, University of Kashmir, Srinagar, 190006, India. https://orcid.org/0000-0003-0930-6391
Adil Hussain Department of Mathematics, University of Kashmir, Srinagar, 190006, India. https://orcid.org/0000-0002-9646-9940

Pages: 541–554
Date Published: 2022-12-21
Vol. 24 No. 3 (2022)

A. Aziz and N. Ahmad, “Inequalities for the derivative of a polynomial”, Proc. Indian Acad. Sci. Math. Sci., vol. 107, no. 2, pp. 189–196, 1997.

A. Aziz and Q. M. Dawood, “Inequalities for a polynomial and its derivative”, J. Approx. Theory, vol. 54, no. 3, pp. 306–313, 1988.

A. Aziz and N. A. Rather, “A refinement of a theorem of Paul Turán concerning polynomials”, Math. Inequal. Appl., vol. 1, no. 2, pp. 231–238, 1998.

S. Bernstein, Sur l‘ordre de la meilleure approximation des functions continues par des polynômes de degré donné, Mémoires de l‘Académie Royale de Belgique 4, Brussels:Hayez, imprimeur des académies royales, 1912.

K. K. Dewan, N. Singh, A. Mir and A. Bhat, “Some inequalities for the polar derivative of a polynomial”, Southeast Asian Bull. Math., vol. 34, no. 1, pp. 69–77, 2010.

K. K. Dewan and C. M. Upadhye, “Inequalities for the polar derivative of a polynomial”, JIPAM. J. Inequal. Pure Appl. Math., vol. 9, no. 4, Article 119, 9 pages, 2008.

R. B. Gardner, N. K. Govil and G. V. MilovanoviÄ‡, Extremal problems and inequalities of Markov-Bernstein type for algebraic polynomials, Mathematical Analysis and Its Applications, London: Elsevier/Academic Press, 2022.

N. K. Govil, “On the derivative of a polynomial”; Proc. Amer. Math. Soc., vol. 41, no. 2, pp. 543–546, 1973.

N. K. Govil, “On a theorem of S. Bernstein”, Proc. Nat. Acad. Sci. India Sec. A., vol. 50, no. 1, pp. 50–52, 1980.

N. K. Govil, “Some inequalities for derivative of polynomials”, J. Approx. Theory, vol. 66, no. 1, pp. 29–35, 1991.

N. K. Govil and P. Kumar, “On sharpening of an inequality of Turán”, Appl. Anal. Discrete Math., vol. 13, no. 3, pp. 711–720, 2019.

N. K. Govil and Q. I. Rahman, “Functions of exponential type not vanishing in a half plane and related polynomials”, Trans. Amer. Math. Soc., vol. 137, pp. 501–517, 1969.

P. Kumar and R. Dhankhar, “Some refinements of inequalities for polynomials”, Bull. Math. Soc. Sci. Math. Roumanie (N.S), vol. 63(111), no. 4, pp. 359–367, 2020.

P. D. Lax, “Proof of a conjecture of P. ErdÅ‘s on the derivative of a polynomial”, Bull. Amer. Math. Soc., vol. 50, no. 8, pp. 509–513, 1944.

M. A. Malik, “On the derivative of a polynomial”, J. London Math. Soc.(2), vol. 1, no. 1, pp. 57–60, 1969.

M. Marden, Geometry of Polynomials, 2nd edition, Mathematical Surveys 3, Providence, R. I.: American Mathematical Society, 1966.

G. V. MilovanoviÄ‡, A. Mir and A. Hussain, “Extremal problems of Bernstein-type and an operator preserving inequalities between polynomials”, Sib. Math. J., vol. 63, no. 1, pp. 138– 148, 2022.

G. V. MilovanoviÄ‡, D. S. MitrinoviÄ‡ and Th. M. Rassias, Topics in polynomials, extremal problems, inequalities, zeros, River Edge, NJ: World Scientific Publishing, 1994.

A. Mir, “On an operator preserving inequalities between polynomials”, Ukrainian Math. J., vol. 69, no. 8, pp. 1234–1247, 2018.

A. Mir and D. Breaz, “Bernstein and Turán-type inequalities for a polynomial with constraints on its zeros”, Rev. R. Acad. Cienc. Exactas F ÌÄ±s. Nat. Ser. A Mat. RACSAM, vol. 115, no. 3, Paper No. 124, 12 pages, 2021.

A. Mir and I. Hussain, “On the ErdÅ‘s-Lax inequality concerning polynomials”, C. R. Math. Acad. Sci. Paris, vol. 355, no. 10, pp. 1055–1062, 2017.

G. Pólya and G. SzegÅ‘, “Aufgaben und Lehrsätze aus der Analysis”, Grundlehren der mathematischen Wissenschaften 19–20, Berlin: Springer, 1925.

Q. I. Rahman and G. Schmeisser, Analytic theory of polynomials, London Mathematical Society Monographs New Series 26, New York: Oxford University Press, Inc., 2002.

T. B. Singh and B. Chanam, “Generalizations and sharpenings of certain Bernstein and Turán types of inequalities for the polar derivative of a polynomial”, J. Math. Inequal., vol. 15, no. 4, pp. 1663–1675, 2021.

P. Turán, "Ìber die Ableitung von Polynomen”, Compositio Math., vol. 7, pp. 89–95, 1940.