International Journal of Nonlinear Analysis and Applications

A note on sharpening of Erdos-Lax and Turan-type inequalities for a constrained polynomial

Document Type : Research Paper

Author

Department of Mathematics, University of Kashmir, Srinagar-190006, India

Abstract

The well-known Erdos-Lax and Turan-type inequalities, which relate the uniform norm of a univariate complex coefficient polynomial to its derivative on the unit circle in the plane, are discussed in this paper. We create some new inequalities here when there is a restriction on its zeros. The obtained results strengthen some recently proved Erdos-Lax and Turan-type inequalities for constrained polynomials and also produce various inequalities that are sharper than the previous ones known in a very rich literature on this subject.

Keywords

[1] A. Aziz and N. Ahmad, Inequalities for the derivative of a polynomial, Proc. Indian Acad. Sci. (Math. Sci.) 107 (1997), 189–196.
[2] A. Aziz and Q. M. Dawood, Inequalities for a polynomial and its derivative, J. Approx. Theory 54 (1988), 306–313.
[3] A. Aziz and N. A. Rather, A refinement of a theorem of Paul Tur´an concerning polynomials, Math. Inequal. Appl. 1 (1998), 231–238.
[4] S. Bernstein, Sur l’ordre de la meilleure approximation des functions continues par des polynˆomes de degr´e donn´e,
Mem. Acad. R. Belg. 4 (1912), 1–103.
[5] V.N. Dubinin, Distortion theorems for polynomials on the circle, Math. Sb. 12 (2000), 51–60.
[6] V.N. Dubinin, Applications of the Schwarz lemma to inequalities for entire functions with constarints on zeros, J. Math. Sci. 143 (2007), 3069–3076.
[7] N.K. Govil, On the derivative of a polynomial, Proc. Amer. Math. Soc. 41 (1973), 543–546.
[8] N.K. Govil, On a theorem of S. Bernstein, Proc. Nat. Acad. Sci. (India) 50 (1980), no. A, 50–52.
[9] N.K. Govil, Some inequalities for derivative of polynomials, J. Approx. Theory 66 (1991), 29–35.
[10] N.K. Govil and P. Kumar, On sharpening of an inequality of Tur´an, Appl. Anal. Discrete Math. 13 (2019), 711–720.
[11] N.K. Govil and G.N. Mctume, Some generalizations involving the polar derivative for an inequality of Paul Turan, Acta Math. Hungar. 104 (2004) 115–126.
[12] N.K. Govil and Q.I. Rahman, Functions of exponential type not vanishing in a half plane and related polynomials, Trans. Amer. Math. Soc. 137 (1969), 501–517.
[13] P. Kumar and R. Dhankhar, Some refinements of inequalities for polynomials, Bull. Math. Soc. Sci. Math. Roumanie 63 (2020), no. 111, 359–367.
[14] P.D. Lax, Proof of a conjecture of P. Erd¨os on the derivative of a polynomial, Bull. Amer. Math. Soc. 50 (1944), 509–513.
[15] M.A. Malik, On the derivative of a polynomial, J. London Math. Soc. 1 (1969), 57–60.
[16] M. Marden, Geometry of Polynomials, Math. Surveys 3 (1966).
[17] G.V. Milovanovi´c, A. Mir and A. Hussain, Extremal problems of Bernstein-type and an operator preserving inequalities between polynomials, Siberian Math. J. 63 (2022), 138–148.
[18] G.V. Milovanovic, D.S. Mitrinovic and Th.M. Rassias, Topics in Polynomials, Extremal Problems, Inequalities, Zeros, World Scientific, Singapore, 1994.
[19] A. Mir, On an operator preserving inequalities between polynomials, Ukrainian Math. J. 69 (2018), 1234–1247.
[20] A. Mir and D. Breaz, Bernstein and Tur´an-type inequalities for a polynomial with constraints on its zeros, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM 115 (2021), Paper No. 124, 1–12.
[21] A. Mir and I. Hussain, On the Erd¨os-Lax inequality concerning polynomials, C. R. Acad. Sci. Paris Ser. I 355 (2017), 1055–1062.
[22] G. P´olya and G. Szeg¨o, Aufgaben und Lehrs¨atze aus der Analysis, Springer-Verlag, Berlin, 1925.
[23] Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press, 2002.
[24] T. B. Singh, M. T. Devi and B. Chanam, Sharpening of Bernstein and Tur´an–type inequalities for polynomials, J. Class. Anal. 18 (2021), 137–148.
[25] P. Tur´an, Uber die Ableitung von Polynomen, ¨ Compositio Math. 7 (1939), 89–95.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 2
July 2022
Pages 3239-3249
  • Receive Date: 06 May 2022
  • Revise Date: 29 July 2022
  • Accept Date: 02 August 2022