Nonlinear Functional Analysis and Applications

  • 제28권4호
  • /
  • Pages.943-956
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  • 2023
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  • 1229-1595(pISSN)
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  • 2466-0973(eISSN)
경남대학교 수학교육과 (Kyungnam University, Department of Mathematics Eduaction)
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INEQUALITIES FOR COMPLEX POLYNOMIAL WITH RESTRICTED ZEROS

  • Istayan Das (Department of Mathematics, National Institute of Technology Manipur) ;
  • Robinson Soraisam (Department of Mathematics, National Institute of Technology Manipur) ;
  • Mayanglambam Singhajit Singh (Department of Mathematics, National Institute of Technology Manipur) ;
  • Nirmal Kumar Singha (Department of Mathematics, National Institute of Technology Manipur) ;
  • Barchand Chanam (Department of Mathematics, National Institute of Technology Manipur)
  • 투고 : 2023.03.02
  • 심사 : 2023.03.31
  • 발행 : 2023.12.15
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초록

Let p(z) be a polynomial of degree n and for any complex number 𝛽, let D𝛽p(z) = np(z) + (𝛽 - z)p'(z) denote the polar derivative of the polynomial with respect to 𝛽. In this paper, we consider the class of polynomial $$p(z)=(z-z_0)^s \left(a_0+\sum\limits_{{\nu}=0}^{n-s}a_{\nu}z^{\nu}\right)$$ of degree n having a zero of order s at z0, |z0| < 1 and the remaining n - s zeros are outside |z| < k, k ≥ 1 and establish upper bound estimates for the maximum of |D𝛽p(z)| as well as |p(Rz) - p(rz)|, R ≥ r ≥ 1 on the unit disk.

키워드

과제정보

We are thankful to NIT, Manipur for providing us financial support. We are also grateful to the referee for his/her useful suggestions.

참고문헌

  1. A. Aziz, A refinement of an inequality of S. Bernstein, J. Math. Anal. Appl., 114 (1989), 226-235, https://doi.org/10.1016/0022-247X(89)90370-3.
  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 W.M. Shah, Inequalities for a polynomial and its derivative, Math. Ineq. Appl., 7(3) (2004), 379-391.
  4. A. Aziz and B.A. Zargar, Inequalities for a polynomial and its derivative, Math. Ineq. Appl., 1(4) (1998), 543-550.
  5. S. Bernstein, Sur lordre de la meilleure approximation des functions continues par des polynomes de degr donn, Mem. Acad. R. Belg., 4 (1912), 1-103.
  6. T.N. Chan and M.A. Malik, On Erdos-Lax theorem, Proc. Indian Acad. Sci., 92(3) (1983), 191-193.
  7. B. Chanam and K.K. Dewan, Inequalities for a polynomial and its derivative, J. Math. Anal. Appl., 336 (2007), 171-179.
  8. B. Chanam, N. Reingachan, K. B. Devi, M. T. Devi and K. Krishnadas, Some Lq inequalities for polynomial, Nonlinear Funct. Anal. Appl., 26(2)(2021), 331-345.
  9. K.K. Dewan and M. Bidkham, Inequalities for a polynomial and its derivative, J. Math. Anal. Appl., 166 (1992), 319-324.
  10. C. Frappier, Q.I. Rahman and S. Ruscheweyh, New inequalities for polynomials, Trans. Amer. Math. Soc., 288 (1985), 6999, https://doi.org/10.1090/S0002-9947-1985-0773048-1.
  11. N.K. Govil, Some inequalities for derivatives of polynomials, J. Approx. Theory, 66 (1991), 29-35.
  12. A. Hussain, A. Mir and A. Ahmad, On Bernstein-type inequalities for polynomials involving the polar derivative, J. Class. Anal., 16 (2020), 9-15, https://doi.org/10.7153/jca-2020-16-02.
  13. P.D. Lax, Proof of a conjecture of P. Erdos on the derivative of a polynomial, Bull. Amer. Math. Soc., 50 (1944), 509-513.
  14. M.A. Malik, On the derivative of a polynomial, J. London Math. Soc., 1 (1969), 57-60.
  15. G.V. Milovanovic and A. Mir, Generalizations of Zygmund-type integral inequalities for the polar derivative of a complex polynomial, J. Ineq. Appl., 2020 (2020), 1-12, https://doi.org/10.1186/s13660-020-02404-x.
  16. G.V. Milovanovic, D.S. Mitrinovic and D.S. Rassias, Topics in Polynomials: Extremal Properties, Inequalities, Zeros, World Scientific Publishing Co. Singapore; 1994.
  17. A. Mir, On an operator preserving inequalities between polynomials, Ukrainian Math. J., 69 (2018), 1234-1247, https://doi.org/10.1007/s11253-017-1427-2.
  18. A. Mir, Some sharp upper bound estimates for the maximal modulus of polar derivative of a polynomial, Annali Dell Universita Di Ferrara. 65 (2019), 327-336, https://doi.org/10.1007/s11565-019-00317-2.
  19. A. Mir and B. Dar, On the polar derivative of a polynomial, J. Ramanujan Math. Soc., 29 (2014), 403-412.
  20. A. Mir and A. Hussain, Generalizations of some Bernstein-type inequalities for the polar derivative of a polynomial, Kragujevac J. Math., 40(1) (2021), 31-41.
  21. A. Mir and I. Hussain, On the Erds-Lax inequality concerning polynomials, C. R. Math. Acad. Sci. Paris, 355 (2017), 1055-1062, https://doi.org/10.1016/j.crma.2017.09.017.
  22. A. Mir and M.A. Malik, Inequalities concerning the rate of growth of polynomials involving the polar derivative, Ann. Univ. Mariae Curie-Sk Lodowska Sect. A, 74 (2020), 67-75, http:dx.doi.org/10.17951/a.2020.74.1.67-75.
  23. A. Mir and A. Wani, A note on two recent results about polynomials with restricted zeros, J. Math. Ineq., 14 (2020), 45-50, https://doi.org/10.7153/jmi-2020-14-04.
  24. K.M. Nakprasit and J. Somsuwan, An upper bound of a derivative for some class of polynomials, J. Math. Ineq., 11 (2017), 143-150.
  25. M.S. Pukhta, Extremal problems for polynomials and on location of zeros of polynomials, Ph.D. Thesis of the Jamia Millia Islamia, New Delhi; 1995.
  26. Q.A. Qazi, On the maximum modulus of polynomials, Proc. Amer. Math. Soc., 115 (1992), 337-343.
  27. Q.I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press, New York, 2002.
  28. A. Zireh and A.M. Bidkham, Inequalities for the polar derivative of a polynomial with restricted zeros, Kragjevac J. Math., 40 (2016), 113-124, https://doi.org/10.5937/KgJMath1601113Z.