Nonlinear Functional Analysis and Applications

Kyungnam University, Department of Mathematics Eduaction (경남대학교 수학교육과)
권호 표지
DOI QR코드

DOI QR Code

TURÁN-TYPE Lr-INEQUALITIES FOR POLAR DERIVATIVE OF A POLYNOMIAL

  • Received : 2022.11.28
  • Accepted : 2023.02.25
  • Published : 2023.09.15
⟨ Previous Next ⟩

Abstract

If p(z) is a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1, then for any complex number α with |α| ≥ k, and r ≥ 1, Aziz [1] proved $$\left{{\int}_{0}^{2{\pi}}\,{\left|1+k^ne^{i{\theta}}\right|^r}\,d{\theta}\right}^{\frac{1}{r}}\;{\max\limits_{{\mid}z{\mid}=1}}\,{\mid}p^{\prime}(z){\mid}\,{\geq}\,n\,\left{{\int}_{0}^{2{\pi}}\,{\left|p(e^{i{\theta}})\right|^r\,d{\theta}\right}^{\frac{1}{r}}.$$ In this paper, we obtain an improved extension of the above inequality into polar derivative. Further, we also extend an inequality on polar derivative recently proved by Rather et al. [20] into Lr-norm. Our results not only extend some known polynomial inequalities, but also reduce to some interesting results as particular cases.

Keywords

Acknowledgement

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

References

  1. A. Aziz, Integral mean estimates for polynomials with restricted zeros, J. Approx. Theory, 55 (1988), 232-239.  https://doi.org/10.1016/0021-9045(88)90089-5
  2. A. Aziz and Q.M. Dawood, Inequalities for a polynomial and its derivatives, J. Approx. Theory, 54 (1988), 306-313.  https://doi.org/10.1016/0021-9045(88)90006-8
  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. A. Aziz and W.M. Shah, Inequalities for a polynomial and its derivative , Math. Ineq. Appl., 7(3) (2004), 379-391.  https://doi.org/10.7153/mia-07-39
  5. A. Aziz and B.A. Zargar, Inequalities for the maximum modulus of the derivative of a polynomial, J. Ineq. Pure Appl. Math., 8(2) (2007), Art. 37. 
  6. Kh.B. Devi, K. Krishnadas and B. Chanam, Some inequalities on polar derivative of a polynomial, Nonlinear Funct Anal. Appl., 27(1) (2022), 141-148.  https://doi.org/10.1007/s41478-021-00332-7
  7. K.K. Dewan, N. Singh, A. Mir and A. Bhat, Some inequalities for the polar derivative of a polynomial, Southeast Asian Bull. Math., 34 (2010), 69-77. 
  8. V.N. Dubinin, Applications of the Schwarz lemma to inequalities for entire functions with constraints on zeros, J. Math. Sci., 143 (2007), 3069-3076.  https://doi.org/10.1007/s10958-007-0192-4
  9. R.B. Gardner, N.K. Govil and S.R. Musukula, Rate of growth of polynomials not vanishing inside a circle, J. Ineq. Pure Appl. Math., 6(2) (2005), Art. 53. 
  10. N.K. Govil, On the Derivative of a Polynomial, Proc. Amer. Math. Soc., 41 (1973), 543-546.  https://doi.org/10.1090/S0002-9939-1973-0325932-8
  11. N.K. Govil, Some inequalities for the derivative of a polynomial, J. Approx. Theory, 66 (1991), 29-35.  https://doi.org/10.1016/0021-9045(91)90052-C
  12. N.K. Govil and P. Kumar, On Sharpning of an inequality of Tur'an, Appl. Anal. Discrete Math., 13 (2019), 711-720.  https://doi.org/10.2298/AADM190326028G
  13. G.H. Hardy, The mean value of the modulus of an analytic function, Proc. Lond. Math. Soc., 14 (1915), 269-277.  https://doi.org/10.1112/plms/s2_14.1.269
  14. R.P. Boas Jr and Q.I. Rahman, Lp inequalities for polynomials and entire functions, Arch. Rational Mech. Anal., 11 (1962), 34-39.  https://doi.org/10.1007/BF00253927
  15. M.A. Malik, An integral mean estimates for polynomials, Proc. Amer. Math. Soc., 91(2) (1984), 281-284.  https://doi.org/10.1090/S0002-9939-1984-0740186-3
  16. M.A. Malik, On the derivative of a polynomial, J. Lond. Math. Soc., 1(2) (1969), 57-60.  https://doi.org/10.1112/jlms/s2-1.1.57
  17. A. Mir, A note on an inequality of Paul Tura'n concerning polynomials, Ramanujan J., 56 (2021), 1061-1071, https://doi.org/10.1007/s11139-021-00494-9. 
  18. Q.I. Rahman and G. Schmeisser, Lp inequalities for polynomials, J. Approx. Theory, 53 (1988), 26-32.  https://doi.org/10.1016/0021-9045(88)90073-1
  19. N.A. Rather, I. Dar and A. Iqbal, Some inequalities for polynomials with restricted zeros, Ann. Univ. Ferrara, 67 (2021), 183-189.  https://doi.org/10.1007/s11565-020-00353-3
  20. N.A. Rather, I. Dar and A. Iqbal, Some Lower Bound Estimates for the Polar Derivatives of Polynomals with Restricted Zeros , J. Anal. Num. Theory, 9(1) (2021), 1-5. 
  21. N. Reingachan and B. Chanam, Bernestien and Turan type inequalities for the polar derivative of a polynomial, Nonlinear Funct Anal. Appl., 28(1) (2023), 287-294.  https://doi.org/10.17654/0972087122013
  22. N. Reingachan, R. Soraisam and B. Chanam, Some inequalities on polar derivative of a polynomial, Nonlinear Funct Anal. Appl., 27(4) (2022), 797-805. 
  23. W. Rudin, Real and Complex Analysis, Tata McGraw-Hill publishing Company, 1977. 
  24. T.B. Singh, K.B. Devi, R. Ngamchui, R. Soraisam and B. Chanam, Lr Inequalities of Generalized Inequalities of Polynomials, Turk. J. Comput. Math. Edu., 12(14) (2021), 3525-3531. 
  25. A.E. Taylor, Introduction to Functional Analysis, John Wiley and Sons, New York, 1958. 
  26. P. Turan, Uber die Ableitung von Polynomen, Compos. Math., 7 (1939), 89-95. 
  27. A. Zireh, Integral mean estimates for polar derivative of polynomials, J. Interdiscip. Math., 21(1) (2018), 29-42.  https://doi.org/10.1080/09720502.2014.962841
  28. A. Zireh, E. Khojastehnezhad and S.R. Musawi, Integral mean estimates for the polar derivative of polynomials whose zeros are within a circle, J. Inequal. Appl., 2013:307 (2013).