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Lobachevskii Journal of Mathematics

, Volume 39, Issue 9, pp 1407–1418 | Cite as

New Integral Mean Estimates for the Polar Derivative of a Polynomial

  • Idrees Qasim
Part 2. Special issue “Actual Problems of Algebra and Analysis” Editors: A. M. Elizarov and E. K. Lipachev
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Abstract

Let p(z) be a polynomial of degree n and for αC, let Dαp(z):= np(z) + (α − z)p′(z) denote the polar derivative of the polynomial p(z) with respect to α. In this paper, we obtain certain integral inequalities concerning the polar derivative of a polynomial, which besides yielding some interesting results, also includes some well-known theorems as special cases. Moreover we refine some Zygmund type inequalities for the polar derivative of a polynomial and present compact generalizations of some prior results.

Keywords and phrases

Polar derivative complex polynomials inequalities zeros 

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References

  1. 1.
    S. N. Bernstein, “Sur l’ordre de la meilleure approximation des functions continues par des polynomes de degrédonné,” Mem. Acad. R. Belg. 4, 100–103 (1912).Google Scholar
  2. 2.
    G. Polya and G. Szego, Problems and Theorems in Analysis (Springer, New York, 1972), Vol. 1.CrossRefzbMATHGoogle Scholar
  3. 3.
    P. D. Lax, “Proof of a conjecture of P. Erd’´ os on the derivative of a polynomial,” Bull. Am. Math. Soc. (N. S.) 50, 509–513 (1944).CrossRefzbMATHGoogle Scholar
  4. 4.
    W. Rudin, Real and Complex Analysis (Tata McGraw Hill, India, 1977).zbMATHGoogle Scholar
  5. 5.
    A. E. Taylor, Introduction to Functional Analysis (Wiley, New York, 1958).zbMATHGoogle Scholar
  6. 6.
    A. Zygmund, “A remark on conjugate series,” Proc. London Math. Soc. 34, 392–400 (1932).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    V. V. Arestov, “On integral inequalities for trigonometric polynomials and their derivatives,” Izv. Akad. Nauk SSSR, Ser. Mat. 45, 3–22 (1981).MathSciNetGoogle Scholar
  8. 8.
    G. H. Hardy, “Themean value of the modulus of an analytic function,” Proc. LondonMath. Soc. 14, 269–277 (1915).CrossRefzbMATHGoogle Scholar
  9. 9.
    N. G. de Bruijn, “Inequalities concerning polynomials in the complex domain,” Nederal. Akad. wetensch., Proc. 50, 1265–1272 (1947).MathSciNetzbMATHGoogle Scholar
  10. 10.
    A. Aziz and N. A. Rather, “Lp inequalities for polynomials,” Appl. Math. 2, 321–328 (2011).MathSciNetCrossRefGoogle Scholar
  11. 11.
    A. Mir and S. A. Baba, “Some integral inequalities for the polar derivative of a polynomial,” Anal. Theory Appl. 27, 340–350 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A. Aziz and N. A. Rather, “On an inequality concerning the polar derivative of a polynomial,” Proc. Indian Acad. Sci. Math. 117, 349–357 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    A. Liman, M. A. Mahapatra, and W. M. Shah, “Inequalities for the polar derivative of polynomial,” Complex Anal. Operat. Theory 6, 1199–1209 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    A. Aziz, “Inequalities for the polar derivative of a polynomial,” J. Approx Theory 55, 183–193 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    G. Singh and W. M. Shah, “Inequalities for the polar derivative of a polynomial,” East J. Approx. 17, 221–228 (2011).MathSciNetGoogle Scholar
  16. 16.
    E. Laguerre, Oeuvres (Gauthier-Villars, Paris, 1898), Vol. 1.zbMATHGoogle Scholar
  17. 17.
    N. A. Rather, “Some integral inequalities for the polar derivative of a polynomial,” Math. Balkan. 22, 207–216 (2008).MathSciNetzbMATHGoogle Scholar
  18. 18.
    N. A. Rather and S. A. Zargar, “New operator preserving Lp inequalities between polynomials,” Adv. Inequal. Appl. 2, 31–60 (2013).Google Scholar
  19. 19.
    N. K. Govil, A. Liman, and W. M. Shah, “Some inequalities concerning derivative and maximum modulus of polynomials,” Austral. J. Math. Anal. Appl. 8, 1–8 (2011).MathSciNetzbMATHGoogle Scholar
  20. 20.
    W. M. Shah, “A generalization of a theorem of Paul Turén,” J. Ramanujan Math. Soc. 1, 67–72 (1996).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Government Girls Higher Secondary School LangateLangate, Jammu and KashmirLangaitIndia

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