Skip to main content
SpringerLink
Log in

Turán type inequalities for the generalized derivative of a polynomial

  • Research
  • Published:
Complex Analysis and its Synergies Aims and scope Submit manuscript

Abstract

Let \(P(z)=\prod _{j=1}^{n}(z-z_j)\) be a polynomial of degree n with \(|z_j|\le k_j\le 1,1\le j\le n\), it is known that

$$\begin{aligned} \max _{|z|=1}|P^\prime (z)|\ge \frac{n}{2}\left\{ 1+\frac{1}{1+\left( \frac{2}{n}\right) \sum _{j=1}^{n}\frac{k_j }{1-k_j}}\right\} \max _{|z|=1}|P(z)|. \end{aligned}$$

In this paper, we shall extend this inequality to the generalized derivative of a polynomial without perturbing the condition on the zeros of underlying polynomial. We shall also present the polar derivative analogue of the result obtained for the generalized derivative of the polynomial. Our results include different relevant inequalities as special cases.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aziz, A., Ahmad, N.: On inequalities for the derivative of a polynomial. Proc. Indian Acad. Sci. (Math. Sci.) 107(2), 189–196 (1997)

    Article  MathSciNet  Google Scholar 

  2. Aziz, A., Rather, N.A.: A refinement of a theorem of Pual Turan concerning polynomials. Math Ineq. Appl. 1, 231–238 (1998)

    Google Scholar 

  3. Bernstein, S.: Sur l’ordre de la meilleure approximation des fonctions continues par des polynômes de degré donné. Mem. Acad. R. Belg. 4, 1–103 (1912)

    Google Scholar 

  4. Bhat, F.A., Rather, N.A., Gulzar, S.: Inequalities for the generalised derivative of a complex polynomial. Publ. Inst. Math. 114(128), 71–80 (2023). https://doi.org/10.2298/PIM2328071B

    Article  Google Scholar 

  5. Diaz-Barrero, J.L., Egozcue, J.J.: A generalisation of Gauss-Lucas theorem. Czech. Math. J. 58, 481–486 (2008)

    Article  Google Scholar 

  6. Gardner, R.B., Govil, N.K., Milovanović, G.V.: Extremal Problems and Inequalities of Markov-Bernstein’s Type for Algebraic Polynomials. Elsevier/Academic Press, London (2022)

    Google Scholar 

  7. Govil, N.K.: On the derivative of a polynomial. Proc. Am. Soc. 41, 543–546 (1973)

    Article  Google Scholar 

  8. Govil, N.K., Labelle, G.: On Bernstein’s inequality. J. Math. Anal. Appl. 126, 494–500 (1987)

    Article  MathSciNet  Google Scholar 

  9. Lax, P.D.: Proof of a conjecture of P. Erdös on the derivative of a polynomial. Bull. Am. Math. Soc. 50, 509–513 (1944)

    Article  Google Scholar 

  10. Malik, M.A.: On the derivative of a polynomial. J. Lond. Math. Soc. Second Ser. 1, 57–60 (1969)

    Article  Google Scholar 

  11. Marden, M.: Geometry of Polynomials. Math Surveys 3. American Mathematical Society. Providence, RI (1996)

  12. Milovanović, G.V., Mitrinović, D.S., Rassias, T.M.: Topics in Polynomials: Extremal Problems, Inequalities, Zeros. World Scientific Publishing Co. Pte. Ltd., Singapore (1994)

    Book  Google Scholar 

  13. Rahman, Q.I., Schmeisser, G.: Analytic Theory of Polynomials. Oxford University Press, Oxford (2002)

    Book  Google Scholar 

  14. Rather, N.A., Ali, L., Shafi, Dar, I.: On the zeros of a class of generalized derivatives. P. J. M. 11(3), 549–557 (2022)

    Google Scholar 

  15. Rather, N.A., Bhat, F.A., Gulzar, S.: On the location of critical points of polynomials. Asian-Eur. J. Math. 12 (2019)

  16. Rather, N.A., Iqbal, A., Dar, I.: On the zeros of a class of generalised derivatives. Rend. Circ. Mat. Palermo. II. Ser (2020)

  17. Sz-Nagy, J.: Verallgemeinerung der Derivierten in der Geometric der Polynome. Acta Univ. Szeged. Sect. Sci. Math. 13, 169–178 (1950)

    MathSciNet  Google Scholar 

  18. Turán, P.: Über die Ableitung von Polynomen. Composito. Math. 7, 89–95 (1939)

    Google Scholar 

  19. Wani, I.A., Mir, M.I., Nazir, I., Dar, I.: Some \(L^p-\) type inequalities for generalized polar derivative. Palest. J. Math. 12(2), 161–168 (2023)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are highly indebted to the referee for his valuable suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Kashmir, South Campus, Anantnag, India

    F. A. Bhat & H. A. Dar

Authors
  1. F. A. Bhat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. A. Dar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. A. Bhat.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bhat, F.A., Dar, H.A. Turán type inequalities for the generalized derivative of a polynomial. Complex Anal Synerg 10, 9 (2024). https://doi.org/10.1007/s40627-024-00135-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40627-024-00135-z

Keywords

Mathematics Subject Classification

Search

Navigation