Skip to main content
Log in

Abstract

This paper aims to set an account of zero-free regions for lacunary type polynomials whose coefficients or their real and imaginary parts are subjected to certain restrictions. We also find bounds concerning the number of zeros in a specific annular region.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

REFERENCES

  1. A. Aziz and B. A. Zargar, ‘‘On the zeros of a class of polynomials and related analytic functions,’’ Anal. Theory Appl. 28 (2), 180–188 (2012).

    MathSciNet  MATH  Google Scholar 

  2. A. Aziz and B. A. Zargar, ‘‘Bounds for the zeros of a polynomial with restricted coefficients,’’ Appl. Math. 3, 30–33 (2012). https://doi.org/10.4236/am.2012.31005

    Article  MathSciNet  Google Scholar 

  3. A. L. Cauchy, Exercices de Mathématiques, vol. 2 (Chez de Bure Fréres, Paris, 1827).

    Google Scholar 

  4. Y. Choo, G. K. Choi, ‘‘On the zero-free regions of polynomials,’’ Int. J. Math. Anal. 5, 975–981 (2011).

    MathSciNet  MATH  Google Scholar 

  5. K. K. Dewan and M. Bidkham, ‘‘On Eneström–Kakeya theorem,’’ J. Math. Anal. Appl. 180, 29–36 (1993). https://doi.org/10.1006/jmaa.1993.1379

    Article  MathSciNet  MATH  Google Scholar 

  6. R. B. Gardner and N. K. Govil, ‘‘On the location of the zeros of a polynomial,’’ J. Approximation Theory 78, 286–292 (1994). https://doi.org/10.1016/0021-9045(78)90037-0

    Article  MathSciNet  MATH  Google Scholar 

  7. R. B. Gardner and B. Shields, ‘‘The number of zeros of a polynomial in a disk,’’ J. Classical Anal. 3, 167–176 (2013). https://doi.org/10.7153/jca-03-15

    Article  MathSciNet  MATH  Google Scholar 

  8. N. K. Govil and Q. I. Rahman, ‘‘On Eneström–Kakeya theorem,’’ Tohoku Math. J. 20, 126–136 (1968). https://doi.org/10.2748/tmj/1178243172

    Article  MathSciNet  MATH  Google Scholar 

  9. M. H. Gulzar, ‘‘Zero-free regions for polynomials with restricted coefficients,’’ Res. Inventy: Int. J. Eng. Sci. 2 (6), 6–10 (2013).

    Google Scholar 

  10. A. Joyal, G. Labelle, and Q. I. Rahman, ‘‘On the location of zeros of polynomials,’’ Can. Math. Bull. 10, 53–63 (1967). https://doi.org/10.4153/CMB-1967-006-3

    Article  MathSciNet  MATH  Google Scholar 

  11. Young-Ju Kim, ‘‘On the zero-free regions of analytic functions,’’ Int. J. Math. Anal. 6, no. 12, 563–571 2012.

    MathSciNet  MATH  Google Scholar 

  12. M. Marden, Geometry of Polynomials, Mathematical Surveys, vol. 3 (American Mathematical Society, Providence, R.I., 1966).

  13. G. V. Milovanović, D. S. Mitrinović, and Th. M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros (World Scientific, Singapore, 1994). https://doi.org/10.1142/1284

  14. A. Mir, A. Ahmad, and A. H. Malik, ‘‘Number of zeros of a polynomial in a specific region with restricted coefficients,’’ J. Math. Appl. 42, 135–146 (2019). https://doi.org/10.7862/rf.2019.9

    Article  MathSciNet  MATH  Google Scholar 

  15. Q. G. Mohammad, ‘‘On the zeros of the polynomials,’’ Am. Math. Monthly 7, 631–633 (1965). https://doi.org/10.2307/2312995

    Article  MATH  Google Scholar 

  16. M. S. Pukhta, ‘‘On the zeros of a polynomial,’’ Appl. Math. 2, 1356–1358 (2011). https://doi.org/10.4236/am.2011.211189

    Article  Google Scholar 

  17. I. Qasim, T. Rasool, and A. Liman, ‘‘Number of zeros of a polynomial (Lacunary-type) in a disk,’’ J. Math. Appl. 41, 181–194 (2018).

    MathSciNet  MATH  Google Scholar 

  18. Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, London Mathematical Society Monographs New Series, vol. 26 (Oxford Univ. Press, Oxford, 2002).

    MATH  Google Scholar 

  19. E. C. Titchmarsh, The Theory of Functions, 2nd ed. (Oxford University Press, London, 1930).

    Google Scholar 

  20. B. A. Zargar, ‘‘Zero-free regions for polynomials with restricted coefficients,’’ Int. J. Math. Sci. Eng. Appl. 6 (4), 33–42 (2012).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to S. Ahmad Malik, A. Kumar or B. Ahmad Zargar.

Ethics declarations

The authors declare that they have no conflicts of interest.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Malik, S.A., Kumar, A. & Zargar, B.A. Zero-Free Regions for Lacunary Type Polynomials. J. Contemp. Mathemat. Anal. 57, 172–182 (2022). https://doi.org/10.3103/S1068362322030062

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1068362322030062

Keywords:

Navigation