Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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ON THE WEAKENED HYPOTHESES-BASED GENERALIZATIONS OF THE ENESTRÖM-KAKEYA THEOREM

  • Shahbaz Mir (Department of Mathematics, National Institute of Technology) ;
  • Abdul Liman (Department of Mathematics, National Institute of Technology)
  • Received : 2024.01.31
  • Accepted : 2024.05.15
  • Published : 2024.06.30
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Abstract

According to the well-known Eneström-Kakeya Theorem, all the zeros of a polynomial $P(z)={\sum_{s=0}^{n}}a_sz^s$ of degree n with real coefficients satisfying an ≥ an-1 ≥ · · · ≥ a1 ≥ a0 > 0 lie in the complex plane |z| ≤ 1. We provide comparable results with hypotheses relating to the real and imaginary parts of the coefficients as well as the coefficients' moduli in response to recent findings about an Eneström-Kakeya "type" condition on real coefficients. Our findings so broadly extend the other previous findings.

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References

  1. Aziz, A. and Zargar, B. A., Bounds for the zeros of a polynomial with restricted coefficients, Applied Mathematics, 3 (2012), 30-33. http://dx.doi.org/10.4236/am.2012.31005
  2. Chan, T. and Malik, M., On Erdos-Lax Theorem, Proceedings of the Indian Academy of Science, 250 (1983), 191-193.
  3. Enestrom, G., Harledning af en allman formel for antalet pensionarer, som vid en godtyeklig tidpunkt forefinnas inom en sluten pensionslcassa, Ovfers. Vetensk.-Akad. Forhh., 50 (1983), 405-415.
  4. Gardner, R. and Govil, N. K., On the Location of the Zeros of a Polynomial, Journal of Approximation Theory, 78 (1994), 286-292 .
  5. Gardner, R. and Govil, N. K., The Enestrom-Kakeya Theorem and Some of Its Generalizations, Current Topics in Pure and Computational Complex Analysis, S. Joshi, M. Dorff, and I. Lahiri, New Delhi: Springer-Verlag (2014), 171-200. https://faculty.etsu.edu/gardnerr/pubs/GardnerGovilF.pdf
  6. Govil, N. K. and Rahman, Q. I., On the Enestrom-Kakeya Theorem, Tˆohoku Math. J. 20 (1968), 126-136.
  7. Joyal, A. ; Labelle, G. and Rahman, Q. I., On the Location of Zeros of Polynomials., Canadian Mathematical Bulletin 10(1) (1967), 53-63 .
  8. Kakeya, S., On the limits of the roots of an algebraic equation with positive coefficient, Tohoku Math. J. 2 (1912-1913), 140-142.
  9. Shah, M. A.; Swroop, R.; Sofi, H. M. and Nisar, I., A generalisation of Enestrom-Kakeya Theorem and a zero free region of a polynomial. Journal of Applied Mathematics and Physics, 9(2021), 1271-1277. https://doi.org/10.4236/jamp.2021.96087
  10. Marden, M., Geometry of Polynomials., Math. Surveys, No. 3, Amer. Math. Soc.