Number of Zeros of a Polynomial in a Specific Region with Restricted Coefficients

This paper focuses on the problem concerning the location and the number of zeros of polynomials in a specific region when their coefficients are restricted with special conditions. We obtain extensions of some classical results concerning the number of zeros of polynomials in a prescribed region by imposing the restrictions on the moduli of the coefficients, the real parts(only) of the coefficients, and the real and imaginary parts of the coefficients.

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1. K.K. Dewan, Extremal properties and coefficient estimates for polynomials with restricted zeros and on location of zeros of polynomials, Ph.D. Thesis, Indian Institute of Technology, Delhi, 1980.
2. K.K. Dewan, M. Bidkham, On the Eneström-Kakeya theorem, J. Math. Anal. Appl. 180 (1993) 29-36.
3. R.B. Gardner, N.K. Govil, On the location of the zeros of a polynomial, J. Approx. Theory 78 (1994) 286-292.
4. R. Gardner, B. Shields, The number of zeros of a polynomial in a disk, J. Class. Anal. 3 (2013) 167-176.
5. N.K. Govil, Q.I. Rahman, On the Eneström-Kakeya theorem, Tôhoku Math. Jour. 20 (1968) 126-136.
6. I. Qasim, T. Rasool, A. Liman, Number of zeros of a polynomial (Lucanarytype) in a disk, J. Math. Appl. 41 (2018) 181-194.
7. A. Joyal, G. Labelle, Q.I. Rahman, On the location of zeros of polynomials, Canad. Math. Bull. 10 (1967) 53-63.
8. M. Marden, Geometry of Polynomials, Math. Surveys, No.3, Amer, Math. Soc., Providence, R.I., 1966.
9. Q.G. Mohammad, On the zeros of the polynomials, Amer. Math. Monthly 72 (1965) 631-633.
10. M.S. Pukhta, On the zeros of a polynomial, Appl. Math. 2 (2011) 1356-1358.
11. Q.I. Rahman, G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press, 2002.
12. T. Rasool, I. Ahmad, A. Liman, On zeros of polynomials with restricted coefficients, Kyungpook Math. J. 55 (2015) 807-816.
13. E.C. Titchmarsh, The Theory of Functions, 2nd Edition, Oxford University Press, London, 1939.