Location of zeros of lacunary-type polynomials

Abstract

Although recently proved generalization of Enestr¨om-Kakey theorem by N.A.Rather et al. are applicable to the larger class of polynomials as compared to all other Enestr¨om- Kakeya type results, but are not applicable to the polynomials whose one or two coefficients are zero. For instance, if we consider the polynomial whose one or two coefficients are zero, then one can note that all Enestr¨om-Kakey type results including the recent generalizations by N.A.Rather et al. are not applicable to this polynomial. So it is interesting to look for the results applicable to such class of polynomials. Motivated by this, here we establish the following results applicable to such class of polynomials called the lacunary-type polynomials. In this paper, by relaxing the hypothesis of well-known Enestr¨om-Kakey theorem, we obtain a result which is applicable to the lacunary-type of polynomials and generalizes several well-known results concerning the location of zeros of polynomials. In addition to this, we show by examples that our results presents better information about the bounds of zeros of polynomials than some known results.

Keywords:

zeros of a polynomial, Lacunary-type polynomial, Enestr¨om-Kakeya theorem