Location of Zeros of Lacunary-type Polynomials

In this paper, we present some interesting results concerning the location of zeros of Lacunary-type of polynomial in the complex plane. By relaxing the hypothesis and putting less restrictive conditions on the coefficients of the polynomial, our results generalize and refines some classical results.

Full text (pdf)

1. R.E. Allardice, On a limit of the roots of an equation that is independent of all but two of the coefficients, Bull. Amer. Math. Soc. 13 (1906-1907) 443-447.
2. A. Aziz, N.A. Rather, Bounds for the zeros of a class of Lacunary-type polynomials, Journal of Mathematical Inequalities 73 (2013) 445-452.
3. A.L. Cauchy, Exercise de mathematique, In Oeuvres 9 (2) (1829) 122.
4. G. Eneström, Remargue sur an theoreme relatif aux racines de 1 equation an x^n+ … + a0 = 0 ou tousles coefficients sont reels et positifs, Tohoku Math. J. 18 (1920), 34-36, Translation of a Swedish article in Ofversigt of Konigl. Vetenskaps Akademiens Forhandlingar, 50 (1893) 405-415.
5. L. Fejér, Sur la racine de moindre module d'une equation algebrique, C. R. Acad. Sci. Paris 145 (1907) 459-461.
6. L. Fejér, Über die Wurzel von kleinsten absoluten Beträge einer algebraischen Gleichung, Math. Ann. 65 (1908) 413-423.
7. A. Hurwitz, Über die Nullslellen des Bessel'schen Function, Math. Ann. 33 (1889) 246-266.
8. A. Joyal, G. Labelle, Q.I. Rahman, On the location of zeros of polynomials, Canad. Math. J. Bull. 10 (1967) 53-63.
9. S. Kakeya, On the limits of the roots of an algebraic equation with positive coefficients, Tohoku Math. J. 2 (1912-13) 140-142.
10. E. Landau, Über den Picardschen Satz, Vierteljahrsschrift Naturforsch. Gesellschaft Zürcher & Furrer 51 (1906) 252-318.
11. E. Landau, Sur quelques generalisations du theoreme de M. Picard, Ann. Ecole Norm. 24 (3) (1907) 179-201.
12. M. Marden, Geometry of Polynomials, Math. Surveys No. 3, Amer. Math. Soc. Providence R. I., 1949.
13. G.V. Milovanovic, D.S. Mitrinovic, Th.M. Rassias, Topics in Polynomials, Extermal Problems, Inequalities, Zeros, World Scientific, Singapore, 1994.
14. E.B. Van Vleck, On the limits to the absolute values of the roots of a polynomial, Bull. Soc. Math. France 53 (1925) 105-125.