On the maximum modulus of a complex polynomial

Abstract

In this paper we impose distinct restrictions on the moduli of the zeros of $p\left(z\right)=\sum _{v=0}^n{a}_v{z}^v$ and investigate the dependence of $\parallel p\left(Rz\right)-p\left(\sigma z\right)\parallel$, $R>\sigma \ge 1$ on $M_{\alpha }$ and $M_{\alpha +\pi }$, where $M_{\alpha }=\underset{1\le k\le n}{max}|p\left(e^{i(\alpha +2k\pi )/n}\right)|$ and on certain coefficients of $p\left(z\right)$. This paper comprises several results, which in particular yields some classical polynomial inequalities as special cases. Moreover, the problem of estimating $p(1-\frac{w}{n})$, $0<w\leq$ given $p(1)=0$ is considered.

Keywords

• Polynomials
• definite integral
• zeros
• inequalities

References

1. Ahmad, I., Some generalizations of Bernstein type inequalities, Int. J. Modern Math. Sci., 4(3) (2012), 133-138.
2. Aziz, A., A refinement of an inequality of S. Bernstein, J. Math. Anal. Appl., 144 (1989), 226-235. https://doi.org/10.1016/0022-247X(89)90370-3
3. Aziz, A., Rather, N. A., New $L^q$ inequalities for polynomials, Mathematical Inequalities and Applications, 2 (1998), 177-191.
4. Bernstein, S.N., EMales Et La Meilleure Approximation Des Fonctions Analytiques D’une Variable REElle, Gauthier-Villars, Paris, 1926.
5. Dewan, K. K., Kaur, J., Mir, A., Inequalities for the derivative of a polynomial, J. Math. Anal. Appl., 269 (2002), 489-499. https://doi.org/10.1016/S0022-247X(02)00030-6
6. Frappier, C., Rahman, Q. I., Ruscheweyh, St., New inequalities for polynomials, Trans. Amer. Math. Soc., 288 (1985), 69-99. https://doi.org/10.2307/2000427
7. Gulzar, M. H., Inequalities for a polynomial and its derivative, International Journal of Mathematical Archive, 3(2) (2012), 528-533.
8. Lax, P. D., Proof of a conjecture of P. Erdos on the derivatives of a polynomial, Bull. Amer. Math. Soc., 50 (1944), 509-513.

9. Marden, M., Geometry of Polynomials, Second Edition, Mathematical Surveys, Amer. Math. Soc. Providence R.I. 1996.
10. Milovanovic, G. V., Mitrinovi´c, D. S., Rassias, Th.M., Topics in Polynomials Extremal Problems, Inequalities, Zeros, World Scientific, Singapore, 1994.
11. Polya, G., Szego, G., Aufgaben und Lehrsatze aus der Analysis, Springer, Berlin, 1925.
12. Rahman, Q. I., Schmeisser, G., Analytic Theory of Polynomials, Oxford University Press, New York, 2002.
13. Rather, N. A., Shah, M. A., On the derivative of a polynomial, Applied Mathematics, 3 (2012), 746-749. http://dx.doi.org/10.4236/am.2012.37110
14. Riesz, M., Eine trigonometrische interpolation formel und einige Ungleichung fur Polynome, Jahresber. Dtsch. Math.Verein, 23 (1914), 354-368.
15. Riesz, M., Über einen Sat’z des Herrn Serge Bernstein, Acta Math., 40 (1916), 337-347.