On the maximum modulus of a complex polynomial

Year 2022, Volume: 71 Issue: 3, 666 - 681, 30.09.2022

Shabir Malik , Ashish Kumar

https://doi.org/10.31801/cfsuasmas.989344

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

• Ahmad, I., Some generalizations of Bernstein type inequalities, Int. J. Modern Math. Sci., 4(3) (2012), 133-138.
• 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
• Aziz, A., Rather, N. A., New $L^q$ inequalities for polynomials, Mathematical Inequalities and Applications, 2 (1998), 177-191.
• Bernstein, S.N., EMales Et La Meilleure Approximation Des Fonctions Analytiques D’une Variable REElle, Gauthier-Villars, Paris, 1926.
• 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
• 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
• Gulzar, M. H., Inequalities for a polynomial and its derivative, International Journal of Mathematical Archive, 3(2) (2012), 528-533.
• Lax, P. D., Proof of a conjecture of P. Erdos on the derivatives of a polynomial, Bull. Amer. Math. Soc., 50 (1944), 509-513.
• Marden, M., Geometry of Polynomials, Second Edition, Mathematical Surveys, Amer. Math. Soc. Providence R.I. 1996.
• Milovanovic, G. V., Mitrinovi´c, D. S., Rassias, Th.M., Topics in Polynomials Extremal Problems, Inequalities, Zeros, World Scientific, Singapore, 1994.
• Polya, G., Szego, G., Aufgaben und Lehrsatze aus der Analysis, Springer, Berlin, 1925.
• Rahman, Q. I., Schmeisser, G., Analytic Theory of Polynomials, Oxford University Press, New York, 2002.
• 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
• Riesz, M., Eine trigonometrische interpolation formel und einige Ungleichung fur Polynome, Jahresber. Dtsch. Math.Verein, 23 (1914), 354-368.
• Riesz, M., Über einen Sat’z des Herrn Serge Bernstein, Acta Math., 40 (1916), 337-347.