×

Inequalities for polynomials not vanishing in a disk. (English) Zbl 1239.30001

Summary: If \(P(z)\) is a polynomial of degree \(n\), having no zeros in the unit disc, then for all \(\alpha , \beta \in \mathbb{C}\) with \(| \alpha | \leq 1\), \(| \beta | \leq 1\) it is known that \[ \begin{split} \left| P(Rz) -\alpha P(z) + \beta \left\{ \left(\frac{R+1}{2}\right)^n - |\alpha|\right\} P(z)\right| \\ \leq \frac12 \left[\left|R^n - \alpha + \beta \left\{\left(\frac{R+1}{2}\right)^n - |\alpha|\right\}\right|| z|^n + \left|1-\alpha + \beta\left\{\left(\frac{R+1}{2}\right)^n - |\alpha|\right\}\right|\right]\max_{| z|=1} | P(z)|\end{split} \] for \(R\geq 1\) and \(| z| \geq 1\).
The present paper contains a generalization and an improvement of this and some other polynomial inequalities of similar nature.

MSC:

30C10 Polynomials and rational functions of one complex variable
30A10 Inequalities in the complex plane
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ankeny, N. C.; Rivlin, T. J., On a theorem of S. Bernstein, Pacific J. Math., 5, 849-852 (1955) · Zbl 0067.01001
[2] Aziz, A.; Dawood, Q. M., Inequalities for a polynomial and its derivative, J. Approx. Theor., 54, 306-313 (1998) · Zbl 0663.41019
[3] Aziz, A.; Rather, N. A., Some compact generalizations of Bernstein-type inequalities for polynomials, J. Math. Inequl. Appl., 7, 3, 393-403 (2004) · Zbl 1057.30003
[4] Aziz, A.; Rather, N. A., On an inequality of S. Bernstein and Gauss-Lucas theorem, (Rassias, Th. M.; Srivastava, H. M., Analytic and Geometric Inequalities and their Applications (2004), Kluwer Academic Press) · Zbl 0982.30003
[5] Aziz, A.; Zargar, B. A., Inequalities for a polynomial and its derivative, Math. Inequl. Appl., 1, 4, 543-550 (1998) · Zbl 0914.30002
[6] Bernstein, S., Sur ĺ ordre de la meilleure approximation des functions continues par des polynomes de degré donné, Memories de ĺ Academie Royale de Belgique, 4, 1-103 (1912) · JFM 45.0633.03
[7] Jain, V. K., Generalization of certain well known inequalities for polynomials, Glasnik Matématick´i, 30, 45-51 (1997) · Zbl 0883.30004
[8] Jain, V. K., Inequalities for a polynomial and its derivative, Proc. Ind. Acad. Sci. (Math. Sci.), 110, 2, 137-146 (2000) · Zbl 0960.30003
[9] Lax, P. D., Proof of a Conjecture of P. Erdös on the derivative of a polynomial, Bull. Amer. Math. Soc. (N.S), 50, 509-513 (1944) · Zbl 0061.01802
[10] Marden, M., Geometry of polynomials. Geometry of polynomials, Mathematical Surveys No. 3 (1966), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0162.37101
[11] Milovanovic, G. V.; Mitrinovic, D. S.; Rassias, Th. M., Topics in Polynomials, Extremal problems, Inequalities, Zeros (1994), World Scientific: World Scientific Singapore · Zbl 0848.26001
[12] Polya, G.; Szego, G., Problems and Theorems in Analysis, vol. 1 (1972), Springer: Springer New York · Zbl 0236.00003
[13] Rahman, Q. I.; Schmeisser, G., Analytic Theory of Polynomials (2002), Oxford University Press: Oxford University Press New York · Zbl 1072.30006
[14] Riesz, M., Über einen satz des Herrn Serge Bernstein, Acta Math., 40, 337-347 (1916) · JFM 46.0472.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.