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Abstract: For rational functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R(z)=P(z)/W(z)$$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P$$\end{document}is a polynomial of degree at the most \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W(z)=\prod_{j=1}^{n}(z-a_j)$$\end{document}, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|a_j|>1,$$\end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\in \{1,2,\dots,n\},$$\end{document}we use simple but elegant techniques to strengthen generalizations of certain results which extend some widely known polynomial inequalities of Erdős-Lax and Turán to rational functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R$$\end{document}. In return these reinforced results, in the limiting case, lead to the corresponding refinements of the said polynomial inequalities. As an illustration and as an application of our results, we obtain some new improvements of the Erdős-Lax and Turán type inequalities for polynomials. These improved results take into account the size of the constant term and the leading coefficient of the given polynomial. As a further factor of consideration, during the course of this paper we will demonstrate how some recently obtained results could have been proved without invoking the results of Dubinin [Distortion theorems for polynomials on the circle,Sb. Math. 191(12) (2000) 1797–1807].

Let P(z) be a polynomial of degree n, then it is known that for α∈Cwith |α|≤n2,max|z|=1|zP′(z)-αP(z)≤n-αmax|z|=1|P(z)|.This inequality includes Bernstein’s inequality, concerning the estimate for |P′(z)|over |z|≤1,as a special case. In this paper, we extend this inequality to Lpnorm which among other things shows that the condition on αcan be relaxed. We also prove similar inequalities for polynomials with restricted zeros.

According to Gauss-Lucas theorem, every convex set containing all the zeros of a polynomial also contains all its critical points. This result is of central importance in the geometry of critical points in the analytic theory of polynomials. In this paper, an extension of Gauss-Lucas theorem is obtained and as an application some generalizations of Bernstein-type polynomial inequalities are also established.

AbstractFor the class of polynomials of degree nhaving all their zeros in where k ≤ 1, Aziz and Shah [4] proved that for each r > 0,In this paper, we extend this inequality to the polar derivative in the sense that we take the polar derivative DαP(z) in place of ordinary derivative P′(z) of polynomial P(z).

In this paper, certain refinements and generalizations of some inequalities concerning the polynomials and their polar derivative are obtained.

In this paper, we present certain results on the bounds for the moduli of the zeros of a polynomial with complex coefficients which among other things contain some generalizations and refinements of classical results due to Cauchy, Tôya, Carmichael and Mason, Williams and others. Nous obtenons quelques raffinements de résultats dûs à Cauchy, Tôya, Carmichael, Mason, Williams et d’autres concernant la location des zéros de polynômes algébriques.