Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T06:01:41.358Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Maximum and Minimum Modulus of Rational Functions

Published online by Cambridge University Press:  20 November 2018

D. S. Lubinsky
D. S. Lubinsky*
Affiliation:
John Knopfmacher Centre, Department of Mathematics, Witwatersrand University, Wits 2050, South Africa email: 036dsl@cosmos.wits.ac.za
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that if $m,n\ge 0,\lambda >1$, and $R$ is a rational function with numerator, denominator of degree $\le m,n$, respectively, then there exists a set $S\subset [0,1]$ of linear measure $\ge \,\frac{1}{4}\,\exp \left( -\frac{13}{\log \,\text{ }\!\!\lambda\!\!\text{ }} \right)$ such that for $r\in S$,

1

$$_{|z|=r\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,|z|=r}^{\max |R(z)|/min|R(z)|\le {{\lambda }^{m+n}}.}$$

Here, one may not replace $\frac{1}{4}\exp (-\frac{13}{\log \lambda })$ by $\exp (-\frac{2-\varepsilon }{\log \lambda })$, for any $\varepsilon >0$. As our motivating application, we prove a convergence result for diagonal Padé approximants for functions meromorphic in the unit ball.

Keywords

30E1030C1531A1541A21
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 52 , Issue 4 , 01 August 2000 , pp. 815 - 832
Copyright
Copyright © Canadian Mathematical Society 2000

References

[1] Baker, G. A. and Graves-Morris, P. R., Padé Approximants. 2nd Edition, Encyclopaedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1996.Google Scholar
[2] Bennett, C. and Sharpley, R., Interpolation of Operators. Academic Press, Orlando, 1988.Google Scholar
[3] Borwein, P. and Erdelyi, T., Polynomials and Polynomial Inequalities. Springer, New York, 1995.Google Scholar
[4] Cuyt, A., Driver, K. A. and Lubinsky, D. S., On the Size of Lemniscates of Polynomials in One and Several Variables. Proc. Amer. Math. Soc. 124(1996), 2123–2136.Google Scholar
[5] Fryntov, A. and Rossi, J., Hyperbolic Symmetrization and an Inequality of Dynkin. to appear.Google Scholar
[6] Garnett, J. B., Bounded Analytic Functions. Academic Press, Orlando, 1981.Google Scholar
[7] Levin, A. L. and Saff, E. B., Optimal Ray Sequences of Rational Functions Connected with the Zolotarev problem. Constr. Approx. 10(1994), 235–273.Google Scholar
[8] Landkof, N. S., Foundations of Modern Potential Theory. Springer, New York, 1972.Google Scholar
[9] Lubinsky, D. S., Diagonal Padé Approximants and Capacity. J. Math. Anal. Appl. 78(1980), 58–67.Google Scholar
[10] Lubinsky, D. S., Convergence of Diagonal Padé Approximants for Functions Analytic Near 0. Trans. Amer.Math. Soc. 8(1995), 3149–3157.Google Scholar
[11] Lubinsky, D. S., Small Values of Polynomials: Cartan, Polya and Others. Inequalities Appl. 1(1997), 199–222.Google Scholar
[12] Lubinsky, D. S., Will Ramanujan kill Baker-Gammel-Wills? In: New Developments in Approximation Theory, (eds. Muller, M. W., et al.), ISNM 132, Birkhäuser, Basel, 1999, 159–174.Google Scholar
[13] Rahmanov, E. A., On the Convergence of Padé Approximants in Classes of Holomorphic Functions. Math. USSR-Sb. 40(1981), 149–155.Google Scholar
[14] Saff, E. B. and Totik, V., Logarithmic Potential with External Fields. Springer, Berlin, 1997.Google Scholar
[15] Stahl, H. B., The Convergence of Padé Approximants to Functions with Branchpoints. J. Approx. Theory 91(1997), 139–204.Google Scholar
[16] Stahl, H. B., Spurious Poles in Padé Approximation. J. Comput. Appl. Math. 99(1998), 511–527.Google Scholar