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Inequalities for Rational Functions With Prescribed Poles

Published online by Cambridge University Press:  20 November 2018

G. Min
G. Min*
Affiliation:
Centre for Experimental & Constructive Mathematics Department of Mathematics and Statistics Simon Fraser University Burnaby, BC V5A 1S6, e-mail: gmin@cecm.sfu.ca
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Abstract

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This paper considers the rational system ${{P}_{n}}({{a}_{1}},{{a}_{2}},...,{{a}_{n}})\,\,:=\,\left\{ \frac{P(x)}{\Pi _{k=1}^{n}(x-{{a}_{k}})},\,P\,\in \,{{P}_{n}} \right\}$ with nonreal elements in $\left\{ {{a}_{k}} \right\}_{k=1}^{n}\,\subset \,\mathbb{C}\,\backslash \,[-1,\,1]$ paired by complex conjugation. It gives a sharp (to constant) Markov-type inequality for real rational functions in ${{P}_{n}}({{a}_{1}},{{a}_{2}},...,{{a}_{n}})$. The corresponding Markov-type inequality for high derivatives is established, as well as Nikolskii-type inequalities. Some sharp Markov- and Bernstein-type inequalities with curved majorants for rational functions in ${{P}_{n}}({{a}_{1}},{{a}_{2}},...,{{a}_{n}})$ are obtained, which generalize some results for the classical polynomials. A sharp Schur-type inequality is also proved and plays a key role in the proofs of our main results

Keywords

41A1726D0726C15Markov-type inequalityBernstein-type inequalityNikolskii-type inequalitySchur-type inequalityrational functions with prescribed polescurved majorantsChebyshev polynomials
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 50 , Issue 1 , 01 February 1998 , pp. 152 - 166
Copyright
Copyright © Canadian Mathematical Society 1998

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