Hostname: page-component-5cf477f64f-rdph2 Total loading time: 0 Render date: 2025-04-02T22:18:24.058Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Properties of Rational Functions with Prescribed Poles

Published online by Cambridge University Press:  20 November 2018

Abdul Aziz-Ul-Auzeem  and
B. A. Zarger
Abdul Aziz-Ul-Auzeem
Affiliation:
Post Graduate Department of Mathematics & Statistics, University of Kashmir, Hazratbal, Srinagar 190006, Kashmir, India
B. A. Zarger
Affiliation:
Post Graduate Department of Mathematics & Statistics, University of Kashmir, Hazratbal, Srinagar 190006, Kashmir, India
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.

Let $P\left( z \right)$ be a polynomial of degree not exceeding $n$ and let $W\left( z \right)\,=\,\prod\nolimits_{j=1}^{n}{\left( z\,-\,{{a}_{j}} \right)}$ where $\left| {{a}_{j}} \right|\,>\,1$, $j\,=\,1,\,2,\,.\,.\,.\,,\,n$. If the rational function $r\left( z \right)\,=\,{P\left( z \right)}/{W\left( z \right)}\;$ does not vanish in $\left| z \right|\,<\,k$, then for $k\,=\,1$ it is known that

$$\left| {{r}^{'}}\left( z \right) \right|\le \frac{1}{2}\left| {{B}^{'}}(z) \right|_{\left| z \right|=1}^{\text{Sup}}\left| r(z) \right|$$

where $B\left( Z \right)\,=\,{{{W}^{*}}\left( z \right)}/{W\left( z \right)}\;$ and ${{W}^{*}}\left( z \right)\,=\,{{z}^{n}}\overline{W\left( {1}/{\overline{z}}\; \right)}$. In the paper we consider the case when $k\,>\,1$ and obtain a sharp result. We also show that

$$\underset{\left| z \right|=1}{\mathop{\text{Sup}}}\,\left\{ \left| \frac{{{r}^{\prime }}\left( z \right)}{{{B}^{\prime }}\left( z \right)} \right|+\left| \frac{{{\left( {{r}^{*}}\left( z \right) \right)}^{\prime }}}{{{B}^{\prime }}\left( z \right)} \right| \right\}=\underset{\left| z \right|=1}{\mathop{\text{Sup}}}\,\left| r\left( z \right) \right|$$

where ${{r}^{*}}\left( z \right)\,=\,B\left( z \right)\overline{r\left( {1}/{\overline{z}}\; \right)}$, and as a consquence of this result, we present a generalization of a theorem of O’Hara and Rodriguez for self-inversive polynomials. Finally, we establish a similar result when supremum is replaced by infimum for a rational function which has all its zeros in the unit circle.

Keywords

26D07
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 42 , Issue 4 , 01 December 1999 , pp. 417 - 426
Copyright
Copyright © Canadian Mathematical Society 1999

References

[1] Aziz, Abdul and Dawood, Q. M., Inequalities for a polynomial and its derivative. J. Approx. Theory (3) 54 (1988), 306–313.Google Scholar
[2] Aziz, Abdul and Mohammad, Q. G., Simple proof of a Theorem of Erdʺos and Lax. Proc. Amer. Math. Soc. 80 (1980), 119–122.Google Scholar
[3] Aziz, Abdul and Shah, W. M., Some refinements of Bernstein type inequalities for rational functions. Glas.Mat. (52) 32 (1997), 29–37.Google Scholar
[4] Lax, P. D., Proof of a conjecture of P. Erdʺos on the derivative of a polynomial. Bull. Amer.Math. Soc. 50 (1944), 509–513.Google Scholar
[5] Malik, M. A., An integral mean estimate for polynomials. Proc. Amer.Math. Soc. 91 (1984), 281–284.Google Scholar
[6] Li, Xin, Mohapatra, R. N. and Rodriguez, R. S., Bernstein-type inequalities for rational functions with prescribed poles. J. London Math. Soc. (51) 20 (1995), 523–531.Google Scholar
[7] O’Hara, P. J. and Rodriguez, R. S., Some properties of self-inversive polynomials. Proc. Amer. Math. Soc. (2) 44 (1974), 331–335.Google Scholar
[8] Schaeffer, A. C., Inequalities of A. Markoff and S. Bernstein for polynomials and related functions. Bull. Amer. Math. Soc. 47 (1941), 565–579.Google Scholar