Generalization of certain well-known inequalities for rational functions
Abstract
Let
be a class of all polynomials of degree at most m and let
where
and
denote the class of rational functions. It is proved that if the rational function
having all its zeros in
, then for
.
The main purpose of this paper is to improve the above inequality for rational functions
having all its zeros in
with
-fold zeros at the origin and some other related inequalities. The obtained results sharpen some well-known estimates for the derivative and polar derivative of polynomials.
![P_m](http://siba-ese.unile.it/plugins/generic/latexRender/cache/6188f42ac8ef04ad0a00bef522a02fc0.png)
![R_{m,n}=R_{m,n}(d_{1}, ..., d_{n})=\{p(z)/w(z);p\in P_{m}, w(z)=\prod_{j={1}}^n(z-d_{j})~](http://siba-ese.unile.it/plugins/generic/latexRender/cache/0102b8c5e6175883e5ab22bfe28d41f9.png)
![~ \vert d_{j}\vert\ >1, j=1, ..., n](http://siba-ese.unile.it/plugins/generic/latexRender/cache/8895b55b99f56252fedef41c3fda929d.png)
![m\leq n\}](http://siba-ese.unile.it/plugins/generic/latexRender/cache/02eed67bfe2d41dad14a05938c3a989a.png)
![r(z)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/2095ff0e698f623d76bcdc68e87f71dc.png)
![\vert z\vert\leq1](http://siba-ese.unile.it/plugins/generic/latexRender/cache/a367c0f2aa7d0fecb71a26e51a7ba187.png)
![\vert z\vert=1](http://siba-ese.unile.it/plugins/generic/latexRender/cache/8b8e7a4d45cc41ae62fd7d615b103ed1.png)
![\vert r^{'}(z)\vert\geq \frac{1}{2}\{\vert B^{'}(z)\vert-(n-m)\} \vert r(z)\vert](http://siba-ese.unile.it/plugins/generic/latexRender/cache/021a24a971febd26fcbdb828a9b03c36.png)
The main purpose of this paper is to improve the above inequality for rational functions
![r(z)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/2095ff0e698f623d76bcdc68e87f71dc.png)
![\vert z\vert\leq k\leq1](http://siba-ese.unile.it/plugins/generic/latexRender/cache/eb8cd0444e0147ab8571758cf3140f4d.png)
![t](http://siba-ese.unile.it/plugins/generic/latexRender/cache/e358efa489f58062f10dd7316b65649e.png)
DOI Code:
10.1285/i15900932v40n1p1
Keywords:
Rational functions; Polynomials; Polar derivative; Inequalities; Poles; Restricted Zeros
Full Text: PDF