Abstract view
Inequalities for rational functions with prescribed poles


Published:19980201
Printed: Feb 1998
Abstract
This paper considers the rational system ${\cal P}_n
(a_1,a_2,\ldots,a_n):= \bigl\{ {P(x) \over \prod_{k=1}^n (xa_k)},
P\in {\cal P}_n\bigr\}$ with nonreal elements in
$\{a_k\}_{k=1}^{n}\subset\Bbb{C}\setminus [1,1]$ paired by complex
conjugation. It gives a sharp (to constant) Markovtype inequality
for real rational functions in ${\cal P}_n (a_1,a_2,\ldots,a_n)$.
The corresponding Markovtype inequality for high derivatives
is established, as well as Nikolskiitype inequalities. Some
sharp Markov and Bernsteintype inequalities with curved majorants
for rational functions in ${\cal P}_n(a_1,a_2,\ldots,a_n)$ are
obtained, which generalize some results for the classical
polynomials. A sharp Schurtype inequality is also proved and
plays a key role in the proofs of our main results.
Keywords: 
Markovtype inequality, Bernsteintype inequality, Nikolskiitype inequality, Schurtype inequality, rational functions with prescribed poles, curved majorants, Chebyshev polynomials
Markovtype inequality, Bernsteintype inequality, Nikolskiitype inequality, Schurtype inequality, rational functions with prescribed poles, curved majorants, Chebyshev polynomials
