
Inequalities for rational functions with prescribed poles
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 Categories:41A17, 26D07, 26C15 