Expand all Collapse all | Results 1 - 2 of 2 |
1. CJM 2002 (vol 54 pp. 648)
Rational Solutions of PainlevÃ© Equations Consider the sixth Painlev\'e equation~(P$_6$) below where $\alpha$,
$\beta$, $\gamma$ and $\delta$ are complex parameters. We prove the
necessary and sufficient conditions for the existence of rational
solutions of equation~(P$_6$) in term of special relations among the
parameters. The number of distinct rational solutions in each case is
exactly one or two or infinite. And each of them may be generated by
means of transformation group found by Okamoto [7] and B\"acklund
transformations found by Fokas and Yortsos [4]. A list of rational
solutions is included in the appendix. For the sake of completeness,
we collected all the corresponding results of other five Painlev\'e
equations (P$_1$)--(P$_5$) below, which have been investigated by many
authors [1]--[7].
Keywords:PainlevÃ© differential equation, rational function, BÃ¤cklund transformation Categories:30D35, 34A20 |
2. CJM 1998 (vol 50 pp. 152)
Inequalities for rational functions with prescribed poles This paper considers the rational system ${\cal P}_n
(a_1,a_2,\ldots,a_n):= \bigl\{ {P(x) \over \prod_{k=1}^n (x-a_k)},
P\in {\cal P}_n\bigr\}$ 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) Markov-type inequality
for real rational functions in ${\cal P}_n (a_1,a_2,\ldots,a_n)$.
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 ${\cal P}_n(a_1,a_2,\ldots,a_n)$ 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:Markov-type inequality, Bernstein-type inequality, Nikolskii-type inequality, Schur-type inequality, rational functions with prescribed poles, curved majorants, Chebyshev polynomials Categories:41A17, 26D07, 26C15 |