Expand all Collapse all | Results 1 - 2 of 2 |
1. CJM 2007 (vol 59 pp. 730)
Large Sieve Inequalities via Subharmonic Methods and the Mahler Measure of the Fekete Polynomials We investigate large sieve inequalities such as
\[
\frac{1}{m}\sum_{j=1}^{m}\psi ( \log | P( e^{i\tau
_{j}}) | ) \leq \frac{C}{2\pi }\int_{0}^{2\pi }\psi \left(
\log [ e| P( e^{i\tau }) | ] \right) \,d\tau
,
\]
where $\psi $ is convex and increasing, $P$ is a polynomial or an
exponential of a potential, and the constant $C$ depends on the degree of $P$,
and the distribution of the points $0\leq \tau _{1}<\tau _{2}<\dots<\tau
_{m}\leq 2\pi $. The method allows greater generality and is in some ways
simpler than earlier ones. We apply our results to estimate the Mahler
measure of Fekete polynomials.
Category:41A17 |
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 |