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- I. E. PRITSKER
- 2007

We study inequalities connecting the product of uniform norms of polynomials with the norm of their product. This circle of problems include the Gelfond-Mahler inequality for the unit disk and the Kneser-Borwein inequality for the segment [−1, 1]. Furthermore, the asymptoti-cally sharp constants are known for such inequalities over arbitrary compact sets in… (More)

- Peter B. Borwein, Christopher Pinner, Igor E. Pritsker
- Math. Comput.
- 2003

We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let Mn(Z) denote the monic polynomials of degree n with integer coefficients. A monic integer Chebyshev polynomial Mn ∈ Mn(Z) satisfies Mn E = inf Pn∈Mn(Z) Pn E. and the monic integer Chebyshev constant is then defined by t M (E) := lim n→∞ Mn 1/n E. This is… (More)

Given a pair (G, W) of an open bounded set G in the complex plane and a weight function W (z) which is analytic and different from zero in G, we consider the problem of the locally uniform approximation of any function f (z), which is analytic in G, by weighted polynomials of the form {W n (z)Pn(z)} ∞ n=0 , where deg Pn ≤ n. The main result of this paper is… (More)

In 1924, Szeg˝ o showed that the zeros of the normalized partial sums, sn(nz), of e z tended to what is now called the Szeg˝ o curve S, where S := z ∈ C : |ze 1−z | = 1 and |z| ≤ 1. Using modern methods of weighted potential theory, these zero distribution results of Szeg˝ o can be essentially recovered, along with an asymptotic formula for the weighted… (More)

- IGOR E. PRITSKER
- 2001

We study inequalities connecting a product of uniform norms of polynomials with the norm of their product. This subject includes the well known Gel'fond-Mahler inequalities for the unit disk and Kneser inequality for the segment [−1, 1]. Using tools of complex analysis and potential theory, we prove a sharp inequality for norms of products of algebraic… (More)

- IGOR E. PRITSKER
- 2006

We study inequalities connecting a product of uniform norms of polynomials with the norm of their product. Generalizing Gel'fond-Mahler inequality for the unit disk and Kneser-Borwein inequality for the segment [−1, 1], we prove an asymptotically sharp inequality for norms of products of algebraic polynomials over an arbitrary compact set in plane. Applying… (More)

- Igor E. Pritsker
- 2007

We consider the problem of finding a measure from the given values of its logarithmic potential on the support. It is well known that a solution to this problem is given by the generalized Laplacian. The case of our main interest is when the support is contained in a rectifiable curve, and the measure is absolutely continuous with respect to the arclength… (More)

- Igor E. Pritsker, Edward B. Saff
- Journal of Approximation Theory
- 2009

We study the reverse triangle inequalities for suprema of logarithmic potentials on compact sets of the plane. This research is motivated by the inequalities for products of supremum norms of polynomials. We find sharp additive constants in the inequalities for potentials, and give applications of our results to the generalized polynomials. We also obtain… (More)

We study inequalities for the infima of Green potentials on a compact subset of an arbitrary domain in the complex plane. The results are based on a new representation of the pseudohyper-bolic farthest-point distance function via a Green potential. We also give applications to sharp inequalities for the supremum norms of Blaschke products.

In this note we give some sharp estimates for norms of polynomials via the products of norms of their linear terms. Different convex norms on the unit disc are considered.