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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)
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)
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)
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)
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)
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)