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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.

- 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)

We study the farthest-point distance function, which measures the distance from z ∈ C to the farthest point or points of a given compact set E in the plane. The logarithm of this distance is subharmonic as a function of z, and equals the logarithmic potential of a unique probability measure with unbounded support. This measure σE has many interesting… (More)

We construct polynomial approximations of Dzjadyk type (in terms of the k -th modulus of continuity, k ≥ 1 ) for analytic functions defined on a continuum E in the complex plane, which simultaneously interpolate at given points of E . Furthermore, the error in this approximation is decaying as e−cn α strictly inside E , where c and α are positive constants… (More)

- IGOR E. PRITSKER
- 2005

Abstract. The original Gelfond-Schnirelman method, proposed in 1936, uses polynomials with integer coefficients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for the integral of Chebyshev’s ψ-function,… (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 tM (E) := lim n→∞ ‖Mn‖ E . This… (More)

- IGOR E PRITSKER
- 2006

We study the asymptotic structure of polynomials with integer coef cients and smallest uniform norms on an interval of the real line Introducing methods of the weighted potential theory into this problem we improve the bounds for the multiplicities of some factors of the integer Chebyshev polynomials Introduction Let Pn C and Pn Z be the sets of algebraic… (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)

Given an open bounded set G in the complex plane and a weight function W (z) which is analytic and di erent from zero in G, we consider the problem of locally uniform rational approximation of any function f(z), which is analytic in G, by particular ray sequences of weighted rational functions of the form Wm+n(z)Rm;n(z), where Rm;n(z) = Pm(z)=Qn(z); with… (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 pseudohyperbolic farthest-point distance function via a Green potential. We also give applications to sharp inequalities for the supremum norms of Blaschke products. 1. Green… (More)