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Inequalities for products of polynomials
 I, Math. Scand
"... Abstract. We study inequalities connecting the product of uniform norms of polynomials with the norm of their product. This circle of problems include the GelfondMahler inequality for the unit disk and the KneserBorwein inequality for the segment [−1, 1]. Furthermore, the asymptotically sharp cons ..."
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Abstract. We study inequalities connecting the product of uniform norms of polynomials with the norm of their product. This circle of problems include the GelfondMahler inequality for the unit disk and the KneserBorwein inequality for the segment [−1, 1]. Furthermore, the asymptotically sharp constants are known for such inequalities over arbitrary compact sets in the complex plane. It is shown here that this best constant is smallest (namely: 2) for a disk. We also conjecture that it takes its largest value for a segment, among all compact connected sets in the plane. 1. The problem and its history Let E be a compact set in the complex plane C. For a function f: E → C define the uniform (sup) norm as follows: �f�E = sup f(z). z∈E Clearly �f1f2 � E ≤ �f1 � E �f2 � E, but this inequality is not reversible, in general, not even with a constant factor in front of the right hand side. Indeed, �f1 � E �f2 � E ≤ C �f1f2 � E does not hold for functions with disjoint supports in E, for example. However, the situation is quite different for algebraic polynomials {pk(z)} m k=1 and their product p(z): = �m k=1 pk(z). Polynomial inequalities of the form m� (1.1) �pk�E ≤ C�p�E, k=1 exist and are readily available. One of the first results in this direction is due to Kneser [19], for E = [−1, 1] and m = 2 (see also Aumann [1]), who proved that (1.2) �p1�[−1,1]�p2�[−1,1] ≤ Kℓ,n�p1p2�[−1,1], deg p1 = ℓ, deg p2 = n − ℓ, 2000 Mathematics Subject Classification. Primary 30C10; Secondary 30C85, 31A15. Key words and phrases. Polynomials, products, factors, uniform norm, logarithmic capacity, equilibrium measure, subharmonic function, Fekete points. Research of I.P. was partially supported by the National Security Agency (grant H982300610055), and by the Alexander von Humboldt Foundation. S.R. acknowledges partial support from the GermanIsraeli Foundation (grant G809234.6/2003), from FONDECYT (grants 1040366 and 7040069) and from DGIPUTFSM (grant 240104). 1
Reverse Triangle Inequalities for Potentials
"... Dedicated to George G. Lorentz, whose works have been a great inspiration Abstract. 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 sha ..."
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Dedicated to George G. Lorentz, whose works have been a great inspiration Abstract. 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 sharp inequalities for products of norms of the weighted polynomials w n Pn, deg(Pn) ≤ n, and for sums of suprema of potentials with external fields. An important part of our work in the weighted case is a Riesz decomposition for the weighted farthestpoint distance function.
MOMENT INEQUALITIES FOR EQUILIBRIUM MEASURES IN THE PLANE
"... Dedicated to our friend Fred Gehring, on the occasion of his 80th birthday ABSTRACT. The equilibrium measure of a compact plane set gives the steady state distribution of charges on the conductor. We show that certain moments of this equilibrium measure, when taken about the electrostatic centroid a ..."
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Dedicated to our friend Fred Gehring, on the occasion of his 80th birthday ABSTRACT. The equilibrium measure of a compact plane set gives the steady state distribution of charges on the conductor. We show that certain moments of this equilibrium measure, when taken about the electrostatic centroid and depending only on the real coordinate, are extremal for an interval centered at the origin. This has consequences for means of zeros of polynomials, and for means of critical points of Green’s functions. We also study moments depending on the distance from the centroid, such as the electrostatic moment of inertia. 1.
On C 2 smooth surfaces of constant width
 Tbilisi Math. Journal
"... Abstract. A number of results for C 2smooth surfaces of constant width in Euclidean 3space E 3 are obtained. In particular, an integral inequality for constant width surfaces is established. This is used to prove that the ratio of volume to cubed width of a constant width surface is reduced by shr ..."
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Abstract. A number of results for C 2smooth surfaces of constant width in Euclidean 3space E 3 are obtained. In particular, an integral inequality for constant width surfaces is established. This is used to prove that the ratio of volume to cubed width of a constant width surface is reduced by shrinking it along its normal lines. We also give a characterization of surfaces of constant width that have rational support function. Our techniques, which are complex differential geometric in nature, allow us to construct explicit smooth surfaces of constant width in E 3, and their focal sets. They also allow for easy construction of tetrahedrally symmetric surfaces of constant width. 1.
Inequalities for Products of Polynomials II
"... Summary. In this paper, we continue the study of inequalities connecting the product of uniform norms of polynomials with the norm of their product, begun in [28]. Asymptotically sharp constants are known for such inequalities over arbitrary compact sets in the complex plane. We show here that such ..."
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Summary. In this paper, we continue the study of inequalities connecting the product of uniform norms of polynomials with the norm of their product, begun in [28]. Asymptotically sharp constants are known for such inequalities over arbitrary compact sets in the complex plane. We show here that such constants can be improved under some natural additional assumptions. Thus we find the best constants for rotationally symmetric sets. In addition, we characterize all sets that allow an improvement in the constant when the number of factors is fixed, and find the improved value.
Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 Inequalities for sums of Green potentials and Blaschke products
"... E. Pritsker 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 farthestpoint distance function via a Green potential. We also give applications to sharp inequ ..."
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E. Pritsker 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 farthestpoint distance function via a Green potential. We also give applications to sharp inequalities for the supremum norms of Blaschke products. 1. Green potentials Let G ⊂ C be a domain possessing the Green function gG(z, ζ) with pole at ζ ∈ G. For the positive Borel measures νk, k = 1,..., m, with compact supports in G, define their Green potentials [1, p. 96] by U νk G (z): = gG(z, ζ) dνk(ζ), z ∈ G. Note that Green potentials are superharmonic and nonnegative in G. Suppose that ν:= ∑ m j=1 νj is a unit measure. We study inequalities of the following type m∑ j=1 inf U E νk G j=1 inf U E νk