A Steiner star for a set P of n points in R connects an arbitrary center point to all points of P , while a star connects a point p ∈ P to the remaining n−1 points of P . All connections are realized by straight line segments. Fekete and Meijer showed that the minimum star is at most √ 2 times longer than the minimum Steiner star for any finite point configuration in R. The maximum ratio betwee...

In this paper, we consider a class L (λ, μ;φ) of analytic functions f defined in the open unit disk U satisfying the subordination condition that q(z) Df(z) Dλf(z) ≺ φ(z) (λ ∈ N0, μ ≥ 0; z ∈ U), where q(z) = ( z Dλf(z) )μ−2 , D is the Sălăgean operator and φ(z) is a convex function with positive real part in U. We obtain some characteristic properties giving the coefficient inequality, radius a...

In this paper we consider the minimal energy problem on the sphere S for Riesz potentials with external …elds. Fundamental existence, uniqueness, and characterization results are derived about the associated equilibrium measure. The discrete problem and the corresponding weighted Fekete points are investigated. As an application we obtain the separation of the minimal senergy points for d 2 < s...

The multivariate integer Chebyshev problem is to find polynomials with integer coefficients that minimize the supremum norm over a compact set in C. We study this problem on general sets, but devote special attention to product sets such as cube and polydisk. We also establish a multivariate analog of the Hilbert-Fekete upper bound for the integer Chebyshev constant, which depends on the dimens...

We generalize the classical Robin formula to higher dimensions. Mathematics Subject Classification (2000) Primary 32U35 · 32U20 · 14G40; Secondary 31B15 · 31C10 The purpose of this note is to give a formula for the Fekete-Leja transfinite diameter on CN , generalizing the classical Robin formula d∞(E) = e−V(E) for the usual transfinite diameter. We will disengage this from a formula for the sec...

For λ ≥ 0 and 0 ≤ α < 1 < β, we denote by K (λ;α, β) the class of normalized analytic functions satisfying the two sided-inequality α << ( z f ′(z) f (z) + λ z2 f ′′(z) f (z) ) < β (z ∈ U), where U is the open unit disk. Let KΣ(λ;α, β) be the class of bi-univalent functions such that f and its inverse f −1 both belong to the class K (λ;α, β). In this paper, we establish bounds for the coefficie...

Motivated by the study of spectral properties of self-adjoint operators in the integral group ring of a sofic group, we define and study integer operators. We establish a relation with classical potential theory and in particular the circle of results obtained by M. Fekete and G. Szegö, see [Fek23,FS55,Sze24]. More concretely, we use results by R. Rumely, see [Rum99], on equidistribution of alg...

Two players are placed on the integer lattice Z (consisting of points in n dimensional space with all coordinates integers) so that their vector difference is of length 2 and parallel to one of the axes. Their aim is to move to an adjacent node in each period, so that they meet (occupy same node) in least expected time R (n) , called the Rendezvous Value. We assume they have no common notion of...