On the zeros of meromorphic functions of the formf(z)=Σ k=1 ∞ a k/z−z k

“…To begin with, in this case the fundamental question dealing with the existence of equilibrium points is already quite delicate and, as a matter of fact, still an open problem (cf. [8,14,24]). …”

Section: Theorem 4 In the Above Notation One Has H(w Z) ≤ H(w E Zmentioning

confidence: 99%

“…Various results concerning the existence of equilibrium points for Newtonian and logarithmic potentials have been obtained in, e.g., [8], [14] and [24]. It should be emphasized though that these deal almost exclusively with unbounded discrete charge configurations.…”

Section: Operator Versions Of the Clunie-eremenko-rossi Conjecturementioning

confidence: 99%

Equilibrium points of logarithmic potentials induced by positive charge distributions. I. Generalized de Bruijn-Springer relations

Borcea

2007

Trans. Amer. Math. Soc.

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Abstract. A notion of weighted multivariate majorization is defined as a preorder on sequences of vectors in Euclidean space induced by the Choquet ordering for atomic probability measures. We characterize this preorder both in terms of stochastic matrices and convex functions and use it to describe the distribution of equilibrium points of logarithmic potentials generated by discrete planar charge configurations. In the case of n positive charges we prove that the equilibrium points satisfy n 2 weighted majorization relations and are uniquely determined by n − 1 such relations. It is further shown that the Hausdorff geometry of the equilibrium points and the charged particles is controlled by the weighted standard deviation of the latter. By using finiterank perturbations of compact normal Hilbert space operators we establish similar relations for infinite charge distributions. We also discuss a hierarchy of weighted de Bruijn-Springer relations and inertia laws, the existence of zeros of Borel series with positive l 1 -coefficients, and an operator version of the Clunie-Eremenko-Rossi conjecture.

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“…The case k = 2 was settled in [13], having been proved by Mues [16] for functions of finite lower order. Simple examples show that Theorem A is not true for k = 1 (see, however, [4]). Now, it is easy to give examples of entire functions f such that f has no zeros, but the following conjecture seems reasonable.…”

Section: Theorem a Suppose That F Is Meromorphic And That F And Fmentioning

confidence: 99%