Critical points of inner functions, nonlinear partial differential equations, and an extension of Liouville's theorem

Kraus, Daniela; Roth, Oliver

doi:10.1112/jlms/jdm095

Cited by 13 publications

(24 citation statements)

References 30 publications

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“…The proofs in this paper are based on conformal (Riemannian) pseudometrics and rely in particular on the results of [23]. We first give a quick account of the relevant facts from conformal geometry and refer to [4,19,21,22,23,24,39] for more information.…”

Section: Auxiliary Resultsmentioning

confidence: 99%

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Maximal Blaschke products

Kraus

Roth

2013

Advances in Mathematics

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We consider the classical problem of maximizing the derivative at a fixed point over the set of all bounded analytic functions in the unit disk with prescribed critical points. We show that the extremal function is essentially unique and always an indestructible Blaschke product. This result extends the Nehari-Schwarz Lemma and leads to a new class of Blaschke products called maximal Blaschke products. We establish a number of properties of maximal Blaschke products, which indicate that maximal Blaschke products constitute an appropriate infinite generalization of the class of finite Blaschke products.

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Section: Auxiliary Resultsmentioning

confidence: 99%

“…The general case of Liouville's Theorem ( ) can be found e.g. in [6,7,8,24,33,41]. Note that in Liouville's theorem, the zeros of the pseudometric ´Þµ Þ are precisely the critical points of its developing map ¾ À ½ .…”

Section: Auxiliary Resultsmentioning

confidence: 99%

Maximal Blaschke products

Kraus

Roth

2013

Advances in Mathematics

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“…they satify (20) for suitable Van Vleck polynomials. Going back from this equation in the calculations in Section 2 we infer they also satisfy equation (19), and hence we have…”

Section: Stieltjes Assumed Thatmentioning

confidence: 90%

“…Stephenson [36,Theorem 21.1] obtains B as the limit of discrete finite Blaschke products, i.e., he considers sequences of circle packings with prescribed branch set. Kraus and Roth [19], [20] describe an approach based on a solution of the Gaussian curvature equation (that works equally well for infinite Blaschke products), and ask for a procedure to actually compute B from its critical points.…”

Section: Introductionmentioning

confidence: 99%

Finite Blaschke products with prescribed critical points, Stieltjes polynomials, and moment problems

Semmler

Wegert

2017

Anal.Math.Phys.

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The determination of a finite Blaschke product from its critical points is a well-known problem with interrelations to several other topics. Though existence and uniqueness of solutions are established for long, we present new aspects which have not yet been explored to their full extent. In particular, we show that the following three problems are equivalent: (i) determining a finite Blaschke product from its critical points, (ii) finding the equilibrium position of moveable point charges interacting with a special configuration of fixed charges, and (iii) solving a moment problem for the canonical representation of power moments on the real axis. These equivalences are not only of theoretical interest, but also open up new perspectives for the design of algorithms. For instance, the second problem is closely linked to the determination of certain Stieltjes and Van Vleck polynomials for a second order ODE and characterizes solutions as global minimizers of an energy functional.

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“…Theorem 2.7 and in particular the special case that ´Þµ Þ is a conformal metric has a number of different proofs, see for instance [7,11,12,39,44,56]. Remark 2.8 is discussed in [32].…”

Section: Definition 26 (Zero Set)mentioning

confidence: 99%

Critical sets of bounded analytic functions, zero sets of Bergman spaces and nonpositive curvature

Kraus¹

2012

Proceedings of the London Mathematical Society

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A classical result due to Blaschke states that for every analytic self‐map f of the open unit disc of the complex plane there exists a Blaschke product B such that the zero sets of f and B agree. Indeed, a sequence is the zero set of an analytic self‐map of the open unit disc if and only if it satisfies the simple geometric condition known as the Blaschke condition. In contrast, the critical sets of analytic self‐maps of the open unit disc have not been completely described yet. In this paper, we show that for every analytic self‐map f of the open unit disc there is even an indestructible Blaschke product B such that the critical sets of f and B coincide. We further relate the problem of describing the critical sets of bounded analytic functions to the problem of characterizing the zero sets of some weighted Bergman space as well as to the Berger–Nirenberg problem from differential geometry. By solving the Berger–Nirenberg problem in a special case, we identify the critical sets of bounded analytic functions with the zero sets of the weighted Bergman space 𝒜12.

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Critical points of inner functions, nonlinear partial differential equations, and an extension of Liouville's theorem

Cited by 13 publications

References 30 publications

Maximal Blaschke products

Maximal Blaschke products

Finite Blaschke products with prescribed critical points, Stieltjes polynomials, and moment problems

Critical sets of bounded analytic functions, zero sets of Bergman spaces and nonpositive curvature

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