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Mathematics > Complex Variables

arXiv:1311.7668 (math)
[Submitted on 29 Nov 2013 (v1), last revised 18 Dec 2014 (this version, v3)]

Title:The Krzyż conjecture revisited

Authors:María J. Martín, Eric T. Sawyer, Ignacio Uriarte-Tuero, Dragan Vukotić
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Abstract:The Krzyż conjecture concerns the largest values of the Taylor coefficients of a non-vanishing analytic function bounded by one in modulus in the unit disk. It has been open since 1968 even though information on the structure of extremal functions is available. The purpose of this paper is to collect various conditions that the coefficients of an extremal function (and various other quantities associated with it) should satisfy if the conjecture is true and to show that each one of these properties is equivalent to the conjecture itself. This may provide several possible starting points for future attempts at solving the problem.
Comments: Condition (d) in the main Theorem of version 1 of this paper was erroneous. This has been corrected in the subsequent versions v2 and v3. Also, a proof of one implication in the Main Theorem has now been made shorter with respect to v2
Subjects: Complex Variables (math.CV)
MSC classes: 30H05, 30C45, 30C50
Cite as: arXiv:1311.7668 [math.CV]
  (or arXiv:1311.7668v3 [math.CV] for this version)
  https://doi.org/10.48550/arXiv.1311.7668
Journal reference: Advances in Mathematics 273 (2015), 716-745
Related DOI: https://doi.org/10.1016/j.aim.2014.12.031

Submission history

From: Dragan Vukotić [view email]
[v1] Fri, 29 Nov 2013 19:08:10 UTC (20 KB)
[v2] Thu, 3 Jul 2014 21:10:21 UTC (27 KB)
[v3] Thu, 18 Dec 2014 18:31:06 UTC (28 KB)
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