Monotonicity of quotients of theta functions related to an extremal problem on harmonic measure

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Abstract

We prove that for fixed u and v such that u, v ∈ [0, 1 / 2), the quotients θj (u | i π t) / θj (v | i π t), j = 1, 2, 3, 4, of the theta functions are monotone on 0 < t < ∞. The case v = 0 has been used by the second author to study a generalization of Gonchar's problem on harmonic measure of radial slits.

Original languageEnglish
Pages (from-to)1042-1053
Number of pages12
JournalJournal of Mathematical Analysis and Applications
Volume336
Issue number2
DOIs
StatePublished - Dec 15 2007

Keywords

  • Harmonic measure
  • Jacobi theta function
  • Monotonicity
  • Weierstrass ℘-function

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