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Bohr radius for locally univalent harmonic mappings
Author(s) -
Kayumov Ilgiz R,
Ponnusamy Saminathan,
Shakirov Nail
Publication year - 2018
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201700068
Subject(s) - bohr radius , bohr model , mathematics , bounded function , infimum and supremum , radius , unit disk , harmonic , mathematical analysis , space (punctuation) , harmonic function , harmonic oscillator , pure mathematics , quantum mechanics , physics , computer security , computer science , quantum dot , linguistics , philosophy
Abstract We consider the class of all sense‐preserving harmonic mappings f = h + g ¯of the unit disk D , where h and g are analytic with g ( 0 ) = 0 , and determine the Bohr radius if any one of the following conditions holds: 1. h is bounded in D . 2. h satisfies the conditionReh ( z ) ≤ 1 in D with h ( 0 ) > 0 . 3. both h and g are bounded in D . 4. h is bounded andg ′ ( 0 ) = 0 . We also consider the problem of determining the Bohr radius when the supremum of the modulus of the dilatation of f in D is strictly less than 1. In addition, we determine the Bohr radius for the space B of analytic Bloch functions and the space B H of harmonic Bloch functions. The paper concludes with two conjectures.