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Bohr radius for locally univalent harmonic mappings

Author

Listed:
  • Ilgiz R Kayumov
  • Saminathan Ponnusamy
  • Nail Shakirov

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 condition Re h(z)≤1 in D with h(0)>0. 3.both h and g are bounded in D. 4.h is bounded and g′(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 BH of harmonic Bloch functions. The paper concludes with two conjectures.

Suggested Citation

  • Ilgiz R Kayumov & Saminathan Ponnusamy & Nail Shakirov, 2018. "Bohr radius for locally univalent harmonic mappings," Mathematische Nachrichten, Wiley Blackwell, vol. 291(11-12), pages 1757-1768, August.
    • Handle: RePEc:bla:mathna:v:291:y:2018:i:11-12:p:1757-1768
    DOI: 10.1002/mana.201700068
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    Cited by:

    1. Gang Liu & Saminathan Ponnusamy, 2023. "Improved Bohr inequality for harmonic mappings," Mathematische Nachrichten, Wiley Blackwell, vol. 296(2), pages 716-731, February.
    2. Hidetaka Hamada, 2023. "Bohr's inequality for holomorphic and pluriharmonic mappings with values in complex Hilbert spaces," Mathematische Nachrichten, Wiley Blackwell, vol. 296(7), pages 2795-2808, July.

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