Uniformly locally univalent harmonic mappings



The primary aim of this paper is to characterize the uniformly locally univalent harmonic mappings in the unit disk. Then, we obtain sharp distortion, growth and covering theorems for one parameter family \({{\mathcal {B}}}_{H}(\lambda )\) of uniformly locally univalent harmonic mappings. Finally, we show that the subclass of k-quasiconformal harmonic mappings in \({{\mathcal {B}}}_{H}(\lambda )\) and the class \({{\mathcal {B}}}_{H}(\lambda )\) are contained in the Hardy space of a specific exponent depending on \(\lambda \), respectively, and we also discuss the growth of coefficients for harmonic mappings in \({{\mathcal {B}}}_{H}(\lambda )\).


Harmonic mapping pre-Schwarzian derivatives uniformly locally univalence growth estimate coefficient estimate harmonic Bloch space Hardy space 

2010 Mathematics Subject Classification

Primary: 30C65 30C45 Secondary: 30C20 30C50 30C80 



The works of Mrs. Jinjing Qiao was supported by National Natural Science Foundation of China (No. 11501159), NSF of Hebei Science Foundation for Young Scientists (No. A2018201033) and was partially supported by INSA JRD-TATA Fellowship of the Centre for International Co-operation in Science (CICS). The third author (XW) was partly supported by NSFs of China (Nos. 11571216, 11671127 and 11720101003) and STU SRFT (No. 130-09400243).


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Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  • Saminathan Ponnusamy
    • 1
  • Jinjing Qiao
    • 2
  • Xiantao Wang
    • 3
  1. 1.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia
  2. 2.Department of MathematicsHebei UniversityBaodingPeople’s Republic of China
  3. 3.Department of MathematicsShantou UniversityShantouPeople’s Republic of China

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