Monatshefte für Mathematik

, Volume 188, Issue 4, pp 703–716 | Cite as

The spherical metric and univalent harmonic mappings

  • Yusuf Abu Muhanna
  • Rosihan M. Ali
  • Saminathan PonnusamyEmail author


Let \(f=h+\overline{g}\) be a harmonic univalent map in the unit disk \(\mathbb {D}\), where h and g are analytic. This paper finds an improved estimate for the second coefficient of h. Indeed, this estimate is the first qualitative improvement since the appearance of the papers by Clunie and Sheil-Small (Ann Acad Sci Fenn Ser A I 9:3–25, 1984), and by Sheil-Small (J Lond Math Soc 42:237–248, 1990). When the sup-norm of the dilatation is less than 1, it is also shown that the spherical area of the covering surface of h is dominated by the spherical area of the covering surface of f.


Harmonic univalent map Subordination Spherical area Hyperbolic metric Hyperbolic domain Modular function 

Mathematics Subject Classification

Primary 30C35 Secondary 30C25 30C45 30F45 31A05 



The authors thank the referee for his/her comments. This work was supported by the American University of Sharjah and by a research university grant from Universiti Sains Malaysia. The work of the third author is supported by Mathematical Research Impact Centric Support of DST, India (MTR/2017/000367). The third author is currently at Indian Statistical Institute (ISI), Chennai Centre, Chennai, India.

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.


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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsAmerican University of SharjahSharjahUnited Arab Emirates
  2. 2.School of Mathematical SciencesUniversiti Sains Malaysia (USM)PenangMalaysia
  3. 3.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia

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