Complex Analysis and Operator Theory

, Volume 12, Issue 5, pp 1291–1301 | Cite as

On the Analytic Part of Univalent Harmonic Mappings

  • Iigiz R. Kayumov
  • Saminthan PonnusamyEmail author
  • Le Anh Xuan


In this article we obtain two sharp results concerning the analytic part of harmonic mappings \(f=h+\overline{g}\) from the class \(\mathcal {S}^0_H(\mathcal {S})\) which was recently introduced by Ponnusamy and Sairam Kaliraj. For example, we get the sharp estimate for \(|\arg h'(z)|\) in the case when \(|z| \le 1/\sqrt{2}\) and obtain the sharp radius of convexity for h. Our approach is applicable to a more general situation. Finally, we determine simple condition on the analytic part of univalent harmonic mappings so that it is in \(H_p\) spaces for \(0<p<1/3\).


Harmonic univalent and convex mappings Rotation theorem Schwarz–Pick Lemma Koebe transform Disk automorphism Harmonic (analytic) Hardy spaces Integral means 

Mathematics Subject Classification

Primary 30C45 30C50 30C55 30H10 31A05 Secondary 30C75 31A20 



The authors thank the referee for his/her careful reading and many useful comments. The research of the first author was supported by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities (1.9773.2017/8.9) and by Russian foundation for basic research, Proj. 17-01-00282, and the research of the second author was supported by the project RUS/RFBR/P-163 under Department of Science and Technology (India). The second author is currently at Indian Statistical Institute (ISI), Chennai Centre, Chennai, India.


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Kazan Federal UniversityKazanRussia
  2. 2.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia

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