Monatshefte für Mathematik

, Volume 188, Issue 4, pp 653–666 | Cite as

The radii of fully starlikeness and fully convexity of a harmonic operator

  • Nirupam Ghosh
  • A. VasudevaraoEmail author


Let \(f = h + \overline{g}\) be a normalized harmonic mapping in the unit disk \({\mathbb {D}}:= \{ z \in {\mathbb {C}}: |z|< 1 \} \). In this paper, we study the radius of fully starlikeness and the radius of fully convexity of the following harmonic operator
$$\begin{aligned} \bigwedge _{0, 1}[f] = \int _{0}^{z}\frac{h(\xi )}{\xi } \,d\xi + \overline{\int _{0}^{z}\frac{g(\xi )}{\xi }\,d\xi }, \end{aligned}$$
where the coefficients of the analytic functions h and g satisfy the conditions of the harmonic Bieberbach coefficient conjecture. We also study the radius of uniform starlikeness, and uniform convexity of harmonic mappings.


Analytic Univalent Starlike Convex Close-to-convex Harmonic functions Operator 

Mathematics Subject Classification

Primary 30C45 30C50 



The author thanks Prof. Daoud Bshouty for the useful discussion. The authors thank Prof. D.K. Thomas for careful reading the paper and suggestions. The first author thanks UGC for financial support. The second author thanks NBHM and SERB.


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Copyright information

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

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology KharagpurKharagpurIndia
  2. 2.NFA-18, IIT CampusIndian Institute of Technology KharagpurKharagpurIndia

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