Starlikeness, Convexity and Close-to-convexity of Harmonic Mappings

  • Sumit Nagpal
  • V. Ravichandran
Part of the Trends in Mathematics book series (TM)


In 1984, Clunie and Sheil-Small proved that a sense-preserving harmonic function whose analytic part is convex, is univalent and close-to-convex. In this chapter, certain cases are discussed under which the conclusion of this result can be strengthened and extended to fully starlike and fully convex harmonic mappings. In addition, we investigate the geometric properties of functions in the class \({\cal M}(\alpha)\) \((|\alpha|\leq 1)\) consisting of harmonic functions \(f=h+\overline{g}\) with \(g'(z)=\alpha zh'(z)\), Re \((1+{zh''(z)}/{h'(z)})>-{1}/{2}\) for \(|z|<1\). The coefficient estimates, growth results, area theorem and bounds for the radius of starlikeness, and convexity of the class \({\cal M}(\alpha)\) are determined. In particular, the bound for the radius of convexity is sharp for the class \({\cal M}(1)\).


Harmonic Function Coefficient Estimate Convex Domain Extremal Function Growth Result 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The research work of the first author is supported by research fellowship from Council of Scientific and Industrial Research (CSIR), New Delhi. The authors are thankful to the referee for his useful comments.


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Copyright information

© Springer India 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of DelhiDelhiIndia

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