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On the Product of Planar Harmonic Mappings

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Abstract

Given two harmonic functions \(f_{1}=h_{1}+{\bar{g}}_{1}\) and \(f_{2}=h_{2}+{\bar{g}}_{2}\) defined on the open unit disk of the complex plane, the geometric properties of the product \(f_{1}\otimes f_{2}\) defined by

$$\begin{aligned} f_{1}\otimes f_{2}=h_{1}*h_{2}-\overline{g_{1}*g_{2}}+ i{\text {Im}}(h_{1}*g_{2}+h_{2}*g_{1}) \end{aligned}$$

are discussed. Here \(*\) denotes the analytic convolution. Sufficient conditions are obtained for the product to be univalent and convex in the direction of the real axis. In addition, a convolution theorem, coefficient inequalities and closure properties for the product \(\otimes \) are proved.

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Acknowledgements

The first author is supported by a Junior Research Fellowship from the Council of Scientific and Industrial Research(CSIR), New Delhi with File No. 09/045 (1515)2017-EMR-I. The authors are thankful to the referee for his/her useful comments.

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Correspondence to Sumit Nagpal.

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Communicated by Kenneth Stephenson.

Dedicated to the memory of Stephan Ruscheweyh.

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Raj, A., Nagpal, S. & Ravichandran, V. On the Product of Planar Harmonic Mappings. Comput. Methods Funct. Theory 21, 427–452 (2021). https://doi.org/10.1007/s40315-021-00367-8

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  • DOI: https://doi.org/10.1007/s40315-021-00367-8

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