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Sharp Coefficient Bounds for Certain p-Valent Functions

  • Nak Eun Cho
  • Virendra KumarEmail author
  • Oh Sang Kwon
  • Young Jae Sim
Article

Abstract

The main aim of this manuscript is to investigate sharp bound on the functional \(|a_{p+1}a_{p+2}-a_{p+3}|\) for functions \(f(z)=z^p+a_{p+1}z^{p+1}+a_{p+2}z^{p+2}+a_{p+3}z^{p+3}+\cdots \) belonging to the class \(\mathcal {R}_p(\alpha )\) associated with the right half-plane. Also sharp bounds on the initial coefficients, bounds on \(|a_{p+1}a_{p+2}-a_{p+3}|\), and \(|a_{p+1}a_{p+3}-a_{p+2}^2|\) for functions in the class \(\mathcal {RL}_p(\alpha )\), related to the lemniscate of Bernoulli, are also derived. Further, these estimates are used to derive a bound on the third Hankel determinant.

Keywords

p-Valent function Coefficient bound Hankel determinant Lemniscate of Bernoulli 

Mathematics Subject Classification

30C45 30C50 

Notes

Acknowledgements

The authors would like to express their gratitude to the referees for many valuable suggestions regarding a previous version of this paper. The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2016R1D1A1A09916450).

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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsPukyong National UniversityBusanKorea
  2. 2.Department of MathematicsKyungsung UniversityBusanKorea

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