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.
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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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Communicated by V. Ravichandran.
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Cho, N.E., Kumar, V., Kwon, O.S. et al. Sharp Coefficient Bounds for Certain p-Valent Functions. Bull. Malays. Math. Sci. Soc. 42, 405–416 (2019). https://doi.org/10.1007/s40840-017-0587-4
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DOI: https://doi.org/10.1007/s40840-017-0587-4