Abstract
We consider analytic functions of the form \(f(z)=\sum _n{a_n z^n}\) with \(|f(z)|\le 1\) defined on the unit disc \(\mathbb {D}:=\{z\in \mathbb {C}:|z|<1\}\). Due to studies on the Bohr phenomenon concerning this class of functions, and recent results on the Bohr inequality of some integral operators, we are interested in Bohr inequalities pertaining to integral transforms. We first obtain a Bohr-type inequality for the (discrete) Fourier transform acting on the functions f defined above, alongside the associated Bohr radius. We find that this inequality is sharp, and that the constant dictating the Bohr radius cannot be improved. We obtain a secondary result by finding an expression for \(a:=|a_0|\) that maintains the Bohr inequality even if \(r:=|z|\) grows past the Bohr radius. We also investigate the behaviour of the Fourier transform of f as \(r\rightarrow 1\), by finding the limiting bound for the aforementioned transform. We prove that this bound is actually also sharp. We then study the (discrete) Laplace transform of f and obtain its relevant Bohr inequality.
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Acknowledgements
The authors would like to thank the anonymous reviewers for the useful comments and suggestions for improving the paper. The second author would also like to acknowledge the Ministry of Higher Education Malaysia (MOHE) for funding under the Fundamental Research Grant Scheme (FRGS) No: FRGS/1/2020/STG06/USM/02/2.
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This work was supported by Ministry of Higher Education Malaysia (MOHE) under the Fundamental Research Grant Scheme (FRGS/1/2020/STG06/USM/02/2).
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The first draft of the manuscript was written by Marcus Wei Loong Ong, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
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Ong, M.W.L., Ng, Z.C. On the Bohr Inequalities for Certain Integral Transforms. Iran J Sci (2024). https://doi.org/10.1007/s40995-024-01607-x
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DOI: https://doi.org/10.1007/s40995-024-01607-x