Abstract
In this paper we first consider another version of the Rogosinski inequality for analytic functions \(f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}\) in the unit disk \(|z|<1\), in which we replace the coefficients \(a_{n}\) \((n=0,1,\ldots,N)\) of the power series by the derivatives \(f^{(n)}(z)/n!\) \((n=0,1,\ldots,N)\). Secondly, we obtain improved versions of the classical Bohr inequality and Bohr’s inequality for the harmonic mappings of the form \(f=h+\overline{g}\), where the analytic part \(h\) is bounded by \(1\) and that \(|g^{\prime}(z)|\leq k|h^{\prime}(z)|\) in \(|z|<1\) and for some \(k\in[0,1]\).
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Funding
The work of S. Alkhaleefah and I. Kayumov is supported by the Russian Science Foundation under grant no. 18-11-00115. The work of the third author is supported by Mathematical Research Impact Centric Support of DST, India (MTR/2017/000367).
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Alkhaleefah, S.A., Kayumov, I.R. & Ponnusamy, S. Bohr–Rogosinski Inequalities for Bounded Analytic Functions. Lobachevskii J Math 41, 2110–2119 (2020). https://doi.org/10.1134/S1995080220110049
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DOI: https://doi.org/10.1134/S1995080220110049