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Approximation of Certain Non-vanishing Analytic Functions in a Parabolic Region

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Abstract

In this work, we consider a class of analytic functions f defined in the unit disk for which the values of \(zf'/f\) lie in a parabolic region of the right-half plane. By using a well-known sufficient condition for functions to be in this class in terms of the Taylor coefficients of z/f, we introduce a subclass \(\mathcal {F}_{\alpha }\) of this class. The aim of the paper is to find the best approximation of non-vanishing analytic functions of the form z/f by functions z/g with \(g\in \mathcal {F}_{\alpha }\). The proof relies on solving a semi-infinite quadratic problem, a problem of independent interest.

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Acknowledgements

The authors would like to express their deep gratitude to the referee for careful checking of the manuscript and constructive suggestions, which greatly improved the exposition.

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  1. Department of Mathematics, Indian Institute of Technology Indore, Indore, 453 552, India

    Vibhuti Arora, Swadesh Kumar Sahoo & Sanjeev Singh

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  1. Vibhuti Arora
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  2. Swadesh Kumar Sahoo
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  3. Sanjeev Singh
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Correspondence to Swadesh Kumar Sahoo.

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Arora, V., Sahoo, S.K. & Singh, S. Approximation of Certain Non-vanishing Analytic Functions in a Parabolic Region. Results Math 76, 163 (2021). https://doi.org/10.1007/s00025-021-01434-1

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