Abstract
In this paper we study sharp estimates of pre-Schwarzian derivatives of functions belonging to the Nehari-type classes by using techniques from differential equations. In the sequel, we also see that a solution of a complex differential equation has a special form in terms of ratio of hypergeometric functions resulting to an integral representation. Finally, we attempt to study those univalent functions in the unit disk for which the image domain is an unbounded John domain.
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Notes
The authors wish to call these functions “John functions” in honor of Professor Fritz John.
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Acknowledgements
The work of the first author was supported by University Grants Commission, New Delhi (grant no. F.2-39/2011 (SA-I)). This research has been carried out from our earlier work when the first author was a PhD student at the Discipline of Mathematics, Indian Institute of Technology Indore. The authors would like to thank the referees for their careful reading of the previous versions of the paper and valuable remarks.
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Communicated by Kaushal Verma
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Agrawal, S., Sahoo, S.K. Nehari’s univalence criteria, pre-Schwarzian derivative and applications. Indian J Pure Appl Math 52, 193–204 (2021). https://doi.org/10.1007/s13226-021-00070-3
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DOI: https://doi.org/10.1007/s13226-021-00070-3
Keywords
- Pre-Schwarzian and Schwarzian derivatives
- The Nehari class
- Hypergeometric function
- Initial value problem
- Grönwall’s inequality
- John domain