Abstract
Let \({\mathcal {S}}\) be the class of analytic and univalent functions in the unit disk \(|z|<1\), that have a series of the form \(f(z)=z+ \sum _{n=2}^{\infty }a_nz^n\). Let F be the inverse of the function \(f\in {\mathcal {S}}\) with the series expansion \(F(w)=f^{-1}(w)=w+ \sum _{n=2}^{\infty }A_nw^n\) for \(|w|<1/4\). The logarithmic inverse coefficients \(\Gamma _n\) of F are defined by the formula \(\log \left( F(w)/w\right) = 2\sum _{n=1}^{\infty }\Gamma _n(F)w^n\). In this paper, we first determine the sharp bound for the absolute value of \(\Gamma _n(F)\) when f belongs to \({\mathcal {S}}\) and for all \(n \ge 1\). This result motivates us to carry forward similar problems for some of its important geometric subclasses. In some cases, we have managed to solve this question completely but in some other cases it is difficult to handle for \(n\ge 4\). For example, in the case of convex functions f, we show that the logarithmic inverse coefficients \(\Gamma _n(F)\) of F satisfy the inequality
and the estimates are sharp for the function \(l(z)=z/(1-z)\). Although this cannot be true for \(n\ge 10\), it is not clear whether this inequality could still be true for \(4\le n\le 9\).
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Acknowledgements
This was work was initiated while the second author was at ISI Chennai centre and was completed during his visit to IIT Madras. The work of the first author is supported by Mathematical Research Impact Centric Support of Department of Science and Technology (DST), India (MTR/2017/000367). The second author thanks Science and Engineering Research Board, DST, India, for its support by providing SERB National Post-Doctoral Fellowship (Grant No. PDF/2016/001274).
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Ponnusamy, S., Sharma, N.L. & Wirths, KJ. Logarithmic Coefficients of the Inverse of Univalent Functions. Results Math 73, 160 (2018). https://doi.org/10.1007/s00025-018-0921-7
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DOI: https://doi.org/10.1007/s00025-018-0921-7
Keywords
- Univalent function
- inverse function
- starlike function
- spirallike function
- close-to-convex function
- convex functions
- subordination
- inverse logarithmic coefficients
- Schwarz’ lemma