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Second Hankel determinant of logarithmic coefficients of inverse functions in certain classes of univalent functions

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Abstract

The Hankel determinant \({H}_{\mathrm{2,1}}\left({F}_{f-1}/2\right)\) of logarithmic coefficients is defined as

\({H}_{\mathrm{2,1}}\left({F}_{f-1}/2\right):=\left|\begin{array}{cc}{\Gamma }_{1}& {\Gamma }_{2}\\ {\Gamma }_{2}& {\Gamma }_{3}\end{array}\right|={\Gamma }_{1}{\Gamma }_{3}-{\Gamma }_{2}^{2},\)

where \({\Gamma }_{1},{\Gamma }_{2},\) and \({\Gamma }_{3}\) are the first, second, and third logarithmic coefficients of inverse functions belonging to the class \(\mathcal{S}\) of normalized univalent functions. In this paper, we establish sharp inequalities \(\left|{H}_{\mathrm{2,1}}\left({F}_{f-1}/2\right)\right|\le 19/288,\) \(\left|{H}_{\mathrm{2,1}}\left({F}_{f-1}/2\right)\right|\le 1/144,\) and \(\left|{H}_{\mathrm{2,1}}\left({F}_{f-1}/2\right)\right|\le 1/36\) for the logarithmic coefficients of inverse functions, considering starlike and convex functions, as well as functions with bounded turning of order 1/2, respectively.

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  1. Department of Mathematics, Jadavpur University, Kolkata, 700032, West Bengal, India

    Sanju Mandal & Molla Basir Ahamed

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  1. Sanju Mandal
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Correspondence to Molla Basir Ahamed.

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Sanju Mandal is supported by CSIR-SRF (file No. 09/0096(12546)/2021-EMR-I, dated: 18/12/2023), Govt. of India, New Delhi.

Molla Basir Ahamed r is supported by SERB, SUR/2022/002244, Govt. India.

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Mandal, S., Ahamed, M.B. Second Hankel determinant of logarithmic coefficients of inverse functions in certain classes of univalent functions. Lith Math J 64, 67–79 (2024). https://doi.org/10.1007/s10986-024-09623-5

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