某些等价函数类中反函数对数系数的第二汉克尔行列式

Pub Date : 2024-02-16 DOI:10.1007/s10986-024-09623-5
Sanju Mandal, Molla Basir Ahamed
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引用次数: 0

摘要

对数系数的汉克尔行列式({H}_{mathrm{2,1}}\left({F}_{f-1}/2\right))定义为: ({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、属于归一化单值函数类 \(\mathcal{S}\) 的反函数的第一、第二和第三对数系数。在本文中,我们建立了尖锐的不等式 ((\left|{H}_{mathrm{2,1}}\left({F}_{f-1}/2\right)\right|le 19/288,\)\(\left|{H}_{mathrm{2,1}\left({F}_{f-1}/2\right)\right|le 1/144,\)和 \(\left|{H}_{mathrm{2,1}\left({F}_{f-1}/2\right)\right|le 1/36\)为反函数的对数系数、分别考虑星形函数和凸函数,以及阶数为 1/2 的有界转折函数。
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Second Hankel determinant of logarithmic coefficients of inverse functions in certain classes of univalent functions

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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