In recent years, the study of Hankel determinants for various subclasses of normalised univalent functions $f\in \mathcal{S}$ given by $f\left(z\right)=z+\sum _{n=2}^{\mathrm{\infty }}{a}_n{z}^n$ for $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ has produced many interesting results. The main focus of interest has been estimating the second Hankel determinant of the form $H_{2,2}\left(f\right)=a_2{a}_4-a_3^2$. A non-sharp bound for $H_{2,2}\left(f\right)$ when $f\in \mathcal{K}\left(\alpha \right)$, $\alpha \in [0,1)$ consisting of convex functions of order α was found by Krishna and Ramreddy (Hankel determinant for starlike and convex functions of order alpha. Tbil Math J. 2012;5:65–76), and later improved by Thomas et al. (Univalent functions: a primer. Berlin: De Gruyter; 2018). In this paper, we give the sharp result. Moreover, we obtain sharp results for $H_{2,2}\left(f^{-1}\right)$ for the inverse functions $f^{-1}$ when $f\in \mathcal{K}\left(\alpha \right)$, and when $f\in {\mathcal{S}}^{\ast }\left(\alpha \right)$, the class of starlike functions of order α. Thus, the results in this paper complete the set of problems for the second Hankel determinants of f and $f^{-1}$ for the classes ${\mathcal{S}}^{\ast }\left(\alpha \right)$, $\mathcal{K}\left(\alpha \right)$, ${\mathcal{S}}_{\text{β}}^{\ast }$ and ${\mathcal{K}}_{\text{β}}$, where ${\mathcal{S}}_{\text{β}}^{\ast }$ and ${\mathcal{K}}_{\text{β}}$ are, respectively, the classes of strongly starlike, and strongly convex functions of order β.

近年来，关于归一化单价函数各种子类的汉克尔行列式的研究$F\in \mathcal{小号}$由$F\left(z\right)=z+\sum _{n=2}^{\mathrm{\infty }}{\mathrm{一个}}_n{z}^n$为了$\mathbb{D}=\{z\in \mathbb{C}：|z|<1\}$产生了许多有趣的结果。主要关注点是估计形式的第二个汉克尔行列式$H_{2,2}\left(F\right)={\mathrm{一个}}_2{\mathrm{一个}}_4-{\mathrm{一个}}_3^2$. 非锐界为$H_{2,2}\left(F\right)$什么时候$F\in \mathcal{ķ}\left(\alpha \right)$,$\alpha \in [0,1)$Krishna 和 Ramreddy 发现了由α阶凸函数组成（Hankel 行列式，用于 α 阶星状和凸函数。Tbil Math J. 2012;5:65-76），后来由 Thomas 等人改进。（单价函数：入门。柏林：De Gruyter；2018）。在本文中，我们给出了尖锐的结果。此外，我们获得了明显的结果$H_{2,2}\left(F^{-1}\right)$对于反函数$F^{-1}$什么时候$F\in \mathcal{ķ}\left(\alpha \right)$， 什么时候$F\in {\mathcal{小号}}^{*}\left(\alpha \right)$, α阶类星函数。因此，本文的结果完成了f和的第二个 Hankel 行列式的问题集$F^{-1}$对于课程${\mathcal{小号}}^{*}\left(\alpha \right)$,$\mathcal{ķ}\left(\alpha \right)$,${\mathcal{小号}}_{\text{β}}^{*}$和${\mathcal{ķ}}_{\text{β}}$， 在哪里${\mathcal{小号}}_{\text{β}}^{*}$和${\mathcal{ķ}}_{\text{β}}$分别是β阶强类星函数和强凸函数类。