In recent years, the problem of estimating Hankel determinants has attracted the attention of many mathematicians. Their research have been focused mainly on deriving the bounds of H2,2{H_{2,2}} or H3,1{H_{3,1}} over different subclasses of 𝒮{\mathcal{S}}. Only in a few papers third Hankel determinants for non-univalent functions were considered. In this paper, we consider two classes of analytic functions with real coefficients. The first one is the class 𝒯{\mathcal{T}} of typically real functions. The second object of our interest is 𝒦ℝ⁢(i){\mathcal{K}_{\mathbb{R}}(i)}, the class of functions with real coefficients which are convex in the direction of the imaginary axis. In both classes, we find lower and upper bounds of the third Hankel determinant. The results are sharp.

中文翻译：

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两类实系数解析函数的第三汉克尔行列式

近年来，估计汉克尔行列式的问题引起了许多数学家的关注。他们的研究主要集中在推导 𝒮{\mathcal{S}} 的不同子类上的 H2,2{H_{2,2}} 或 H3,1{H_{3,1}} 的边界。仅在少数论文中考虑了非单价函数的第三个 Hankel 行列式。在本文中，我们考虑两类具有实系数的解析函数。第一个是典型实函数的 𝒯{\mathcal{T}} 类。我们感兴趣的第二个对象是 𝒦ℝ⁢(i){\mathcal{K}_{\mathbb{R}}(i)}，这是一类具有在虚轴方向上凸出的实系数的函数。在这两个类中，我们都找到了第三个 Hankel 行列式的下限和上限。结果是尖锐的。