In this work, we consider a class of analytic functions f defined in the unit disk for which the values of \(zf'/f\) lie in a parabolic region of the right-half plane. By using a well-known sufficient condition for functions to be in this class in terms of the Taylor coefficients of z/f, we introduce a subclass \(\mathcal {F}_{\alpha }\) of this class. The aim of the paper is to find the best approximation of non-vanishing analytic functions of the form z/f by functions z/g with \(g\in \mathcal {F}_{\alpha }\). The proof relies on solving a semi-infinite quadratic problem, a problem of independent interest.

中文翻译：

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抛物线区域内某些非零解析函数的逼近

在这项工作中，我们考虑定义在单位圆盘中的一类解析函数f，其中\(zf'/f\) 的值位于右半平面的抛物线区域。通过使用众所周知的函数在z / f的泰勒系数方面的充分条件，我们引入了这个类的子类\(\mathcal {F}_{\alpha }\)。该论文的目的是通过函数z / g与\(g\in \mathcal {F}_{\alpha }\)找到形式为z / f的非零解析函数的最佳近似值. 证明依赖于解决一个半无限二次问题，一个独立感兴趣的问题。