${\mathcal U}(λ)$类的几何研究
Geometric studies on the class ${\mathcal U}(λ)$
摘要：
本文讨论了单位圆上所有正规化解析函数$f$的族${\mathcal U}(λ)$，使得$\big |\big (z/f(z)\big )^{2}f'(z)-1\big |<λ$对某些$0<λ\leq 1$ . ${\mathcal U}(λ)$家族在最近的过去已经被广泛研究，并且已知该家族中的功能在$\ID$中是单价的。然而，在这个家庭中的第二个系数的函数的确定尖锐的界限的问题解决了最近在\cite{VY2013}由Vasudevarao和Yanagihara，但证明是复杂的。在这篇文章中，我们首先给出一个简单的证明。我们得到了一些新的从属结果，这个家庭和他们的后果。此外，我们表明，家庭${\mathcal U}(λ)$下的一些初等变换，如旋转，共轭，膨胀和省略值变换，但令人惊讶的是，这个家庭不保存下的$n$ -th根变换的任何$n\geq 2$。这是一个基本的在这里，这有助于产生一些新的定理，特别是提供了一种方法，建设的职能，从家庭${\mathcal U}(λ)$。最后，我们讨论一个半径问题。
英文摘要：
The article deals with the family ${\mathcal U}(λ)$ of all functions $f$ normalized and analytic in the unit disk such that $\big |\big (z/f(z)\big )^{2}f'(z)-1\big |<λ$ for some $0<λ\leq 1$. The family ${\mathcal U}(λ)$ has been studied extensively in the recent past and functions in this family are known to be univalent in $\ID$. However, the problem of determining sharp bounds for the second coefficients of functions in this family was solved recently in \cite{VY2013} by Vasudevarao and Yanagihara but the proof was complicated. In this article, we first present a simpler proof. We obtain a number of new subordination results for this family and their consequences. In addition, we show that the family ${\mathcal U}(λ)$ is preserved under a number of elementary transformations such as rotation, conjugation, dilation and omitted value transformations, but surprisingly this family is not preserved under the $n$-th root transformation for any $n\geq 2$. This is a basic here which helps to generate a number of new theorems and in particular provides a way for constructions of functions from the family ${\mathcal U}(λ)$. Finally, we deal with a radius problem.
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