类$U(λ)$上的系数问题
Coefficient problems on the class $U(λ)$
摘要：
对于$0<λ\leq 1$，令${\mathcal U}(λ)$表示在$\ID$中满足$\ID$中的条件$\left |\left (\frac{z}{f(z)}\right )^{2}f'(z)-1\right |<λ$的单位圆盘$\ID$中解析的函数族$f(z)=z+\sum_{n=2}^{\infty}a_nz^n$。虽然在这个家庭中的功能被称为是在$\ID$的单价，系数猜想$a_n$的$n\geq 5$仍然是一个悬而未决的问题。在这篇文章中，我们将首先提出一个非尖锐的界限$|a_n|$。${\mathcal U}(λ)$族的一些成员由$$ \frac{z}{f(z)}=1-(1+λ)ϕ(z) + λ(ϕ(z))^2 $$和$ϕ(z)=e^{iθ}z$给出，它们解决了${\mathcal U}(λ)$中的许多极值问题。其次，我们将考虑以下问题：是否存在函数$ϕ$分析$\ID$与$|ϕ(z)|<1$，不的形式$ϕ(z)=e^{iθ}z$，其中相应的功能$f$的上述形式的家庭成员${\mathcal U}(λ)$？最后，我们将以形式为$$f(z) =\frac{z}{1-a_2z+λz\int_0^zω(t)\,dt}, $$的$f\in {\mathcal U}(λ)$的显式形式求解第二系数（$a_2$）问题，其中$ω$在$\ID$中是解析的，使得$|ω(z)|\leq 1$和$ω(0)=a$，其中$a\in \overline{\ID}$。
英文摘要：
For $0<λ\leq 1$, let ${\mathcal U}(λ)$ denote the family of functions $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ analytic in the unit disk $\ID$ satisfying the condition $\left |\left (\frac{z}{f(z)}\right )^{2}f'(z)-1\right |<λ$ in $\ID$. Although functions in this family are known to be univalent in $\ID$, the coefficient conjecture about $a_n$ for $n\geq 5$ remains an open problem. In this article, we shall first present a non-sharp bound for $|a_n|$. Some members of the family ${\mathcal U}(λ)$ are given by $$ \frac{z}{f(z)}=1-(1+λ)ϕ(z) + λ(ϕ(z))^2 $$ with $ϕ(z)=e^{iθ}z$, that solve many extremal problems in ${\mathcal U}(λ)$. Secondly, we shall consider the following question: Do there exist functions $ϕ$ analytic in $\ID$ with $|ϕ(z)|<1$ that are not of the form $ϕ(z)=e^{iθ}z$ for which the corresponding functions $f$ of the above form are members of the family ${\mathcal U}(λ)$? Finally, we shall solve the second coefficient ($a_2$) problem in an explicit form for $f\in {\mathcal U}(λ)$ of the form $$f(z) =\frac{z}{1-a_2z+λz\int_0^zω(t)\,dt}, $$ where $ω$ is analytic in $\ID$ such that $|ω(z)|\leq 1$ and $ω(0)=a$, where $a\in \overline{\ID}$.
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