论文标题
在一系列由差分不平等定义的单价函数上
On a class of univalent functions defined by a differential inequality
论文作者
论文摘要
对于$ 0<λ\ le 1 $，让$ \ m nrycal {u}（λ）$是类分析函数$ f（z）= z+\ sum_ {n = 2}^{\ infty} a_n z^n $在单位磁盘$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f'（z）中（z）（z）（z）（z）（z）（z）（z）（z）（z）（z）（z）（ $ \ MATHCAL {U}：= \ MATHCAL {U}（1）$。在本文中，我们证明了类$ \ mathcal {u} $包含在类似Starlike功能类的封闭凸面中，并且使用了这一事实，我们解决了一些极端问题，例如积分均值问题和弧度长度问题，用于$ \ Mathcal {u} $中的函数的函数。通过所谓的恒星函数理论，我们还解决了$ \ Mathcal {u}（λ）$中功能的整体平均问题。我们还获得了fekete-szegö函数的估计值和$ \ Mathcal {u}（λ）$中函数某些非线性积分变换的固定范围。此外，对于以$Δ的定义的类函数类别的类别，= \ {ζ\ in \ mathbb {\ wideHat {c}}}：| e |ζ|> 1 \} $并与类$ \ mathcal {u}（U}（λ）$相关联，我们获得了一个足够的条件，我们可以获得一个极端的$ g $ g $ g点。
For $0<λ\le 1$, let $\mathcal{U}(λ)$ be the class analytic functions $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in the unit disk $\mathbb{D}$ satisfying $|f'(z)(z/f(z))^2-1|<λ$ and $\mathcal{U}:=\mathcal{U}(1)$. In the present article, we prove that the class $\mathcal{U}$ is contained in the closed convex hull of the class of starlike functions and using this fact, we solve some extremal problems such as integral mean problem and arc length problem for functions in $\mathcal{U}$. By means of the so-called theory of star functions, we also solve the integral mean problem for functions in $\mathcal{U}(λ)$. We also obtain the estimate of the Fekete-Szegö functional and the pre-Schwarzian norm of certain nonlinear integral transform of functions in $\mathcal{U}(λ)$. Further, for the class of meromorphic functions which are defined in $Δ:=\{ζ\in\mathbb{\widehat{C}}:|ζ|>1\}$ and associated with the class $\mathcal{U}(λ)$, we obtain a sufficient condition for a function $g$ to be an extreme point of this class.
