论文标题
在一系列某些非单数函数上
On a Class of certain Non-Univalent Functions
论文作者
论文摘要
在本文中，我们介绍了一个由$$ψ_{a，b}（z）给出的一系列分析功能：= \ dfrac {1} {1} {a-b} \ log {\ dfrac {\ dfrac {1+az} {1+az} {1+bz}}}}}美元 \ begin {equation*} \ Mathcal {f} [a，b]：= \ left \ {f \ in \ mathcal {a}：\ left（\ dfrac {Zf'（z）} {f（z）} {f（z）} - 1 \ right）此外，我们研究了$ψ_{a，b}（z）$的各种特征属性，以及在类$ \ Mathcal {f} [a，b] $中的功能，并获得了$δ$的Starlikesiles of Coldcal $Δ$的敏锐度，并且在$ \ nathcal calcal {f} $ the $ \ nathcal {a n a n a n a n a $ the $ \ nathcal {另外，我们在$ \ Mathcal {bs}（α）中找到了功能的尖锐半径：= \ {f \ in \ Mathcal {a}：zf'（z）/f（z）/f（z）-1 \ prec z/（1-αz^2），\;α\;α\ in（0,1）\ in（0,1） $ \ MATHCAL {s} _ {cs}（α）：= \ {f \ in \ Mathcal {a}：zf'（z）/f（z）/f（z）-1 \ prec z/（（1-z）（1+αz）（1+αz）），\;α\ in（0,1）\ in（0,1） $ \ Mathcal {f} [a，b]。$
In this paper, we introduce a family of analytic functions given by $$ψ_{A,B}(z):= \dfrac{1}{A-B}\log{\dfrac{1+Az}{1+Bz}},$$ which maps univalently the unit disk onto either elliptical or strip domains, where either $A=-B=α$ or $A=αe^{iγ}$ and $B=αe^{-iγ}$ ($α\in(0,1]$ and $γ\in(0,π/2]$). We study a class of non-univalent analytic functions defined by \begin{equation*} \mathcal{F}[A,B]:=\left\{f\in\mathcal{A}:\left( \dfrac{zf'(z)}{f(z)}-1\right)\precψ_{A,B}(z)\right \}. \end{equation*} Further, we investigate various characteristic properties of $ψ_{A,B}(z)$ as well as functions in the class $\mathcal{F}[A,B]$ and obtain the sharp radius of starlikeness of order $δ$ and univalence for the functions in $\mathcal{F}[A,B]$. Also, we find the sharp radii for functions in $\mathcal{BS}(α):=\{f\in\mathcal{A}:zf'(z)/f(z)-1\prec z/(1-αz^2),\;α\in(0,1)\}$, $\mathcal{S}_{cs}(α):=\{f\in\mathcal{A}:zf'(z)/f(z)-1\prec z/((1-z)(1+αz)),\;α\in(0,1)\}$ and others to be in the class $\mathcal{F}[A,B].$
