Let \(\Delta \) be the unit disk on the complex plane \(\mathbb{C}\) and \(\mathcal{A}\) be the class of normalized analytic functions in \(\Delta \). We denote by \(\mathcal{S}_{\alpha }(\beta )\) the class of \(\alpha \)-spirallike functions f of order \(\beta \) as follows$$\begin{aligned} \mathcal{S}_{\alpha }(\beta ):=\left\{ f\in \mathcal{A}: \mathrm{Re}\left\{ e^{i\alpha }\frac{zf'(z)}{f(z)}\right\} >\beta \cos \alpha , \, z\in \Delta \right\} , \end{aligned}$$where \(|\alpha |<\pi /2\) and \(\beta \in [0,1)\). In the present paper, some properties of this certain subclass of analytic functions including, subordination relations, estimates of logarithmic coefficients \(f\in \mathcal{S}_{\alpha }(\beta )\), coefficients inequality and Fekete–Szegö inequality for the kth root transform of \(f\in \mathcal{S}_{\alpha }(\beta )\) are investigated.

中文翻译：

---

β阶α-螺旋状函数的进一步结果

令\（\ Delta \）为复平面\（\ mathbb {C} \）上的单位磁盘，而\（\ mathcal {A} \）为\（\ Delta \）中的归一化解析函数的类。我们用\（\ mathcal {S} _ {\ alpha}（\ beta）\）表示\（\ alpha \） -阶\（\ beta \）的螺旋状函数f的类，如下$$ \ begin {aligned } \ mathcal {S} _ {\ alpha}（\ beta）：= \ left \ {f \ in \ mathcal {A}：\ mathrm {Re} \ left \ {e ^ {i \ alpha} \ frac {zf '（z）} {f（z）} \ right \}> \ beta \ cos \ alpha，\，z \ in \ Delta \ right \}，\ end {aligned} $$其中\（| \ alpha | < \ pi / 2 \）和\（\ beta \ in [0,1）\）。在本文中，该解析函数的某些子类的一些属性包括从属关系，对数系数的估计\（f \ in \ mathcal {S} _ {\ alpha}（\ beta）\），系数不等式和Fekete–研究了\（f \ in \ mathcal {S} _ {\ alpha}（\ beta）\）的第k个根变换的Szegö不等式。