Further Results for α-Spirallike Functions of Order β

Abstract

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–Szego inequality for the kth root transform of $$f\in \mathcal{S}_{\alpha }(\beta )$$ are investigated.