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Spirallike functions of complex order. (English) Zbl 0966.30010
In 1932 L. Špaček [Casopis Pest, Math. 12-19 (1932; Zbl 0082.01207)] defined the class $$S^\lambda_p$$ of spirallike functions univalent in the unit disc $$\Delta=\{z:|z|<1\}$$. Obviously $$S^0_p=S^*$$ where $$S^*$$ is the class of starlike univalent functions $$f$$, $$f(0)=0$$, $$f'(0)=1$$. R. Libera [Duke Math. 31, 143-158 (1964; Zbl 0129.29403)] considered the class of starlike univalent functions of order $$\rho\in(0,1)$$. Let $$S_p^\lambda(1-b)$$, $$b\neq 0$$, $$b\in\mathbb{C}$$ denote the class of functions $$f(z)- z+a_2z^2 +\dots$$ holomorphic in $$\Delta$$ and satisfying the condition $\text{Re}\left\{{1 \over b\cos\lambda} \left[e^{i\lambda} {zf'(z)\over f(z)}-(1-b)\cos \lambda-i\sin \lambda\right] \right\}>0,\;z\in\Delta.$ In this paper the authors obtain a few properties of the class $$S_p^\lambda(1-b)$$.

##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)