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 \(S_p^\lambda(1-b)\).


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