On certain subclass of analytic and univalent functions in the unit disk (Q788858)

The motivation for writing this paper seems to be two papers [Nieuw Arch. Wiskunde, III. Ser. 15, 55-63 (1967; Zbl 0172.375); and Indian J. Math. 13, 141-145 (1971; Zbl 0261.30011)] by the reviewer. Let S denote the class of functions \(f(z)=z+\sum^\infty_{n=2}a_ nz^ n\) analytic and univalent in the unit disc U. Let \(S_{\alpha}(\beta)\) denote the subclass of functions in S which satisfy the condition \[ | \frac{D^{\alpha}_ zf(z)}{z^{1-\alpha}}-\beta |<\beta,\quad z\in U,\quad 0<\alpha<1 \] and \(\beta>\frac{1}{2\Gamma(2-\alpha)}\) where \(D_ z^{\alpha}f(z)\) denotes the fractional derivative of order \(\alpha\) of f(z). By following the techniques used by the reviewer in the papers referred above, the author proves a distortion theorem and determines the coefficient estimates of functions in the class \(S_{\alpha}(\beta)\). It is further shown that the class \(S_{\alpha}(\beta)\) is closed under convolution and the functions in this class form a convex set.