ON CERTAIN GENERALIZATIONS OF THE SPIRAL-LIKE AND ROBERTSON FUNCTIONS

Let $S^\lambda(\alpha, \beta, A, B)$ denote the class of functions $f(z)=z+\sum_{n=2}^\infty a_nz^n$ which are analytic in the unit disc $U=\{z:|z|<1\}$ and satisfy the inequality\[\left|\frac{F(z)}{(B-A)(F(z)+(1-\alpha)e^{-i\lambda}\cos \lambda)+AF(z)}\right|<1, \ \]where $F(z)=zf'(z)/f(z)-1$ for some $\lambda, \alpha, \beta, A, B$ ($|\lambda|<\pi/2, 0\le \alpha< 1, 0<\beta\le 1, -1\le A< B\le 1$ and $0<B\le 1$) and for all $z\in U$. Further $f(z)$ is said to belong to the class $C^\lambda(\alpha, \beta, A, B)$ ($|\lambda|<\pi/2, 0\le \alpha< 1, 0<\beta\le 1, -1\le A< B\le 1$ and $0<B\le 1$) if and only if $zf'(z) \in S^\lambda(\alpha, \beta, A, B)$. In the present paper, the authors give several representation formulas, distortion theorems, and coefficient bounds for functons belonging to these classes. They also obtain the sharp radius of $\gamma$-spiral and starlikeness for the class $S^\lambda(\alpha, \beta, A, B)$ and the sharp radius of $\gamma$-convex and convexity for the class $C^\lambda(\alpha, \beta, A, B)$.