ON SUBCLASSES OF P-VALENT CLOSE-TO-CONVEX FUNCTIONS

Abstract

Let $K[C,D,p, \alpha]$, $- 1 \le D <C \le 1$ and $0\le \alpha <p$ denote the class of functions \[ g(z) =z^p+\sum_{n=p+1}^\infty b_nz^n \] analytic in the unit disc $U =\{z:|z|<1\}$ and satisfying the condition $1+\frac{zg''(z)}{g'(z)}$ is subcoordinate to $\frac{p+[pD+(C-D)(p-\alpha)]z}{1+Dz}$. We investigate the subclass of p-valent close-to-convex functions \[ f(z) =z^p+\sum_{n=p+1}^\infty a_nz^n, \] for which there exists $g(z)\in K[C,D,p, \alpha]$ such that $\frac{pf'(z)}{g'(z)}$ is subcoordinate to $\frac{p+[pB+(A-B)(p-\beta)]z}{1+Bz}$, $- 1 \le B <A \le 1$ and $0\le \beta <p$ . Distortion and rotation theorems and coefficient bounds are obtained.