On a new Subclass of Analytic P-valent Functions

Shigeyoshi Owa

Publications de l'Institut Mathématique, _N_S_35 (1984) no. 49, p. 53 .

Résumé

There are many classes of analytic and $p$-valent functions in the unit disk U.N.S. Sohi studied a class $S_p(\alpha)$ of analytic and $p$-valent functions $ f(z)= z^p+ \sum_{n=1}^\infty a_{p+n}z^{p+n},\qquad (p\in N) $ in the unit disk $U$ satisfying the condition $ |f'(z)/pz^{p-1}-\alpha|\alpha,\qquad (z\in U) $ for $\alpha >1/2$. In this paper, we consider a new subclass $S_{p,k}(\alpha)$ of analytic and $p$-valent functions $ f(z)= z^p+\sum a_{p+n}z^{p+n},\qquad (p\in N) $ in the unit disk $U$ satisfying the condition $ łeft|\frac{\Gamma(p+1-k)D^k_z(z)}{\Gamma(p+1)z^{p-k}}\right|\alpha, \qquad (z\in U) $ for $01/2$ and $p\in N$, where $D^k_zf(z)$ means the fractional derivative of order $k$ of $f(z)$. It is the purpose of this paper to show a distortion theorem, the coefficient estimates and a convolution theorem for the class $S_{p,k}(\alpha)$. Further we give a theorem about convex set of functions in the class $S_{p,k}(\alpha)$.

Détail

Classification : 26A24