An Application of the Ruscheweyh Derivatives II

Shigeyoshi Owa

Publications de l'Institut Mathématique, _N_S_38 (1985) no. 52, p. 99

Résumé

We introduce the class $R(\alpha)$ of functions of the form $ f(z)= z- \sum_{k=2}^\infty a_kz^k \qquad (a_k\geq 0) $ satisfying the condition $ \text{Re}\{D^{\alpha+1}f(z)/D^\alpha f(z)\}>\alpha/(\alpha+1) $ for some $(\alpha\geq 0)$ and for all $z\in U=\{z: |z| 1\}$, where $D^\alpha f(z)$ denotes the Hadamard product of $z/(1-z)^{\alpha+1}$ and $f(z)$. The object of the present paper is to prove some distortion and closure theorems for functions $f(z)$ in $R(\alpha)$, and to give the result for the modified Hadamard product of functions $f(z)$ belonging to the class $R(\alpha)$. Furthermore, we determine the radii of starlikeness and convexity of functions $f(z)$ in the class $R(\alpha)$.

Détail

Classification : 26A24 30C45