zbMATH — the first resource for mathematics

A certain fractional derivative operator and its applications to a new class of analytic and multivalent functions with negative coefficients. II. (English) Zbl 0831.30008
Recently, the authors [J. Math. Anal. Appl. 171, 1-13 (1992; Zbl 0760.30006)] made use of a certain operator of fractional derivatives in order to introduce (and initiate a systematic study of) a novel subclass $$T_p (\alpha, \beta, \lambda)$$ of analytic and $$p$$-valent functions with negative coefficients. In this sequel to the aforementioned work, they prove a number of closure and inclusion theorems and determine the radii of $$p$$-valent close-to-convexity, starlikeness, and convexity for the class $$T_p (\alpha, \beta, \lambda)$$. They also obtain a class- preserving integral operator of the form: $F(z) = (J_{\gamma, p} f) (z) : = {\gamma + p \over z^\gamma} \int^z_0 t^{\gamma - 1} f(t) dt\;(\gamma > - p)$ for the class studied here.
Reviewer: H.M.Srivastava

MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 26A33 Fractional derivatives and integrals
Keywords:
fractional derivatives
Zbl 0760.30006
Full Text: