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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

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