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An application of a certain fractional derivative operator. (English) Zbl 0728.30013

The object of the present paper is to introduce and study a linear operator \(N_{0,z}^{\alpha,\beta,\eta}\) which is defined in terms of a certain fractional derivative operator. Various interesting properties of the operator \(N_{0,z}^{\alpha,\beta,\eta}\), including its connection with the Carlson-Shaffer operator L(a,c), are given. It is also shown how these operators can be applied successfully with a view to proving a number of inclusion and connection theorems involving starlike, convex, and prestarlike functions in the open unit disk U.
Reviewer: S.Owa (Osaka)

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
26A33 Fractional derivatives and integrals
33C20 Generalized hypergeometric series, \({}_pF_q\)
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