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Univalent and starlike generalized hypergeometric functions. (English) Zbl 0611.33007
Recently, B. C. Carlson and D. B. Shaffer [SIAM J. Math. Anal. 15, 737-745 (1984; Zbl 0567.30009)] introduced a linear operator and showed how this operator can be applied to the study of certain interesting classes of starlike, convex, and prestarlike hypergeometric functions. The object of the present paper is to prove some basic results involving univalent generalized hypergeometric functions, starlike generalized hypergeometric functions of order $\alpha$, and convex generalized hypergeometric functions of order $\alpha$. Moreover, a certain linear operator involving fractional calculus is introduced, and several general characterization theorems are given in terms of fractional calculus of functions belonging to some of the aforementioned classes of analytic functions.

##### MSC:
 33C05 Classical hypergeometric functions, ${}_{2}{F}_{1}$ 30C45 Special classes of univalent and multivalent functions 26A33 Fractional derivatives and integrals (real functions)