Geodesic
On Two Saigo’s Fractional Integral Operators in the Class of Univalent Functions
Fractional calculus and applied analysis, Tome 9 (2006) no. 2, pp. 159-176.

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Recently, many papers in the theory of univalent functions have been devoted to mapping and characterization properties of various linear integral or integro-differential operators in the class S (of normalized analytic and univalent functions in the open unit disk U), and in its subclasses (as the classes S∗ of the starlike functions and K of the convex functions in U). Among these operators, two operators introduced by Saigo, one involving the Gauss hypergeometric function, and the other - the Appell (or Horn) F3-function, are rather popular. Here we view on these Saigo’s operators as cases of generalized fractional integration operators, and show that the techniques of the generalized fractional calculus and special functions are helpful to obtain explicit sufficient conditions that guarantee mappings as: S → S and K → S, that is, preserving the univalency of functions.
Keywords: Generalized Fractional Integrals, Saigo Operators, Classes of Univalent, Starlike and Convex Functions, Gauss and Generalized Hypergeometric Functions, 26A33, 30C45, 33A35 
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Kiryakova, Virginia. On Two Saigo’s Fractional Integral Operators in the Class of Univalent Functions. Fractional calculus and applied analysis, Tome 9 (2006) no. 2, pp. 159-176. http://geodesic.mathdoc.fr/item/FCAA_2006_9_2_a4/