Generalized Fractional Integrals Saigo Operators Classes of Univalent Starlike and Convex Functions Gauss and Generalized Hypergeometric Functions 26A33 30C45 33A35
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Fractional Calculus and Applied Analysis, Vol. 9, No 2, (2006), 159p-176p
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.