There are different approaches to the fractional calculus which, not being all
equivalent, have lead to a certain degree of confusion and several misunderstandings
in the literature. Probably for this the fractional calculus is in some way the "black
sheep" of the analysis. In spite of the numerous eminent mathematicians who have
worked on it, still now the fractional calculus is object of so many prejudices. In these
review lectures we essentially consider and develop two different approaches to the
fractional calculus in the framework of the real analysis: the continuous one, based
integral operators and the discrete one, based on infinite series of finite differences
with increments tendig to zero. Both approaches turn out to be useful in treating our
generalized diffusion processes in the theory of probability and stochastic processes.We obtain basic properties like coefficient
inequality, distortion and covering theorem, radii of starlikeness, convexity and closeto-convexity, extreme points, Hadamard product, and closure theorems for functions
belonging to our class