This paper gives a unified
account of a body of work on mean values of functions regular in the unit
disc, relating particularly to the fractional derivatives and integrals of such
functions.
Two types of fractional derivative and integral are discussed. For each of the
two types of fractional derivative considered, a function analogous to the
Littlewood-Paley g-function is defined, and the properties of these two g-type
functions are discussed. The results obtained here include several new inequalities,
and, in particular, an extension (Theorem 5) of a theorem of Hirschman for indices
less than or equal to 1.
