On an F-Algebra of Holomorphic Functions

Abstract

The complex maximal theorem of Hardy and Little-wood states:(Mp). For 0 < p < ∞, there exists a positive constant Cp such that if f is holomorphic in the unit disc U of the complex plane thenwhereThe corresponding statement to the limiting case p = 0 can be stated as follows:(M0) There exists a positive constant C0 such that if f is holomorphic in Uwhere log+t = max(log t, 0).The statement (M0) is false as the following example shows.