Two Elementary Analytic Functions and Their Relationship with Hardy and Bergman Spaces

Abstract

We prove that the function f(z)=1/(1-z) which is holomorphic in the open unit disc  centered at the origin, is an element of a Hardy space H^p if and only if  p<1. Here we give a new proof for a known result. Moreover, the present work provides two different new proofs for one of the implications mentioned above. One proves that the same function f is an element of a Bergman space A^p if and only if p<1. This is the first completely new result of this work. From these theorems we deduce the behavior of the function g(z)=1/(1-z^2)^(1/2)  in the half – open unit disc {z; |z|<1, Re(z)>0}. Although the assertions claimed above refer to complex analytic functions, and the involved spaces are function spaces of analytic complex functions, the proofs from below are based on results and methods of real analysis