Bounded Functions

Let \(\mathcal {B}\) denote the class of all analytic functions \(w:\mathbb {D}\rightarrow \overline{ \mathbb {D}}\) , and let \(\mathcal {B}_0\) be its subclass consisting of those functions w satisfying \(w(0)=0\) . The functions in the class \(\mathcal {B}_0\) are known as bounded or Schur functions. They are also referred to as Schwarz functions, as they satisfy the conditions of the Schwarz’s lemma. This chapter is devoted to the study of various properties of these functions, including sharp estimates for their Taylor coefficients.