Bloch space

In the mathematical field of complex analysis, the Bloch space, named after French mathematician André Bloch and denoted ${\mathcal {B}}$ or ℬ, is the space of holomorphic functions f defined on the open unit disc D in the complex plane, such that the function

$(1-|z|^{2})|f^{\prime }(z)|$

is bounded. ${\mathcal {B}}$ is a type of Banach space, with the norm defined by

$\|f\|_{\mathcal {B}}=|f(0)|+\sup _{z\in \mathbf {D} }(1-|z|^{2})|f'(z)|.$

This is referred to as the Bloch norm and the elements of the Bloch space are called Bloch functions.