An extremal function for the Chang-Marshall inequality over the Beurling functions

Abstract:

S.-Y. A. Chang and D. E. Marshall showed that the functional $\Lambda (f) =(1/2\pi ) \int _0^{2\pi }\exp \{ |f(e^{i\theta })|^2\}d\theta$ is bounded on the unit ball $\mathcal {B}$ of the space $\mathcal {D}$ of analytic functions in the unit disk with $f(0)=0$ and Dirichlet integral not exceeding one. Andreev and Matheson conjectured that the identity function $f(z)=z$ is a global maximum on $\mathcal {B}$ for the functional $\Lambda$. We prove that $\Lambda$ attains its maximum at $f(z)=z$ over a subset of $\mathcal {B}$ determined by kernel functions, which provides a positive answer to a conjecture of Cima and Matheson.