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