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S.-Y. A. Chang and D. E. Marshall showed that the functional Λ(f) = (1/2π) 2π 0 exp{|f(e iθ )| 2 }dθ is bounded on the unit ball B of the space V 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 B for the functional A. We prove that A attains its maximum at f(z) = z over a subset of B determined by kernel functions, which provides a positive answer to a conjecture of Cirna and Matheson.

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