Functions Invariant under the Berezin Transform

We show that a bounded function f satisfies Bf = f, where B is the Berezin tranform on the unit disc (defined in (2) below), if and only if f is harmonic. There is an equivalent formulation of this result [S. Axler and Ž. Čučković, Integral Equations Operator Theory14 (1991), 1-12; W. Rudin, "Function Theory in the Unit Ball of CN," Springer-Verlag, New York/Berlin, 1980]: If f is bounded and satifies the invariant version of the area mean value property, then f is harmonic. The main tool employed is Fourier analysis on the Lie group of all Möbius transformations.