= Conjugate holomorphic functions do not exhaust the Bergman orthogonal complement
In the area <Hilbert space> of the unit disk, conjugates of <Bergman space> functions vanishing at zero are orthogonal to all holomorphic functions, since $\int_Dhg\,dA=\pi h(0)g(0)$. This follows from the angular mean-value property and the <Cauchy-Schwarz inequality>, which makes the product integrable. But the radial function $|z|^2-1/2$ is also orthogonal to every holomorphic function: its radial integral against the constant angular mean is zero. It is not conjugate holomorphic, since it is real-valued and nonconstant. Thus the conjugate-holomorphic family is a proper subset of the <orthogonal complement>.
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