 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 ${\int }_{D}hgdA=\pi h(0)g(0)$. 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.