Moduli of bounded holomorphic functions in the ball

We prove that there is a continuous non-negative function $g$ on the unit sphere in $\cd$, $d \geq 2$, whose logarithm is integrable with respect to Lebesgue measure, and which vanishes at only one point, but such that no non-zero bounded analytic function $m$ in the unit ball, with boundary values $m^\star$, has $|m^\star| \leq g$ almost everywhere. The proof analyzes the common range of co-analytic Toeplitz operators in the Hardy space of the ball.