The Douglas question for functions of the form $\bar{z}+h$ with $h$ in the disk algebra

We prove that the Toeplitz operator $T_{\bar{z}+h}$ is invertible on the Bergman space provided that $|\bar{z}+h|$ is bounded below by some positive constant on the open unit disk, where $h$ is in the disk algebra. This provides a general class of harmonic functions for which the answer to the D