A refined Luecking's theorem and finite-rank products of Toeplitz operators

For any function $f$ in $L^{\mathrm{\infty }}(\mathbb{D})$, let $T_f$ denote the corresponding Toeplitz operator the Bergman space $A^2(\mathbb{D})$. A recent result of D. Luecking shows that if $T_f$ has finite rank then $f$ must be the zero function. Using a refined version of this result, we show that if all except possibly one of the functions $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_m$ are radial and $T_{f_1}\mathrm{.}\mathrm{.}\mathrm{.}{T}_{f_m}$ has finite rank, then one of these functions must be zero.