Hyponormal Toeplitz Operators on the Bergman Space of the Disk

We consider Toeplitz operators with bounded symbol acting on the Bergman space of the unit disk and assess their hyponormality. We will mainly be concerned with the symbol $\phi (z)=z^n\mathrm{\mid }z{\mathrm{\mid }}^{2s}+a(t){\bar{z}}^m\mathrm{\mid }z{\mathrm{\mid }}^{2t}$, where $s$ and $t$ are positive real numbers and $m$ and $n$ are natural numbers. The main goal is to understand how large $\mathrm{\mid }a(t)\mathrm{\mid }$ can be for this operator to be hyponormal and we will answer this question for large values of $t$. We also correct a typo from a 2019 paper of Fleeman and Liaw concerning the norm of the commutator of the Toeplitz operator with symbol $z^m{\bar{z}}^n$ when $m>n$.