Hyponormality of Toeplitz Operators with Polynomial Symbols: An Extremal Case

If T φ is a hyponormal Toeplitz operator with polynomial symbol φ = ḡ + f ( f , g ∈ H ∞ ( )) such that g divides f , and if ψ := $f \over g$ then$$\left\vert \sum_{\zeta \in {\cal Z} (\psi)} \zeta \right\vert \le \vert \mu \vert - {{1} \over {\vert \mu \vert}},$$where μ is the leading coefficient of ψ and (ψ) denotes the set of zeros of ψ. In this paper we present a necessary and sufficient condition for T φ to be hyponormal when φ enjoys an extremal case in the above inequality, that is, equality holds in the above inequality.