For f ∈ H2, let Gf := {g ∈ zH : f + g ∈ L∞ and Tf+g is hyponormal}. In 1988, C. Cowen posed the following question: If g ∈ Gf is such that λ g / ∈ Gf (all λ ∈ C, |λ| > 1), is g an extreme point of Gf ? In this note we answer this question in the negative. At the same time, we obtain a general sufficient condition for the answer to be affirmative; that is, when f ∈ H∞ is such that rankHf < ∞.
