  The main result of this note is that, for each $n\in \{1,2,3,\ldots\}$, there
exists a Hodge metric on the $n$-th Hirzebruch surface whose positive
holomorphic sectional curvature is $\frac{1}{(1+2n)^2}$-pinched. The type of
metric under consideration was first studied by Hitchin in this context. In
order to address the case $n=0$, we prove a general result on the pinching of
the holomorphic sectional curvature of the product metric on the product of two
Hermitian manifolds $M$ and $N$ of positive holomorphic sectional curvature.
