  Harmonic manifolds of hypergeometric type form a class of non-compact
harmonic manifolds that includes rank one symmetric spaces of non-compact type
and Damek-Ricci spaces. When normalizing the metric of a harmonic manifold of
hypergeometric type to satisfy the Ricci curvature $\mathrm{Ric} = -(n-1)$, we
show that the volume entropy of this manifold satisfies a certain inequality.
Additionally, we show that manifolds yielding the upper bound of volume entropy
are only real hyperbolic spaces with sectional curvature $-1$, while examples
of Damek-Ricci spaces yielding the lower bound exist in only four cases.
