  QCD bounds on the leading-order (LO) hadronic vacuum polarization (HVP)
contribution to the anomalous magnetic moment of the muon
($a_\mu^{\mathrm{HVP,LO}}$, $a_\mu=\left(g-2\right)_\mu/2$) are determined by
imposing H\"older inequalities and related inequality constraints on systems of
Finite-Energy QCD sum-rules. This novel methodology is complementary to lattice
QCD and data-driven approaches to determining $a_\mu^{\mathrm{HVP,LO}}$. For
the light-quark ($u,d,s$) contributions up to five-loop order in perturbation
theory in the chiral limit, LO in light-quark mass corrections, next-to-leading
order in dimension-four QCD condensates, and to LO in dimension-six QCD
condensates, we find that $\left(657.0\pm 34.8\right)\times 10^{-10}\leq
a_\mu^{\mathrm{HVP,LO}} \leq \left(788.4\pm 41.8\right)\times10^{-10}\,$,
bridging the range between lattice QCD and data-driven values.
