QCD bounds on the leading-order (LO) hadronic vacuum polarization (HVP) contribution to the anomalous magnetic moment of the muon [, ] are determined by imposing Hölder 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 . For the light-quark (, , ) 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 , bridging the range between lattice QCD and data-driven values.