2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/179244QCD Laplace sum-rules must satisfy a fundamental (Holder) inequality if they are to consistently represent an integrated hadronic cross-section. After subtraction of the pion-pole, the Laplace sum-rule of pion currents is shown to violate this fundamental inequality unless the up and down quark masses are sufficiently large, placing a lower bound on the 1.0 GeV MS-bar running masses.latex2e, 8 pages, 2 figures. Error corrected in equation (41)High Energy Physics - Phenomenologytext