Hölder inequalities and bounds on the masses of light quarks

QCD Laplace sum-rules must satisfy a fundamental (Hölder) 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 u and d quark masses (respectively denoted by mu and md) are sufficiently large, placing a lower bound on the 1 eV MS running masses.