  QCD Laplace Sum-Rules must satisfy a fundamental Hoelder inequality if they
are to consistently represent an integrated hadronic spectral function. The
Laplace sum-rules of pion currents is shown to violate this inequality unless
the $u$ and $d$ quark masses are sufficiently large, placing a lower bound on
$m_u+m_d$, the SU(2)-invariant combination of the light-quark masses.
