We consider bounds on light quark masses that follow from positivity of thepseudoscalar correlator spectral function plus the assumption that perturbativeQCD is valid for the correlator and its derivatives up to order $N$ for momenta$t\geq\hat{t}$. We find that the bounds vary a lot depending on the assumedvalue of $\hat{t}$ and even, if it is too small ($\hat{t}\simeq 1.5, 1.6, 2.1Gev^2$ for respectively $N=0, 1, 2$), that there is incompatibility between theasumption of validity of perturbative QCD and positivity. This allows us toestablish a criterion for the values of $\hat{t}$ admissible, and to get upperand lower bounds for $m_s$ and upper bounds for $m_d+m_u, m_d-m_u$. The upperbounds are not particularly interesting, but the lower ones are very tight;specifically we find $$240 Mev\leq m_s; 16 Mev\leq m_d+m_u$$ if we assumeperturbative QCD to give a valid description of the correlator for $t\geq 2.2Gev^2$; or, if it only holds for $t\geq 4.5 Gev^2$ then $$150 Mev\leq m_s; 10Mev\leq m_d+m_u.$$ Here the masses are running masses defined at $1 Gev$. Wealso show reasonable models where the bounds are saturated. The results suggestthat some of the current estimates of the light quark masses are less precisethan ordinarily claimed.
