  The bounds on the light quark masses are obtained by fitting the squares of
pseudoscalar meson masses $m_\pi^2$ and $m_K^2$ to second order in $1/N_c$
expansion. The result is an algebraic cubic curve whose coefficients are the
known Weinberg values for the quark mass ratios $m_u/m_d$ and $m_s/m_d$.
Additional restrictions arise when using the ratio $m_s/m_{ud}=27.23(10)$
quoted by FLAG for lattice simulations with four quark flavors. This provides a
tight constraint on the ratio $m_u/m_d = 0.455(8)$.
