  We study the higher order corrections of quark masses to the
Gell-Mann$-$Oakes$-$Renner (GOR) relation by constructing QCD sum rules
exclusively for pseudoscalar mesons from the axial-vector correlation function,
$i \int d^4x~ e^{ip\cdot x} < 0| T[A_\mu (x) A_\nu (0)] | 0>$. To project out
the pseudoscalar meson contributions, we apply $-p^\mu p^\nu/p^2$ to this
correlation function and construct sum rules for the decay constants of
pseudoscalar mesons, $f_\pi, f_k$ and $f_{\eta_8}$. The OPE is proportional to
quark masses due to PCAC. To leading order in quark mass, each sum rule
reproduces the corresponding GOR relation. For kaon and $\eta_8$, the deviation
from the GOR relation due to higher orders in quark mass is found to be
substantial. But the deviation gives better agreements with the phenomenology.
Our sum rule provides a sensitive relation between $f_K$ and $m_s$, which
stringently constrain the value for $m_s$. To reproduce the experimental value
for $f_K$, $m_s$ is found to be 186 MeV at 1 GeV scale. The $f_{\eta_8}$ sum
rule also supports this finding.
