  An analysis is given of the $b-\tau$ Yukawa coupling unification in view of
the recent result from Brookhaven on $g_{\mu}-2$ and under the constraint of
$b\to s+\gamma$. We explore $b-\tau$ unification under the above constraints
for nonuniversal boundary conditions for the soft SUSY breaking parameters. We
find new regions of the parameter space where significant negative
supersymmetric contribution to the b quark mass can arise and $b-\tau$
unification within SU(5) framework can occur with nonuniversal gaugino masses.
Specifically we find that for the case where the gaugino mass matrix transforms
like a 24 plet one finds a negative contribution to the b quark mass
irrespective of the sign of the Higgs mixing parameter $\mu$ when the
supersymmetric contribution to $g_{\mu}-2$ is positive. We exhibit regions of
the parameter space where $b-\tau$ unification occurs for $\mu>0$ satisfying
the constraints of $b\to s+\gamma$ and $g_\mu-2$. The $\mu<0$ case is also
explored. The dependence of the accuracy of $b-\tau$ unification defined by
$|\lambda_b-\lambda_{\tau}|/\lambda_{\tau}$ on the parameter space is also
investigated and it is shown that unification with an accuracy of a few percent
can be achieved in significant regions of the parameter space. The allowed
parameter space is consistent with the naturalness constraints and the
corresponding sparticle spectrum is accessible at the Large Hadron Collider.
