  Recently, a new set of positivity bounds with $t$ derivatives have been
discovered. We explore the generic features of these generalized positivity
bounds with loop amplitudes and apply these bounds to constrain the parameters
in chiral perturbation theory up to the next-to-next-to-leading order. We show
that the generalized positivity bounds give rise to stronger constraints on the
$\bar l_i$ constants, compared to the existing axiomatic bounds. The parameter
space of the $b_i$ constants is constrained by the generalized positivity
bounds to be a convex region that is enclosed for many sections of the total
space. We also show that the improved version of these positivity bounds can
further enhance the constraints on the parameters. The often used Pad\'e
unitarization method however does not improve the analyticity of the amplitudes
in the chiral perturbation theory at low energies.

High Energy Physics - Phenomenology;High Energy Physics - Lattice;High Energy Physics - Theory