  Analyticity and unitarity techniques are employed to obtain bounds on the
shape parameters of the scalar and vector form factors of semileptonic $K_{l3}$
decays. For this purpose we use vector and scalar correlators evaluated in
pQCD, a low energy theorem for scalar form factor, lattice results for the
ratio of kaon and pion decay constants, chiral perturbation theory calculations
for the scalar form factor at the Callan-Treiman point and experimental
information on the phase and modulus of $K\pi $ form factors up to an energy
$\tin=1 {\rm GeV}^2$. We further derive regions on the real axis and in the
complex-energy plane where the form factors cannot have zeros.
