  The recently evaluated two-pion contribution to the muon g-2 and the phase of
the pion electromagnetic form factor in the elastic region, known from \pi\pi
scattering by Fermi-Watson theorem, are exploited by analytic techniques for
finding correlations between the coefficients of the Taylor expansion at t=0
and the values of the form factor at several points in the spacelike region. We
do not use specific parametrizations and the results are fully independent of
the unknown phase in the inelastic region. Using for instance, from recent
determinations, <r^2_\pi> = (0.435 \pm 0.005) fm^2 and F(-1.6
GeV^2)=0.243^{+0.022}_{-0.014}, we obtain the allowed ranges 3.75
GeV^{-4}\lesssim c \lesssim 3.98 GeV^{-4} and 9.91 GeV^{-6}\lesssim d \lesssim
10.46 GeV^{-6} for the curvature and the next Taylor coefficient, with a strong
correlation between them. We also predict a large region in the complex plane
where the form factor can not have zeros.
