Bounds on pi pi amplitudes in terms of the D-wave scattering lengths

Rigorous bounds on physical region averages of real parts of pi pi amplitudes are obtained in terms of the D-wave scattering lengths. An example of the bounds is (in units with h(cross)=c=m/sub pi /=1) -.013<or=g identical to (3/(32)/sup 3/) integral /sub 4//sup 8/ds (s +4)(8-s)(-2(s-4)ReF/sup (o)/(s, o)+3(3s-4) ReF/sup (1)/(s, o)+5(s-4) ReF/sup (2)/(s, o))<or=.019 where F/sup (I)/(s, t) are definite isospin pi pi amplitudes. From the CERN-Munich experiment the value of g would be in the range .018+or-.008 to .029+or-.007 depending upon the value of I=2, S-wave scattering length. The bounds thus provide consistency checks on the pi pi scattering data, experimental information on which is not in a satisfactory state.