It is proved that a two-body reaction amplitude involving scalar particles and satisfying Mandelstam's representation is bounded by expressions of the form Csln²s at the forward and backward angles, and Cs3/4ln3/2s at any other fixed angle in the physical region, C being a constant, s being the total squared c.m. energy. This corresponds to cross sections increasing at most like ln²s. These restrictions limit the freedom of choice of the subtraction terms to six arbitrary single spectral functions and one subtraction constant.
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M. Froissart (1961) studied this question.