Within the range of elastic scattering of two particles the unitarity equation can be formulated as an integral equation for the phase of the analytic scattering amplitude if the cross-section is given. Since the publications of \textit{R. G. Newton} [Determination of the amplitude from the differential cross-section by unitarity, J. Math. Phys. 9, 2050-2055 (1968)] and \textit{A. Martin} [Construction of the scattering amplitude from the differential cross-sections, Nuovo Cimento 59 A, 131-152 (1969) and Scattering theory: Unitarity, analyticity and crossing, Lect. Notes Phys. 3 (1969)] this nonlinear equation has been investigated as a fixed- point problem in a real Banach space with the Banach contraction mapping principle or the Schauder fixed-point theorem. In the present publication this mapping is reexamined in a complex Banach space and the real integral equation is extended to a fixed-point problem in a complex Banach space. The fixed point theorem of Earle and Hamilton for holomorphic mappings leads then to the determination of a unique scattering amplitude if the parameter sin \(\mu\) of Newton and Martin (which characterizes the cross-section) is bounded by the improved limit sin \(\mu\) \(<0,86\).