The electronic structure of Pb_1−xSn_xTe is studied by using the relativistic Korringa-Kohn-Rostoker Green function method in the framework of density functional theory. For all concentrations x, Pb_1−xSn_xTe is a direct semiconductor with a narrow band gap. In contrast to pure lead telluride, tin telluride shows an inverted band characteristic close to the Fermi energy. It will be shown that this particular property can be tuned, first, by alloying PbTe and SnTe and, second, by applying hydrostatic pressure or uniaxial strain. Furthermore, the magnitude of strain needed to switch between the regular and inverted band gap can be tuned by the alloy composition. Thus there is a range of potential usage of Pb_1−xSn_xTe for spintronic applications.