If on , open and connected, there exists a unique quasiextension on so that on and on , on , on , where
[cf. Z. Daróczy and L. Losonczi, Publ. Math. 14, 239–245 (1967; Zbl 0175.15305)]. If on there exist unique extensions , , on , so that on and on , on , on [cf. F. Radó and J. A. Baker, Aequationes Math. 32, 227–239 (1987; Zbl 0625.39007)].
In the present paper the authors study whether the similar results hold for the restricted exponential Cauchy functional equation and for the Pexider variant of this equation on (both). First they show by counterexamples that in general this is not the case and further determine the general solutions, with and without regularity assumptions, of these restricted equations on .