Generalized Vincze’s functional equations on any group in connection with the maximum functional equation

In this research paper, we investigate a generalization of Vincze’s type functional equations involving several (up to four) unknown functions in connection with the maximum functional equation as

$$\begin{aligned} \max \{\psi (xy), \psi (xy^{-1})\}&= \psi (x)\eta (y)+\psi (y), \\ \max \{\psi (xy), \psi (xy^{-1})\}&= \psi (x)\eta (y)+\chi (y), \\ \max \{\psi (xy), \psi (xy^{-1})\}&= \phi (x)\eta (y), \\ \max \{\psi (xy), \psi (xy^{-1})\}&= \phi (x)\eta (y)+\chi (y), \end{aligned}$$

where G is an arbitrary group, \(x, y \in G\), and \(\psi , \eta , \chi , \phi :G \rightarrow \mathbb {R}\) are unknown functions.