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 \(\begin{aligned} \max \{\psi (xy), \psi (xy^{-1})\}&= \psi (x)\eta (y)+\psi (y), \\ \max \{\psi (xy), \psi (xy^{-1})\}&= \psi (x)\eta (y)+\chi (y), \\ \max \{\psi (xy), \psi (xy^{-1})\}&= \phi (x)\eta (y), \\ \max \{\psi (xy), \psi (xy^{-1})\}&= \phi (x)\eta (y)+\chi (y), \end{aligned}\) $$\begin{array}{cc}\hfill max\{\psi (xy),\psi (x{y}^{-1})\}& =\psi (x)\eta (y)+\psi (y),\hfill \\ \hfill max\{\psi (xy),\psi (x{y}^{-1})\}& =\psi (x)\eta (y)+\chi (y),\hfill \\ \hfill max\{\psi (xy),\psi (x{y}^{-1})\}& =\varphi (x)\eta (y),\hfill \\ \hfill max\{\psi (xy),\psi (x{y}^{-1})\}& =\varphi (x)\eta (y)+\chi (y),\hfill \end{array}$$ where G is an arbitrary group, \(x, y \in G\) $x,y\in G$ , and \(\psi , \eta , \chi , \phi :G \rightarrow \mathbb {R}\) $\psi ,\eta ,\chi ,\varphi :G\to \mathbb{R}$ are unknown functions.