Generalization of functional equation for the square root spiral (Q858799)

Applying the method in \textit{K. J. Heuvers, D. S. Moak} and \textit{B. Boursaw} [Math. Appl., Dordr. 518, 111--117 (2000; Zbl 0976.39018)], the authors solve the functional equation \(\varphi(p^{-1}(p(x)+c))=\varphi(x)+h(x)\), where \(p, h\) are given functions, \(p^{-1}\) denotes the inverse of \(p\), \(\varphi\) is the unknown function and \(c \neq 0\) is a constant. For some technical reasons, they prove the further generalized stability and the stability in the sense of Ger of the special case of the above functional equation, where \(h=0\). Applying their results to the \(n\)-th root spiral \(\varphi(\root n\of{x^n+1})=\varphi(x)+\arctan\frac{1}{x}\), they generalize the results of the cited paper above and \textit{S. M. Jung} and \textit{P. K. Sahoo} [Appl. Math. Lett. 15, No. 4, 435--438 (2002; Zbl 1016.39020)].