Ratti, J. S.; Lin, Y.-F. A functional equation involving f and \(f^{-1}\). (English) Zbl 0731.39006 Colloq. Math. 60/61, No. 2, 519-523 (1990). Motivated by a more special problem, the authors show first that for an arbitrary bijection f on a compact interval (1) \(f(2x-f(x))=x\) implies that \(f(x)=x\). Equivalent forms of (1) are (2) \(1/2(f(x)+f^{-1}(x))=x\) and (3) \(f^ 2(x)-2f(x)+x=0.\) Generalizing (3), it is shown that \((3)'\exists n\sum^{n}_{k=0}(-1)^ k\left( \begin{matrix} n\\ k\end{matrix} \right)f^{n-k}(x)=0\) also implies \(f(x)=x.\) The same result is shown to be true for the following generalization of (2): \((2)'\exists \alpha \quad \quad \alpha f(x)+(1-\alpha)f^{-1}(x)=x\), \(0<\alpha <1.\) So far f was a real function. In another direction the authors prove a result motivated by the result on (3), about linear self-mappings A of a vector space. A is invertible and \(\sum^{n}_{k=0}(-1)^ k\left( \begin{matrix} n\\ k\end{matrix} \right)A^{n-k}=0\) if and only if \(A=I+T+...+T^{n-1}\) and \(T^ n=0\) for some linear self-map T of the vector space. Reviewer: G.Targonski (Marburg) Cited in 6 Documents MSC: 39B12 Iteration theory, iterative and composite equations 39B22 Functional equations for real functions 39B42 Matrix and operator functional equations Keywords:bijections; weighted arithmetic means; operator equation PDFBibTeX XMLCite \textit{J. S. Ratti} and \textit{Y. F. Lin}, Colloq. Math. 60/61, No. 2, 519--523 (1990; Zbl 0731.39006) Full Text: DOI