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Orthosymmetrical monotone functions. (English) Zbl 1142.26007

The authors introduce a new type of inverse of a set \(F\subseteq \mathbb{R} ^{2}\) with respect to a monotone \(\mathbb{R}\rightarrow \mathbb{R}\) bijection \(\phi \). Consider a point \(\left( x_{0},y_{0}\right) \) on \(F\). Due to the strict monotonicity of \(\phi \), the triplet \(\left( \left( x_{0},\phi \left( x_{0}\right) \right) ,\left( x_{0},y_{0}\right) ,\left( x_{0},y_{0}\right) ,\left( \phi ^{-1}\left( y_{0}\right) ,y_{0}\right) \right) \) determines a unique rectangle through the point \(\left( x_{0},y_{0}\right) ,\) with each side parallel to one of the axes and having at least two vertices on \(\phi \). The fourth point \(\left( \phi ^{-1}\left( y_{0}\right) ,\phi \left( x_{0}\right) \right) \) of the rectangle belongs to the set \[ F^{\phi }:=\{(x,y)\in \mathbb{R}^{2}| \left( \phi ^{-1}\left( y\right) ,\phi \left( x\right) \right) \in F\}. \] Call \(F^{\phi }\) the \(\phi \)-inverse of \(F\). The \(\phi \)-inverse of a function \(f\) is again a function if and only if \(f\) is injective. To other monotone functions, the authors associate a set of \(\phi \)-inverse functions.

MSC:

26A48 Monotonic functions, generalizations
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Full Text: Euclid