Halfconvex functions and equality of Daróczy-Páles conjugate means (Q372384)

Let \(I\subset \mathbb{R}\) be an interval. By \(\mathcal{CM}(I)\) we denote the family of all continuous and strictly monotonic functions \(\varphi : I \rightarrow \mathbb{R}\). The functions \(\varphi, \psi \in \mathcal{CM}(I)\) are called equivalent, in writing \(\psi \sim \varphi\), if \(\psi =a\varphi +b\) for some \(a,b\in \mathbb{R}\), \(a\not= 0\). Let \(K:I^2 \rightarrow I\) be a mean in \(I\), \(p,q\in [0,1]\) fixed numbers and \(\varphi \in \mathcal{CM}(I)\). The function \(K_\varphi ^{(p,q)} :I^2\rightarrow \mathbb{R}\) defined by \[ K_\varphi ^{(p,q)}(x,y) =\varphi^{-1}(p\varphi(x)+q\varphi(y)+(1-p-q)\varphi(K(x,y))), \quad \quad x,y\in I, \] is a mean and it is called the \(K\)-conjugate mean of the quasi-generator \(\varphi\), and the weights \(p\), \(q\). The author considers the question: when two conjugate means of different generators and weights are equal? The main result is the following theorem. Theorem. Let \(K,L : I^2 \rightarrow I\) be means in the interval \(I\), \(p,q,r,s \in [0,1]\) be such that \(p+q \not= 1\not= r+s\), and \(\varphi, \psi \in \mathcal{CM}(I)\). Then \[ L_\psi ^{(r,s)}=K_\varphi ^{(p,q)} \] if and only if \(\psi \sim \varphi\) and \[ L(x,y)=\varphi^{-1}\left( \frac{(p-r)\varphi(x)+(q-s)\varphi(y)+(1-p-q)\varphi(K(x,y))}{1-r-s}\right) \] for all \(x,y\in I\).