Modified version of Jensen equation and orthogonal additivity (Q2754947)

The equation NEWLINE\[NEWLINEf \left(\frac {x+y}{2} \right) = \gamma \left( \frac{\{x-y\}}{\{x+y\}}\right) [f(x)+f(y)] \qquad \qquad \qquad (1)NEWLINE\]NEWLINE is considered in two cases. NEWLINENEWLINENEWLINEThe first when \(f:\mathbb R \rightarrow \mathbb R\), \ \(\gamma:[0, \infty] \rightarrow \mathbb R\) and \(\{x\} = |x|\). In this one-dimensional case, the author shows: NEWLINENEWLINENEWLINETheorem 1. If \(f : \mathbb R \rightarrow \mathbb R\) is a solution of equation (1) where \(\gamma : [0, \infty) \rightarrow \mathbb R\) is a given injection, then either \(f(x) = \beta \text{sgn}(x)\phi(|x|), \;x \in \mathbb R \backslash \{0 \}, \;\text{or} \;f(x) = \beta \phi (|x|), x\in\mathbb R \backslash \{0 \}\), \ for some real number \(\beta \neq 0\), where \(\phi : \mathbb R^+ \rightarrow \mathbb R\) is a function satisfying the equation NEWLINE\[NEWLINE\phi (uv) = \phi(u)\phi(v), \;\;\;u, v \in \mathbb R^+.NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEIn the second, multidimensional case, \(f : X \rightarrow \mathbb R\) where \(X\) is a real linear space with \(dim X \geq 2\) and \(\{ x \} = \parallel x \parallel\), the form of solutions \(f\) to equation (1) depends on whether the norm does or does not come from an inner product. In particular: NEWLINENEWLINENEWLINETheorem 2. Let \(X\) be an inner product space. Then \(f : X \rightarrow \mathbb R\) satisfies equation (1) if and only if either \(\gamma (\alpha) = \frac 12\) for \(\alpha \geq 0\) and NEWLINE\[NEWLINEf(x) = b(x) + c, \qquad x \in X,NEWLINE\]NEWLINE \noindent for some additive function \(b : X \rightarrow \mathbb R\) and some real constant \(c\), or \(\gamma (\alpha) = 1/(2(1+\alpha ^2))\) for \(\alpha \geq 0\) and NEWLINE\[NEWLINEf (x) = k \parallel x \parallel ^2, \qquad x \in X,NEWLINE\]NEWLINE \noindent for some constant \(k \in \mathbb R\).