On a generalized orthogonal additivity (Q2777735)

In real normed spaces \((E,\|\;\|)\) one considers the conditional functional equation of Cauchy: \(f(x+y)= f(x)+f(y)\) whenever \(x \perp y\). Since there are various notions of orthogonality \(\perp\), many results in this area have been proved. The author considers, among other questions, the conditional equation NEWLINE\[NEWLINE{\|x-y|\over\|x+y\|} =\alpha\Rightarrow f(x+y)= \gamma\bigl [f(x)+f(y)\bigr] \tag{*}NEWLINE\]NEWLINE where \(\gamma,\alpha\) are given real numbers, \(\alpha\neq 1\). Thus equations of the form NEWLINE\[NEWLINEf(x+y)=g\left({\|x-y \|\over \|x+y \|}\right) \bigl[ f(x)+f(y) \bigr]NEWLINE\]NEWLINE are considered. Some general results on inner product spaces are proven, extending previous studies in this area.