Characterization of Jensen differences and related forms (Q2378035)

A \textit{Jensen difference} for a given mapping \(f\) is defined as \[ \Delta(x,y):=f(x)+f(y)-\lambda f(\mu(x+y)) \] with given parameters \(\lambda,\mu\) (in particular \(\lambda=2\) and \(\mu=\frac{1}{2}\)). Provided commutativity of the addition, \(\Delta\) satisfies a functional equation: \[ \Delta(x,y)+\Delta(z,w)+\lambda\Delta(\mu(x+y),\mu(z+w)) =\Delta(x,z)+\Delta(y,w)+\lambda\Delta(\mu(x+z),\mu(y+w)) \tag{1} \] The main result of the paper is the general solution of (1). In the particular case \(\lambda=0\) or \(\mu=0\) it is proved that \(\Delta: G\to H\) (for \(G\) being a nonempty set and \(H\) an abelian group) satisfies \[ \Delta(x,y)+\Delta(z,w)=\Delta(x,z)+\Delta (y,w) \] if and only if there is a mapping \(f: G\to H\) and \(\eta\in H\) (vanishing if \(H\) is 2-divisible) such that \[ \Delta(x,y)=f(x)+f(y)-\eta. \] In general case, for a commutative ring \(R\), \(\mu\in R\setminus\{0\}\), \(V\) being an \(R\)-module such that \(\mu V=V\), \(X\) being a vector space over a field \(K\) containing \(\mathbb{Q}\) and \(\lambda\in K\setminus \{0\}\), it is proved that \(\Delta: V\times V\to X\) satisfying (1) is of the form \[ \Delta(x,y)=f(x)+f(y)-\lambda f(\mu(x+y))+h_1(x)+h_1(y)+h_2(x)+c \] where \(h_1,h_2: V\to X\) are group homomorphisms, \(h_2(x)=\lambda h_2(\mu x)\), \(c\in X\), \(f: V\to X\).