Characterization of quadratic mappings through a functional inequality
Motivated by some results of \textit{K. Baron} and \textit{Z. Kominek} [Bull. Pol. Acad. Sci., Math. 51, No. 3, 301--307 (2003; Zbl 1046.39021)] the author studies the functional inequality \[ f(x+y)-\frac{f(2x)+f(2y)}{4} \geq \phi(x,y) \tag{I} \] where \(f:X \to \mathbb R\), \(\phi:X \times X \to \mathbb R\), \(X\) is either an abelian group or a real inner product space. Among others, the following two theorems are proved. Theorem 1. Assume that \((X,+)\) is a uniquely \(2\)-divisible abelian group, and \(f\) and \(\phi\) satisfy (I). If \(f(0)\leq 0\) and \[ \begin{aligned} \phi(x,-y) &\geq -\phi(x,y),\\ \limsup_{k \to +\infty} 4^k \phi(2^{-k}x,2^{-k}x)&< +\infty,\\ \limsup_{k \to +\infty} 4^k \phi(2^{-k}x,2^{-k}y) &\geq \phi(x,y), \end{aligned} \] then \(f(x)=\frac{1}{2}\phi(x,x)\), \(\phi\) is biadditive and symmetric. Theorem 2. Assume that \(X\) is a real inner product space with \(\dim X \geq 3\), and \(f\) and \(\phi\) satisfy (I). If \(f(0)\leq 0\) and \[ \begin{aligned} &\phi(2x,2x) \geq 4\phi(x,x),\\ &\phi(x,-x) \geq -\phi(x,x),\\ &\phi(x,y) \geq 0 \quad \text{if} \quad x \perp y, \end{aligned} \] then there exists a sublinear map \(P:\mathbb R_+ \to \mathbb R\) such that \(f(x)=\frac{1}{2}\phi(x,x)=-P(\| x \|^2)\).
scientific article; zbMATH DE number 5078074