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On functionals with the Cauchy difference bounded by a homogeneous functional. (English) Zbl 1046.39021
The authors consider the functional inequality \[ f(x+y)-f(x)-f(y) \geq \Phi(x,y) \] where \(x,y\) are in a real vector space \(V\), \(f:V \to \mathbb R\), \(\Phi:V \times V \to \mathbb R\) and \(\Phi(x,\cdot)\) is homogeneous for every \(x \in V\). After defining the class \(\mathcal F\) of functions from \(\mathbb R\) into \(\mathbb R\) which are Lebesgue integrable on every compact interval, differentiable at zero and at a negative point and at a positive point, and non negative in zero, they prove the following
Theorem: If \(f\) satisfies the previous inequality and the function \(t \mapsto f(tx)\) belongs to \(\mathcal F\), then there exist a linear \(L:V\to\mathbb R\) and a bilinear and symmetric \(B:V\times V \to\mathbb R\) such that \(\Phi=2B\) and \[ f(x)=L(x)+B(x,x),\quad x\in V. \]

39B62 Functional inequalities, including subadditivity, convexity, etc.