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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 $f(x)=L(x)+B(x,x),\quad x\in V.$

##### MSC:
 39B62 Functional inequalities, including subadditivity, convexity, etc.