## Hyperstability of a functional equation.(English)Zbl 1212.39044

The main result of the paper is the following. Let $$\alpha,\varepsilon \in\mathbb{R}$$ be fixed, $$\alpha<0$$ and $$\varepsilon\geq 0$$. Then, the function $$f:\,]0,1[\,\to\mathbb{R}$$ satisfies the inequality $\left| f(x)+(1-x)^{\alpha}f\left(\frac{y}{1-x}\right)-f(y)-(1-y)^{\alpha} f\left(\frac{x}{1-y}\right)\right| \leq \varepsilon \tag{1}$ for all $$(x,y)\in D\doteq \{(x,y)\in\mathbb{R}^{2}:x,y,x+y\in \,]0,1[\,\}$$ if, and only if, there exist $$a,b \in\mathbb{R}$$ such that $f(x)= ax^{\alpha}+b((1-x)^{\alpha}-1). \qquad\qquad (x\in \,]0,1[\,)$ This result has the somewhat surprising consequence that the parametric fundamental equation of information (that can be obtained from $$(1)$$ with $$\varepsilon=0$$ and plays an important role in characterizing information measures) is hyperstable, that is, the solutions of the stability inequality $$(1)$$ and the parametric fundamental equation of information are the same. As a corollary, it is also proved that the system of equations that defines $$\alpha$$-recursive and semi-symmetric information measures is stable.

### MSC:

 39B82 Stability, separation, extension, and related topics for functional equations 94A17 Measures of information, entropy
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### References:

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