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Hyperstability of a functional equation. (English) Zbl 1212.39044

The main result of the paper is the following. Let $\alpha ,\epsilon \in ℝ$ be fixed, $\alpha <0$ and $\epsilon \ge 0$. Then, the function $f:\phantom{\rule{0.166667em}{0ex}}\right]0,1\left[\phantom{\rule{0.166667em}{0ex}}\to ℝ$ satisfies the inequality

$\left|f\left(x\right)+{\left(1-x\right)}^{\alpha }f\left(\frac{y}{1-x}\right)-f\left(y\right)-{\left(1-y\right)}^{\alpha }f\left(\frac{x}{1-y}\right)\right|\le \epsilon \phantom{\rule{2.em}{0ex}}\left(1\right)$

for all $\left(x,y\right)\in D\doteq \left\{\left(x,y\right)\in {ℝ}^{2}:x,y,x+y\in \phantom{\rule{0.166667em}{0ex}}\right]0,1\left[\phantom{\rule{0.166667em}{0ex}}\right\}$ if, and only if, there exist $a,b\in ℝ$ such that

$f\left(x\right)=a{x}^{\alpha }+b\left({\left(1-x\right)}^{\alpha }-1\right)·\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\left(x\in \phantom{\rule{0.166667em}{0ex}}\right]0,1\left[\phantom{\rule{0.166667em}{0ex}}\right)$

This result has the somewhat surprising consequence that the parametric fundamental equation of information (that can be obtained from $\left(1\right)$ with $\epsilon =0$ and plays an important role in characterizing information measures) is hyperstable, that is, the solutions of the stability inequality $\left(1\right)$ 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 94A17 Measures of information, entropy
##### References:
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