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The stability of the entropy of degree alpha. (English) Zbl 1149.39024
The functional equation \[ g(x)+(1-x)^{\alpha}g\left(\frac{y}{1-x}\right)=g(y)+(1-y)^{\alpha}g \left(\frac{x}{1-y}\right)\tag{1} \] with unknown mapping \(g:[0,1]\to\mathbb R\) and with \(\alpha=1\) is known as a fundamental equation of information. For \(0<\alpha\neq 1\) it has been considered by Z. Daróczy [Inf. Control 16, 36–51 (1970; Zbl 0205.46901)]. In the present paper, the author proves the Hyers-Ulam stability of the equation (1) (for \(\alpha\neq 1\)) and generalizes the result of Daróczy. Then, the stability of (1) is applied in the proof of the stability of some system of functional equations characterizing the entropy of degree alpha (Havrda-Charvát or Tsallis entropy). Some open problems are posed, in particular the one concerning the stability of equation (1) for \(\alpha=1\).

39B82 Stability, separation, extension, and related topics for functional equations
94A17 Measures of information, entropy
39B72 Systems of functional equations and inequalities
Full Text: DOI
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