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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 \(\alpha=1\).

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
94A17 Measures of information, entropy
39B72 Systems of functional equations and inequalities
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[1] Aczél, J.; Daróczy, Z., On measures of information and their characterizations, (1975), Academic Press New York, San Francisco · Zbl 0345.94022
[2] Daróczy, Z., Generalized information functions, Inform. and control, 16, 36-51, (1970) · Zbl 0205.46901
[3] Ebanks, B.; Sahoo, P.; Sander, W., Characterizations of information measures, (1998), Word Scientific Publishing Co. Inc. River Edge, NJ
[4] Forti, G.L., Hyers – ulam stability of functional equations in several variables, Aequationes math., 50, 1-2, 143-190, (1995) · Zbl 0836.39007
[5] R. Ger, A survey of recent results on stability of functional equations, in: Proceedings of the 4th International Conference on Functional Equations and Inequalities, Pedagogical University in Cracow, 1994, pp. 5-36
[6] Havrda, J.; Charvát, F., Quantification method of classification processes. concept of structural α-entropy, Kybernetika, 3, 30-35, (1967) · Zbl 0178.22401
[7] Hyers, D.H., On the stability of the linear functional equations, Proc. nat. acad. sci. USA, 27, 222-224, (1941) · Zbl 0061.26403
[8] Maksa, Gy., Solution on the open triangle of the generalized fundamental equation of information with four unknown functions, Util. math., 21, 267-282, (1982) · Zbl 0497.94003
[9] Maksa, Gy.; Ng, C.T., The fundamental equation of information on open domain, Publ. math. debrecen, 33, 1-2, 9-11, (1986) · Zbl 0618.94004
[10] Morando, A., A stability result concerning Shannon entropy, Aequationes math., 62, 286-296, (2001) · Zbl 0991.39021
[11] Moszner, Z., Sur LES définitions différentes de la stabilitédes équation fonctionelles, Aequationes math., 68, 260-274, (2004)
[12] Shannon, C.E., A mathematical theory of communication, Bell system tech. J., 27, 379-423, (1948), and 623-656 · Zbl 1154.94303
[13] Székelyhidi, L., 38. problem, Aequationes math., 41, 302, (1991)
[14] Tsallis, C., Possible generalization of boltzmann – gibbs statistics, J. stat phys., 52, 1-2, 479-487, (1988) · Zbl 1082.82501
[15] Ulam, S.M.; Ulam, S.M., Problems in modern mathematics, (1964), Wiley New York · Zbl 0137.24201
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