zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Hyperstability of a functional equation. (English) Zbl 1212.39044

The main result of the paper is the following. Let α,ε be fixed, α<0 and ε0. Then, the function f:]0,1[ satisfies the inequality

f(x)+(1-x) α fy 1-x-f(y)-(1-y) α fx 1-yε(1)

for all (x,y)D{(x,y) 2 :x,y,x+y]0,1[} if, and only if, there exist a,b such that

f(x)=ax α +b((1-x) α -1)·(x]0,1[)

This result has the somewhat surprising consequence that the parametric fundamental equation of information (that can be obtained from (1) with ε=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 α-recursive and semi-symmetric information measures is stable.

MSC:
39B82Stability, separation, extension, and related topics
94A17Measures of information, entropy
References:
[1]J. Aczél and Z. Daróczy, On Measures of Information and their Characterization, Academic Press (New York – San Francisco – London, 1975).
[2]Z. Daróczy, Generalized information functions, Information and Control, 16 (1970), 36–51. · Zbl 0205.46901 · doi:10.1016/S0019-9958(70)80040-7
[3]B. R. Ebanks, P. Sahoo and W. Sander, Characterizations of Information Measures, World Scientific Publishing Co., Inc. (River Edge, NJ, 1998).
[4]Z. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math., 50 (1995), 143–190. · Zbl 0836.39007 · doi:10.1007/BF01831117
[5]E. Gselmann, Recent results on the stability of the parametric fundamental equation of information, to appear in Acta Math. Acad. Paedagog. Nyházi.
[6]E. Gselmann and Gy. Maksa, Stability of the parmetric fundamental equation of information for nonpositive parameters, to appear in Aequationes Math.
[7]J. Havrda and F. Charvát, Quantification method of classification processes, concept of structural α-entropy, Kybernetika, 3 (1967), 30–35.
[8]Gy. Maksa, Solution on the open triangle of the generalized fundamental equation of information with four unknown functions, Utilitas Math., 21 (1982), 267–282.
[9]Gy. Maksa, The stability of the entropy of degree alpha, J. Math. Anal. Appl. 346 (2008), 17–21. · Zbl 1149.39024 · doi:10.1016/j.jmaa.2008.05.034
[10]Gy. Maksa and Zs. Páles, Hyperstability of a class of linear functional equations, Acta Math. Acad. Paedagog. Nyházi, 17 (2001), 107–112.
[11]Z. Moszner, Sur les définitions différentes de la stabilité des équations fonctionnelles, Aequationes Math., 68 (2004), 260–274. · Zbl 1060.39031 · doi:10.1007/s00010-004-2749-3
[12]L. Székelyhidi, Problem 38 (in Report of Meeting), Aequationes Math., 41 (1991), 302.
[13]C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. of Statistical Physics, 52 (1988), 479–487. · Zbl 1082.82501 · doi:10.1007/BF01016429