zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
On the modified entropy equation. (English) Zbl 1152.39019
The author presents solutions (both general and continuous) of the functional equation $$f(x,y,z)=f(x,y+z,0)+\mu(y+z)f(0,y/(y+z),z/(y+z))$$ for $x,y,z$ in the positive cone of a $k$-dimensional Euclidean space, where $\mu$ is a given multiplicative function on this cone. The statements of the main results, Theorems 2.1 and 3.1, can be enhanced so that their converses are also true. For example, in Theorem 2.1 in the case $\mu$ is a projection one can find in the proof (using $f(x,y,0)=F(x,y)$) that $f(x,y,0)=\mu(x)l(x)+\mu(y)l(y)+\psi_{1}(x+y)$. Substitution into the functional equation (1.1) yields also $\psi_{1}(1)=0$. If one adds these two statements to this case of Theorem 2.1, then the converse is also true. Similar comments apply to the other two cases, which can be combined into a single case. There one can deduce the additional conditions $f(x,y,0)=b\mu(x)+b\mu(y)+\psi_{3}(x+y)$ and $\psi_{3}(1)=-b$, and with these the converse is again true. Note that the terms $-b\mu(y+z)$, $-b\mu(y)$ are missing from the right hand sides of equations (2.8), respectively (2.9).

39B22Functional equations for real functions
94A17Measures of information, entropy
Full Text: EMIS EuDML arXiv