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Advances in Mathematical Physics
Volume 2011 (2011), Article ID 126108, 12 pages
Characterizations of Generalized Entropy Functions by Functional Equations
Department of Computer Science and System Analysis, College of Humanities and Sciences, Nihon University, 3-25-40, Sakurajyousui, Setagaya-ku, Tokyo 156-8550, Japan
Received 3 March 2011; Revised 22 May 2011; Accepted 23 May 2011
Academic Editor: Giorgio Kaniadakis
Copyright © 2011 Shigeru Furuichi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We will show that a two-parameter extended entropy function is characterized by a functional equation. As a corollary of this result, we obtain that Tsallis entropy function is characterized by a functional equation, which is a different form that used by Suyari and Tsukada, 2009, that is, in a proposition 2.1 in the present paper. We give an interpretation of the functional equation in our main theorem.
Recently, generalized entropies have been studied from the mathematical point of view. The typical generalizations of Shannon entropy  are Rényi entropy  and Tsallis entropy . The recent comprehensive book  and the review  support to understand the Tsallis statistics for the readers. Rényi entropy and Tsallis entropy are defined by
Throughout this paper, we call a parametric extended entropy, such as Rényi entropy and Tsallis entropy, a generalized entropy. If we take in (1.2), we have the so-called binary entropy . Also we take in (1.2), and we have the Shannon's entropy function . In this paper, we treat the entropy function with two parameters. We note that we can produce the relative entropic function by the use of the Shannon's entropy function .
We note that Rényi entropy has the additivity but Tsallis entropy has the nonadditivity where means that and are independent random variables. Therefore, we have a definitive difference for these entropies although we have the simple relation between them where -exponential function is defined if . Note that we have .
Tsallis entropy is rewritten by where -logarithmic function (which is an inverse function of ) is defined by which converges to in the limit .
Since Shannon entropy can be regarded as the expectation value for each value , we may consider that Tsallis entropy can be regarded as the -expectation value for each value , as an analogy to the Shannon entropy, where -expectation value is defined by
However, the -expectation value lacks the fundamental property such as , so that it was considered to be inadequate to adopt as a generalized definition of the usual expectation value. Then the normalized -expectation value was introduced and by using this, the normalized Tsallis entropy was defined by
We easily find that we have the following nonadditivity relation for the normalized Tsallis entropy:
As for the details on the mathematical properties of the normalized Tsallis entropy, see , for example. See also  for the role of Tsallis entropy and the normalized Tsallis entropy in statistical physics. The difference between two non-additivity relations (1.4) and (1.11) is the signature of the coefficient in the third term of the right-hand sides.
We note that Tsallis entropy is also rewritten by so that we may regard it as the expectation value such as where means the usual expectation value . However, if we adopt this formulation in the definition of Tsallis conditional entropy, we do not have an important property such as a chain rule (see  for details). Therefore, we often adopt the formulation using the -expectation value.
As a further generalization, a two-parameter extended entropy was recently introduced in [11, 12] and systematically studied with the generalized exponential function and the generalized logarithmic function . In the present paper, we treat a two-parameter extended entropy defined in the following form:
for two positive numbers and . This form can be obtained by putting and in (1.13), and it coincides with the two-parameter extended entropy studied in . In addition, the two-parameter extended entropy (1.14) was axiomatically characterized in . Furthermore, a two-parameter extended relative entropy was also axiomatically characterized in .
In the paper , a characterization of Tsallis entropy function was proven by using the functional equation. In the present paper, we will show that the two-parameter extended entropy function can be characterized by the simple functional equation.
2. A Review of the Characterization of Tsallis Entropy Function by the Functional Equation
Proposition 2.1 (see ). If the differentiable nonnegative function with positive parameter satisfies the following functional equation: then the function is uniquely given by where is a nonnegative constant depending only on the parameter .
Proof. If we put in (2.1), then we have . From here, we assume that . We also put then we have
Putting in (2.3), we have
Substituting into , we have
By repeating similar substitutions, we have
Then, we have
due to . Differentiating (2.3) by , we have
Putting in the above equation, we have
By integrating (2.3) from to 1 with respect to and performing the conversion of the variables, we have By differentiating the above equation with respect to , we have Taking the limit in the above, we have thanks to (2.7). From (2.9) and (2.12), we have the following differential equation: This differential equation has the following general solution: where is an integral constant depending on . From , we have . Thus, we have Finally, we have From , we have .
If we take the limit as in Proposition 2.1, we have the following corollary.
Corollary 2.2 (see ). If the differentiable nonnegative function satisfies the following functional equation: then the function is uniquely given by where is a nonnegative constant.
3. Main Results
In this section, we give a characterization of a two-parameter extended entropy function by the functional equation. Before we give our main theorem, we review the following result given by Kannappan [18, 19].
Equation (3.3) was used to prove the uniqueness theorem for two-parameter extended entropy in . As for (3.3), a tree-graphical interpretation was given in . The condition (3.1) can be read as the independent case () in (3.3).
If is an entropic function, then it vanishes at 0 or 1, since the entropy has no informational quantity for the deterministic cases, then the above identity is reduced in the following:
In the following theorem, we adopt a simpler condition than (3.1).
Theorem 3.2. If the differentiable nonnegative function with two positive parameters satisfies the following functional equation: then the function is uniquely given by where and are nonnegative constants depending only on the parameters (and ).
Proof. If we put , then we have due to . By differentiating (3.7) with respect to , we have
Putting in (3.9), we have the following differential equation:
where we put . Equation (3.10) can be deformed as follows:
that is, we have
Integrating both sides on the above equation with respect to , we have
where is a integral constant depending on and . Therefore, we have
By , we have . Thus, we have
Also by , we have .
As for the case of , we can prove by the similar way.
Remark 3.3. We can derive (3.6) from our condition (3.7). Firstly, we easily have from our condition equation (3.7). In addition, we have for ,
Thus, we may interpret that our condition (3.7) contains an essential part of the two-parameter generalized Shannon additivity.
Note that we can reproduce the two-parameter entropic function by the use of as with for simplicity. This leads to two-parameter extended relative entropy  See also  on the first appearance of the Tsallis relative entopy (generalized Kullback-Leibler information).
If we take or in Theorem 3.2, we have the following corollary.
Corollary 3.4. If the differentiable nonnegative function with a positive parameter satisfies the following functional equation: then the function is uniquely given by where is a nonnegative constant depending only on the parameter .
Here, we give an interpretation of the functional equation (3.19) from the view of Tsallis statistics.
Remark 3.5. We assume that we have the following two functional equations for , :
These equations lead to the following equations for :
where and . Taking the summation on and in both sides, we have
under the condition . If the function is given by (3.20), then two above functional equations coincide with two nonadditivity relations given in (1.4) and (1.11).
On the other hand, we have the following equation from (23) and (3.21): By a similar way to the proof of Theorem 3.2, we can show that the functional equation (3.25) uniquely determines the function by the form given in (3.20). Therefore, we can conclude that two functional equations (23) and (3.21), which correspond to the non-additivity relations (1.4) and (1.11), also characterize Tsallis entropy function.
If we again take the limit as in Corollary 3.4, we have the following corollary.
Corollary 3.6. If the differentiable nonnegative function satisfies the following functional equation: then the function is uniquely given by where is a nonnegative constant.
As we have seen, the two-parameter extended entropy function can be uniquely determined by a simple functional equation. Also an interpretation related to a tree-graphical structure was given as a remark.
Recently, the extensive behaviours of generalized entropies were studied in [21–23]. Our condition given in (3.7) may be seen as extensive form. However, I have not yet found any relation between our functional (3.7) and the extensive behaviours of the generalized entropies. This problem is not the purpose of the present paper, but it is quite interesting to study this problem as a future work.
This paper is dedicated to Professor Kenjiro Yanagi on his 60th birthday. The author would like to thank the anonymous reviewers for providing valuable comments to improve the paper. The author was partially supported by the Japanese Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists (B) 20740067.
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