Generalizations of Hölder inequalities for Csiszar’s f-divergence
Journal of Inequalities and Applications volume 2013, Article number: 151 (2013)
In this paper, we establish some new generalizations of the Hölder’s inequality involving Csiszar’s f-divergence of two probability measures. Some related inequalities are also presented.
MSC:26D15, 28A25, 60E15.
Let , assume that and are continuous real-valued functions on . Then
for , we have the following Hölder inequality (see ):(1.1)
for , we have the following reverse Hölder inequality (see ):(1.2)
The above inequalities play an important role in many areas of pure and applied mathematics. A large number of generalizations, refinements, variations and applications of (1.1) and (1.2) have been investigated in the literature (see [3–11] and references therein). Recently, G.A. Anastassiou  established some Hölder’s type inequalities regarding Csiszar’s f-divergence of two probability measures as follows.
Theorem 1.1 (see )
Let such that . Then
Theorem 1.2 (see )
Let , , . Then
which is a generalization of Theorem 1.1.
It follows the counterpart of Theorem 1.1.
Theorem 1.3 (see )
Let and such that , we assume that a.e. . Then we have
The aim of this paper is to give new generalizations of inequalities (1.4) and (1.5). Some related inequalities are also considered. The paper is organized as follows. In Section 2, we recall some basic facts about the Csiszar’s f-divergence of two probability measures. In Section 3, we will give the main result and its proof.
Suppose that is an arbitrary measure space with λ being a finite or σ-finite measure. Let , be probability measures on X such that (absolutely continuous).
The Radon-Nikodym derivatives (densities) of with respect to λ is expressed by :
Definition 2.1 (see )
The f-divergence of the probability measures and is defined as follows:
where the function f is named the base function. From Lemma 1.1 of , is always well-defined and with equality only for . From , we know that does not depend on the choice of λ. If , then can be considered as the most general measure of difference between probability measures. For arbitrary convex function f, we notice that .
The Csiszar’s f-divergence incorporated most of special cases of probability measure distances, including the variation distance, -divergence, information for discrimination or generalized entropy, information gain, mutual information, mean square contingency, etc. has many applications to almost all applied sciences where stochastics enters. For more references, one can see [12–22].
In this paper, we assume that the base function f appearing in the function have all the above properties of f.
3 Main results
In the section, we establish some new generalizations of the Hölder inequality involving Csiszar’s f-divergence of two probability measures.
Theorem 3.1 Let , (), , . Then
Proof Here, we use the generalized Hölder’s inequality (see ). We obtain
Hence, we get the desired inequality. □
Theorem 3.2 Let (, ), , . Then
for , we have the following inequality:(3.3)
for , (), we have the following reverse inequality:(3.4)
Applying the assumptions and , we have
Then we find
By the inequality (1.4), we obtain
In view of (3.5), we have
By (3.6), (3.7) and (3.8), we obtain inequality (3.3).
Similar to the proof of inequality (3.3), by (3.5), (3.6), (3.8) and the inequality (3.1), we have inequality (3.4) immediately. □
Corrollary 3.1 Under the assumptions of Theorem 3.2, taking , for and with , then we have
for , we have the following inequality:(3.9)
for , (), we have the following reverse inequality:(3.10)
Theorem 3.3 Let (, ), , . Then
for , we have the following inequality:(3.11)
for , (), we have the following reverse inequality:(3.12)
Proof (1) Since and , we get . Then by (3.3), we immediately obtain the inequality (3.11).
Since , and , we have , by (3.4), we immediately have the inequality (3.12). This completes the proof. □
Theorem 3.4 Under the assumptions of Theorem 3.3, and let , , , , then
for , we have the following inequality:(3.13)
for , we have the following reverse inequality:(3.14)
Proof (1) By inequality (1.3), we get
Similar to the proof of inequality (3.13), by inequality (1.5), we obtain inequality (3.14). □
Remark Assume that X is a finite or countable discrete set, A is its power set and λ has mass 1 for each , then becomes a finite or infinite sum, respectively. As a consequence, all the above obtained integral inequalities are discretized and become summation inequalities.
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Dedicated to Professor Hari M Srivastava.
The authors thank the editor and the referees for their valuable suggestions to improve the quality of this paper.
The authors declare that they have no competing interests.
All the authors contributed to the writing of the present article. They also read and approved the final manuscript.
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Chen, GS., Shi, XJ. Generalizations of Hölder inequalities for Csiszar’s f-divergence. J Inequal Appl 2013, 151 (2013). https://doi.org/10.1186/1029-242X-2013-151