Generalizations of Hölder inequalities for Csiszar’s f-divergence
© Chen and Shi; licensee Springer. 2013
Received: 12 December 2012
Accepted: 21 March 2013
Published: 3 April 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.
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 )
Theorem 1.2 (see )
which is a generalization of Theorem 1.1.
It follows the counterpart of Theorem 1.1.
Theorem 1.3 (see )
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).
Definition 2.1 (see )
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.
Hence, we get the desired inequality. □
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. □
Since , and , we have , by (3.4), we immediately have the inequality (3.12). This completes the proof. □
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.
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.
- Mitrinović DS: Analytic Inequalities. Springer, New York; 1970.View ArticleGoogle Scholar
- Kuang J: Applied Inequalities. Shandong Science Press, Jinan; 2003.Google Scholar
- Hardy G, Littlewood JE, Pólya G: Inequalities. 2nd edition. Cambridge University Press, Cambridge; 1952.Google Scholar
- Yang X: A generalization of Hölder inequality. J. Math. Anal. Appl. 2000, 247: 328–330.MathSciNetView ArticleGoogle Scholar
- Yang X: Refinement of Hölder inequality and application to Ostrowski inequality. Appl. Math. Comput. 2003, 138: 455–461.MathSciNetView ArticleGoogle Scholar
- Yang X: A note on Hölder inequality. Appl. Math. Comput. 2003, 134: 319–322.MathSciNetView ArticleGoogle Scholar
- Yang X: Hölder’s inequality. Appl. Math. Lett. 2003, 16: 897–903.MathSciNetView ArticleGoogle Scholar
- Wu S, Debnath L: Generalizations of Aczél’s inequality and Popoviciu’s inequality. Indian J. Pure Appl. Math. 2005, 36(2):49–62.MathSciNetGoogle Scholar
- He WS: Generalization of a sharp Hölder’s inequality and its application. J. Math. Anal. Appl. 2007, 332: 741–750.MathSciNetView ArticleGoogle Scholar
- Wu S: A new sharpened and generalized version of Hölder’s inequality and its applications. Appl. Math. Comput. 2008, 197: 708–714.MathSciNetView ArticleGoogle Scholar
- Kwon EG, Bae EK: On a continuous form of Hölder inequality. J. Math. Anal. Appl. 2008, 343: 585–592.MathSciNetView ArticleGoogle Scholar
- Anastassiou GA: Hölder-like Csiszar’s type inequalities. Int. J. Pure Appl. Math. 2004, 1: 9–14. www.gbspublisher.comGoogle Scholar
- Csiszar I: Information-type measures of difference of probability distributions and indirect observations. Studia Sci. Math. Hung. 1967, 2: 299–318.MathSciNetGoogle Scholar
- Anastassiou GA, et al.: Basic optimal approximation of Csiszar’s f -divergence. In Proceedings of 11th Internat. Conf. Approx. Th Edited by: Chui CK. 2004, 15–23.Google Scholar
- Anastassiou GA: Fractional and other approximation of Csiszar’s f -divergence. Rend. Circ. Mat. Palermo Suppl. 2005, 76: 197–212.MathSciNetGoogle Scholar
- Anastassiou GA: Representations and estimates to Csiszar’s f -divergence. Panam. Math. J. 2006, 16: 83–106.MathSciNetGoogle Scholar
- Anastassiou GA: Higher order optimal approximation of Csiszar’s f -divergence. Nonlinear Anal. 2005, 61: 309–339.MathSciNetView ArticleGoogle Scholar
- Csiszar I: Eine Informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten. Magy. Tud. Akad. Mat. Kut. Intéz. Közl. 1963, 8: 85–108.MathSciNetGoogle Scholar
- Csiszar I: On topological properties of f -divergences. Studia Sci. Math. Hung. 1967, 2: 329–339.MathSciNetGoogle Scholar
- Dragomir SS: Inequalities for Csiszar f-Divergence in Information Theory. Victoria University, Melbourne; 2000. Edited monograph. On line: http://rgmia.vu.edu.auGoogle Scholar
- Anwar M, Hussain S, Pečarić J: Some inequalities for Csiszár-divergence measures. Int. J. Math. Anal. 2009, 3(26):1295–1304.Google Scholar
- Kafka P, Österreicher F, Vincze I: On powers of f -divergences defining a distance. Studia Sci. Math. Hung. 1991, 26(4):415–422.Google Scholar
- Cheung W-S: Genegralizations of Hölder’s inequality. Int. J. Math. Math. Sci. 2001, 26(1):7–10.MathSciNetView ArticleGoogle Scholar
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