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Information 2013, 4(2), 198-239; doi:10.3390/info4020198

Article
Seven Means, Generalized Triangular Discrimination, and Generating Divergence Measures
Inder Jeet Taneja
Department of Mathematics, Federal University of Santa Catarina, Florianópolis, SC 88040-900, Brazil; E-Mail: ijtaneja@gmail.com; Tel.: +55-48-8831-6616
Retired. Current Address: R. Desembargador Vitor Lima, 354, Apt. 402, Bl. A, Carvoeira, Florianópolis, SC 88040-400, Brazil.
Received: 4 January 2013; in revised form: 17 February 2013 / Accepted: 28 March 2013 /
Published: 23 April 2013

Abstract

: Jensen-Shannon, J-divergence and Arithmetic-Geometric mean divergences are three classical divergence measures known in the information theory and statistics literature. These three divergence measures bear interesting inequality among the three non-logarithmic measures known as triangular discrimination, Hellingar’s divergence and symmetric chi-square divergence. However, in 2003, Eve studied seven means from a geometrical point of view, which are Harmonic, Geometric, Arithmetic, Heronian, Contra-harmonic, Root-mean square and Centroidal. In this paper, we have obtained new inequalities among non-negative differences arising from these seven means. Correlations with generalized triangular discrimination and some new generating measures with their exponential representations are also presented.
Keywords:
arithmetic mean; geometric mean; Heronian mean; Hellinger’s distance; triangular discrimination; information inequalities

1. Introduction

Let

Information 04 00198 i001, Information 04 00198 i002,

be the set of all complete finite discrete probability distributions. For all Information 04 00198 i003, let us consider two groups of measures:

• Logarithmic Measures

Information 04 00198 i004,

Information 04 00198 i005,

Information 04 00198 i006.

The above three measures are classical divergence measures in the literature on information theory and statistics known as Jensen-Shannon divergence, J-divergence and Arithmetic-Geometric mean divergence respectively. These three measures bear the following two relations:

(i)

Information 04 00198 i007;

(ii)

Information 04 00198 i008.

• Non-logarithmic Measures

Information 04 00198 i009,

Information 04 00198 i010

and

Information 04 00198 i011.

The above three measures Information 04 00198 i012, Information 04 00198 i013 and Information 04 00198 i014 are respectively known as triangular discrimination, Hellingar’s divergence and symmetric chi-square divergence. These measures allow the following inequalities among the measures.

Information 04 00198 i015.

If we consider all the six measures, they satisfy [1] the following inequalities:

Information 04 00198 i016 Information 04 00198 i017.

• Generalized Symmetric Divergence Measures

Let us consider the measure

Information 04 00198 i018

for all Information 04 00198 i003. The measure Information 04 00198 i019 is generalized J-divergence extensively studied in Taneja [2,3]. It admits the following particular cases:

(i)

Information 04 00198 i020;

(ii)

Information 04 00198 i021;

(iii)

Information 04 00198 i022.

Again consider another generalized measure

Information 04 00198 i023 Information 04 00198 i024

for all Information 04 00198 i003. The measure Information 04 00198 i025 known as generalized arithmetic and geometric mean divergence. It also admits the following particular cases:

(i)

Information 04 00198 i026;

(ii)

Information 04 00198 i027;

(iii)

Information 04 00198 i028;

(iv)

Information 04 00198 i029.

We observe that the six measures given in above inequality appear as particular cases of the above two generalized measures. These two generalizations are mainly the generalizations of the logarithmic measures Information 04 00198 i030, Information 04 00198 i031 and Information 04 00198 i032. The non-negativity of the arithmetic-geometric mean divergence, Information 04 00198 i033 is based on the well-known arithmetic and geometric means, i.e., we can write it as

Information 04 00198 i034,

where Information 04 00198 i035 and Information 04 00198 i036 are arithmetic and geometric means respectively. Moreover, the measure Information 04 00198 i037 can also be written in terms of arithmetic mean

Information 04 00198 i038.

On the other side, these means plays important roles, being applied in different areas, especially in information theory and statistics. Eve [4] studied some interesting geometrical interpretation of some means, famous as Eve’s seven means.

Our aim here is to present generalizations of non-logarithmic measures, starting from triangular discrimination. Also connections Eve’s seven means with the non-logarithmic measures are given. We performed this through inequalities, where some new generalized means are also presented.

2. Seven Means

Let Information 04 00198 i039 be two positive numbers. Eves [4] studied the geometrical interpretation of the following seven means:

  • Arithmetic mean: Information 04 00198 i040;

  • Geometric mean: Information 04 00198 i041;

  • Harmonic mean: Information 04 00198 i042;

  • Heronian mean: Information 04 00198 i043;

  • Contra-harmonic mean: Information 04 00198 i044;

  • Root-mean-square: Information 04 00198 i045;

  • Centroidal mean: Information 04 00198 i046.

We can easily verify that the following inequality having the above seven means:

Information 04 00198 i047

Let us write, Information 04 00198 i048, where Information 04 00198 i049stands for any of the above seven means, then we have

Information 04 00198 i050
where Information 04 00198 i052 Information 04 00198 i053 Information 04 00198 i054 Information 04 00198 i055 Information 04 00198 i056, Information 04 00198 i057 and Information 04 00198 i058, Information 04 00198 i059, Information 04 00198 i060. In all cases, we would have equality sign if Information 04 00198 i061, i.e., Information 04 00198 i062.

As Information 04 00198 i037 and Information 04 00198 i033, the means Information 04 00198 i063, Information 04 00198 i064, Information 04 00198 i065, Information 04 00198 i066 and Information 04 00198 i067 may also be written in terms of the means Information 04 00198 i035 and Information 04 00198 i036.

2.1. Inequalities among Differences of Means

For simplicity, let us write

Information 04 00198 i068
where Information 04 00198 i069, with Information 04 00198 i070. Thus, according to (3), the inequality (1) admits 21 non-negative differences. These differences satisfy some simple inequalities given by the following pyramid:
Information 04 00198 i071
where, for example, Information 04 00198 i077, Information 04 00198 i078, etc. After simplifications, we have the following equalities among some of these measures:
(i)

Information 04 00198 i079;

(ii)

Information 04 00198 i080;

(iii)

Information 04 00198 i081.

The measures Information 04 00198 i082 and Information 04 00198 i083 are the well know triangular discsrimination [5] and Hellinger’s distance [6] given by Information 04 00198 i084 and Information 04 00198 i085 respectively. Not all the measures appearing in the above pyramid (4) are convex in the pair Information 04 00198 i086. Recently, the author [7] has proved the following theorem for the convex measures.

Theorem 2.1. The following inequalities hold:

Information 04 00198 i087

The proof of the above theorem is based on the following two lemmas [8,9].

Lemma 2.1. Let Information 04 00198 i088 be a convex and differentiable function satisfying Information 04 00198 i089. Consider a function

Information 04 00198 i090, Information 04 00198 i091,

then the function Information 04 00198 i092 is convex in Information 04 00198 i093. Additionally, if Information 04 00198 i094, then the following inequality hold:

Information 04 00198 i095.

Lemma 2.2. Let Information 04 00198 i096 be two convex functions satisfying the assumptions:

(i)

Information 04 00198 i097, Information 04 00198 i098;

(ii)

Information 04 00198 i099 and Information 04 00198 i100 are twice differentiable in Information 04 00198 i101;

(iii)

there exists the real constants Information 04 00198 i102 such that Information 04 00198 i103 and

Information 04 00198 i104, Information 04 00198 i105,

for all Information 04 00198 i106 then we have the inequalities:

Information 04 00198 i107,

for all Information 04 00198 i108, where the function Information 04 00198 i109 is as defined in Lemma 2.1.

2.2. Generalized Triangular Discrimination

For all Information 04 00198 i039 consider the following measure generalizing triangular discrimination

Information 04 00198 i110

In particular, we have

Information 04 00198 i111,

Information 04 00198 i112,

Information 04 00198 i113,

Information 04 00198 i114

and

Information 04 00198 i115

From above, we observe that the expression (6) contains some well-known measures such as Information 04 00198 i116 (reference Jain and Srivastava [10]), Information 04 00198 i117 (reference Kumar and Johnson [11]) and Information 04 00198 i118, the latter being symmetric to Information 04 00198 i119measure [12]. Information 04 00198 i120 will be considered here for the first time. The generalization (6) considered above is little different from the one considered by Topsoe [13]:

Information 04 00198 i121.

Furthermore, there are more particular cases as known measures.

Convexity: Let us prove now the convexity of the measure (6). We can write Information 04 00198 i122, Information 04 00198 i123, where

Information 04 00198 i124.

The second order derivative of the function Information 04 00198 i125 is given by

Information 04 00198 i126,

where

Information 04 00198 i127

From (7) we observe that we are unable to find a unique value of Information 04 00198 i129 when the function is positive. But for at least Information 04 00198 i130, Information 04 00198 i106, Information 04 00198 i060, we have Information 04 00198 i131. Also, we have Information 04 00198 i132. Thus according to Lemma 1.1, the measure Information 04 00198 i133 is convex for all Information 04 00198 i086, Information 04 00198 i134. Testing individually to fix Information 04 00198 i135, we can check the convexity for other measures as well, for example Information 04 00198 i136, Information 04 00198 i137is convex.

Monotonicity: Calculating the first order derivative of the function Information 04 00198 i125 with respect to Information 04 00198 i129, we have

Information 04 00198 i138.

We can easily check that for all Information 04 00198 i139, Information 04 00198 i140. This proves that the function Information 04 00198 i125 is monotonically increasing with respect to Information 04 00198 i129. This gives

Information 04 00198 i141

Also we know that Information 04 00198 i142 and Information 04 00198 i143. Thus combining Equations (5) and (8), we have

Information 04 00198 i144

As a part of (9), let us consider the following inequalities:

Information 04 00198 i145

Our purpose is to study further inequalities by considering possible nonnegative differences from (10).

3. New Inequalities

In this section we will bring inequalities in different stages. In the first stage the measures considered are the nonnegative differences arising from (10). This will be done many times until one final measure.

3.1. First Stage

For simplicity, let us write the expression (10) as

Information 04 00198 i146
where for example Information 04 00198 i147, Information 04 00198 i148, Information 04 00198 i149, etc. We can write
Information 04 00198 i150
where
  • Information 04 00198 i152,

  • Information 04 00198 i153,

  • Information 04 00198 i154,

  • Information 04 00198 i155,

  • Information 04 00198 i156,

  • Information 04 00198 i157,

  • Information 04 00198 i158,

  • Information 04 00198 i159,

    and

  • Information 04 00198 i160.

Calculating the second order derivative of above functions we have

  • Information 04 00198 i161,

  • Information 04 00198 i162,

  • Information 04 00198 i163,

  • Information 04 00198 i164,

  • Information 04 00198 i165,

  • Information 04 00198 i166;

  • Information 04 00198 i167,

  • Information 04 00198 i168,

    and

  • Information 04 00198 i169.

The Inequalities (11) again admit 45 nonnegative differences. These differences satisfy some natural inequalities given in a pyramid below:

Information 04 00198 i170
where Information 04 00198 i178, Information 04 00198 i179, etc. After simplifications, we have equalities among first four lines of the pyramid:
Information 04 00198 i180

In the view of above equalities, we are left only with 27 nonnegative convex measures and these are connected with each other by the inequalities given in the theorem below.

Theorem 3.1. The following sequences of inequalities hold:

Information 04 00198 i182

Proof. We will prove the above theorem by parts.

1. For Information 04 00198 i186: We shall apply two approaches to prove this result.

1st Approach: Let us consider a function

Information 04 00198 i187 Information 04 00198 i188,

After simplifications, we have

Information 04 00198 i189, Information 04 00198 i190 Information 04 00198 i191

and

Information 04 00198 i192.

By the application Lemma 2.2 we get the required result.

2nd Approach: We shall use an alternative approach to prove the above result. We know that Information 04 00198 i193. In order to prove the result, we need to show that Information 04 00198 i194. By considering the difference Information 04 00198 i195, we have

Information 04 00198 i196 Information 04 00198 i197 Information 04 00198 i198,

where

Information 04 00198 i199

Since Information 04 00198 i201, we get the required result.

For simplicity, from now onward we shall use only the second approach.

2. For Information 04 00198 i202: Let us consider a function Information 04 00198 i203. After simplifications, we have

Information 04 00198 i204, Information 04 00198 i205

and

Information 04 00198 i206 Information 04 00198 i207 Information 04 00198 i208

3. For Information 04 00198 i209: Let us consider a function Information 04 00198 i210. After simplifications, we have

Information 04 00198 i211, Information 04 00198 i212

and

Information 04 00198 i213 Information 04 00198 i214 Information 04 00198 i215.

4. For Information 04 00198 i216: Let us consider a function Information 04 00198 i217. After simplifications, we have

Information 04 00198 i218, Information 04 00198 i219

and

Information 04 00198 i220 Information 04 00198 i221 Information 04 00198 i222.

5. For Information 04 00198 i223: Let us consider a function Information 04 00198 i224. After simplifications, we have

Information 04 00198 i225, Information 04 00198 i226

and

Information 04 00198 i227 Information 04 00198 i228 Information 04 00198 i229.

6. For Information 04 00198 i230: Let us consider a function Information 04 00198 i231. After simplifications, we have

Information 04 00198 i232, Information 04 00198 i233

and

Information 04 00198 i234 Information 04 00198 i235 Information 04 00198 i236,

where

Information 04 00198 i237

7. For Information 04 00198 i238: Let us consider a function Information 04 00198 i239. After simplifications, we have

Information 04 00198 i240, Information 04 00198 i241

and

Information 04 00198 i242 Information 04 00198 i243 Information 04 00198 i244,

where

Information 04 00198 i245

8. For Information 04 00198 i247: Let us consider a function Information 04 00198 i248. After simplifications, we have

Information 04 00198 i249, Information 04 00198 i250

and

Information 04 00198 i251 Information 04 00198 i252.

9. For Information 04 00198 i253: Let us consider a function Information 04 00198 i254. After simplifications, we have

Information 04 00198 i255, Information 04 00198 i256

and

Information 04 00198 i257 Information 04 00198 i258 Information 04 00198 i259.

10. For Information 04 00198 i260: Let us consider a function Information 04 00198 i261. After simplifications, we have

Information 04 00198 i262, Information 04 00198 i263

and

Information 04 00198 i264 Information 04 00198 i265.

11. For Information 04 00198 i266: Let us consider a function Information 04 00198 i267. After simplifications, we have

Information 04 00198 i268, Information 04 00198 i269

and

Information 04 00198 i270 Information 04 00198 i271,

where

Information 04 00198 i272

12. For Information 04 00198 i274: Let us consider a function Information 04 00198 i275. After simplifications, we have

Information 04 00198 i276, Information 04 00198 i277

and

Information 04 00198 i278 Information 04 00198 i279 Information 04 00198 i280,

where

Information 04 00198 i281

13. For Information 04 00198 i282: Let us consider a function Information 04 00198 i283. After simplifications, we have

Information 04 00198 i284, Information 04 00198 i285

and

Information 04 00198 i286 Information 04 00198 i287,

where

Information 04 00198 i288

14. For Information 04 00198 i289: Let us consider a function Information 04 00198 i290. After simplifications, we have

Information 04 00198 i291, Information 04 00198 i292

and

Information 04 00198 i293 Information 04 00198 i294.

15. For Information 04 00198 i295: Let us consider a function Information 04 00198 i296. After simplifications, we have

Information 04 00198 i297, Information 04 00198 i298

and

Information 04 00198 i299 Information 04 00198 i300.

16. For Information 04 00198 i301: Let us consider a function Information 04 00198 i302. After simplifications, we have

Information 04 00198 i303, Information 04 00198 i304

and

Information 04 00198 i305 Information 04 00198 i306.

17. For Information 04 00198 i307: Let us consider a function Information 04 00198 i308. After simplifications, we have

Information 04 00198 i309, Information 04 00198 i310

and

Information 04 00198 i311 Information 04 00198 i312,

where

Information 04 00198 i313

18. For Information 04 00198 i314: Let us consider a function Information 04 00198 i315. After simplifications, we have

Information 04 00198 i316, Information 04 00198 i317

and

Information 04 00198 i318 Information 04 00198 i319,

where

Information 04 00198 i320

19. For Information 04 00198 i321: Let us consider a function Information 04 00198 i322. After simplifications, we have

Information 04 00198 i323, Information 04 00198 i324

and

Information 04 00198 i325 Information 04 00198 i326,

where

Information 04 00198 i327

20. For Information 04 00198 i328: Let us consider a function Information 04 00198 i329. After simplifications, we have

Information 04 00198 i330, Information 04 00198 i331

and

Information 04 00198 i332 Information 04 00198 i333,

where

Information 04 00198 i334

21. For Information 04 00198 i335: Let us consider a function Information 04 00198 i336. After simplifications, we have

Information 04 00198 i337, Information 04 00198 i338

and

Information 04 00198 i339 Information 04 00198 i340.

22. For Information 04 00198 i341: Let us consider a function Information 04 00198 i342. After simplifications, we have

Information 04 00198 i343, Information 04 00198 i344

and

Information 04 00198 i345 Information 04 00198 i346.

23. For Information 04 00198 i347: Let us consider a function Information 04 00198 i348. After simplifications, we have

Information 04 00198 i349, Information 04 00198 i350

and

Information 04 00198 i351 Information 04 00198 i352.

24. For Information 04 00198 i353: Let us consider a function Information 04 00198 i354. After simplifications, we have

Information 04 00198 i355, Information 04 00198 i356

and

Information 04 00198 i357 Information 04 00198 i358,

where

Information 04 00198 i359

25. For Information 04 00198 i360: Let us consider a function Information 04 00198 i361. After simplifications, we have

Information 04 00198 i362, Information 04 00198 i363

and

Information 04 00198 i364 Information 04 00198 i365,

where

Information 04 00198 i366

26. For Information 04 00198 i367: Let us consider a function Information 04 00198 i368. After simplifications, we have

Information 04 00198 i369, Information 04 00198 i370

and

Information 04 00198 i371 Information 04 00198 i372,

with

Information 04 00198 i373

27. For Information 04 00198 i374: Let us consider a function Information 04 00198 i375. After simplifications, we have

Information 04 00198 i376, Information 04 00198 i377

and

Information 04 00198 i378 Information 04 00198 i379,

where

Information 04 00198 i380

Combining the results 1–27, we get the proof of (15).

Remarks. Based on the equalities given in (14), we have the following proportionality relations:

(i)

Information 04 00198 i381;

(ii)

Information 04 00198 i382;

(iii)

Information 04 00198 i383;

(iv)

Information 04 00198 i384;

(v)

Information 04 00198 i385;

(vi)

Information 04 00198 i386;

(vii)

Information 04 00198 i387;

(viii)

Information 04 00198 i388;

(ix)

Information 04 00198 i389;

(x)

Information 04 00198 i390;

(xi)

Information 04 00198 i391;

(xii)

Information 04 00198 i392.

3.1.1. Reverse Inequalities

We observe from the above results that the first four inequalities appearing in pyramid are equal with some multiplicative constants. The other four inequalities satisfies reverse inequalities given by

(i)

Information 04 00198 i393

Information 04 00198 i394

(ii)

Information 04 00198 i395

Information 04 00198 i396

(iii)

Information 04 00198 i397

Information 04 00198 i398;

(iv)

Information 04 00198 i399

Information 04 00198 i400.

3.2. Second Stage

In this stage we shall bring inequalities based on measures arising from the stage. The above 27 parts generate some new measures given by

Information 04 00198 i401
where Information 04 00198 i403, Information 04 00198 i402 are given by (16)–(29) respectively. In all the cases we have Information 04 00198 i404, Information 04 00198 i402. By the application of Lemma 2.1, we can say that the above 14 measures are convex. We may try to connect 14 measures given in (30) through inequalities.

Theorem 3.2. The following inequalities hold:

Information 04 00198 i405
and
Information 04 00198 i406

Proof. We shall prove the above theorem following similar lines to Theorem 3.1. Since we need the second derivatives of the functions given by (16)–(29) to prove the theorem, their values are as follows:

  • Information 04 00198 i407,

  • Information 04 00198 i408,

  • Information 04 00198 i409,

  • Information 04 00198 i410,

  • Information 04 00198 i411,

  • Information 04 00198 i412,

  • Information 04 00198 i413,

  • Information 04 00198 i414,

  • Information 04 00198 i415

  • Information 04 00198 i416,

  • Information 04 00198 i417,

  • Information 04 00198 i418,

  • Information 04 00198 i419

    and

  • Information 04 00198 i420.

We will prove the above theorem by parts. In view of procedure used in Theorem 3.1, we shall write the proof of each part in summarized way.

1. For Information 04 00198 i421: Let us consider a function Information 04 00198 i422. After simplifications, we have

Information 04 00198 i423, Information 04 00198 i424

and

Information 04 00198 i425,

where

Information 04 00198 i426

2. For Information 04 00198 i427: Let us consider a function Information 04 00198 i428. After simplifications, we have

Information 04 00198 i429, Information 04 00198 i430

and

Information 04 00198 i431.

3. For Information 04 00198 i432: Let us consider a function Information 04 00198 i433. After simplifications, we have

Information 04 00198 i434, Information 04 00198 i435

and

Information 04 00198 i436,

where

Information 04 00198 i437

4. For Information 04 00198 i438: Let us consider a function Information 04 00198 i439. After simplifications, we have

Information 04 00198 i440, Information 04 00198 i441

and

Information 04 00198 i442,

where

Information 04 00198 i443

5. For Information 04 00198 i444: Let us consider a function Information 04 00198 i445. After simplifications, we have

Information 04 00198 i446.

This gives Information 04 00198 i447. Let us consider now, and

Information 04 00198 i448,

where

Information 04 00198 i449

6. For Information 04 00198 i450 : Let us consider a function Information 04 00198 i451. After simplifications, we have

Information 04 00198 i452, Information 04 00198 i453

and

Information 04 00198 i454,

where

Information 04 00198 i455

7. For Information 04 00198 i456: Let us consider a function Information 04 00198 i457. After simplifications, we have

Information 04 00198 i458, Information 04 00198 i459

and

Information 04 00198 i460,

where

Information 04 00198 i461

8. For Information 04 00198 i462: Let us consider a function Information 04 00198 i463. After simplifications, we have

Information 04 00198 i464, Information 04 00198 i465

and

Information 04 00198 i466.

9. For Information 04 00198 i467: Let us consider a function Information 04 00198 i468. After simplifications, we have

Information 04 00198 i469, Information 04 00198 i470

and

Information 04 00198 i471,

where

Information 04 00198 i472

10. For Information 04 00198 i473: Let us consider a function Information 04 00198 i474. After simplifications, we have

Information 04 00198 i475, Information 04 00198 i476

and

Information 04 00198 i477,

where

Information 04 00198 i478

11. For Information 04 00198 i479: Let us consider a function Information 04 00198 i480. After simplifications, we have

Information 04 00198 i481, Information 04 00198 i482

and

Information 04 00198 i483,

where

Information 04 00198 i484

12. For Information 04 00198 i485: Let us consider a function Information 04 00198 i486. After simplifications, we have

Information 04 00198 i487, Information 04 00198 i488

and

Information 04 00198 i489,

where

Information 04 00198 i490

13. For Information 04 00198 i491: Let us consider a function Information 04 00198 i492. After simplifications, we have

Information 04 00198 i493, Information 04 00198 i494

and

Information 04 00198 i495,

where

Information 04 00198 i496

14. For Information 04 00198 i497: Let us consider a function Information 04 00198 i498. After simplifications, we have

Information 04 00198 i499, Information 04 00198 i500

and

Information 04 00198 i501.

3.3. Third Stage

The proof of above 14 parts gives us some new measures. These are given by

Information 04 00198 i502
where the functions Information 04 00198 i504, Information 04 00198 i503 are given by (33)–(43) respectively. In all the cases, we have Information 04 00198 i505, Information 04 00198 i503. By the application of Lemma 2.1, we can say that the above 11 measures are convex. Here follows the second derivatives of the functions (33)–(43), applied frequently in the next theorem.
  • Information 04 00198 i506,

  • Information 04 00198 i507,

  • Information 04 00198 i508,

  • Information 04 00198 i509,

  • Information 04 00198 i510,

  • Information 04 00198 i511,

  • Information 04 00198 i512,

  • Information 04 00198 i513,

  • Information 04 00198 i514,

  • Information 04 00198 i515

    and

  • Information 04 00198 i516.

The theorem below connects only the first nine measures. The other two will be given later.

Theorem 3.3. The following inequalities hold:

Information 04 00198 i517

Proof. We will prove the inequalities (45) by parts and shall use the same approach applied in the above theorems. Without specifying, we will frequently use the second derivatives Information 04 00198 i518, Information 04 00198 i503.

1. For Information 04 00198 i519: Let us consider a function Information 04 00198 i520. After simplifications, we have

Information 04 00198 i521, Information 04 00198 i522

and

Information 04 00198 i523.

2. For Information 04 00198 i524: Let us consider a function Information 04 00198 i525. After simplifications, we have

Information 04 00198 i526, Information 04 00198 i527

and

Information 04 00198 i528.

3. For Information 04 00198 i529: Let us consider a function Information 04 00198 i530. After simplifications, we have

Information 04 00198 i531, Information 04 00198 i532

and

Information 04 00198 i533,

where

Information 04 00198 i534

4. For Information 04 00198 i535: Let us consider a function Information 04 00198 i536. After simplifications, we have

Information 04 00198 i537, Information 04 00198 i538

and

Information 04 00198 i539.

5. For Information 04 00198 i540: Let us consider a function Information 04 00198 i541. After simplifications, we have

Information 04 00198 i542,

and

Information 04 00198 i543,

where

Information 04 00198 i544

6. For Information 04 00198 i545: Let us consider a function Information 04 00198 i546. After simplifications, we have

Information 04 00198 i547, Information 04 00198 i548

and

Information 04 00198 i549,

where

Information 04 00198 i550

7. For Information 04 00198 i551: Let us consider a function Information 04 00198 i552. After simplifications, we have

Information 04 00198 i553, Information 04 00198 i554

and

Information 04 00198 i555.

8. For Information 04 00198 i556: Let us consider a function Information 04 00198 i557. After simplifications, we have

Information 04 00198 i558, Information 04 00198 i559

and

Information 04 00198 i560.

Combining the parts 1–8, we get the proof of the Inequalities (45).

3.4. Forth Stage

Still, we have more measures to compare, i.e., Information 04 00198 i561 to Information 04 00198 i562. This comparison is given in the theorem below. Here below are the second derivatives of the functions given by (46)–(58).

Information 04 00198 i563,

Information 04 00198 i564,

and

Information 04 00198 i565.

Theorem 3.4. The following inequalities hold:

Information 04 00198 i566

Proof. We shall prove the above theorem by parts.

1. For Information 04 00198 i567: Let us consider a function Information 04 00198 i568. After simplifications, we have

Information 04 00198 i569, Information 04 00198 i570

and

Information 04 00198 i571,

where

Information 04 00198 i572

2. For Information 04 00198 i573: Let us consider a function Information 04 00198 i574. After simplifications, we have

Information 04 00198 i575, Information 04 00198 i576

and

Information 04 00198 i577.

3. For Information 04 00198 i578: Let us consider a function Information 04 00198 i579. After simplifications, we have

Information 04 00198 i580, Information 04 00198 i581

and

Information 04 00198 i582.

4. For Information 04 00198 i583: Let us consider a function Information 04 00198 i584. After simplifications, we have

Information 04 00198 i585, Information 04 00198 i586

and

Information 04 00198 i587.

Remark: Interestingly, in all the four cases only with a single measure is left, i.e., Information 04 00198 i588 given by

Information 04 00198 i590

3.5. Equivalent Expressions

The measures appearing in the proof of Theorems 3.2–3.4 can be written in terms of the measures appearing in the Inequalities (9). Here follow equivalent versions of these measures.

  • Measures appearing in Theorem 3.2. We can write

    • Information 04 00198 i591

      Information 04 00198 i592,

    • Information 04 00198 i593,

    • Information 04 00198 i594

      Information 04 00198 i595

    • Information 04 00198 i596,

    • Information 04 00198 i597,

    • Information 04 00198 i598

      Information 04 00198 i599

    • Information 04 00198 i600,

    • Information 04 00198 i601,

    • Information 04 00198 i602,

    • Information 04 00198 i603

      Information 04 00198 i604

    • Information 04 00198 i605,

    • Information 04 00198 i606,

    • Information 04 00198 i607,

    • Information 04 00198 i608.

  • Measures appearing in Theorems 3.3 and 3.4. We could write

    • Information 04 00198 i609,

    • Information 04 00198 i610,

    • Information 04 00198 i611,

    • Information 04 00198 i612,

    • Information 04 00198 i613,

    • Information 04 00198 i614,

    • Information 04 00198 i615,

    • Information 04 00198 i616,

    • Information 04 00198 i617,

    • Information 04 00198 i618,

    • Information 04 00198 i619,

    • Information 04 00198 i620,

    • Information 04 00198 i621,

    • Information 04 00198 i622,

    • Information 04 00198 i623.

4. Generating Divergence Measures and Exponential Representations

Some of the measures given in Section 2 can be written in generating forms. Below are the generating measures.

4.1. First Generalization of Triangular Discrimination

For all Information 04 00198 i086, let us consider the following measures

Information 04 00198 i624

In particular, we have

Information 04 00198 i625,

Information 04 00198 i626,

Information 04 00198 i627

and

Information 04 00198 i628 Information 04 00198 i629.

The Expression (52) gives first generalization of the measure Information 04 00198 i082. Now we will prove its convexity. We can write Information 04 00198 i630, Information 04 00198 i135, where

Information 04 00198 i631

The second order derivative of the function Information 04 00198 i632 is given by

Information 04 00198 i633,

where

Information 04 00198 i634.

For all Information 04 00198 i635, Information 04 00198 i106, Information 04 00198 i060, we have Information 04 00198 i636. Also we have Information 04 00198 i637. In view of Lemma 1.1, the measure Information 04 00198 i638 is convex for all Information 04 00198 i086, Information 04 00198 i135.

Now, we shall present exponential representation of the measure (52) based on the function given by (53). Let us consider a linear combination of convex functions,

Information 04 00198 i639

i.e.,

Information 04 00198 i640

Information 04 00198 i641,

where Information 04 00198 i642 are the constants. For simplicity we will choose,

Information 04 00198 i643

Thus, we have

Information 04 00198 i644

Information 04 00198 i645

Information 04 00198 i646

Which will give us

Information 04 00198 i647

As a consequence of (54), we will have the following exponential triangular discrimination

Information 04 00198 i648

4.2. Second Generalization of Triangular Discrimination

For all Information 04 00198 i086, let us consider the following measures

Information 04 00198 i649

In particular, we have

Information 04 00198 i650

and

Information 04 00198 i651.

The Expression (33) gives the second generalization of the measure Information 04 00198 i082. Now we will prove its convexity. We can write Information 04 00198 i652, Information 04 00198 i135, where

Information 04 00198 i653.

The second order derivative of the function Information 04 00198 i654 is given by

Information 04 00198 i655,

where

Information 04 00198 i656.

For all Information 04 00198 i635, Information 04 00198 i106, Information 04 00198 i060, we have Information 04 00198 i657. Also we have Information 04 00198 i658. In view of Lemma 2.1, the measure Information 04 00198 i659 is convex for all Information 04 00198 i086, Information 04 00198 i135.

Following the similar lines of (54) and (55), the exponential representation of the measure Information 04 00198 i659 is given by

Information 04 00198 i660.

4.3. First Generalization of the Measure Information 04 00198 i116

For all Information 04 00198 i086, let us consider the following measures

Information 04 00198 i661

In particular, we have

  • Information 04 00198 i662,

  • Information 04 00198 i663,

  • Information 04 00198 i664 Information 04 00198 i665

    and

  • Information 04 00198 i666 Information 04 00198 i667.

The expression (57) gives the first parametric generalization of the measure Information 04 00198 i116 given by (3). We will prove now its convexity. We might write Information 04 00198 i668, Information 04 00198 i135, where

Information 04 00198 i669.

The second order derivative of the function Information 04 00198 i670 is given by

Information 04 00198 i671,

where

Information 04 00198 i672.

For all Information 04 00198 i635, Information 04 00198 i106, Information 04 00198 i060, we have Information 04 00198 i673. Also we have Information 04 00198 i674. In view of Lemma 1.1, the measure Information 04 00198 i675 is convex for all Information 04 00198 i086, Information 04 00198 i135.

Following the similar lines of (54) and (55), the exponential representation of the measure Information 04 00198 i675 is given by

Information 04 00198 i676.

4.4. Second Generalization of the Measure Information 04 00198 i116

For all Information 04 00198 i086, let us consider the following measures

Information 04 00198 i677

In particular, we will have

Information 04 00198 i678

and

Information 04 00198 i679.

The Expression (58) gives the second generalization of the measure Information 04 00198 i116 given by (1.3). We will now prove its convexity. We can write Information 04 00198 i680, Information 04 00198 i135, where

Information 04 00198 i681.

The second order derivative of the function Information 04 00198 i682 is given by

Information 04 00198 i683,

where

Information 04 00198 i684.

For all Information 04 00198 i635, Information 04 00198 i106, Information 04 00198 i060, we have Information 04 00198 i685. Also we have Information 04 00198 i686. In view of Lemma 1.1, the measure Information 04 00198 i687 is convex for all Information 04 00198 i086, Information 04 00198 i135.

Following the similar lines of (54) and (55), the exponential representation of the measure Information 04 00198 i687 is given by

Information 04 00198 i688.

4.5. Generalization of Hellingar’s Discrimination

For all Information 04 00198 i086, let us consider the following measures

Information 04 00198 i689

In particular, we have

  • Information 04 00198 i690,

  • Information 04 00198 i691,

  • Information 04 00198 i692,

  • Information 04 00198 i693

    and

  • Information 04 00198 i694.

The measure (59) give generalized Hellingar’s discrimination. Let us now prove its convexity. We might write Information 04 00198 i695, Information 04 00198 i135, where

Information 04 00198 i696.

The second order derivative of the function Information 04 00198 i697 is given by

Information 04 00198 i698,

where

Information 04 00198 i699.

For all Information 04 00198 i635, Information 04 00198 i106, Information 04 00198 i060, we have Information 04 00198 i700. Also Information 04 00198 i701. In view of Lemma 2.1, the measure Information 04 00198 i702 is convex for all Information 04 00198 i086, Information 04 00198 i135.

Following the similar lines of (54) and (55), the exponential representation of the measure Information 04 00198 i702 is given by

Information 04 00198 i703.

4.6. New Measure

For all Information 04 00198 i704, let us consider the following measures

Information 04 00198 i705

In particular, we will have

  • Information 04 00198 i706,

  • Information 04 00198 i707,

  • Information 04 00198 i708,

  • Information 04 00198 i709

    and

  • Information 04 00198 i710 Information 04 00198 i711.

We will prove now the convexity of the measure (60). We can write Information 04 00198 i712, Information 04 00198 i135, where

Information 04 00198 i713.

The second order derivative of the function Information 04 00198 i714 is given by

Information 04 00198 i715,

where

Information 04 00198 i716.

For all Information 04 00198 i635, Information 04 00198 i106, Information 04 00198 i060, we have Information 04 00198 i717. Also we have Information 04 00198 i718. In view of Lemma 2.1, the measure Information 04 00198 i719 is convex for all Information 04 00198 i086, Information 04 00198 i135.

Following the similar lines of (54) and (55), the exponential representation of the measure Information 04 00198 i719 is given by

Information 04 00198 i720.

Remarks:

(i)

The first 10 measures appearing in the second pyramid (13) represents the same measure (14) and is same as Information 04 00198 i721. The last measure given by (51) is the same as Information 04 00198 i722. The measure (51) is the only one that appears in all the four parts of the Theorem 3.4. Both these measures generate the interesting measure shown in (60).

(ii)

The measure Information 04 00198 i723 appears in the work of Dragomir et al. [14]. An improvement over his work can be seen in Taneja [9].

(iii)

Following the similar lines of (54) and (55), the exponential representation of the principal measure Information 04 00198 i133 appearing in (6) is given by

Information 04 00198 i724

We observe that the expression (61) is different from the one obtained above in six parts. Applications of the generating measures (6), (52), (56), (57), (58), (59) and (60) along with their exponential representations should be encouraged in further studies.

Acknowledgements

Author is thankful to anonymous reviewers for their valuable comments and suggestions on an earlier version the paper. The author also thanks Atul Kumar Taneja for English grammar review.

References and notes

  1. Taneja, I.J. Refinement inequalities among symmetric divergence measures. Austr. J. Math. Anal. Appl. 2005, 2, 1–23. [Google Scholar]
  2. Taneja, I.J. New developments in generalized information measures. In Advances in Imaging and Electron Physics; Hawkes, P.W., Ed.; Elsevier Publisher: New York, NY, USA, 1995; Volume 91, pp. 37–135. [Google Scholar]
  3. Taneja, I.J. On symmetric and non-symmetric divergence measures and their generalizations. In Advances in Imaging and Electron Physics; Hawkes, P.W., Ed.; Elsevier Publisher: New York, NY, USA, 2005; Volume 138, pp. 177–250. [Google Scholar]
  4. Eves, H. Means appearing in geometrical figures. Math. Mag. 2003, 76, 292–294. [Google Scholar]
  5. LeCam, L. Asymptotic Methods in Statistical Decision Theory; Springer: New York, NY, USA, 1986. [Google Scholar]
  6. Hellinger, E. Neue Begründung der Theorie der quadratischen Formen von unendlichen vielen Veränderlichen. J. Reine Aug. Math. 1909, 136, 210–271. [Google Scholar]
  7. Taneja, I.J. Inequalities having seven means and proportionality relations. 2012. Available online: http://arxiv.org/abs/1203.2288/ (accessed on 7 April 2013). [Google Scholar]
  8. Taneja, I.J.; Kumar, P. Relative information of type s, Csiszar’s f-divergence, and information inequalities. Inf. Sci. 2004, 166, 105–125. [Google Scholar] [CrossRef]
  9. Taneja, I.J. Refinement of inequalities among means. J. Combin. Inf. Syst. Sci. 2006, 31, 357–378. [Google Scholar]
  10. Jain, K.C.; Srivastava, A. On symmetric information divergence measures of Csiszar’s f-divergence class. J. Appl. Math. Stat. Inf. 2007, 3, 85–102. [Google Scholar]
  11. Kumar, P.; Johnson, A. On a symmetric divergence measure and information inequalities. J. Inequal. Pure Appl. Math. 2005, 6, 1–13. [Google Scholar]
  12. Taneja, I.J. Bounds on triangular discrimination, harmonic mean and symmetric chi-square divergences. J. Concr. Appl. Math. 2006, 4, 91–111. [Google Scholar]
  13. Topsoe, F. Some inequalities for information divergence and related measures of discrimination. IEEE Trans. Inf. Theory 2000, 46, 1602–1609. [Google Scholar] [CrossRef]
  14. Dragomir, S.S.; Sunde, J.; Buse, C. New inequalities for jefferys divergence measure. Tamsui Oxf. J. Math. Sci. 2000, 16, 295–309. [Google Scholar]
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