Nested Inequalities Among Divergence Measures

Transcription

1 Appl Math Inf Sci 7, No, Applied Mathematics & Information Sciences An International Journal c 0 NSP Natural Sciences Publishing Cor Nested Inequalities Among Divergence Measures Inder J Taneja Departamento de Matemática, Universidade Federal de Santa Catarina, Florianópolis, SC, Brazil Received: Jan 0 Revised 7 Apr 0 Accepted Apr 0 Published online: Jan 0 Abstract: In this paper we considered an inequality having divergence measures Out of them three are logarithmic such as Jeffryes-Kullback-Leiber [4] [5] J-divergence Burbea-Rao [] Jensen-Shannon divergence Taneja [7] arithmetic-geometric mean divergence The other three are non-logarithmic such as Hellinger discrimination, symmetric χ divergence, triangular discrimination Three more are considered are due to mean divergences Pranesh Johnson [6] Jain Srivastava [] studied different kind of divergence measures We considered measures arising due to differences of single inequality having divergence measures in terms of a sequence Based on these differences we obtained many inequalities These inequalities are kept as nested or sequential forms Some reverse inequalities equivalent versions are also studied Keywords: J-divergence Jensen-Shannon divergence Arithmetic-Geometric divergence Mean divergence measures Information inequalities Introduction Let } Γ n P p, p,, p n p i > 0, p i, n, i be the set of all complete finite discrete probability distributions Let the two groups of divergence measures: Logarithmic divergence measures [ n i IP Q pi p i ln p i q i JP Q T P Q i i i ] qi q i ln, p i q i p i q i ln p i q i pi q i ln Corresponding author: ijtaneja@gmailcom pi q i p i q i Non-logarithmic divergence measures P Q hp Q ΨP Q i i p i q i p i q i, p i q i i p i q i p i q i p i q i The logarithmic measures IP Q, JP Q T P Q are three classical divergence measures known in the literature on information theory statistics are Jensen-Shannon divergence, J-divergence Arithmetic- Geometric mean divergence respectively [7] [8] The nonlogarithmic measures P Q, hp Q ΨP Q are respectively known as triangular discrimination, Hellingar s divergence symmetric chi-square divergence In 005, the author [9] proved the following inequality among these si symmetric divergence measures: 4 I h 8 J T Ψ 6 c 0 NSP Natural Sciences Publishing Cor

2 50 Inder J Taneja: Nested Inequalities The above inequality admits many nonnegative differences among the divergence measures Based on these non-negative differences, the author [9] proved the following result: } D I D h DhI D J D T D T h D Jh 6 D Ψ D T J 5 D ΨI 9 D Ψh 4 D ΨJ D ΨT,, for eample D I : I 4, D T J : T 8 J, D F Ψ : 6 F 8Ψ, etc The proof of the inequalities is based on the following two lemmas: Lemma If the function f : [0, R is conve normalized, ie, f 0, then the f-divergence, C f P Q given by pi C f P Q q i f, i is nonnegative conve in the pair of probability distribution P, Q Γ n Γ n Lemma Let f, f : I R R two generating mappings are normalized, ie, f f 0 satisfy the assumptions: i f f are twice differentiable on a, b ii there eists the real constants α, βsuch that α < β α f β, f > 0, a, b, f then we the inequalities: α C f P Q C f P Q β C f P Q The Lemma is due to Csiszár [] the Lemma is due to author [8] The aim of this paper is to consider more measures in improve the inequalities given in These measures are based on the some well-known mean divergences Mean Divergence Measures Author [0] studied the following inequality GP Q N P Q N P Q AP Q, 4 GP Q N P Q q i pi q i, i pi q i, i N P Q pi q i pi q i i AP Q i p i q i The above inequality admits non-negative differences given by i M P Q D N N P Q pi q i pi q i pi q i, i i M P Q D NGP Q [ pi q i pi q i [ pi q i New Measures M P Q D AN P Q pi q i hp Q D AN P Q p i q i ], pi q i D AG P Q D N GP Q ] Jain Srivastava [] Kumar Johnson [6] respectively studied the measures K 0 P Q F P Q i i p i q i pi q i p i q i pi q i In total we divergence measures By the application of Lemmas we can put them in a single inequality as 4 I 4M 4 M h 4 M 8 J T 8 K 0 6 Ψ 6F 5 c 0 NSP Natural Sciences Publishing Cor

3 Appl Math Inf Sci 7, No, / wwwnaturalspublishingcom/journalsasp 5 Pyramid The measures appearing in the inequalities 5 admits 55 non-negative differences These 55 non-negative difference satisfies some obvious inequalities given below in the form of pyramid or triangular: D I D M I D M D 4 M M D 5 M I D6 M 4D 7 hm D 8 hm D 9 hi D0 h 5D M h D M M D M M D 4 M I D5 M 6D 6 JM D 7 Jh D8 JM D 9 JM D 0 JI D J 7DT J D T M DT 4 h D5 T M DT 6 M D 7 D 8 T T I 8DK 9 0T D0 K 0J D K 0M DK 0 h D K 0M DK 4 0M DK 5 0I D6 K 0 9DΨK 7 0 DΨT 8 D9 ΨJ D40 ΨM DΨh 4 D4 ΨM DΨM 4 DΨI 44 D45 Ψ 0DF 46 Ψ D47 F K 0 DF 48 T D49 F J D50 F M D 5 DF 5 M DF 5 M DF 54 I D55 F The following equalities hold: D 8 hm D 7 hm D4 M M DM h D M M 4 D M M D T J D7 T I D0 JI F h In view of Lemmas, the measures appearing in the above pyramid are conve in a pair of probability distributions can be written as D AB : i pi q i f AB, 6 q i f AB f A f B, A B with the property that f AB 0, > 0 In this paper our aim is to etend the results given by by taking all possible nonnegative differences given in the above pyramid These inequalities we put in nested or sequential form Nested Inequalities In this section, we shall try to put the measures appearing in above pyramid in terms of nested or sequential form This we done in a theorem below Theorem The following inequalities hold: DI 8 9 D M 8 D J 8 D8 hm D6 M D0 h 8 5 D5 M D8 T 8 D8 hm 8 7 D4 M I D9 hi D6 K 0 D T J 8DM I 8 5 D6 T M 8 7 D9 JM 5 D8 JM D 7 Jh 8 9 D T M 8 D K 0 M D K 0 h 8 D5 M I 8 D5 T M D4 T h D5 K 8 0I 5 D4 K 0M D 0 K 0 J 6 D45 Ψ 8 9 D K 0 M 8 9 D4 ΨM 8 7 D4 ΨM 9 D4 Ψh D7 ΨK 0 D8 ΨT 9 D55 F 8 6 D5 F M 5 D5 F h 5 D44 ΨI 4 D9 ΨJ 8 D40 ΨM 8 D54 F I 8 6 D5 F M 7 D49 F J 8 57 D50 F M 6 D47 F K 0 6 D48 F T D46 F Ψ 7 4 D7 Jh D M I 9 D T M DJM 6 4 D46 F Ψ 8 c 0 NSP Natural Sciences Publishing Cor

4 5 Inder J Taneja: Nested Inequalities ProofIn view of 6 we shall prove the theorem just writing the epressions for f AB The rest part is understood obviously For D I P Q 8 9 D M P Q: After simplification, we observe that equivalently, we to show the following: I 6 [8M ] 9 Let the function g I M f then we g I M I f M, 0 / 6 / Here we f M 8f M f Calculating the first order derivative of the function g I M, we get g I M / k 6, / / k / By the application Lemma over we get 9, proving the required result Argument: Let a b two positive numbers, ie, a > 0 b > 0 If a b > 0, then we can conclude that a > b because a b a b / a b We used this argument to prove k > 0, > 0, We shall use frequently this argument to prove the other parts of the theorem RemarkFrom the above proof we observe that it is sufficient to write epressions similar to 0, The rest part of the proof follows by the application of Lemma In view of this we shall avoid details for the proof of other parts From now onward, throughout it is understood that > 0, For D M P Q 9 D6 M g M M f M / f M, then we g M M 4 / / ] [ 6 / / lim g M M 9 Equivalently, we to show that Ω 9 D6 M D M g I M > 0, < < 0, > 4M 88M 0 We can write Ω : n i q if Ω q i /p i, f Ω k, with Epression is valid only when k > 0, > 0, Now, we shall show that k > 0, > 0, Let h [ / ] 4 After simplifications, we h / Since h > 0, > 0, proving that k > 0, > 0, Also we sup g I M lim g I M 6 k 0 / Let h 0 / 0 4 [ 6 ] After simplifications, we get h 6 [ 6 6 ] Since h > 0, proving that k > 0 This proves the required result c 0 NSP Natural Sciences Publishing Cor

5 Appl Math Inf Sci 7, No, / wwwnaturalspublishingcom/journalsasp 5 For D 6 M P Q D0 h g M h f M / f h, then we g M h ] [ 6 / / / 6 lim g M h Equivalently, we to show that Ω D0 h D6 M 48 44h 64M 0 We can write Ω : n i q if Ω q i /p i, f Ω k 48, with k k > 0 This proves the required result 4For D 0 h P Q 4 5 D5 M g h M f h / f M, then we g h M / 6 [ / 4 / ] lim g h M 4 5 Equivalently, we to show that Ω 4 5 D5 M D0 h 0 64M 0h 0 We can write Ω : n i q if Ω q i /p i, f Ω k4 0, with k 4 k > 0 This proves the required result 5For D 5 M P Q 5D8 hm g M hm f M / f hm, then we g M hm [ / 4 / ] [ / ] lim g M hm 5 Equivalently, we to show that Ω D5 M D0 h 4 0h 80M 64M 0 We can write Ω 4 : n i q if Ω4 q i /p i, f Ω4 k 5 4, with k 5 k > 0 This proves the required result 6For D 5 M P Q 5 6 D J g M J f M / f J, then we 4 / 4 / g M J 6 g M J k 6 6, k 6 6 / 4 9 9/ 4 7/ 8 8 / g M > 0, < J < 0, >, provided k 6 > 0 Now we shall show that k 6 > 0 Let [ h 6 6 / ] 4 9 9/ 4 7/ 8 8 / Simplifying the above epression, we get h / 6 / 460 7/ 96 9/ 58 / / 6 5/ 460 / c 0 NSP Natural Sciences Publishing Cor

6 54 Inder J Taneja: Nested Inequalities Since h 6 > 0, proving that k 6 > 0 Also we sup g M J lim g M J 5 6 7For D J P Q D8 T P Q: It holds in view of 8For D 8 hm P Q 8 D8 T g hm T f hm / f T, then we g hm T [ / ] g hm 4 k 7 T 4, k / This give g hm > 0, < T < 0, >, provided k 7 > 0 Now we shall show that k 7 > 0 Let h / 0 0 [ ] After simplifications, we get h / / / 04 6 / 4 7/ 00 9/ 7/ / 44 4 / 7 4 Since h 7 > 0, this gives that k 7 > 0 Also we sup g hm T lim g hm T 8 9For D M P Q h 7 D4 M I g Mh M I f M h / f M I, then we g Mh M I / [ / ] g M h M I k 8 [ / ], k 8 / 4 g M > 0, < h M I < 0, >, provided k 8 > 0 In order to prove k 8 > 0, let h 8 / 4 [ ] After simplifications, we h 8 4 Since h 8 > 0, this gives that k 8 > 0 Also we sup g Mh M I lim g Mh M I 7 0For D 4 M I P Q 7 4 D9 hi g M I hi f M I / f M h, then we g M I hi [ / ] g M k 9 I hi, k 9 k 8 > 0 This give g M > 0, < I hi < 0, > Also we sup g M I hi lim g M I hi 7 4 c 0 NSP Natural Sciences Publishing Cor

7 Appl Math Inf Sci 7, No, / wwwnaturalspublishingcom/journalsasp 55 For D 9 hi P Q 4 D5 M I g hi M I f hi / f M I, then we g hi MI g M > 0, < I K 0 < 0, >, [ / ] g hi M I k 0 [ /, ] k 0 k 8 > 0 g hi > 0, < M I < 0, > Also we sup g hi MI lim g hi MI 4 For D 5 M I P Q 8 D6 K 0 g MI K 0 f M I, then we f K 0 g MI K 0 6 [ / ] 4 6 7/ 0 4 5/ 66 4 / 0 6 g M I K 0 8 k 4 6 7/ 0 4 5/, 66 4 / 0 6 k u 7 / 7 6 / / / / / 7, with u 46 9 / / / / 60 4 / provided k > 0 In order to prove k > 0, let vt ut 4t 9t 4t 0 4t 9 60t 8 50t 7 48t 6 50t 5 60t 4 4t 4t 9t 4 Solving the polynomial equation vt 0, we observe that there are no real solutions All the twelve solutions are comple are given by ± I, ± 578 I, ± I, ± I, ± I, 656 ± 449 I This means that for all t > 0, either vt > 0 or vt < 0 Calculating a particular value of vt, for eample for t, we get v 8 > 0 This means that vt > 0, for all t > 0, hence u > 0, > 0 Let h [ u ] 7 / 7 6 / / / / / 7 After simplifications, we h / / 56 7/ 40 9/ 4 / 4 / / 56 7/ / 4 / / Since h > 0, this gives that k > 0 Also we sup g M I K 0 lim g M I K 0 8 For D 5 M P Q I 8 D TJ g M I T J f M I / f T J, then we g M I T J c 0 NSP Natural Sciences Publishing Cor

8 56 Inder J Taneja: Nested Inequalities 4 [ / ] g M k I T J, k 7/ 5/ 4 / 4 g M > 0 < I T J < 0 >, provided k > 0 In order to prove k > 0, let h 7/ 5/ 4 / 4 After simplifications, we h Since h > 0, this gives that k > 0 Also we sup g M I T J lim g M I T J 8 4For D 5 M I P Q 5 D6 TM g M I T M f M I / f T M, then we g M I T M / / / g M I T M k [ /, / ] k 4 [ / ] 4 7/ 7 5 5/ 0 5 / 7 g M > 0, < I T M < 0, >, provided k > 0 In order to prove k > 0, let [ h 4 ] [ / ] } 4 7/ 7 5 5/ 0 5 / 7 After simplifications, we h / / / 60 6 / 0 6 Since h > 0, this gives that k > 0 Also we sup g MI T M lim g MI T M 5 5For D 5 M I P Q 7 D9 JM g M I JM f M I / f JM, then we 4 / g M I JM 4 4 / g M I JM k 4 [ 4 /, 4 ] k 4 c 0 NSP Natural Sciences Publishing Cor

9 Appl Math Inf Sci 7, No, / wwwnaturalspublishingcom/journalsasp 57 [ ] 4 7/ 5/ / g M > 0 < I JM < 0 >, provided k 4 > 0 In order to prove k 4 > 0, let [ ] h 4 4 7/ 5/ / After simplifications, we h 4 5 5/ 48 9/ 6 7/ 6 5/ 48 / Since h 4 > 0, this gives that k 4 > 0 Also we sup g MI JM lim g MI JM 7 6For D 5 M I P Q D M I g M I M I f M I / f M I, then we g MI M I / [ / ] g M I M I k 5 [ /, ] Where k 5 k 8 > 0 This give g M > 0, < I M I < 0, > Also we sup g M I M I lim g M I M I 7For D TJ P Q 8 D5 TM g T J T M f T J / f T M, then we g T J T M / 4 / 4 g T J T M k 6 [ /, 4 / 4 ] k 6 [ 4 ] This give g T > 0 < J T M < 0 >, provided k 6 > 0 In order to prove k 6 > 0, let h 6 [ 4 ] After simplifications, we h Since h 6 > 0, this gives that k 6 > 0 Also we sup g T J T M lim g T J T M 8 8For D 6 TM P Q 5 D5 TM g T M T M f T M / f T M, then we g T M T M / / 4 / 4 / c 0 NSP Natural Sciences Publishing Cor

10 58 Inder J Taneja: Nested Inequalities g T M T M 6 k 7 [ /, 4 / 4 ] This give k / 8 5/ 8 / g T M > 0, < T M < 0, >, provided k 7 > 0 In order to show k 7 > 0, let h 7 4 [ ] 4 7/ 8 5/ 8 / After simplifications, we h / 7 5/ / 7 7/ 0 9/ Since h 7 > 0, proving that k 7 > 0 Also we sup g T M T M lim g T M T M 5 9For D 5 TM P Q D4 Th g T M T h f T M / f T h, then we g T M g T M T h 4 / 4 / k 8 T h 6, k 8 k 7 > 0 g T > 0, < M T h < 0, > Also we sup g T M T h lim g T M T h 0For D 4 Th P Q 4 D TM g T h T M f T h / f T M, then we g T h T M / g T h T M k 9 [, / ] k 9 k 7 > 0 g T > 0, < h T M < 0, > Also we For D 4 Th sup g T h T M lim g T h T M 4 P Q 5 D8 g T h JM f T h / f JM JM, then we g T h JM / g T h JM k 0 [, 8 4 / ] k / 8 5/ 8 / c 0 NSP Natural Sciences Publishing Cor

11 Appl Math Inf Sci 7, No, / wwwnaturalspublishingcom/journalsasp 59 g T > 0 < h JM < 0 >, provided k 0 > 0 In order to show k 0 > 0, let h 0 [ ] 4 0 [ ] 4 4 7/ 8 5/ 8 / After simplifications, we h / / / / 0 9/ Since h 0 > 0, proving that k 0 > 0 Also we sup g T h JM lim g T h JM 5 For D 8 JM P Q 5 4 D7 Jh g JM Jh f JM / f Jh, then we g JM g JM Jh 8 4 / k Jh, k [ ] 6 / g JM > 0, < Jh < 0, > provided k > 0 In order to prove k > 0, let h 4 6 / [ ] After simplifications, we h / 6 6 5/ 8 6 / 6 6 Since h > 0, this gives that k > 0 Also we β JM Jh sup g JM Jh lim g JM Jh 5 4 For D 6 K P Q 0 D5 K 0I g K0 K 0I f K / 0 f K 0I, then we g K0 K 0 I 4 6 7/ 0 4 5/ 66 4 / / 4 6 Also we g K 0 K 0 I / / 4 6 g K > 0, < 0 K 0 I < 0, > sup g K0 K 0I lim g K0 K 0I 4For D TM P Q 9 6 D5 K 0 I g T M K 0 I f T M / f K 0 I, then we g T M K 0I [ / ] 8 6 / 4 6 g T M K 0 I 4 k, 6 / 4 6 c 0 NSP Natural Sciences Publishing Cor

12 60 Inder J Taneja: Nested Inequalities k / 8 / / 8 7/ 9/ 6 40 / 40 7/ / / 5 6 g T M > 0, < K 0 I < 0, >, provided k > 0 In order to prove k > 0, let h / 8 / / 8 7/ 9/ 6 40 / 40 7/ / / 5 6 After simplification, we h / 46 7/ 4 9/ 4 / 95 5/ 808 / / / 46 / Since h > 0, proving that k > 0 Also we sup g T M K 0 I lim g T M K 0 I 9 6 5For D M I P Q 6 D5 K 0 I g M I K 0 I f M I / f K 0 I, then we g MI K 0I 6 [ / ] 6 / 4 6 g M I K 0I 8 k, 6 / 4 6 k 6 / / 40 7/ 84 5/ 40 / g M > 0, < I K 0 I < 0, >, provided k > 0 In order to prove k > 0, let h [ ] 6 / / 40 7/ 84 5/ 40 / After simplification, we get h / 96 / 4 9/ 96 / / 7 / Since h > 0, this gives that k > 0 Also we sup g MI K 0I lim g MI K 0I 6 6For D 7 Jh P Q 4 D5 K 0 I g Jh K0 I f Jh / f K 0 I, then we g Jh K0 I 4 6 / 4 6 g Jh K 0 I [ ] 6 / 4 6 Also we g Jh > 0, < K 0I < 0, > sup g Jh K0 I lim g Jh K0 I 4 c 0 NSP Natural Sciences Publishing Cor

13 Appl Math Inf Sci 7, No, / wwwnaturalspublishingcom/journalsasp 6 7For D 5 6 K 0I P Q 5 D4 K 0M g K0 I K 0 M f K 0 I / f K 0 M, then we g K0 I K 0 M 6 / / g K 0I K 0M 6 k 4 [ 6 /, 4 ] k 4 k > 0 g K > 0 < 0 I K 0 M < 0 > Also we sup g K0 I K 0 M lim g K0 I K 0 M 6 5 8ForD 4 K 0 M P Q 5 D K 0 M g K0M K 0M f K 0 M / f K 0 M, then we g K0M K 0M 4 6 / / g K 0 M K 0 M 88 k , 6 / k 5 7/ 5/ 5 5 / g K 0 M > 0 < K 0 M < 0 >, provided k 5 > 0 In order to prove k 5 > 0, let a function h 5 7/ 5/ 5 5 / [ ] After simplifications, we get h / / 9 4 5/ 9 0 / 8 0 Since h 5 > 0, this gives that k 5 > 0 Also we sup g K0 M K 0 M lim g K0 M K 0 M 5 9For D K 0 M P Q D K 0 h g K0 M K 0 h f K 0 M / f K 0 h, then we g K0 M K 0 h / 9 g K 0 M 8 k 6 K 0 h 9, k 6 k 5 > 0 g K > 0, < 0 M K 0 h < 0, > Also, we sup g K0 M K 0 h lim g K0M K 0h 0For D K P Q 4 0h D K 0M g K0h K 0M f K / 0h f K 0M, then we g K0 h K 0 M 6 / g K 0h K 0M 4 k 7 [, 6 / ] c 0 NSP Natural Sciences Publishing Cor

14 6 Inder J Taneja: Nested Inequalities k 7 k 5 > 0 Also we g K 0 h K 0 M > 0, < < 0, >, sup g K0 h K 0 M lim g K0 h K 0 M 4 For D K P Q 0h D0 K 0 J g K0 h K 0 J f K / 0h f K 0 J, then we g K0 h K 0 J, g K 0 h K 0 J 6 > 0, < < 0, > sup g K0 h K 0 J lim g K0 h K 0 J For D K 0 h P Q 4 D45 Ψ g K0 h Ψ f K 0 h / f Ψ, then we g K0 h Ψ , g K 0h Ψ 8 5 > 0, < 4 5 < 0, > 5 sup g K0h Ψ lim g K0h Ψ 4 For D 0 K P Q 0J 5 D44 ΨI g K0J Ψ f K / 0J f Ψ, then we g K0 J ΨI 4 g K 0J ΨI 8 5/ Also we g K > 0, < 0J ΨI < 0, > sup g K0J ΨI lim g K0J ΨI 5 4For D 45 Ψ P Q 6 5 D44 ΨI P Q: It is true in view of 5For D K 0 M P Q D4 ΨM g K0M ΨM f K 0 M / f ΨM, then we g K0M ΨM 6 / 4 4 / 4 / / g K 0 M ΨM k 8 8, 4 5 / k / 8 8 / / 6 4 7/ 8 4 5/ 8 / g K 0 M > 0, < ΨM < 0, >, provided k 8 > 0 In order to prove k 8 > 0, let h 8 [ ] [ ] 4 8 5/ 8 8 / / 6 4 7/ 8 4 5/ 8 / After simplification, we h 8 4 c 0 NSP Natural Sciences Publishing Cor

15 Appl Math Inf Sci 7, No, / wwwnaturalspublishingcom/journalsasp / / 646 7/ 47 9/ 08 / 47 / 760 7/ 646 5/ 08 9/ 0 / Since h 8 > 0, proving that k 8 > 0 Also we sup g K0 M ΨM 6For D 44 ΨI lim g K0M ΨM P Q 40 9 D4 g ΨI ΨM f ΨI / f ΨM ΨM, then we g ΨI ΨM 4 / 4 / / g ΨI ΨM / k 9 4 /, 4 / / k 9 7/ 0 / 6 9/ / 5 5 5/ 0 6 / 6 g ΨI > 0 < ΨM < 0 >, provides k 9 > 0 In order to prove k 9 > 0, let h 9 [ ] 7/ 0 / 6 9/ / 5 5 5/ 0 6 / 6 After simplifications, we h 9 4 s, s We know that, this allows us to conclude that s ] After simplifications, we s us h 9 > 0 Hence k 9 > 0 Also we sup g ΨI ΨM lim g ΨI ΨM For D 4 ΨM P Q 9 7 D4 ΨM g ΨM ΨM f ΨM / f ΨM, then we g ΨM ΨM 4 / 4 / / 8 / 4 / g ΨM ΨM 6 / k 0 [, 8 / 4 / / ] k 0 4 5/ / / g ΨM > 0, < ΨM < 0, >, c 0 NSP Natural Sciences Publishing Cor

16 64 Inder J Taneja: Nested Inequalities provided k 0 > 0 In order to prove k 0 > 0, let [ h 0 4 5/ / ] [ / ] After simplifications, we h / / / 4 8 5/ 5 0 / 5 4 Since h 0 > 0, this gives that k 0 > 0 Also we sup g ΨM ΨM lim g ΨM ΨM 9 7 8For DΨM 4 P Q 7 6 D4 Ψh P Q: Let g ΨM Ψh f ΨM / f Ψh, then we g ΨM g ΨM Ψh 8 / 4 / / [ / ] k Ψh /, k k 0 > 0 g ΨM > 0, < Ψh < 0, > Also we sup g ΨM Ψh lim g ΨM Ψh 7 6 9For D 4 Ψh P Q 9 8 D9 ΨJ P Q: It is true in view of 40ForD 4 Ψh P Q D40 g Ψh ΨM f Ψh / f ΨM ΨM, then we g Ψh ΨM / 4 / / g Ψh ΨM / / k, 4 / / k k 0 > 0 g Ψh > 0, < ΨM < 0, > Also we sup g Ψh ΨM lim g Ψh ΨM 4For D 40 ΨM P Q 8 D7 ΨK 0 g ΨM ΨK 0 f ΨM / f ΨK 0, then we g ΨM ΨK / / 4 5 / g ΨM ΨK 0 6 k, 4 5 / k / 4 5/ 6 7/ 4 5 9/ g ΨM > 0 < ΨK 0 < 0 >, provided k > 0 In order to prove k > 0, let [ ] h / 4 5/ 6 7/ 4 5 9/ After simplifications, we h 4 c 0 NSP Natural Sciences Publishing Cor

17 Appl Math Inf Sci 7, No, / wwwnaturalspublishingcom/journalsasp / 0 / 056 7/ / / 056 7/ / 0 / 66 / 66 / 874 5/ Since h > 0, this gives that k > 0 Also we sup g ΨM ΨK 0 lim g ΨM ΨK 0 8 4For D 9 ΨJ P Q 4 D7 ΨK 0 g ΨJ ΨK0 f ΨJ / f ΨK 0, then we g ΨJ ΨK / 6 5 4, g ΨJ ΨK 0 4 / 0 4 > 0, < 4 5 / 6 5 < 0, > 4 sup g ΨJ ΨK0 lim g ΨJ ΨK0 4 4For D 7 ΨK 0 P Q D 8 ΨT P Q: It is true in view of pyramid 44For D 7 ΨK 0 P Q D55 F g ΨK0 F f ΨK 0 / f F, then we g ΨK0 F / , / 64 5/ 64 7/ 50 9/ 0 / g ΨK 0 F / 64 5/ 64 7/ 50 9/ 0 / > 0, < 05 / 586 5/ < 0, > 586 7/ 05 9/ 5 / sup g ΨK0 F lim g ΨK0 F 45For D 8 ΨT P Q 8 D54 FI g ΨT F I f ΨT /f F I, then we g ΨT F I / / 90 / g ΨT F I / / 90 / / / / Also, we > 0, < g ΨT F I < 0, > sup g ΨT F I lim g ΨT F I 8 46For D 55 F P Q 9 8 D54 FI g Ψ F I f Ψ /f F I, then we g F F I / 64 5/ 64 7/ 50 9/ 0 / / / 90 / g F F I / / / 90 / / / / / c 0 NSP Natural Sciences Publishing Cor

18 66 Inder J Taneja: Nested Inequalities Also, we 47For D 54 FI g F > 0, < F I < 0, > sup g F F I lim g F F I 9 8 P Q 64 6 D5 g F I F M f F I / f FM F M, then we g F I F M / / / 90 / g F I F M 64 k , 64 / k / / / 60 / 5 0 7/ 65 / / 40 / 67 7/ / / 40 / 60 7 g F > 0, < I F M < 0, >, provided k 4 > 0 In order to prove k 4 > 0, let h / / / 60 / 5 0 7/ 65 / / 40 / 67 7/ / / 40 / 60 7 After simplification, we get h / 600 / / / / / / 600 / / / / Since h 4 > 0, this gives that k 4 > 0 Also we sup g F I F M lim g F I F M For D 5 FM P Q 6 6 D5 FM g F M F M f F M / f F M, then we g F M F M / / g F 5760 M F M k , 64 / k 5 [ 6 / ] g F M > 0, < F M < 0, >, provided k 5 > 0 In order to prove k 5 > 0, let h 5 c 0 NSP Natural Sciences Publishing Cor

19 Appl Math Inf Sci 7, No, / wwwnaturalspublishingcom/journalsasp 67 [ 6 / ] [ ] After simplifications, we h / 4 4 7/ / 49 4 / 8 Since h 5 > 0, this gives that k 5 > 0 Also we sup g F M F M lim g F M F M For D 5 FM P Q 6 60 D5 Fh g F M F h f F M / f F h, then we consider g F M g F M F h / 45 k 6 F h 45 4 k 6 k 5 > 0 g F > 0 < M F h < 0 > Also we sup g F M F h lim g F M F h For D 5 5 Fh P Q g F h F J f F h /f g F h F J 4 D49 FJ F J, then we 5, 5 0 5/ / g F h F J / / Also, we g F > 0, < h F J < 0, > sup g F h F J lim g F h F J 5 4 5For D 5 0 Fh P Q 9 D50 FM g F h F M f F h / f F M, then we g F h F M / g F h F M 960 k 7 [ 64 /, ] Where k 7 k 5 > 0 g F > 0, < h F M < 0, > Also we sup g F h F M lim g F h F M 0 9 5For D 50 FM P Q 9 6 D47 FK 0 g F M F K 0 f F M / f F K 0, then we g F M g F M F K / k 8 F K 0, k / 6 0 g F M > 0, < F K 0 < 0, >, provided k 8 > 0 In order to prove k 8 > 0, let [ ] h 8 c 0 NSP Natural Sciences Publishing Cor

20 68 Inder J Taneja: Nested Inequalities / 6 0 After simplifications, we h / / / / / / / / 900 5/ 4960 / / / Since h 8 > 0, proving that k 8 > 0 Also we sup g F M F K 0 lim g F M F K For D 49 FJ P Q 7 6 D47 FK 0 g F J F K0 f F J / f F K 0, then we g F J F K / / g F J F K / / Also, we g F > 0, < J F K 0 < 0, > sup g F J F K0 lim g F J F K For D 47 FK 0 P Q D 48 FT P Q: It holds in view of pyramid 55For D 48 FT P Q D46 FΨ g F K0 F T f F K 0 / f F T, then we g F T F Ψ / / / / / 4 5 g F T F Ψ / / / / / / / Also,we g F > 0, < T F Ψ < 0, > sup g F T F Ψ lim g F T F Ψ 56For D 7 Jh P Q 4D6 JM g Jh JM f Jh / f JM, then we g Jh JM 4 / g Jh JM k 9 [, 4 / ] k 9 k > 0 Also, we g Jh > 0, < JM < 0, > sup g Jh JM lim g Jh JM 4 c 0 NSP Natural Sciences Publishing Cor

21 Appl Math Inf Sci 7, No, / wwwnaturalspublishingcom/journalsasp 69 57For D M I P Q D6 JM g M I JM f M I / f JM, then we g MI JM 4 [ / ] 4 / g M I JM k 40 [ 4 / ], k 40 5/ 8 / g M > 0, < I JM < 0, >, provided k 40 > 0 In order to prove k 40 > 0, let [ ] h 40 5/ 8 / After simplifications, we h / / 4 4 5/ 4 0 / 8 8 Since h 40 > 0, proving that k 40 > 0 Also we sup g M I JM lim g M I JM 58For D TM P Q 9D 6 JM g T M JM f T M / f JM, then we g T M JM [ / ] 4 / g T M JM k 4 k 4 [, 4 / ] 7/ 5/ 4 4 / g T M > 0, < JM < 0, >, provided k 4 > 0 In order to prove k 4 > 0, let h 4 7/ 5/ 4 4 / [ ] After simplifications, we h / / 4 4 5/ 4 0 / 8 4 Since h 4 > 0, proving that k 4 > 0 Also we sup g T M JM lim g T M JM 9 59For D 6 JM P Q 4 D46 FΨ g JM F Ψ f JM / f F Ψ, then we g JM F Ψ [ 6 / 4 / ] / / 4 g JM 8 F Ψ k 4, / / 4 k / / 4 5/ 4 / c 0 NSP Natural Sciences Publishing Cor

22 70 Inder J Taneja: Nested Inequalities 06 4 / / 4 7/ 4 5/ 44 / g JM > 0, < F Ψ < 0, >, provided k 4 > 0 In order to prove k 4 > 0, let [ ] h 4 459/ / 4 5/ 4 / / / 4 7/ 4 5/ 44 / After simplifications, we h 4 4 v, v / / / / / / / / / / / Now we shall show that v > 0 Let mt vt 05t 4 970t 444t 864t 7498t 0 506t 9 995t 8 04t 7 900t 6 980t 5 776t 4 444t 646t 444t 776t 0 980t 9 900t 8 04t 7 995t 6 506t t 4 864t 444t 970t 05 The polynomial equation mt 0 of 4 th degree admits 4 solutions Out of them are comple not written here two of them are real given by Both these solutions are negative Since we are working with t > 0, this means that there are no real positive solutions of the equation mt 0 Thus we conclude that either mt > 0 or mt < 0, for all t > 0 In order to check it is sufficient to see for any particular value of mt, for eample when t m 778, hereby proving that mt > 0 for all t > 0, consequently, v > 0, for all > 0, proving that h 4 > 0, > 0, Since h 4 > 0, proving that k 4 > 0 Also we sup g JM F Ψ lim g JM F Ψ 4 Parts -55 refers to the proof of the inequalities given in 7 the parts give the proof of 8 Combining the parts -59 we get the proof of the Theorem Remark Theorem connects 54 members out of 55 appearing in the pyramid Since some them are equals by multiplicative constants, the Theorem contains 47 different measures In this way we can make a sequential inequality connecting 4 divergence measures From the inequalities given in 7 8, it is interesting to observe that all the measures remain between DI D46 F Ψ, ie, in between the first members of first last line of the pyramid The last members of each line corners members of the pyramid are connected in an increasing order, ie, D I 8 9 D M 8 D6 M D0 h 8 5 D5 M D J D8 T D6 K 0 6 D45 Ψ 9 D55 F Equivalent Inequalities As a consequence of Theorem, the sequences of inequalities appearing in 7 8 can be written in an individual form This means that the 59 results proving the Theorem can be written in an equivalent form This we done below in two groups The first group is with three measures in each case the second group is with four measures c 0 NSP Natural Sciences Publishing Cor

23 Appl Math Inf Sci 7, No, / wwwnaturalspublishingcom/journalsasp 7 Group I 8M 6 h F 80M 976 I 4 K0 4 h I6M 7 I 0 Ψ 96 M 5J 5 4 I F 44 4 M J8h 8 5 M 4M 88 5 M T h 6 6 M 0IK0 5 6 M K04h 8 7 M 64IΨ M F 04h 80 8 M 008IF M Ψ76h M IM 8 9 J K 06h 0 M 44h 64 0 J 6T M T 6M 0 J Ψ8h 8 M K008M 80 J F 4h 0 M Ψ84M 46 J 0T 56M 9 4 M F 95M 67 4 J 8T 56M 9 5 M I9h 6 5 K 0 6JΨ 8 6 h 64M 0 6 K 0 JF 4 7 h J8M 0 9 K 0 Ψ5M 8 h T 6M 8 K 0 F 04M 8 9 h K 08M 04 9 Ψ F 6T 0 h Ψ768M 59 Group 80M 6M 0h h 4T 8M 6 56M 9I K M 4M 68I 9J 5 M 0M 5I T 6 9J 56M 9I 7T 7 0T M J 0h 8 7I 8T 9K 0 5M 9 8I 4J K 0 h 0 4 8K 0 Ψ 64h 6I 0K 0 Ψ 0J 6K 0 9M Ψ 8M M M J 8T 4 4 Ψ F 6K I 8Ψ F 8T 6 48J Ψ F 56M Direct relations of the inequalities given in Groups to the inequalities given in 5 shall be dealt else 4 Reverse Inequalities In view of Theorem, we shall derive some inequalities in reverse order for the last three lines of the pyramid Combining the inequalities given in the 0 th line of the pyramid the one given in 7 having the measure F P Q, we the following etended inequality D 46 F Ψ D 47 F K 0 D 48 F T D 49 F J D 50 F M D 5 F h D 5 F M D 5 F M D 54 F I D 55 F 9 8 D D5 F M 7 6 D5 F M 6 5 D5 F h 7 D49 4 F I F J 9 D50 F M } D47 F K 0 D48 F T D46 F Ψ 4 According to inequalities given in pyramid we DF 49 J D50 F M but according to our approach we don t reverse relation among the measures DF 49 J DF 50 M Also DF 5 h is related to D49 F J D50 F M with different multiplicative constants We call the epression 0 as reverse inequalities Combining the inequalities given in the 9 th line of the pyramid the one given in 7 having the measure ΨP Q, we the following etended inequality DΨK 7 0 DΨT 8 DΨJ 9 DΨM 40 DΨh 4 DΨM 4 D 4 ΨM D 44 ΨI D 45 Ψ 6 5 D D4 ΨM 4 D4 Ψh D9 ΨJ 6 ΨI 6 5 D4 ΨM D40 ΨM } DΨK 7 0 DΨT 8 5 According to inequalities given in pyramid we DΨJ 9 D40 ΨM but according to our approach we don t reverse relation among the measures D ΨJ D ΨM Also DΨh 4 is related to D9 ΨJ D40 ΨM with different multiplicative constants Again we call the epression 5 as reverse inequalities Combining the inequalities given in the 8 th line of the pyramid the one given in 7 having the measure ΨP Q, we the following etended inequality D 9 K 0T D 0 K 0J D K 0M D K 0h D K 0 M D 4 K 0 M D 5 K 0 I D 6 K 0 D5 K 0 I 8 5 D4 K 0 M 4 D K 0 M D K 0h D 0 K 0J 8 D K 0 M } 6 We observe that the measure DK 9 0 T don t appears in the reverse side Moreover, it don t appears in Theorem too Similarly we can write reverse inequalities for the other lines of the pyramid c 0 NSP Natural Sciences Publishing Cor

24 7 Inder J Taneja: Nested Inequalities References [] J BURBEA, CR RAO, On the conveity of some divergence measures based on entropy functions, IEEE Trans on Inform Theory, IT-898, [] I CSISZÁR, Information type measures of differences of probability distribution indirect observations, Studia Math Hungarica, 967, 99-8 [] KC JAIN AND A SRIVASTAVA, On Symmetric Information Divergence Measures of Csiszars f - Divergence Class, Journal of Applied Mathematics, Statistics Informatics JAMSI, 007, 85-0 [4] H JEFFREYS, An invariant form for the prior probability in estimation problems, Proc Roy Soc Lon, Ser A, 86946, [5] S KULLBACK RA LEIBLER, On information sufficiency, Ann Math Statist, 95, [6] P KUMAR A JOHNSON, On A Symmetric Divergence Measure Information Inequalities, Journal of Inequalities in Pure Appl Math, 6005, - [7] IJ TANEJA, New developments in generalized information measures, Chapter in: Advances in Imaging Electron Physics, Ed PW Hawkes, 9995, 7-5 [8] IJ TANEJA, On symmetric non-symmetric divergence measures their generalizations, Chapter in: Advances in Imaging Electron Physics, Ed PW Hawkes, 8005, [9] IJ TANEJA, Refinement Inequalities Among Symmetric Divergence Measures, The Australian Journal of Mathematical Analysis Applications, 005, Art 8, pp - [0] IJ TANEJA, Refinement of Inequalities among Means, Journal of Combinatorics, Information Systems Sciences, 006, [] IJ TANEJA P KUMAR, Relative Information of Type s, Csiszar s f-divergence, Information Inequalities, Information Sciences, 66004, 05-5 Inder J Taneja, Professor at the Department of Mathematics, Federal University of Santa Catarina, Florianópolis, Brazil, since 978 MSc PhD degrees in Mathematics from Delhi University, India, in respectively Post-doctoral research in Italy Spain, in respectively Research interests in information theory specially on information measures, probability of error, noiseless coding, fuzzy set theory, inequalities, etc Recent interest are on magic squares applications of information measures to genetic code, DNA, etc More than 00 research papers in journals of international reputations Five chapters in information measures in book series One online book on information measures c 0 NSP Natural Sciences Publishing Cor