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Establishing Relations among Various Measures by Using Well  Known Inequalities
Establishing Relations among Various Measures by Using Well  Known Inequalities
Establishing Relations among Various Measures by Using Well  Known Inequalities
Establishing Relations among Various Measures by Using Well  Known Inequalities
Establishing Relations among Various Measures by Using Well  Known Inequalities
Establishing Relations among Various Measures by Using Well  Known Inequalities
Establishing Relations among Various Measures by Using Well  Known Inequalities
Establishing Relations among Various Measures by Using Well  Known Inequalities
Establishing Relations among Various Measures by Using Well  Known Inequalities
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Establishing Relations among Various Measures by Using Well Known Inequalities

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  • 1. International OPEN Journal ACCESS Of Modern Engineering Research (IJMER) Establishing Relations among Various Measures by Using Well Known Inequalities K. C. Jain1, Praphull Chhabra2 1,2 Department of Mathematics, Malaviya National Institute of Technology Jaipur- 302017 (Rajasthan), INDIA ABSTRACT: In this paper, we are establishing many interesting and important relations among several divergence measures by using known inequalities. Actually this work is application of well known inequalities in information theory. Except various relations, we tried to get bounds of * * * Nk  P, Q  , J k  P, Q  , k  P, Q , Ek  P, Q  , S *  P, Q  , L  P, Q  ,  M  P, Q  , R2  P, Q  terms of standard divergence measures. Some relations in terms of Arithmetic Mean A  P, Q  , Geometric  P, Q  , Harmonic Mean H  P, Q  , Heronian Mean N  P, Q  , Contra Harmonic C  P, Q  , Root Mean Square S  P, Q  and Centroidal Mean R  P, Q  , are also obtained. Mean G in * Mean Mathematics Subject Classification 2000: 62B- 10, 94A17, 26D15 Keywords: Standard Inequalities, Divergence Measures, Convex and Normalized function, Csiszar’s Generalized f- Divergence Measure, Seven Standard Means. I. Introduction n    n   P   p1 , p2 , p3 ........., pn  : pi  0,  pi  1 , n  2 be the set of all complete finite discrete i 1   probability distributions. If we take pi ≥0 for some i  1, 2, 3,..., n , then we have to suppose that Let 0 0 f  0   0 f    0 . Csiszar’s f- divergence [1] is a generalized information divergence measure, which is 0 given by: n p  C f  P, Q    qi f  i  i 1  qi  (1) Where f: (0,)  R (set of real no.) is a convex function and P, Q  Γn. Many known divergences can be obtained from these generalized measures by suitably defining the convex function f. By (1), we obtain the following divergence measures: Following measures are due to (Jain and Srivastava [7]). n E  P, Q    * k i 1 n * J k  P, Q    i 1  pi  qi  k 1 k 2 , k  1, 3, 5, 7, ...  pi qi  k 1 2  pi  qi  exp   pi  qi   , k  1, 3, 5, 7, ...   k  pi qi  2    pi qi    (2) (3) Following measures are due to Kumar P. and others. n S  P, Q    *  pi  qi   pi  qi  i 1 | IJMER | ISSN: 2249–6645 | pi qi 2  p q log  i i 2 p q i i  www.ijmer.com   (Kumar P. and Chhina [9])   (4) | Vol. 4 | Iss. 1 | Jan. 2014 |238|
  • 2. Establishing Relations among Various Measures by Using Well Known Inequalities n  M  P, Q    i 1 n L  P, Q    p 2 i  qi 2  2  pi qi   pi  qi  2 pi  qi i 1 2 (Kumar P. and Johnson [11]) 3 2  p q log  i i 2 p q i i  (5)   (Kumar P. and Hunter [10])   (6) Renyi’s second order entropy (Renyi A [12]). pi 2 qi n R2  P, Q    i 1 (7) Puri and Vineze Divergence Measures (Kafka,Osterreicher and Vincze [8]).  k  P, Q    i 1 k 1 pi  qi n  pi  qi  k , k   0,   (8) Relative Jensen- Shannon divergence (Sibson [13]).  2 pi  F  P, Q    pi log   i 1  pi  qi  n (9) Relative Arithmetic- Geometric Divergence (Taneja [14]). n  pi  qi   p  qi  G  P, Q     i   log  2  i 1   2 pi  (10) Arithmetic- Geometric Mean divergence Measure (Taneja [14]). T  P, Q   Where n  p q p q 1 G  P, Q   G  Q, P     i i log  i i  2 p q 2 2 i 1 i i  G  P, Q  is given by (10).     (11) Symmetric Chi- square Divergence (Dragomir, Sunde and Buse [4]). n  pi  qi   pi  qi  i 1 pi qi   P, Q    2 (12) Relative J- Divergence (Dragomir, Gluscevic and Pearce [3]). n  p  qi  J R  P, Q   2  F  Q, P   G  Q, P      pi  qi  log  i    i 1  2qi  Where F  P, Q  and G  P, Q  are given by (9) and (10) respectively. (13) Hellinger Discrimination (Hellinger [5]). h  P, Q   1 n  2 i 1  pi  qi  2 (14) Triangular Discrimination (Dacunha- Castelle [2]). n   P, Q    i 1  pi  qi  2 (15) pi  qi Except above, we obtain the following divergence measures (Due to Jain and Saraswat [6]).  p  q   p q   P, Q    i i 2k 1 exp  i i 2 , k  1, 2, 3,...   i 1  pi  qi    pi  qi     n N * k | IJMER | ISSN: 2249–6645 | 2k 2 www.ijmer.com (16) | Vol. 4 | Iss. 1 | Jan. 2014 |239|
  • 3. Establishing Relations among Various Measures by Using Well Known Inequalities II. Well Known Inequalities The following inequalities are famous in literature of pure and applied mathematics, which are important tools to prove many interesting and important results in information theory. 1  t  et  1  t et , t  0 t  log 1  t   t , t  0 1 t (17) (18) III. Relations Among Various Divergence Measures Now, we shall obtain bounds of some measures in terms of other divergence measures and many important and interesting relations among several divergence measures by using inequalities (17) and (18) respectively. Proposition 1: Let (P,Q)Γn×Γn, then we have the inequalities: * * Nk  P, Q   Nk 1  P, Q   2k 1  P, Q  (19) And 2k 1  P, Q   N  P, Q  (20) * Where k  1, 2, 3, ... , and N k  P, Q  ,  k  P, Q  are given by (16) and (8) respectively. 2  pi  qi  in inequalities (17), we get Proof: Put t  2  pi  qi  2 2 2 2  pi  qi   exp  pi  qi   1   pi  qi  exp  pi  qi  , 1 2 2 2 2  pi  qi   pi  qi   pi  qi   pi  qi  2k  pi  qi  , k  1, 2, 3, ... and sum over all i=1, 2, 3…n, we get now multiply the above expression by 2 k 1  pi  qi  2k 2k 2 2k n n n   pi  qi 2   pi  qi   pi  qi   pi  qi     p  q 2k 1   p  q 2k 1   p  q 2k 1 exp  p  q 2  i 1  i i 1  i i 1  i  i i i i i    i. e. n   p  qi    p  qi   p  qi     i  i exp  i 2 k 1 2 k 1 2  i 1  pi  qi  i 1  pi  qi    pi  qi     * * 2k 1  P, Q   2k 1  P, Q   Nk  P, Q   2k 1  P, Q   Nk 1  P, Q  n * k 1 2k 2 2k 2 (21) From second and third part of (21), we get inequality (19) and from first and third part, we get (20). Now at k=1, 2, 3 … we get the followings [from inequalities (19) and (20)]: At k=1  * N1*  P, Q   N2  P, Q     P, Q  and * 3  P, Q   N2  P, Q  At k=2     P, Q     P, Q  (22) 1 * * N2  P, Q   N3  P, Q   3  P, Q  and * 5  P, Q   N3  P, Q  At k=3  * * N3  P, Q   N4  P, Q   5  P, Q  and * 7  P, Q   N4  P, Q  and so on… Proposition 2: Let (P,Q)Γn×Γn, then we have the inequalities: * * J k  P, Q   J k 2  P, Q   E*k  P, Q   P, Q   J  P, Q  * Where k  1, 3, 5, ... , and Ek  P, Q  , And E * k 2 (23) * k 2 | IJMER | ISSN: 2249–6645 | J * k  P, Q  are given by (2) and (3) respectively. www.ijmer.com (24) | Vol. 4 | Iss. 1 | Jan. 2014 |240|
  • 4. Establishing Relations among Various Measures by Using Well Known Inequalities Proof: Put t  pi  qi  2 in inequalities (17), we get pi qi  p  qi  1 i  pi qi   p  qi   p  qi   p  qi  ,  exp i  1 i exp i  pi qi   pi qi   pi qi  k 1  pi  qi  Now multiply the above expression by , k  1, 3, 5, ... and sum over all i=1, 2, 3…n, we get k /2  pi qi  k 1 k 3 k 1 n n n   pi  qi 2   pi  qi   pi  qi   pi  qi     exp    p q k /2   k k /2 1 i 1  i i 1 i 1  pi qi    pi qi   2 i    pi qi  2 2  p  qi   i k /2 i 1  pi qi  k 1 n n  2  pi  qi  k k 3 2   pi  qi 2    exp     pi qi      pi qi  2 * * * * * Ek  P, Q   Ek 2  P, Q   J k  P, Q   Ek  P, Q   J k 2  P, Q  i. e. i 1 1 (25) From second and third part of (25), we get inequality (23) and from first and third part, we get (24). Now at k=1, 3, 5 … we get the followings [from inequalities (23) and (24)]: At k=1  * J1*  P, Q   J 3  P, Q   E1*  P, Q  and * E*3  P, Q   J 3  P, Q  At k=3  * * J 3  P, Q   J 5  P, Q   E*3  P, Q  and * E*5  P, Q   J 5  P, Q  and so on… Except these, from first and second part of the inequalities (25), we can easily see that E1*  P, Q   J1*  P, Q  (26) Proposition 3: Let (P,Q)Γn×Γn, then we have the inequalities:   P, Q   2E1*  P, Q   S *  P, Q  (27) And S *  P, Q     P, Q    M  P, Q  Where   P, Q  , E  P, Q  , S  P, Q  and  M  P, Q  are given by (12), (2), (4) and (5) respectively. * 1  Proof: Put t   pi  qi 2  1 i. e.  pi qi pi  qi 2  2 in inequalities (18), we get pi qi pi  qi 2 (28) * 2  2   log 1      pi  qi 2 pi qi  2       pi  qi 2  2 pi qi pi qi pi  qi  2 pi  qi pi qi  p  qi  pi  qi  2 pi qi  log  i  2 pq  2 pi qi i i   | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 1 | Jan. 2014 |241|
  • 5. Establishing Relations among Various Measures by Using Well Known Inequalities Now multiply the above expression by  pi  qi  pi  qi  n   pi  qi  pi  qi  pi qi pi  qi  2 2  pi  qi  pi  qi  n  2 2  pi  qi  pi  qi     pi  qi   pi  qi  2 pi qi i 1 n  2  pi  qi  pi qi 2 pi qi i 1  p  qi  log  i  2 pq  i i   pi qi i.e. n 2 pi qi i 1 pi  qi  2 pi qi i 1 and sum over all i=1, 2, 3…n, we get n pi  qi pi qi i 1 pi qi 2 S n *  P, Q    i 1 p 2 i  qi 2  2  pi qi  3 2 2 n  pi  qi   pi  qi  i 1 pi qi  2   P, Q   2E1*  P, Q   S *  P, Q    M  P, Q    P, Q  i. e. (29) From first and second part of (29), we get inequality (27) and from second and third part, we get (28). Except these, if we add (27) and (28), we get the following 2   P, Q    M  P, Q   2 E1*  P, Q  (30) From second and third part of the inequalities (29), we can easily see that S *  P, Q    M  P, Q  (31) By taking both (27) and (31), we can write   P, Q   2E1*  P, Q   S *  P, Q    M  P, Q  (32) Proposition 4: Let (P,Q)Γn×Γn, then we have the inequalities: 1 * E1  P, Q  2 2 n pi qi  pi  qi    P , Q   L  P , Q   2 2 i 1  pi  qi  L  P, Q     P, Q   And Where pi  qi 2  2 in inequalities (18), we get pi qi pi  qi  2 pi qi pi  qi  p  qi  pi  qi  2 pi qi  log  i  2 pq  2 pi qi i i   Now multiply the above expression by  i 1  pi  qi   pi  qi  2 pi qi pi  qi  pi  qi  2 n  i 1 n  2  pi  qi  pi  qi  pi  qi  pi  qi pi qi  pi  qi  2 2 2 and sum over all i=1, 2, 3…n, we get  p  qi  n log  i  2 pq   i 1 i i    pi  qi  pi  qi n 1 n  p  qi   L  P, Q    i  2 i 1 pi qi i 1 2  pi  qi  2 n pi qi  pi  qi  1   P, Q   2  L  P, Q   E1*  P, Q     P, Q  2 2 i 1  pi  qi  i 1 i. e. 2 pi  qi n i. e. (34) L  P, Q  , E1*  P, Q  ,   P, Q  are given by (6), (2) and (15) respectively.  Proof: Put t  n (33) pi  qi i 1 | IJMER | ISSN: 2249–6645 | 2 www.ijmer.com 2 pi  qi  2 2  pi  qi  pi qi 2 pi  qi (35) | Vol. 4 | Iss. 1 | Jan. 2014 |242| pi qi
  • 6. Establishing Relations among Various Measures by Using Well Known Inequalities From second and third part of (35), we get inequality (33) and from first and second part, we get (34). From inequality (33), we can easily see that 1 * E1  P, Q  2   P, Q   (36) n Proposition 5: Let (P,Q)Γn×Γn and n  p  q i 1 i A  P, Q   h  P, Q   T  P, Q  A  P, Q   h  P , Q   And i i 1  1 , then we have the inequalities: (37) 1 n  pi  qi   pq 4 i 1 i i 2 1 n  p  qi  A  P, Q   T  P , Q    i 4 i 1 pi qi (38) 2 (39) n Where T  P, Q  , h  P, Q  are given by (11) and (14) respectively and A  P, Q    i 1 pi  qi  1 is well 2 known Arithmetic Mean Divergence.  Proof: Put t  pi  qi 2  2 in inequalities (18), we get pi qi  p  qi  pi  qi  2 pi qi  log  i  2 pq  pi  qi 2 pi qi i i   pi  qi Now multiply the above expression by and sum over all i=1, 2, 3…n, we get 2 n  p  q  p  qi  2 p q n  p  q   p  q  n  p  q  p  q  2 p q  i 2 i i p  q i i   i 2 i log  2 i p qi    i 2 i i 2i p q i i   i 1 i 1 i 1 i i i i  i i  pi  qi  2 pi qi n i 1 i. e. pi  qi  2 pi qi 2  n i. e.  i 1  pi  qi 2 n  T  P, Q    i 1  2 n  T  P, Q    i. e. h  P, Q   T  P , Q     pi  qi  4 i 1 pi qi 4 pi qi  pi  qi  i 1 n  pi  qi  4 pi qi 2 1 2 1 2 1 (40) From first and third part of (40), we get inequality (38) and from second and third part, we get (39). Except these, from (38) and (40), we can easily see the followings 1 n  p  qi  A  P, Q    i 4 i 1 pi qi 1 n  p  qi  h  P, Q    i 4 i 1 pi qi And 2 (41) 2 (42) h  P, Q   T  P, Q  (43) Now do (41)-(42), we get | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 1 | Jan. 2014 |243|
  • 7. Establishing Relations among Various Measures by Using Well Known Inequalities A  P, Q   h  P, Q   0  A  P, Q   h  P, Q  (44) By taking both (43) and (44), we get the inequalities (37). n n i 1 Proposition 6: Let (P,Q)Γn×Γn and i 1  pi   qi  1 , then we have the inequalities: 1  log 2 2 1 log 2  G  Q, P    R2  P, Q   1 And  2 Where R2  P, Q  , G  Q, P  are given by (7) and (10) respectively. p Proof: Put t  i in inequalities (18), we get qi G  Q, P   (45) (46)  p q pi  log  i i pi  qi  qi  pi   qi pi  qi Now multiply the above expression by and sum over all i=1, 2, 3…n, we get 2 n n  2 pi  qi  n pi  qi pi pi  qi pi p q   i i log    2 p  q i1 2 i 1 i i  2 qi  i 1 2 qi n  pi  qi  n pi 2 n pi pi pi  qi n pi  qi  2  log 2 2   2 log  2q    2q   2 i 1 i 1 i 1 i   i 1 i i 1 1 1  log 2  G  Q, P    R2  P, Q   1  2 2 n i. e. i. e. (47) From first and second part of (47), we get inequality (45) and from second and third part, we get (46). n n i 1 Proposition 7: Let (P,Q)Γn×Γn and i 1  pi   qi  1 , then we have the inequalities: log 2  F  P, Q   A  P, Q  And (48) 1 H  P, Q   F  P, Q   log 2 2 Where F  P, Q  is given by (9), (49) n A  P, Q    i 1 pi  qi 2 pi qi  1 and H  P, Q    are Arithmetic 2 i 1 pi  qi n Mean and Harmonic Mean Divergences respectively. Proof: Put t  pi in inequalities (18), we get qi  p q  p pi  log  i i   i pi  qi  qi  qi Now multiply the above expression by 2qi and sum over all i=1, 2, 3…n, we get  n pi    2qi qi i 1  i 1 n n n  2qi  H  P, Q   2log 2 qi  2 qi log   2 pi  i 1 i 1 i 1  pi  qi  n  2qi i. e. n  2 pi  qi pi   2qi log  pi  qi i 1  2 qi | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 1 | Jan. 2014 |244|
  • 8. Establishing Relations among Various Measures by Using Well Known Inequalities H  P, Q   2log 2  2F  Q, P   2 i. e. After interchanging P and Q, we get the following H  P, Q   2log 2  2F  P, Q   2 (50) from second and third part of (50), we get inequality (48) and from first and second part, we get (49). Some Relations:  H  P, Q   G*  P, Q   N  P, Q   A  P, Q   R  P, Q   S  P, Q   C  P, Q  (Taneja [15]). (51) The above inequalities (51) is a famous relation among seven means, where H  P, Q  , G  P, Q  , N  P, Q  , A  P, Q  , R  P, Q  , S  P, Q  , C  P, Q  are mentioned in abstract. * Now we can get some other important relations among various divergences with the help of above inequalities, these are as follows.  from (37) and (51), we get H  P, Q   G*  P, Q   N  P, Q   A  P, Q   h  P, Q   T  P, Q   (52) from (48) and (51), we get log 2  F  P, Q   A  P, Q   R  P, Q   S  P, Q   C  P, Q   (53) from (37) and (48), we get log 2  F  P, Q   A  P, Q   h  P, Q   T  P, Q   (54) do (46) - (48), we get 1  R2  P, Q   1  A  P, Q   2 i. e. 2 A  P, Q   2 G  Q, P   F  Q, P    R2  P, Q   1   G  Q, P   F  Q, P   i. e.  2 A  P, Q   J R  P, Q   R2  P, Q   1 (55) from (22), (26) and (36), we get * N1*  P, Q   N 2  P, Q     P, Q    1 * 1 E1  P, Q   J1*  P, Q  2 2 (56) 1 * E1  P, Q   L  P, Q  2 (57) from (22) and (23), we get * N1*  P, Q   N 2  P, Q     P, Q   REFERENCES [1]. [2]. [3]. [4]. [5]. [6]. [7]. [8]. [9]. [10]. Csiszar I., Information type measures of differences of probability distribution and indirect observations, Studia Math. Hungarica, 2(1967), 299-318. Dacunha- Castelle D., Ecole d’Ete de Probabilites de Saint-Flour VII-1977, Berlin, Heidelberg, New York: Springer, 1978. Dragomir S.S., Gluscevic V. and Pearce C.E.M, Approximation for the Csiszar f-divergence via midpoint inequalities, in inequality theory and applications - Y.J. Cho, J.K. Kim and S.S. Dragomir (Eds.), Nova Science Publishers, Inc., Huntington, New York, Vol. 1, 2001, pp. 139-154. Dragomir S.S., Sunde J. and Buse C., “New inequalities for Jeffreys divergence measure”, Tamusi Oxford Journal of Mathematical Sciences, 16(2) (2000), 295-309. Hellinger E., Neue begrundung der theorie der quadratischen formen von unendlichen vielen veranderlichen, J. Rein.Aug. Math., 136(1909), 210-271. Jain K.C. and Saraswat R. N., Series of information divergence measures using new f- divergences, convex properties and inequalities, International Journal of Modern Engineering Research (IJMER), vol. 2(2012), pp- 32263231. Jain K.C. and Srivastava A., On symmetric information divergence measures of Csiszar’s f- divergence class, Journal of Applied Mathematics, Statistics and Informatics (JAMSI), 3 (2007), no.1, pp- 85- 102. Kafka P., Osterreicher F. and Vincze I., On powers of f− divergence defining a distance, Studia Sci. Math. Hungar., 26 (1991), 415-422. Kumar P. and Chhina S., A symmetric information divergence measure of the Csiszar’s f-divergence class and its bounds, Computers and Mathematics with Applications, 49(2005),575-588. Kumar P. and Hunter L., On an information divergence measure and information inequalities, Carpathian Journal of Mathematics, 20(1) (2004), 51-66. | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 1 | Jan. 2014 |245|
  • 9. Establishing Relations among Various Measures by Using Well Known Inequalities [11]. [12]. [13]. [14]. [15]. Kumar P. and Johnson A., On a symmetric divergence measure and information inequalities, Journal of Inequalities in Pure and Applied Mathematics, 6(3) (2005), Article 65, 1-13. Renyi A., On measures of entropy and information, Proc. 4th Berkeley Symposium on Math. Statist. and Prob., 1(1961), 547-561. Sibson R., Information radius, Z. Wahrs. Undverw. Geb., (14) (1969),149-160. Taneja I.J., New developments in generalized information measures, Chapter in: Advances in Imaging and Electron Physics, Ed. P.W. Hawkes, 91(1995), 37-135. Taneja, I.J. Inequalities having seven means and proportionality relations, 2012. Available online: http://arxiv.org/abs/1203.2288/ (accessed on 7 April 2013). | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 1 | Jan. 2014 |246|

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