Establishing Relations among Various Measures by Using Well Known Inequalities

Transcription

1 Iteratoal OPEN ACCESS Joural Of Moder Egeerg Research (IJMER) Establshg Relatos amog Varous Measures by Usg Well Kow Ieualtes K. C. Ja, Prahull Chhabra, Deartmet of Mathematcs, Malavya Natoal Isttute of Techology Jaur (Rajastha), INDIA ABSTRACT: I ths aer, we are establshg may terestg ad mortat relatos amog several dvergece measures by usg ow eualtes. Actually ths wor s alcato of well ow eualtes formato theory. Excet varous relatos, we tred to get bouds of N P,, J P,, P,, E P,, S P,, L P,, M P,, R P, terms of stadard dvergece measures. Some relatos terms of Arthmetc Mea AP,, Geometrc Mea G P,, Harmoc Mea H P,, Heroa Mea N P,, Cotra Harmoc Mea C P,, Root Mea Suare S P, ad Cetrodal Mea, Mathematcs Subject Classfcato 000: 6B- 0, 94A7, 6D5 R P, are also obtaed. Keywords: Stadard Ieualtes, Dvergece Measures, Covex ad Normalzed fucto, Csszar s Geeralzed f- Dvergece Measure, Seve Stadard Meas. I. Itroducto Let P,, 3..., : 0,, be the set of all comlete fte dscrete robablty dstrbutos. If we tae 0 for some,, 3,...,, the we have to suose that 0 0 f 0 0 f 0. Csszar s f- dvergece [] s a geeralzed formato dvergece measure, whch s 0 gve by: C f P, f () Where f: (0,) R (set of real o.) s a covex fucto ad P, Γ. May ow dvergeces ca be obtaed from these geeralzed measures by sutably defg the covex fucto f. By (), we obta the followg dvergece measures: Followg measures are due to (Ja ad Srvastava [7]). E P,,, 3, 5, 7,... () J P, ex,, 3, 5, 7,... (3) Followg measures are due to Kumar P. ad others. S P, log (Kumar P. ad Chha [9]) (4) IJMER ISSN: Vol. 4 Iss. Ja

2 M P, Establshg Relatos amog Varous Measures by Usg Well Kow Ieualtes 3 (Kumar P. ad Johso []) (5) LP, log (Kumar P. ad Huter [0]) (6) Rey s secod order etroy (Rey A []). R P, (7) Pur ad Veze Dvergece Measures (Kafa,Osterrecher ad Vcze [8]). P,, 0, (8) Relatve Jese- Shao dvergece (Sbso [3]). F P, log (9) Relatve Arthmetc- Geometrc Dvergece (Taeja [4]). GP, log (0) Arthmetc- Geometrc Mea dvergece Measure (Taeja [4]). T P, G P, G, P log G P, s gve by (0). Where Symmetrc Ch- suare Dvergece (Dragomr, Sude ad Buse [4]). () P, () Relatve J- Dvergece (Dragomr, Gluscevc ad Pearce [3]). J R P, F, P G, P log (3) F P, ad G P, are gve by (9) ad (0) resectvely. Where Hellger Dscrmato (Hellger [5]). h P (4), Tragular Dscrmato (Dacuha- Castelle []). P, (5) Excet above, we obta the followg dvergece measures (Due to Ja ad Saraswat [6]). N P, ex,,, 3,... (6) IJMER ISSN: Vol. 4 Iss. Ja

3 Establshg Relatos amog Varous Measures by Usg Well Kow Ieualtes II. Well Kow Ieualtes The followg eualtes are famous lterature of ure ad aled mathematcs, whch are mortat tools to rove may terestg ad mortat results formato theory. t t t e t e, t 0 (7) t log t t, t 0 t (8) III. Relatos Amog Varous Dvergece Measures Now, we shall obta bouds of some measures terms of other dvergece measures ad may mortat ad terestg relatos amog several dvergece measures by usg eualtes (7) ad (8) resectvely. Proosto : Let (P,)Γ Γ, the we have the eualtes: N P, N P, P, (9) Ad P, N P, (0) Where,, 3,..., P, are gve by (6) ad (8) resectvely. Proof: Put, ad N P, t eualtes (7), we get ex ex ow multly the above exresso by,,, 3,... ex. e. P, P, N P, P, N P,, ad sum over all =,, 3, we get ex () From secod ad thrd art of (), we get eualty (9) ad from frst ad thrd art, we get (0). Now at =,, 3 we get the followgs [from eualtes (9) ad (0)]: N P, N P, P, ad P, P, () At = 3P, NP, At = N P, N3 P, 3 P, ad 5P, N3 P, At =3 N 3 P, N4 P, 5 P, ad P, N P, ad so o 7 4 Proosto : Let (P,)Γ Γ, the we have the eualtes: J P, J P, E P, Ad E P, J P,, ad E P,, J, Where, 3, 5,... P are gve by () ad (3) resectvely. (3) (4) IJMER ISSN: Vol. 4 Iss. Ja

4 Proof: Put t Establshg Relatos amog Varous Measures by Usg Well Kow Ieualtes eualtes (7), we get ex ex, Now multly the above exresso by,, 3, 5,... / 3 ex /. e. E P, E P, J P, E P, J P, ad sum over all =,, 3, we get 3 ex / / (5) From secod ad thrd art of (5), we get eualty (3) ad from frst ad thrd art, we get (4). Now at =, 3, 5 we get the followgs [from eualtes (3) ad (4)]: At =, 3,, E 3 P, J3 P, At =3 3, 5, 3, E P, J P, J P J P E P ad J P J P E P ad 5 5 ad so o Excet these, from frst ad secod art of the eualtes (5), we ca easly see that E P, J P, (6) Proosto 3: Let (P,)Γ Γ, the we have the eualtes: P, E P, S P, Ad S P, P, M P, Where P,, E P,, S P, ad M P, (7) (8) are gve by (), (), (4) ad (5) resectvely. Proof: Put t eualtes (8), we get. e. log log IJMER ISSN: Vol. 4 Iss. Ja. 04 4

5 Now multly the above exresso by.e. Establshg Relatos amog Varous Measures by Usg Well Kow Ieualtes ad sum over all =,, 3, we get log S P, 3. e. P, E P, S P, M P, P, (9) From frst ad secod art of (9), we get eualty (7) ad from secod ad thrd art, we get (8). Excet these, f we add (7) ad (8), we get the followg P M P E P,,, (30) From secod ad thrd art of the eualtes (9), we ca easly see that S P, M P, (3) By tag both (7) ad (3), we ca wrte P, E P, S P, M P, (3) Proosto 4: Let (P,)Γ Γ, the we have the eualtes: L P P E P,,, Ad P LP,, Where LP,, E P,, P, Proof: Put t eualtes (8), we get (33) (34) are gve by (6), () ad (5) resectvely. log Now multly the above exresso by ad sum over all =,, 3, we get log. e. L P,. e. P, L P, E P, P, (35) IJMER ISSN: Vol. 4 Iss. Ja. 04 4

6 Establshg Relatos amog Varous Measures by Usg Well Kow Ieualtes From secod ad thrd art of (35), we get eualty (33) ad from frst ad secod art, we get (34). From eualty (33), we ca easly see that P, E P, (36) Proosto 5: Let (P,)Γ Γ ad, the we have the eualtes:,,, A P h P T P, hp, A P AP, T P, Ad 4 (38) 4 (39) Where T P,, hp, are gve by () ad (4) resectvely ad AP ow Arthmetc Mea Dvergece. Proof: Put t eualtes (8), we get log Now multly the above exresso by ad sum over all =,, 3, we get IJMER ISSN: Vol. 4 Iss. Ja (37), s well log. e. T P. e., 4 T P, 4. e. hp T P,, 4 (40) From frst ad thrd art of (40), we get eualty (38) ad from secod ad thrd art, we get (39). Excet these, from (38) ad (40), we ca easly see the followgs A P, h P, (4) 4 (4) 4 Ad hp, T P, Now do (4)-(4), we get (43)

7 Establshg Relatos amog Varous Measures by Usg Well Kow Ieualtes, hp, 0 AP, hp, A P By tag both (43) ad (44), we get the eualtes (37). (44) Proosto 6: Let (P,)Γ Γ ad, the we have the eualtes: G, P log (45) log G, P R P, (46) R P,, G, P are gve by (7) ad (0) resectvely. t eualtes (8), we get Ad Where Proof: Put log Now multly the above exresso by. e. ad sum over all =,, 3, we get log log log log G, P R P, (47). e. From frst ad secod art of (47), we get eualty (45) ad from secod ad thrd art, we get (46). Proosto 7: Let (P,)Γ Γ ad, the we have the eualtes: AP log F P,, Ad Where,,, log H P F P (49) A P, ad H P, F P s gve by (9), Mea ad Harmoc Mea Dvergeces resectvely. Proof: Put t eualtes (8), we get log (48) are Arthmetc Now multly the above exresso by ad sum over all =,, 3, we get H P, log log log. e. IJMER ISSN: Vol. 4 Iss. Ja

8 Establshg Relatos amog Varous Measures by Usg Well Kow Ieualtes. e. H P F P, log, After terchagg P ad, we get the followg F P H P, log, (50) from secod ad thrd art of (50), we get eualty (48) ad from frst ad secod art, we get (49). Some Relatos:,,,,,,, H P G P N P A P R P S P C P (Taeja [5]). (5) The above eualtes (5) s a famous relato amog seve meas, where,,,,,,,,,,,,, H P G P N P A P R P S P C P are metoed abstract. Now we ca get some other mortat relatos amog varous dvergeces wth the hel of above eualtes, these are as follows. from (37) ad (5), we get,,,,,, H P G P N P A P h P T P (5) from (48) ad (5), we get log F P, A P, R P, S P, C P, (53) from (37) ad (48), we get log F P, A P, h P, T P, (54) do (46) - (48), we get G, P F, P R P, AP, A P, G, P F, P R P, A P, J P, R P, (55). e.. e. R from (), (6) ad (36), we get N P, N P, P, E P, J P, (56) from () ad (3), we get N P, N P, P, E P, LP, (57) REFERENCES []. Csszar I., Iformato tye measures of dffereces of robablty dstrbuto ad drect observatos, Studa Math. Hugarca, (967), []. Dacuha- Castelle D., Ecole d Ete de Probabltes de Sat-Flour VII-977, Berl, Hedelberg, New Yor: Srger, 978. [3]. Dragomr S.S., Gluscevc V. ad Pearce C.E.M, Aroxmato for the Csszar f-dvergece va mdot eualtes, eualty theory ad alcatos - Y.J. Cho, J.K. Km ad S.S. Dragomr (Eds.), Nova Scece Publshers, Ic., Hutgto, New Yor, Vol., 00, [4]. Dragomr S.S., Sude J. ad Buse C., New eualtes for Jeffreys dvergece measure, Tamus Oxford Joural of Mathematcal Sceces, 6() (000), [5]. Hellger E., Neue begrudug der theore der uadratsche forme vo uedlche vele veraderlche, J. Re.Aug. Math., 36(909), 0-7. [6]. Ja K.C. ad Saraswat R. N., Seres of formato dvergece measures usg ew f- dvergeces, covex roertes ad eualtes, Iteratoal Joural of Moder Egeerg Research (IJMER), vol. (0), [7]. Ja K.C. ad Srvastava A., O symmetrc formato dvergece measures of Csszar s f- dvergece class, Joural of Aled Mathematcs, Statstcs ad Iformatcs (JAMSI), 3 (007), o., [8]. Kafa P., Osterrecher F. ad Vcze I., O owers of f dvergece defg a dstace, Studa Sc. Math. Hugar., 6 (99), [9]. Kumar P. ad Chha S., A symmetrc formato dvergece measure of the Csszar s f-dvergece class ad ts bouds, Comuters ad Mathematcs wth Alcatos, 49(005), [0]. Kumar P. ad Huter L., O a formato dvergece measure ad formato eualtes, Caratha Joural of Mathematcs, 0() (004), IJMER ISSN: Vol. 4 Iss. Ja

9 Establshg Relatos amog Varous Measures by Usg Well Kow Ieualtes []. Kumar P. ad Johso A., O a symmetrc dvergece measure ad formato eualtes, Joural of Ieualtes Pure ad Aled Mathematcs, 6(3) (005), Artcle 65, -3. []. Rey A., O measures of etroy ad formato, Proc. 4th Bereley Symosum o Math. Statst. ad Prob., (96), [3]. Sbso R., Iformato radus, Z. Wahrs. Udverw. Geb., (4) (969), [4]. Taeja I.J., New develomets geeralzed formato measures, Chater : Advaces Imagg ad Electro Physcs, Ed. P.W. Hawes, 9(995), [5]. Taeja, I.J. Ieualtes havg seve meas ad roortoalty relatos, 0. Avalable ole: htt://arxv.org/abs/03.88/ (accessed o 7 Arl 03). IJMER ISSN: Vol. 4 Iss. Ja