Research Article Bounds on Nonsymmetric Divergence Measure in terms of Other Symmetric and Nonsymmetric Divergence Measures

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1 Iteratioal Scholarly Research Notices Volume 04, Article ID 8037, 9 pages Research Article Bouds o Nosymmetric Divergece Measure i terms of Other Symmetric ad Nosymmetric Divergece Measures K. C. Jai, ad Praphull Chhabra Departmet of Mathematics, Malaviya Natioal Istitute of Techology, Jaipur, Rajastha 3007, Idia B-, Staff Coloy, MNIT, Jaipur, Rajastha 3007, Idia Correspodece should be addressed to K. C. Jai; jaikc 003@yahoo.com Received Jue 04; Accepted September 04; Published 9 October 04 Academic Editor: Agelo De Satis Copyright 04 K. C. Jai ad P. Chhabra. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Vajda (97)studiedageeralized divergece measureofcsiszar sclass, socalled Chi-m divergece measure. Variatioal distace ad Chi-square divergece are the special cases of this geeralized divergece measure at ad,respectively.i this work, oparametric osymmetric measure of divergece, a particular part of Vajda geeralized divergece at 4, is take ad characterized. Its bouds are studied i terms of some well-kow symmetric ad osymmetric divergece measures of Csiszar s class by usig well-kow iformatio iequalities. Compariso of this divergece with others is doe. Numerical illustratios (verificatio) regardig bouds of this divergece are preseted as well.. Itroductio Let Γ = {P = (p,p,p 3,...,p ) : p i > 0, p i = }, be the set of all complete fiite discrete probability distributios. If we take p i 0for some,,3,...,,the we have to suppose that 0f(0) = 0f(0/0) = 0. Csiszar [] itroduced a geeralized f-divergece measure, which is give by distributios P, Q Γ. These measures are as follows (see [ 7]): Triagular discrimiatio [] =Δ (p i ), p i + () C f f( p i ), () where f : (0, ) R (set of real umbers) is a real, cotiuous, ad covex fuctio ad P=(p,p,p 3,...,p ), Q=(q,q,q 3,...,q ) Γ,wherep i ad are probability massfuctios.maykowdivergecescabeobtaied from this geeralized measure by suitably defiig the covex fuctio f.someofthoseareasfollows... Symmetric Divergece Measures. Symmetric measures are those that are symmetric with respect to probability Helliger discrimiatio [3] =h(p, Q) JS divergece [4, ] = ( p i ), =I(P, Q) = [ p i log ( p i )+ q p i + log ( )], i p i + (Jai ad Mathur [6]) =P (p i ) 4 (p i + )(p i +q i ), p 3 i q3 i (3) (4) ()

2 Iteratioal Scholarly Research Notices (Jai ad Srivastava [7]) =E (p i ) 4. (6) 3/ (p i ).. Nosymmetric Divergece Measures. Nosymmetric measures are those that are ot symmetric with respect to probability distributios P, Q Γ. These measures are as follows (see [8 0]): Relative J-divergece [8] =J R (P, Q) = Relative iformatio [9] =K Relative AG divergece [0] =G(P, Q) = ( p i + ) log ( p i + ). p i (p i ) log ( p i + ), (7) p i log ( p i ), (8) We ca see that J R [F(Q, P) + G(Q, P)], Δ [ W(P, Q)], h B(P, Q), ad I (/)[F(P, Q) + F(Q, P)], wherew (p i /(p i + )) is the harmoic mea divergece, B p i is the geometric mea divergece, ad F p i log(p i /(p i + )) is the relative JS divergece []. Equatios (3), (4), ad (8) are also kow as Kolmogorov s measure, iformatio radius, ad directed divergece, respectively.. Nosymmetric Divergece Measure ad Its Properties I this sectio, we obtai osymmetric divergece measure for covex fuctio ad further defie the properties of fuctio ad divergece. Firstly, Theorem is well kow i literature []. Theorem. If the fuctio f is covex ad ormalized, that is, f() = 0, thec f (P, Q) ad its adjoit C f (Q, P) are both oegative ad covex i the pair of probability distributio (P, Q) Γ Γ. Now, let f : (0, ) R be a fuctio defied as f (t) =f (t) = (t )4 t 3, t (0, ), f () =0, (9) f (t) = (t )3 (t+3) t 4, (t ) (t) = t. (0) Properties of the fuctio defied by (0) are as follows. (a) Sice f () = 0, f (t) is a ormalized fuctio. (b) Sice (t) 0 for all t (0, ) f (t) is a covex fuctio as well. (c) Sice f (t) < 0 at (0, ) ad f (t) > 0 at (, ) f (t) is mootoically decreasig i (0, ) ad mootoically icreasig i (, ) ad f () = 0, () = 0. Now put f (t) i (); the followig ew divergece measure: χ 4 (Q, P) =C f (p i ) 4. () χ 4 (Q, P) is called adjoit of χ 4 ((p i ) 4 /q 3 i ) (after puttig cojugate covex fuctio of χ 4 (Q, P) i ()) ad it is a particular case of Chi-m divergece measure [] at m = 4, which is give by χ m ( p i m /q m i ), m. Properties of the divergece measure defied i () are as follows. (a) I view of Theorem, wecasaythatχ 4 (Q, P) is covex ad oegative i the pair of probability distributio (P, Q) Γ Γ. (b) χ 4 (Q, P) = 0 if P=Qor p i = (attaiig its miimum value). (c) Sice χ 4 (Q, P) =χ 4 (P,Q) χ 4 (Q, P) is a osymmetric divergece measure. 3. Iformatio Iequalities of Csiszar s Class I this sectio, we are takig well-kow iformatio iequalities o C f (P, Q), whichisgivebytheorem. Such iequalities are, for istace, eeded i order to calculate the relative efficiecy of two divergeces. This theorem is due to literature [], which relates two geeralizedf-divergece measures. Theorem. Let f,f : I R + R be two covex ad ormalized fuctios, that is, f () = f () = 0, adsuppose the followig assumptios. (a) f ad f are twice differetiable o (α, β) where 0< α β<, α =β. (b) There exist the real costats m, M such that m<m ad m f (t) M, () (t) where (t) > 0 for all t (α, β). p 3 i

3 Iteratioal Scholarly Research Notices 3 If P, Q Γ ad satisfyig the assumptio 0<α p i / β<, the oe has the followig iequalities: mc f (P, Q) C f (P, Q) MC f (P, Q), (3) where C f (P, Q) is give by (). 4. Bouds i terms of Symmetric Divergece Measures Nowithissectio,weobtaiboudsofdivergecemeasure () i terms of other symmetric divergece measures by usig Theorem. Propositio 3. Let Δ(P, Q) ad χ 4 (Q, P) be defied as i () ad (),respectively.forp,q Γ, oe has the followig. (a) If 0<α<,the 0 χ 4 (P, Q) 3 max [(α ) (α+) 3 α, (β ) (β + ) 3 β ]Δ(P, Q). (b) If α=,the 0 χ 4 (P, Q) 3(β ) (β + ) 3 Proof. Let us cosider (4) β Δ (P, Q). () f (t) = (t ), t (0, ), t+ f () =0, f (t) = (t )(t+3) (t+), (t) = 8 (t+) 3. (6) Sice (t) > 0 for all t>0ad f () = 0, f (t) is a covex ad ormalized fuctio, respectively. Now put f (t) i (); C f (p i ) p i + =Δ(P, Q). (7) Now, let g(t) = (t)/f (t) = 3(t ) (t + ) 3 /t,where (t) ad f (t) are give by (0)ad(6), respectively, ad g (t) = 3 (t )(t+) ( t) t 6, g (t) = 3(t4 6t 3 t + 0t + ) t 7. If g (t) = 0 t =, t=,adt=. (8) It is clear that g(t) is decreasig i (0, ) ad [, ) but icreasig i [, ). Also g(t) has a miimum ad maximum value at t= ad t=,respectively,becauseg () = 4 > 0 ad g () = 6/6 < 0,so Ad (a) if 0<α<,the if t (0, ) g (t) =g() =0. (9) M= sup g (t) = max {g (α), g (β)} = max [ 3(α ) (α+) 3 α, 3(β ) (β + ) 3 β ]; (b) if α=,the (0) M= sup g (t) =g(β)= 3(β ) (β + ) 3 t (,β) β. () Results (4)ad() are obtaied by usig (), (7), (9), (0), ad () i(3), after iterchagig P ad Q. Propositio 4. Let h(p, Q) ad χ 4 (Q, P) be defied as i (3) ad (),respectively.forp, Q Γ, oe has the followig. (a) If 0<α<,the 0 χ 4 (P, Q) 48max [ (α ) (β ), ]h(p, Q). α 7/ β 7/ (b) If α=,the 0 χ 4 (P, Q) Proof. Let us cosider () 48(β ) β 7/ h (P, Q). (3) ( t) f (t) =, t (0, ), f () =0, f (t) = (t / ), (t) = 4t 3/. (4) Sice (t) > 0 for all t>0ad f () = 0, f (t) is a covex ad ormalized fuctio, respectively. Now put f (t) i (); C f ( p i ) =h(p, Q). ()

4 4 Iteratioal Scholarly Research Notices Now, let g(t) = (t)/f (t) = 48(t ) /t 7/,where (t) ad (t) are give by (0)ad(4), respectively, ad g 4 (t )(7 3t) (t) =, t 9/ g (t) = (6) (t 70t + 63). t / If g (t) = 0 t = ad t=7/3. It is clear that g(t) is decreasig i (0, ) ad [7/3, ) but icreasig i [, 7/3). Also g(t) has a miimum ad maximum value at t= ad t = 7/3,respectively,becauseg () = 96 > 0 ad g (.33) = 9/93 < 0,so if g (t) =g() =0. (7) t (0, ) Ad (a) if 0<α<,the M= sup g (t) = max {g (α),g(β)} = max [ 48(α ) (8) 48(β ), ]; α 7/ β 7/ (b) if α=,the 48(β ) M= sup g (t) =g(β)=. t (,β) β 7/ (9) Results () ad(3) are obtaied by usig (), (), (7), (8), ad (9)i(3), after iterchagig P ad Q. Sice (t) > 0 for all t>0ad f () = 0, f (t) is a covex ad ormalized fuctio, respectively. Now put f (t) i (); C f [ p i log ( p i )+ q p i + log ( )] i p i + =I(P, Q). (33) Now, let g(t) = (t)/f (t) = 4(t ) (t + )/t 4,where (t) ad f (t) are give by (0)ad(3), respectively, ad g (t) = 4 (t ) ( t +t+4) t, g (t) =48( t 3 3 t 4 6 t + 0 t 6 ). (34) If g (t) = 0 t =, t = ( 7)/,adt = ( + 7)/. It is clear that g(t) is decreasig i (0, ) ad [( + 7)/, ) but icreasig i [, ( + 7)/). Also g(t) has a miimum ad maximum value at t= ad t = ( + 7)/, respectively,becauseg () = 96 > 0 ad g (( + 7)/) = 6/86 < 0,so Ad (a) if 0<α<,the if t (0, ) g (t) =g() =0. (3) Propositio. Let I(P, Q) ad χ 4 (Q, P) be defied as i (4) ad (),respectively.forp, Q Γ, oe has the followig. (a) If 0<α<,the 0 χ 4 (P, Q) M= sup g (t) = max {g (α), g (β)} = max [ 4(α ) (α+) α 4, 4(β ) (β + ) β 4 ]; (36) 4max [ (α ) (α+) α 4, (β ) (β + ) β 4 ]I(P, Q). (b) If α=,the (30) 0 χ 4 (P, Q) 4(β ) (β + ) β 4 I (P, Q). (3) Proof. Let us cosider f (t) = t log t+t+ f () =0, f (t) = t log ( t+ ), (t) = t (t+). log ( ), t (0, ), t+ (3) (b) if α=,the M= sup g (t) =g(β)= 4(β ) (β + ) t (,β) β 4. (37) Results (30)ad(3) are obtaied by usig (), (33), (3), (36), ad (37)i(3), after iterchagig P ad Q. Propositio 6. Let P (P, Q) ad χ 4 (Q, P) be defied as i () ad (),respectively. For P, Q Γ,oehas β +β 4 +β 3 +β +β+ P (P, Q) χ 4 (P, Q) α +α 4 +α 3 +α +α+ P (P, Q). (38)

5 Iteratioal Scholarly Research Notices Proof. Let us cosider f (t) = (t )4 (t+)(t +) t 3, t (0, ), f () =0, f (t) = (t )3 (4t 4 +3t 3 +3t +3t+3) t 4, (t ) (t) = t [t +6t 4 +6t 3 +6t + 6t + ]. (39) Sice (t) 0 for all t>0ad f () = 0, f (t) is a covex ad ormalized fuctio, respectively. Now put f (t) i (); C f (p i ) 4 (p i + )(p i +q i ) =P (P, Q). p 3 i q3 i (40) Now, let g(t) = (t)/f (t) = /(t +t 4 +t 3 +t +t+), where (t) ad f (t) are give by (0)ad(39), respectively, ad g (t) = (0t4 +4t 3 +3t +t+) (t +t 4 +t 3 +t +t+) <0. (4) It is clear that g(t) is always decreasig i (0, ),so if g (t) =g(β)= β +β 4 +β 3 +β +β+, M= sup g (t) =g(α) = α +α 4 +α 3 +α +α+. (4) Result (38) isobtaiedbyusig(), (40), ad (4) i(3), after iterchagig P ad Q. Propositio 7. Let E (P, Q) ad χ 4 (Q, P) be defied as i (6) ad (),respectively.forp, Q Γ,oehas 6 β 3/ (β +6β+) E (P, Q) χ 4 (P, Q) Proof. Let us cosider 6 α 3/ (α +6α+) E (P, Q). f (t) = (t )4, t (0, ), t 3/ f () =0, f (t) = (t )3 (t + 3) t /, (t) = 3(t ) (t +6t+) 4t 7/. (43) (44) Sice (t) 0 for all t>0ad f () = 0, f (t) is a covex ad ormalized fuctio, respectively. Now put f (t) i (); C f (p i ) 4 (p i ) 3/ =E (P, Q). (4) Now, let g(t) = (t)/f (t) = 6/t3/ (t +6t+),where (t) ad f (t) are give by (0)ad(44), respectively, ad g (t) = 40 (7t +6t+3) <0. (46) t / (t +6t+) It is clear that g(t) is always decreasig i (0, ),so if g (t) =g(β)= 6 β 3/ (β +6β+), 6 M= sup g (t) =g(α) = α 3/ (α +6α+). (47) Result (43) isobtaiedbyusig(), (4), ad (47) i(3), after iterchagig P ad Q.. Bouds i terms of Nosymmetric Divergece Measures Nowithissectio,weobtaiboudsofdivergecemeasure () i terms of other osymmetric divergece measures by usig Theorem. Propositio 8. Let J R (P, Q) ad χ 4 (Q, P) be defied as i (7) ad (),respectively.forp, Q Γ, oe has the followig. (a) If 0<α<,the 0 χ 4 (α ) (P, Q) max [ (α+3) α, (β ) (β + 3) β ] J R (Q, P). [ ] (48) (b) If α=,the Proof. Let us cosider 0 χ 4 (P, Q) (β ) (β + 3) β J R (Q, P). (49) f (t) = (t ) log ( t+ ), t (0, ), f () =0, f (t) = (t )+log (t+ t+ ), (t+3) (t) = (t+). (0)

6 6 Iteratioal Scholarly Research Notices Sice (t) > 0 for all t>0ad f () = 0, f (t) is a covex ad ormalized fuctio, respectively. Now put f (t) i (); C f (p i ) log ( p i + )=J R (P, Q). () Proof. Let us cosider f (t) =tlog t, t (0, ), f () =0, f (t) =+log t, (8) Now, let g(t) = (t)/f (t) = (t ) /(t + 3)t,where (t) ad f (t) are give by (0)ad(0), respectively, ad g (t) = (t )(t 3 +3t 6t ) (t+3) t 6, g (t) = 4 ( 3t6 9t +t 4 +90t 3 +87t 0t 3) (t+3) t 7. () If g (t) = 0 t =, t=,adt 6/9. It is clear that g(t) is decreasig i (0, ) ad [6/9, ) but icreasig i [, 6/9). Also g(t) has a miimum ad maximum value at t= ad t = 6/9, respectively,becauseg () = 4 > 0 ad g (6/9) 7/0 < 0,so Ad (a) if 0<α<,the if t (0, ) g (t) =g() =0. (3) M= sup g (t) = max {g (α),g(β)} (α ) = max [ (α+3) α, (β ) (β + 3) β ] ; [ ] (b) if α=,the (4) M= sup g (t) =g(β) = (β ) t (,β) (β + 3) β. () Results (48) ad(49) are obtaied by usig (), (), (3), (4), ad ()i(3), after iterchagig P ad Q. Propositio 9. Let K(P, Q) ad χ 4 (Q, P) be defied as i (8) ad (),respectively.forp, Q Γ, oe has the followig. (a) If 0<α<,the 0 χ 4 (P, Q) max [ (α ) (β ) α 4, β 4 ]K(Q, P). (b) If α=,the 0 χ 4 (P, Q) (6) (β ) β 4 K (Q, P). (7) (t) = t. Sice (t) > 0 for all t>0ad f () = 0, f (t) is a covex ad ormalized fuctio, respectively. Now put f (t) i (); C f p i log ( p i )=K(P, Q). (9) Now, let g(t) = (t)/f (t) = (t ) /t 4,where (t) ad (t) are give by (0)ad(8), respectively, ad g 4 (t )( t) (t) = t, g (t) =4( 3 t 4 t + 0 (60) t 6 ). If g (t) = 0 t = ad t=. It is clear that g(t) is decreasig i (0, ) ad [, ) but icreasig i [, ). Also g(t) has a miimum ad maximum value at t= ad t=,respectively,becauseg () = 4 > 0 ad g () = 3/4 < 0,so if g (t) =g() =0. (6) t (0, ) Ad (a) if 0<α<,the M= sup g (t) = max {g (α),g(β)} (6) = max [ (α ) (β ) α 4, β 4 ]; (b) if α=,the (β ) M= sup g (t) =g(β)= t (,β) β 4. (63) Results (6) ad(7) are obtaied by usig (), (9), (6), (6), ad (63) i(3), after iterchagig P ad Q. Propositio 0. Let G(P, Q) ad χ 4 (Q, P) be defied as i (9) ad (),respectively.forp, Q Γ, oe has the followig. (a) If 0<α<,the 0 χ 4 (P, Q) 4max [ (α ) (α+) α 3, (β ) (β + ) β 3 ]G(Q, P). (64)

7 Iteratioal Scholarly Research Notices 7 Table : (=0,p=/,q=/). x i p(x i )=p i q(x i )= p i (b) If α=,the 0 χ 4 (P, Q) 4(β ) (β + ) β 3 G (Q, P). (6) Proof. Let us cosider f (t) = t+ f () =0, log ( t+ ), t (0, ), t f (t) = [log (t+ t ) t ], (t) = t (t+). (66) Sice (t) > 0 for all t>0ad f () = 0, f (t) is a covex ad ormalized fuctio, respectively. Now put f (t) i (); C f ( p i + ) log ( p i + )=G(P, Q). (67) p i Now, let g(t) = (t)/f (t) = 4(t ) (t + )/t 3,where (t) ad f (t) are give by (0)ad(66), respectively, ad g 4 (t )(t+3) (t) = t 4, g (t) =4( t 3 6 t 4 + t ). (68) If g (t) = 0 t = ad t= 3. It is clear that g(t) is decreasig i (0, ) ad icreasig i [, ). Also g(t) has a miimum value at t=,becauseg () = 0 > 0,so Ad (a) if 0<α<,the M= sup g (t) = max {g (α), g (β)} if t (0, ) g (t) =g() =0. (69) = max [ 4(α ) (α+) α 3, 4(β ) (β + ) β 3 ]; (70) (b) if α=,the M= sup g (t) =g(β)= 4(β ) (β + ) t (,β) β 3. (7) Results (64) ad(6) are obtaied by usig (), (67), (69), (70), ad (7) i(3), after iterchagig P ad Q. 6. Numerical Illustratio I this sectio, we give two examples for calculatig the divergeces Δ(P, Q), h(p, Q), J R (Q, P), K(Q, P),adχ 4 (P, Q) ad verify iequalities (4), (), (48), ad (6) orverify bouds of χ 4 (P, Q). Example. Let P be the biomial probability distributio with parameters ( = 0, p = /) ad Q its approximated Poisso probability distributio with parameter (λ = p = ), for the radom variable X;the we have Table. By usig Table, the followig: α(= 6 ) p i β(= ), Δ h J R (Q, P) = K (Q, P) = χ 4 (p i ) p i + 9, ( p i ) 3 0, ( p i ) log ( p i + p i ) 3 397, log ( ) 9 p i 70, (p i ) 4 q 3 i 9 7. (7) Put the approximated umerical values from (7) i(4), (), (48), ad (6) ad verify iequalities (4), (), (48), ad (6) for p=/.

8 8 Iteratioal Scholarly Research Notices Table : ( = 0, p = 7/0, q = 3/0). x i p(x i )=p i q(x i )= p i Example. Let P be the biomial probability distributio with parameters ( =0, p = 7/0) adq its approximated Poisso probability distributio with parameter (λ = p = 7), for the radom variable X;the we have Table. By usig Table,wegetthefollowig: α(= 77 ) p i β(= ), Δ h J R (Q, P) = K (Q, P) = χ 4 (p i ) p i , ( p i ) 498, ( p i ) log ( p i + p i ) , log ( ) 4 p i 77, (p i ) 4 q 3 i (73) Put the approximated umerical values from (73) i(4), (), (48), ad (6)adverifyiequalities(4), (), (48), ad (6) for p = 7/0. Similarly, we ca verify iequalities (or verify bouds of χ 4 (P, Q))(30), (38), (43), ad (64). Figure shows the behavior of covex fuctio i (0, ). Fuctio is decreasig i (0, ) ad icreasig i (, ). Figure shows the behavior of χ 4 (P, Q), E (P, Q), K(P, Q), Δ(P, Q), adg(p, Q). Wehavecosideredp i = (a, a), = ( a, a), wherea (0, ). ItisclearfromFigure that the divergece χ 4 (P, Q) has a steeper slope tha E (P, Q), K(P, Q), Δ(P, Q),adG(P, Q) Figure : Covex fuctio f (t) Chi divergece () Divergece (6) Relative etropy a Triagular discrimiatio Relative AG divergece Figure : Compariso of divergeces. 7. Coclusios May research papers have bee studied by Taeja, Kumar, Dragomir, Jai, ad others, who gave the idea of divergece measures, their properties, their bouds, ad relatios with other measures. Taeja especially did a lot of quality work i this field: for istace, i [3] he derived bouds o differet osymmetric divergeces i terms of differet symmetric divergeces, i [4] he itroduced ew geeralized divergeces ad ew divergeces as a result of differece of meas ad characterized their properties ad bouds, ad i [] ew iequalities amog oegative differeces arisig from seve meas have bee itroduced ad correlatios with geeralized triagular discrimiatio ad some ew geeratig measures with their expoetial represetatios have also bee preseted.

9 Iteratioal Scholarly Research Notices 9 Divergece measures have bee demostrated to be very useful i a variety of disciplies such as athropology, geetics, fiace, ecoomics ad political sciece, biology, aalysis of cotigecy tables, approximatio of probability distributios, sigal processig, patter recogitio, sesor etworks, testig of the order i a Markov chai, risk for biary experimets, regio segmetatio, ad estimatio. This paper also defies the properties ad bouds of Vajda s divergece ad derives ew relatios with other symmetric ad osymmetric well-kow divergece measures. [4] I. J. Taeja, O symmetric ad o symmetric divergece measures ad their geeralizatios, Advaces i Imagig ad Electro Physics,vol.38,pp.77 0,00. [] I. J. Taeja, Seve meas, geeralized triagular discrimiatio, ad geeratig divergece measures, Iformatio, vol.4, o., pp , 03. Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper. Refereces [] I. Csiszar, Iformatio type measures of differeces of probability distributio ad idirect observatios, Studia Scietiarum Mathematicarum Hugarica, vol., pp , 967. [] D. Dacuha-Castelle, Ecole d Ete de Probabilites de Sait-Flour VII 977, Spriger, Berli, Germay, 978. [3] E. Helliger, Neue begrüdug der theorie der quadratische forme vo uedliche viele veräderliche, Joural für Die Reie ud Agewadte Mathematik,vol.36,pp.0 7,909. [4] J. Burbea ad C. R. Rao, O the covexity of some divergece measures based o etropy fuctios, IEEE Trasactios o Iformatio Theory,vol.8,o.3,pp ,98. [] R. Sibso, Iformatio radius, Zeitschrift für Wahrscheilichkeitstheorie ud Verwadte Gebiete, vol.4,o.,pp.49 60, 969. [6] K. C. Jai ad R. Mathur, A symmetric divergece measure ad its bouds, Tamkag Joural of Mathematics, vol. 4, o. 4, pp , 0. [7] K. C. Jai ad A. Srivastava, O symmetric iformatio divergece measures of Csiszar s f-divergece class, Joural of Applied Mathematics, Statistics ad Iformatics, vol.3,o., 007. [8] S. S. Dragomir, V. Gluscevic, ad C. E. M. Pearce, Approximatio for the Csiszar f-divergece via midpoit iequalities, i Iequality Theory ad Applicatios, Y.J.Cho,J.K.Kim,adS. S.Dragomir,Eds.,vol.,pp.39 4,NovaSciecePublishers, Hutigto, NY, USA, 00. [9] S. Kullback ad R. A. Leibler, O iformatio ad sufficiecy, Aals of Mathematical Statistics,vol.,pp.79 86,9. [0] I. J. Taeja, New developmets i geeralized iformatio measures, Advaces i Imagig ad Electro Physics,vol.9,pp. 37 3, 99. [] I. Vajda, O the f-divergece ad sigularity of probability measures, Periodica Mathematica Hugarica, vol., pp. 3 34, 97. [] I. J. Taeja, Geeralized symmetric divergece measures ad iequalities, RGMIA Research Report Collectio, vol.7,o.4, article 9, 004. [3] I. J. Taeja, Bouds o o-symmetric divergece measures i terms of symmetric divergece measures, Joural of Combiatorics, Iformatio & System Scieces, vol.9,o. 4,pp. 34, 00.

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