ONE SILVESTRU SEVER DRAGOMIR 1;2

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1 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE SILVESTRU SEVER DRAGOMIR ; Abstract. In this paper we introuce some -ivergence measures that are relate to the Jensen s ivergence introuce by Burbea an Rao in 98. We establish their joint convexity an provie some inequalities between these measures an a combination o Csiszár s -ivergence, -mipoint ivergence an -integral ivergence measures.. Introuction Let (; A) be a measurable space satisying jaj > an be a - nite measure on (; A) : Let P be the set o all probability measures on (; A) which are absolutely continuous with respect to : For P; Q P, let p = P Q an q = enote the Raon-Nikoym erivatives o P an Q with respect to : Two probability measures P; Q P are sai to be orthogonal an we enote this by Q? P i P (q = g) = Q (p = g) = : Let : [; )! ( ; ] be a convex unction that is continuous at ; i.e., () = lim u# (u) : In 963, I. Csiszár [4] introuce the concept o -ivergence as ollows. De nition. Let P; Q P. Then (.) I (Q; P ) = (x) ; is calle the -ivergence o the probability istributions Q an P: Remark. Observe that, the integran in the ormula (.) is une ne when = : The way to overcome this problem is to postulate or as above that (.) = lim u ; x : u# u We now give some examples o -ivergences that are well-known an oten use in the literature (see also [3]). 99 Mathematics Subject Classi cation. 94A7, 6D5. Key wors an phrases. Jensen ivergence, -ivergence measures, HH -ivergence measures, -ivergence.

2 S. S. DRAGOMIR.. The Class o -Divergences. The -ivergences o this class, which is generate by the unction ; [; ); e ne by have the orm (.3) I (Q; P ) = (u) = ju j ; u [; ) p q p = p jq pj : From this class only the parameter = provies a istance in the topological sense, namely the total variation istance V (Q; P ) = R jq pj : The most prominent special case o this class is, however, Karl Pearson s -ivergence q (Q; P ) = p that is obtaine or = :.. Dichotomy Class. From this class, generate by the unction : [; )! R 8 u ln u or = ; >< (u) = ( ) [u + u ] or Rn ; g ; >: u + u ln u or = ; only the parameter = (u) = (p u ) provies a istance, namely, the Hellinger istance H (Q; P ) = ( p p q p) : Another important ivergence is the Kullback-Leibler ivergence obtaine or = ; q KL (Q; P ) = q ln : p.3. Matsushita s Divergences. The elements o this class, which is generate by the unction ' ; (; ] given by ' (u) := j u j ; u [; ); are prototypes o metric ivergences, proviing the istances I ' (Q; P ) :.4. Puri-Vincze Divergences. This class is generate by the unctions ; [; ) given by j uj (u) := ; u [; ): (u + ) It has been shown in [6] that this class provies the istances [I (Q; P )] :

3 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE 3.5. Divergences o Arimoto-type. This class is generate by the unctions 8 h i ( + u ) ( + u) or (; ) n g ; >< (u) := ( + u) ln + u ln u ( + u) ln ( + u) or = ; >: j uj or = : It has been shown in [8] that this class provies the istances [I (Q; P ) )]min(; or (; ) an V (Q; P ) or = : For continuous convex on [; ) we obtain the -conjugate unction o by (u) = u ; u (; ) u an () = lim (u) : u# It is also known that i is continuous convex on [; ) then so is : The ollowing two theorems contain the most basic properties o -ivergences. For their proos we reer the reaer to Chapter o [7] (see also [3]). Theorem (Uniqueness an Symmetry Theorem). Let ; be continuous convex on [; ): We have I (Q; P ) = I (Q; P ) ; or all P; Q P i an only i there exists a constant c R such that or any u [; ): (u) = (u) + c (u ) ; Theorem (Range o Values Theorem). Let : [; )! R be a continuous convex unction on [; ): For any P; Q P, we have the ouble inequality (.4) () I (Q; P ) () + () : (i) I P = Q; then the equality hols in the rst part o (.4). I is strictly convex at ; then the equality hols in the rst part o (.4) i an only i P = Q; (ii) I Q? P; then the equality hols in the secon part o (.4). I () + () < ; then equality hols in the secon part o (.4) i an only i Q? P: The ollowing result is a re nement o the secon inequality in Theorem (see [3, Theorem 3]). Theorem 3. Let be a continuous convex unction on [; ) with () = ( is normalise) an () + () < : Then (.5) I (Q; P ) [ () + ()] V (Q; P ) or any Q; P P. For other inequalities or -ivergence see [], [7]-[].

4 4 S. S. DRAGOMIR. Some Preliminary Facts For a unction e ne on an interval I o the real line R, by ollowing the paper by Burbea & Rao [], we consier the J -ivergence between the elements t; s I given by J (t; s) := [ (t) + (s)] : As important examples o such ivergences, we can consier [], 8 < ( ) ( t+s ) ; 6= ; J (t; s) := : t ln (t) + s ln (s) () ln t+s ; = : I is convex on I; then J (t; s) or all (t; s) I I: The ollowing result concerning the joint convexity o J also hols: Theorem 4 (Burbea-Rao, 98 []). Let be a C unction on an interval I: Then J is convex (concave) on I I; i an only i is convex (concave) an is concave (convex) on I: We e ne the Hermite-Haamar trapezoi an mi-point ivergences (.) T (t; s) := [ (t) + (s)] (( an (.) M (t; s) := or all (t; s) I I: We observe that ) ) (( ) ) (.3) J (t; s) = T (t; s) + M (t; s) or all (t; s) I I: I is convex on I; then by Hermite-Haamar inequalities (a) + (b) a + b (( ) a + b) or all a; b I; we have the ollowing unamental acts (.4) T (t; s) an M (t; s) or all (t; s) I I: Using Bullen s inequality, see or instance [, p. ], a + b (( ) a + b) we also have (a) + (b) (( (.5) M (t; s) T (t; s) : Let us recall the ollowing special means: ) a + b)

5 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE 5 a) The arithmetic mean A (a; b) := a + b ; a; b > ; b) The geometric mean c) The harmonic mean ) The ientric mean 8 >< I (a; b) := >: G (a; b) := p ab; a; b ; H (a; b) := a + ; a; b > ; b b b e a a b a i b 6= a a i b = a ; a; b > e) The logarithmic mean 8 b a >< i b 6= a L (a; b) := ln b ln a ; a; b > >: a i b = a ) The p-logarithmic mean 8 b p+ a >< p+ p i b 6= a; p Rn ; g L p (a; b) := (p + ) (b a) ; a; b > : >: a i b = a I we put L (a; b) := I (a; b) an L (a; b) := L (a; b) ; then it is well known that the unction R 3p 7! L p (a; b) is monotonic increasing on R. We observe that or p Rn ; g we have an [( ) a + b] p = L p p (a; b) ; [( ) a + b] ln [( ) a + b] = ln I (a; b) : = L (a; b) Using these notations we can e ne the ollowing ivergences or (t; s) I n I n where I is an interval o positive numbers: an or all p Rn ; g ; an T p (t; s) := A (t p ; s p ) L p p (t; s) M p (t; s) := L p p (t; s) A p (t; s) T (t; s) := H (t; s) L (t; s) M (t; s) := L (t; s) A (t; s)

6 6 S. S. DRAGOMIR or p = an an T (t; s) := ln M (t; s) := ln G (t; s) I (t; s) I (t; s) A (t; s) or p = : Since the unction () = p ; > is convex or p ( have ; ) [ (; ); then we (.6) T p (t; s) ; M p (t; s) or all (t; s) I I: For p (; ) the unction () = p ; > an or p = ; the unction () = ln are concave, then we have or p [; ) that (.7) T p (t; s) ; M p (t; s) or all (t; s) I I: Finally or p = we have both T (t; s) = M (t; s) = or all (t; s) I I: We nee the ollowing convexity result that is a consequence o Burbea-Rao s theorem above: Lemma. Let be a C unction on an interval I: Then T an M are convex (concave) on I I; i an only i is convex (concave) an is concave (convex) on I: Proo. I T an M are convex on I I then the sum T + M = J is convex on I I, which, by Burbea-Rao theorem implies that is convex an is concave on I: Now, i is convex an is concave on I; then by the same theorem we have that the unction J : I I! R J (t; s) := [ (t) + (s)] is convex. Let t; s; u; v I. We e ne ' () := J (( ) (t; s) + (u; v)) = J ((( ) t + u; ( ) s + v)) or [; ] : = [ (( ) t + u) + (( ) s + v)] ( ) t + u + ( ) s + v = [ (( ) t + u) + (( ) s + v)] ( ) + u + v

7 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE 7 Let ; [; ] an ; with + = : By the convexity o J we have ' ( + ) = J (( ) (t; s) + ( + ) (u; v)) = J (( + ) (t; s) + ( + ) (u; v)) = J ( ( ) (t; s) + ( ) (t; s) + (u; v) + (u; v)) = J ( [( ) (t; s) + (u; v)] + [( ) (t; s) + (u; v)]) J (( ) (t; s) + (u; v)) + J (( ) (t; s) + (u; v)) = ' ( ) + ' ( ) ; which proves that ' is convex on [; ] or all t; s; u; v I: Applying the Hermite-Haamar inequality or ' we get (.8) an since an [' () + ' ()] ' () = [ (t) + (s)] ' () = [ (u) + (v)] ' () ; u + v ' () = (( ) t + u) + (( ( ) + u + v ; hence by (.8) we get [ (t) + (s)] (( ) t + u) + ( ) + u + v ) s + v) + u + v [ (u) + (v)] : Re-arranging this inequality, we get (t) + (u) (( + (s) + (v) (( u + v + (( ) t + u) ) s + v) ) s + v) ( ) + u + v ;

8 8 S. S. DRAGOMIR which is equivalent to [T (t; u) + T (s; v)] T ; u + v = T (t; u) + (s; v) ; or all (t; u) ; (s; v) I I, which shows that T is Jensen s convex on I I: Since T is continuous on I I, hence T is convex in the usual sense on I I: Now, i we use the secon Hermite-Haamar inequality or ' on [; ] ; we have (.9) Since ' = hence by (.9) we have t + u ' () ' + (( ) t + u) + s + v : (( ( ) + u + v t + u s + v + + ) s + v) u + v + u + v ; which is equivalent to t + u (( ) t + u) + s + v (( ) s + v) ( ) + u + v + u + v that can be written as [M (t; u) + M (s; v)] M ; u + v = M (t; u) + (s; v) or all (t; u) ; (s; v) I I, which shows that M is Jensen s convex on I I: Since M is continuous on I I, hence M is convex in the usual sense on I I: The ollowing reverses o the Hermite-Haamar inequality hol:

9 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE 9 Lemma (Dragomir, [] an []). Let h : [a; b]! R be a convex unction on [a; b] : Then a + b a + b (.) h + h (b a) 8 an (.) The constant 8 We also have: h (a) + h (b) b a b 8 [h (b) h + (a)] (b a) 8 b a + b h + a b a a h () a + b h (b a) a + b h () h 8 [h (b) h + (a)] (b a) : is best possible in all inequalities rom (.) an (.). Lemma 3. Let be a C convex unction on an interval I: I I is the interior o I; then or all (t; s) I I we have (.) M (t; s) T (t; s) 8 C (t; s) where (.3) C (t; s) := [ (t) (s)] (t s) : Proo. Since or b 6= a b a then rom (.) we get (t) + (s) or all (t; s) I I: Remark. I b a (t) t = (( t) a + tb) t; (( ) ) t 8 [ (t) (s)] (t s) = in (t) an t I = sup (t) t I are nite, then C (t; s) ( ) jt sj an by (.) we get the simpler upper boun M (t; s) T (t; s) ( ) jt sj : 8 Moreover, i t; s [a; b] I an since is increasing on I; then we have the inequalities (.4) M (t; s) T (t; s) 8 [ (b) (a)] jt sj :

10 S. S. DRAGOMIR Since J (t; s) = T (t; s) + M (t; s) ; hence J (t; s) 4 [ (b) (a)] jt sj : Corollary. With the assumptions o Lemma 3 an i the erivative is Lipschitzian with the constant K > ; namely then we have the inequality j (t) (s)j K jt sj or all t; s I; (.5) M (t; s) T (t; s) 8 K (t s) : 3. Main Results Let P; Q; W P an : (; )! R. We e ne the ollowing -ivergence (3.) J (P; Q; W ) := J ; (x) = (x) + (x) + : I we consier the mi-point ivergence measure M e ne by + M (Q; P; W ) := (x) or any Q; P; W P, then rom (3.) we get (3.) J (P; Q; W ) = [I (P; W ) + I (Q; W )] M (Q; P; W ) : We can also consier the integral ivergence measure ( t) + t A (Q; P; W ) := t (x) : We introuce the relate -ivergences (3.3) T (P; Q; W ) := an (3.4) We observe that M (P; Q; W ) := T ; (x) = [I (P; W ) + I (Q; W )] A (Q; P; W ) M ; (x) = A (Q; P; W ) M (Q; P; W ) : J (P; Q; W ) = T (P; Q; W ) + M (P; Q; W ) : I is convex on (; ) then by the Hermite-Haamar an Bullen s inequalities we have the positivity properties M (P; Q; W ) T (P; Q; W )

11 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE an or P; Q; W P. We have the ollowing result: J (P; Q; W ) Theorem 5. Let be a C unction on an interval (; ) : I is convex on (; ) an is concave on (; ) ; then or all W P, the mappings are convex. P P 3 (P; Q) 7! J (P; Q; W ) ; M (P; Q; W ) ; T (P; Q; W ) Proo. Let (P ; Q ) ; (P ; Q ) P P an ; with + =. We have J ( (P ; Q ; W ) + (P ; Q ; W )) = J (P + P ; Q + Q ; W ) p (x) + p (x) = J ; q (x) + q (x) (x) = J p (x) + p (x) ; q (x) + q (x) (x) p (x) = J ; q (x) p (x) + ; q (x) (x) p (x) J ; q (x) p (x) + J ; q (x) (x) p (x) = J ; q (x) p (x) (x) + J ; q (x) (x) = J (P ; Q ; W ) + J (P ; Q ; W ) ; which proves the convexity o P P 3 (P; Q) 7! J (P; Q; W ) or all W P. The convexity o the other two mappings ollows in a similar way an we omit the etails. Theorem 6. Let be a C (; ) ; then or all W P unction on an interval (; ) : I is convex on (3.5) M (P; Q; W ) T (P; Q; W ) 8 (Q; P; W ) where (3.6) (Q; P; W ) := Proo. From the inequality (.) we have + 8 or all x : ( ) (x) : ( t) + t t

12 S. S. DRAGOMIR I we multiply by > an integrate on we get [I (P; W ) + I (P; W )] A (Q; P; W ) 8 = 8 which implies the esire inequality. ( ) (x) ; (x) Corollary. With the assumptions o Theorem 6 an i is Lipschitzian with the constant K > ; namely then j (s) (t)j K js tj or all t; s (; ) ; (3.7) M (P; Q; W ) T (P; Q; W ) 8 K (Q; P; W ) ; where (3.8) (Q; P; W ) := ( ) (x) : Remark 3. I there exists < r < < R < such that the ollowing conition hols ((r,r)) r ; R or -a.e. x ; then (3.9) M (P; Q; W ) T (P; Q; W ) 8 [ (R) (r)] (Q; P ) where (Q; P ) := Moreover, i is twice i erentiable an j j (x) : (3.) k k [r;r]; := sup j (t)j < t[r;r] then (3.) M (P; Q; W ) T (P; Q; W ) 8 k k [r;r]; (Q; P; W ) : We also have: Theorem 7. Let be a C unction on an interval (; ) : I is convex on (; ) an is concave on (; ) ; then or all W P, (3.) J (P; Q; W ) [ (P; Q; W ) + (Q; P; W )] ; where (P; Q; W ) := + ( ) (x) :

13 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE 3 Proo. It is well known that i the unction o two inepenent variables F : D R R! R is convex on the convex omain D an has an on D then or all (t; s) ; (u; v) D we have the (t; (t; s) (t u) + F (t; s) F (u; (u; (t u) (u; (s v) : Now, i we take F : (; ) (; )! R given by an observe that an F (t; s) = [ (t) + (t; = = (t) (s) an since F is convex on (; ) (; ) ; then by (3.3) we get (t) (t u) + (3.4) (s) (s v) u + v [ (t) + (s)] [ (u) + (v)] + u + v (u) (t u) + u + v (v) (s v) : I we take u = v = in (3.4), then we have (t) (t ) + (3.5) (s) [ (t) + (s)] (s ) or all (t; s) (; ) (; ) : I we take t = p(x) q(x) an s = in (3.5) then we obtain + (3.6) :

14 4 S. S. DRAGOMIR By multiplying this inequality with > we get ( ) + + ( ) or all x : Corollary 3. With the assumptions o Theorem 6 an i is Lipschitzian with the constant K > ; then (3.7) J (P; Q; W ) Proo. We have that an similarly 4 K j j + (P; Q; W ) + j j (x) K + j j (x) = K j j (x) = K j j j j (x) = K j j (x) (P; Q; W ) K j j Finally, by the use o (3.) we get the esire result. (x) : (x) : Remark 4. I there exist < r < < R < such that the ollowing conition (r; R) hols an i is twice i erentiable an (3.) is vali, then (3.8) J (P; Q; W ) 4 k k [r;r]; j j Since an + ; max R ; rg R r; (x) :

15 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE 5 hence by (3.8) we get the simpler boun (3.9) J (P; Q; W ) k k [r;r]; (R r) max R ; rg : We also have: Theorem 8. With the assumptions o Theorem 6 an i is Lipschitzian with the constant K > ; then (3.) T (P; Q; W ) 6 K j j + (x) : Proo. Let (x; y) ; (u; v) (; )(; ) : I we take F : (; )(; )! R given by then an F (t; s) (t; (t) + (s) = (t) (t; (( ) ) ( ) (( ) ) ( ) [ (t) (( ) )] = (s) = (( ) ) [ (s) (( ) )] an since F is convex on (; ) (; ) ; then by (3.) we get (3.) (t u) + (s v) ( ) [ (t) (( ) )] (t) + (s) (u) + (v) + (t u) + (s v) [ (s) (( ) )] or all (t; s) ; (u; v) (; ) (; ) : (( (( ) ) ) u + v) ( ) [ (u) (( ) u + v)] [ (v) (( ) u + v)]

16 6 S. S. DRAGOMIR I we take u = v = in (3.), then we have (3.) (t ) + (s ) ( ) [ (t) (( ) )] (t) + (s) [ (s) (( ) )] (( ) ) or all (u; v) (; ) (; ) : I we take t = p(x) q(x) an s = (3.3) + p(x) ( ) + q(x) in (3.) then we get ( ) + ( ) + ( ) + : Since is Lipschitzian with the constant K >, hence p(x) + q(x) ( ) + ( ) ( ) + + ( ) + K ( ) + K ( ) = 6 K + : I we multiply this inequality by > an integrate, then we get the esire result (3.). Corollary 4. I there exist < r < < R < such that the conition (r; R) hols an i is twice i erentiable an (3.) is vali, then (3.4) T (P; Q; W ) 3 k k [r;r]; (R r) max R ; rg : Finally, we also have:

17 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE 7 Theorem 9. With the assumptions o Theorem 6 an i is Lipschitzian with the constant K > ; then (3.5) M (P; Q; W ) 8 K j j + (x) : Proo. Let (t; s) ; (u; v) (; ) (; ) : I we take F : (; ) (; )! R given by F (t; s) = (( ) ) (t; s) = ( ) (( ) = ( ) (( ) ) (t; = = (( ) ) (( ) ) an since F is convex on (; ) (; ) ; then by (3.) we get (3.6) (t u) ( ) (( ) ) + (s v) (( ) ) (( ) ) u + v (( ) u + v) + u + v (t u) ( ) (( ) u + v) u + v + (s v) (( ) u + v) : I we take u = v = in (3.6), then we have (3.7) (t ) ( ) + (s ) or all (t; s) (; ) (; ) : (( ) ) (( ) ) (( ) )

18 8 S. S. DRAGOMIR I we take t = p(x) (3.8) ( ) an s = q(x) ( ) + in (3.7) then we get ( ) + + ( ) ( ) ( ) ( ) + + K ( ) + K ( ) : Since hence ( ) = 8 ; ( ) + 8 K + + or all x : I we multiply this inequality by > an integrate, then we get the esire result (3.). Corollary 5. I there exist < r < < R < such that the conition (r; R) hols an i is twice i erentiable an (3.) is vali, then (3.9) M (P; Q; W ) 4 k k [r;r]; (R r) max R ; rg :

19 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE 9 4. Some Examples The Dichotomy Class o -ivergences are generate by the unctions : [; )! R e ne as 8 u ln u or = ; >< (u) = ( ) [u + u ] or Rn ; g ; >: u + u ln u or = : Observe that 8 >< u or = ; (u) = u or Rn ; g ; >: u or = : In this amily o unctions only the unctions with [; ) are both convex an with concave on (; ) : We have I (P; W ) = (x) an M (Q; P; W ) = 8 >< = >: 8 >< = >: R ( ) w (x) p (x) (x) ; (; ) ; R ln p(x) (x) ; = ; + h R ( ) R (x) h q(x)+p(x) i w (x) (x) i ; (; ) h i h i q(x)+p(x) ln q(x)+p(x) (x) ; = : We also have [( t) a + tb] ln [( t) a + tb] t = 4 (b + a) ln I a ; b = A (a; b) ln I a ; b : Thereore ( A (Q; P; W ) := 8 h R q(x) >< ( ) L ; p(x) = >: R A q(x) ; p(x) ln I t i t) + t (x) q(x) (x) ; (; ) ; p(x) (x) ; = :

20 S. S. DRAGOMIR We have J (P; Q; W ) = [I (P; W ) + I (Q; W )] M (Q; P; W ) ; T (P; Q; W ) = [I (P; W ) + I (Q; W )] A (Q; P; W ) an M (P; Q; W ) = A (Q; P; W ) M (Q; P; W ) : Accoring to Theorem 5, or all [; ), the mappings P P 3 (P; Q) 7! J (P; Q; W ) ; M (P; Q; W ) ; T (P; Q; W ) are convex or all W P. I < r < < R, then kk [r;r]; = sup (t) = or [; ): t[r;r] r I there exists < r < < R < such that the ollowing conition hols ((r,r)) then by (3.9), (3.4) an (3.9) we get r ; R or -a.e. x ; (4.) J (P; Q; W ) k k [r;r]; (R r) max R ; rg ; (4.) T (P; Q; W ) 3 an (R r) max R ; rg r (4.3) M (P; Q; W ) (R r) max R 4 r ; rg ; or all [; ) an W P. The intereste reaer may apply the above general results or other particular ivergences o interest generate by the convex unctions provie in the introuction. We omit the etails. Reerences [] J. Burbea an C. R. Rao, On the convexity o some ivergence measures base on entropy unctions, IEEE Tran. In. Theor., Vol. IT-8, No. 3, 98, [] P. Cerone an S. S. Dragomir, Approximation o the integral mean ivergence an - ivergence via mean results. Math. Comput. Moelling 4 (5), no. -, 7 9. [3] P. Cerone, S. S. Dragomir an F. Österreicher, Bouns on extene -ivergences or a variety o classes, Kybernetika (Prague) 4 (4), no. 6, Preprint, RGMIA Res. Rep. Coll. 6(3), No., Article 5. [ONLINE: [4] I. Csiszár, Eine inormationstheoretische Ungleichung un ihre Anwenung au en Beweis er Ergoizität von Marko schen Ketten. (German) Magyar Tu. Aka. Mat. Kutató Int. Közl. 8 (963) [5] I. Csiszár, Inormation-type measures o i erence o probability istributions an inirect observations, Stuia Math. Hungarica, (967), [6] I. Csiszár, On topological properties o -ivergences, Stuia Math. Hungarica, (967), [7] S. S. Dragomir, Some inequalities or (m; M)-convex mappings an applications or the Csiszár -ivergence in inormation theory. Math. J. Ibaraki Univ. 33 (), [8] S. S. Dragomir, Some inequalities or two Csiszár ivergences an applications. Mat. Bilten No. 5 (), 73 9.

21 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE [9] S. S. Dragomir, An upper boun or the Csiszár -ivergence in terms o the variational istance an applications. Panamer. Math. J. (), no. 4, [] S. S. Dragomir, An inequality improving the rst Hermite-Haamar inequality or convex unctions e ne on linear spaces an applications or semi-inner proucts, J. Inequal. Pure an Appl. Math., 3 () (), Art. 3. [] S. S. Dragomir, An inequality improving the secon Hermite-Haamar inequality or convex unctions e ne on linear spaces an applications or semi-inner proucts, J. Inequal. Pure an Appl. Math., 3 (3) (), Art. 35. [] S. S. Dragomir, Upper an lower bouns or Csiszár -ivergence in terms o Hellinger iscrimination an applications. Nonlinear Anal. Forum 7 (), no., 3 [3] S. S. Dragomir, Bouns or -ivergences uner likelihoo ratio constraints. Appl. Math. 48 (3), no. 3, 5 3. [4] S. S. Dragomir, New inequalities or Csiszár ivergence an applications. Acta Math. Vietnam. 8 (3), no., [5] S. S. Dragomir, A generalize -ivergence or probability vectors an applications. Panamer. Math. J. 3 (3), no. 4, [6] S. S. Dragomir, Some inequalities or the Csiszár '-ivergence when ' is an L-Lipschitzian unction an applications. Ital. J. Pure Appl. Math. No. 5 (4), [7] S. S. Dragomir, A converse inequality or the Csiszár -ivergence. Tamsui Ox. J. Math. Sci. (4), no., [8] S. S. Dragomir, Some general ivergence measures or probability istributions. Acta Math. Hungar. 9 (5), no. 4, [9] S. S. Dragomir, Bouns or the normalize Jensen unctional, Bull. Austral. Math. Soc. 74(3)(6), [] S. S. Dragomir, A re nement o Jensen s inequality with applications or -ivergence measures. Taiwanese J. Math. 4 (), no., [] S. S. Dragomir, A generalization o -ivergence measure to convex unctions e ne on linear spaces. Commun. Math. Anal. 5 (3), no., 4. [] S. S. Dragomir an C. E. M. Pearce, Selecte Topics on Hermite- Haamar Inequalities an Applications, RGMIA Monographs,. [Online [3] S. Kullback an R. A. Leibler, On inormation an su ciency, Ann. Math. Stat., (95), [4] J. Lin an S. K. M. Wong, A new irecte ivergence measure an its characterization, Int. J. General Systems, 7 (99), [5] H. Shioya an T. Da-Te, A generalisation o Lin ivergence an the erivative o a new inormation ivergence, Elec. an Comm. in Japan, 78 (7) (995), [6] P. Kaka, F. Österreicher an I. Vincze, On powers o -ivergence e ning a istance, Stuia Sci. Math. Hungar., 6 (99), [7] F. Liese an I. Vaja, Convex Statistical Distances, Teubuer Texte zur Mathematik, Ban 95, Leipzig, 987. [8] F. Österreicher an I. Vaja, A new class o metric ivergences on probability spaces an its applicability in statistics. Ann. Inst. Statist. Math. 55 (3), no. 3, Mathematics, College o Engineering & Science, Victoria University, PO Box 448, Melbourne City, MC 8, Australia. aress: sever.ragomir@vu.eu.au URL: DST-NRF Centre o Excellence in the Mathematical, an Statistical Sciences, School o Computer Science, & Applie Mathematics, University o the Witwatersran,, Private Bag 3, Johannesburg 5, South Arica