A New Refinement of Jensen's Inequality in Linear Spaces with Applications

A New Refinement of Jensen's Inequality in Linear Spaces with Applications

This is the Published version of the following publication

Dragomir, Sever S (2009) A New Refinement of Jensen's Inequality in Linear Spaces with Applications. Research report collection, 12 (Supp).

The publisher’s official version can be found at

Note that access to this version may require subscription.

Downloaded from VU Research Repository https://vuir.vu.edu.au/17965/

SPACES WITH APPLICATIONS

S.S. DRAGOMIR

Abstract. A new refinement of Jensen’s celebrated inequality for functions defined on convex sets in linear spaces is given. Applications for norms, mean f-deviation andf-divergences are provided as well.

1. Introduction

LetC be a convex subset of the linear space X andf a convex function on C.

Ifp= (p₁, . . . , p_n) is a probability sequence andx= (x₁, . . . , x_n)∈Cⁿ,then

(1.1) f

n

X

i=1

pixi

!

≤

n

X

i=1

pif(xi),

is well known in the literature asJensen’s inequality.

The Jensen inequality for convex functions plays a crucial role in the Theory of Inequalities due to the fact that other inequalities such as the arithmetic mean- geometric mean inequality, H¨older and Minkowski inequalities, Ky Fan’s inequality etc. can be obtained as particular cases of it.

In 1989, J. Peˇcari´c and the author obtained the following refinement of (1.1) (see [13]):

f

n

X

i=1

pixi

!

≤

n

X

i1,...,ik+1=1

pi₁. . . pi_k+1f

xi₁+· · ·+xi_k+1

k+ 1 (1.2)

≤

n

X

i₁,...,i_k=1

p_i1. . . p_ikf

x_i1+· · ·+x_ik k

≤ · · · ≤

n

X

i=1

p_if(x_i),

fork≥1 andp,xas above.

Date: January 07, 2009.

1991Mathematics Subject Classification. 26D15; 94.

Key words and phrases. Convex functions, Jensen’s inequality, Norms, Meanf-deviation,f- Divergences.

1

If q1, . . . , qk ≥ 0 withPk

j=1qj = 1, then the following refinement obtained in 1994 by the author also holds (see [6]):

f

n

X

i=1

pixi

!

≤

n

X

i1,...,ik=1

pi₁. . . pi_kf

xi₁+· · ·+xi_k

k

(1.3)

≤

n

X

i₁,...,i_k=1

pi₁. . . pi_kf(q1xi₁+· · ·+qkxi_k)

≤

n

X

i=1

p_if(x_i), where 1≤k≤nandp,x are as above.

More recently the author obtained a different refinement of Jensen’s inequality incorporated in (see [8]):

f





n

X

j=1

p_jx_j



≤ min

k∈{1,...,n}

"

(1−p_k)f Pn

j=1pjxj−pkxk

1−pk

!

+p_kf(x_k)

# (1.4)

≤ 1 n

" _n X

k=1

(1−pk)f Pn

j=1p_jx_j−p_kx_k 1−pk

! +

n

X

k=1

pkf(xk)

#

≤ max

k∈{1,...,n}

"

(1−p_k)f Pn

j=1pjxj−pkxk

1−p_k

!

+p_kf(x_k)

#

≤

n

X

j=1

pjf(xj), wheref, xk andpk are as above.

For other refinements and applications related to Ky Fan’s inequality, the arith- metic mean-geometric mean inequality, the generalized triangle inequality, the f- Divergence measure etc., see [3]-[12].

In this paper, a new refinement of Jensen’s celebrated inequality for functions defined on convex sets in linear spaces is given. Applications for norms, mean f-deviation andf-divergences are provided as well.

2. General Results

LetC be a convex subset in the real linear spaceX and assume thatf :C→R is a convex function on C. If xi ∈ C and pi >0, i∈ {1, ..., n} with Pn

i=1pi = 1, then for any nonempty subset J of {1, ..., n} we put ¯J := {1, ..., n} \J(6=∅) and define P_J := P

i∈Jp_i and ¯P_J := PJ¯ = P

j∈J¯p_j = 1−P

i∈Jp_i. For the convex functionf and then-tuples x:= (x₁, ..., x_n) andp:= (p₁, ..., p_n) as above, we can define the following functional

(2.1) D(f,p,x;J) :=PJf 1 P_J

X

i∈J

pixi

! + ¯PJf



 1 P¯J

X

j∈J¯

pjxj





where here and everywhere belowJ ⊂ {1, ..., n}withJ 6=∅and J6={1, ..., n}.

It is worth to observe that forJ ={k}, k∈ {1, ..., n}we have the functional Dk(f,p,x) :=D(f,p,x;{k})

(2.2)

=p_kf(x_k) + (1−p_k)f Pn

i=1pixi−pkxk

1−pk

that has been investigated in the earlier paper [8].

Theorem 1. LetC be a convex subset in the real linear spaceX and assume that f : C → R is a convex function on C. If xi ∈ C and pi > 0, i ∈ {1, ..., n} with Pn

i=1p_i= 1 then for any nonempty subsetJ of {1, ..., n} we have (2.3)

n

X

k=1

pkf(xk)≥D(f,p,x;J)≥f

n

X

k=1

pkxk

! . Proof. By the convexity of the functionf we have

D(f,p,x;J) =PJf 1 PJ

X

i∈J

pixi

! + ¯PJf



 1 P¯_J

X

j∈J¯

pjxj





≥f



P_J 1 PJ

X

i∈J

p_ix_i

! + ¯P_J



 1 P¯J

X

j∈J¯

p_jx_j









=f

n

X

k=1

pkxk

!

for anyJ,which proves the second inequality in (2.3).

By the Jensen inequality we also have

n

X

k=1

pkf(xk) =X

i∈J

pif(xi) +X

j∈J¯

pjf(xj)

≥PJf 1 P_J

X

i∈J

pixi

! + ¯PJf



 1 P¯J

X

j∈J¯

pjxj





=D(f,p,x;J)

for anyJ,which proves the first inequality in (2.3).

Remark 1. We observe that the inequality (2.3) can be written in an equivalent form as

(2.4)

n

X

k=1

pkf(xk)≥ max

∅6=J⊂{1,...,n}D(f,p,x;J) and

(2.5) min

∅6=J⊂{1,...,n}D(f,p,x;J)≥f

n

X

k=1

p_kx_k

! .

These inequalities imply the following results that have been obtained earlier by the author in[8]utilising a different method of proof slightly more complicated:

(2.6)

n

X

k=1

pkf(xk)≥ max

k∈{1,...,n}Dk(f,p,x)

and

(2.7) min

k∈{1,...,n}Dk(f,p,x)≥f

n

X

k=1

pkxk

! . Moreover, since

max

∅6=J⊂{1,...,n}D(f,p,x;J)≥ max

k∈{1,...,n}Dk(f,p,x) and

min

k∈{1,...,n}Dk(f,p,x)≥ min

∅6=J⊂{1,...,n}D(f,p,x;J),

then the new inequalities (2.4) and (2.4) are better than the earlier results developed in[8].

The case of uniform distribution, namely, when pi = _n¹ for all {1, ..., n} is of interest as well. If we consider a natural numbermwith 1≤m≤n−1 and if we define

(2.8) Dm(f,x) :=m nf 1

m

m

X

i=1

xi

!

+n−m n f



 1 n−m

n

X

j=m+1

xj





then we can state the following result:

Corollary 1. LetC be a convex subset in the real linear spaceX and assume that f :C→Ris a convex function on C.If xi∈C, then for anym∈ {1, ..., n−1}we have

(2.9) 1

n

n

X

k=1

f(xk)≥Dm(f,x)≥f 1 n

n

X

k=1

xk

! . In particular, we have the bounds

(2.10) 1 n

n

X

k=1

f(xk)

≥ max

m∈{1,...,n−1}



 m

nf 1 m

m

X

i=1

xi

!

+n−m n f



 1 n−m

n

X

j=m+1

xj









and

(2.11) min

m∈{1,...,n−1}



 m

nf 1 m

m

X

i=1

x_i

!

+n−m n f



 1 n−m

n

X

j=m+1

x_j









≥f 1 n

n

X

k=1

xk

! . The following version of the inequality (2.3) may be useful for symmetric convex functions:

Corollary 2. Let C be a convex function with the property that 0∈C. If yj ∈X such that forpi>0, i∈ {1, ..., n} withPn

i=1pi= 1we haveyj−Pn

i=1piyi ∈Cfor any j∈ {1, ..., n},then for any nonempty subset J of{1, ..., n} we have

(2.12)

n

X

k=1

p_kf y_k−

n

X

i=1

p_iy_i

!

≥P_Jf



P¯_J



 1 PJ

X

i∈J

p_iy_i− 1 P¯J

X

j∈J¯

p_jy_j









+ ¯PJf



PJ



 1 P¯J

X

j∈J¯

pjyj− 1 P_J

X

i∈J

piyi







≥f(0). Remark 2. IfC is as in Corollary 2 andyj∈X such thatyj−_n¹Pn

i=1yi∈Cfor any j∈ {1, ..., n}then for any m∈ {1, ..., n−1} we have

(2.13) 1 n

n

X

k=1

f yk−1 n

n

X

i=1

yi

!

≥ m nf



 n−m

n



 1 m

m

X

i=1

yi− 1 n−m

n

X

j=m+1

yj









+n−m n f



 m

n



 1 n−m

n

X

j=m+1

yj− 1 m

m

X

i=1

yi







≥f(0). Remark 3. It is also useful to remark that if J ={k} where k∈ {1, ..., n} then the particular form we can derive from (2.12) can be written as

(2.14)

n

X

`=1

p_{`</sub>f y<sub>`}−

n

X

i=1

p_iy_i

!

≥p_kf



(1−p_k)



y_k− 1 1−pk





n

X

j=1

p_jy_j−p_ky_k













+ (1−p_k)f



p_k



 1 1−p_k





n

X

j=1

p_jy_j−p_ky_k



−y_k







≥f(0) which is equivalent with

(2.15)

n

X

`=1

p`f y`−

n

X

i=1

piyi

!

≥pkf



yk−

n

X

j=1

pjyj





+ (1−p_k)f



 p_k 1−pk





n

X

j=1

p_jy_j−y_k







≥f(0) for any k∈ {1, ..., n}.

Remark 4. Continuous versions for the Lebesgue integral are considered in [9].

3. A Lower Bound for Mean f-Deviation

Let X be a real linear space. For a convex function f : X → R with the properties that f(0) = 0, define the mean f-deviation of an n-tuple of vectors

x = (x1, ..., xn) ∈ Xⁿ with the probability distribution p = (p1, ..., pn) by the non-negative quantity

(3.1) Kf(p,x) :=

n

X

i=1

pif xi−

n

X

k=1

pkxk

! .

The fact thatKf(p,x) is non-negative follows by Jensen’s inequality, namely K_f(p,x)≥f

n

X

i=1

p_i x_i−

n

X

k=1

p_kx_k

!!

=f(0) = 0.

A natural example of such deviations can be provided by the convex function f(x) :=kxk^rwithr≥1 defined on a normed linear space (X,k·k).We denote this by

(3.2) Kr(p,x) :=

n

X

i=1

pi

xi−

n

X

k=1

pkxk

r

and call it themean r-absolute deviationof then-tuple of vectorsx= (x₁, ..., x_n)∈ Xⁿ with the probability distributionp= (p1, ..., pn).

The following result that provides a lower bound for the meanf-deviation holds:

Theorem 2. Let f : X → [0,∞) be a convex function with f(0) = 0. If x = (x₁, ..., x_n) ∈ Xⁿ and p = (p₁, ..., p_n) is a probability distribution with all p_i nonzero, then

(3.3) Kf(p,x)≥ max

∅6=J⊂{1,...,n}





 PJf



P¯J



 1 P_J

X

i∈J

pixi− 1 P¯J

X

j∈J¯

pjxj









+PJf



 1 P¯_J

X

j∈J¯

pjyj− 1 PJ

X

i∈J

piyi









 (≥0). In particular, we have

(3.4) Kf(p,x)

≥ max

k∈{1,...,n}

(

(1−p_k)f

"

pk

1−pk n

X

l=1

p_lx_l−x_k

!#

+p_kf x_k−

n

X

l=1

p_lx_l

!) (≥0). The proof follows from Corollary 2 and Remark 3.

As a particular case of interest, we have the following:

Corollary 3. Let (X,k·k) be a normed linear space. Ifx= (x1, ..., xn)∈Xⁿ and p= (p1, ..., pn)is a probability distribution with all pi nonzero, then for r≥1 we have

(3.5) Kr(p,x)

≥ max

∅6=J⊂{1,...,n}







P_JP¯_J P¯_J^r−1+P_J^r−1

1 PJ

X

i∈J

p_ix_i− 1 P¯J

X

j∈J¯

p_jx_j

r



 (≥0).

Remark 5. By the convexity of the power functionf(t) =t^r, r≥1 we have P_JP¯_J P¯_J^r−1+P_J^r−1

=P_JP¯_J^r+ ¯P_JP_J^r

≥ PJP¯J+ ¯PJPJ^r

= 2^rP_J^rP¯_J^r therefore

(3.6) P_JP¯_J P¯_J^r−1+P_J^r−1

1 PJ

X

i∈J

p_ix_i− 1 P¯J

X

j∈J¯

p_jx_j

r

≥2^rP_J^rP¯_J^r

1 P_J

X

i∈J

pixi− 1 P¯J

X

j∈J¯

pjxj

r

= 2^r

P¯J

X

i∈J

pixi−PJ

X

j∈J¯

pjxj

r

.

Since

P¯_JX

i∈J

p_ix_i−P_JX

j∈J¯

p_jx_j = (1−P_J)X

i∈J

p_ix_i−P_J

n

X

k=1

p_kx_k−X

i∈J

p_ix_i

! (3.7)

=X

i∈J

pixi−PJ n

X

k=1

pkxk,

then by (3.5)-(3.7) we deduce the coarser but perhaps more useful lower bound

(3.8) Kr(p,x)≥2^r max

∅6=J⊂{1,...,n}

(

X

i∈J

pixi−PJ n

X

k=1

pkxk

r) (≥0). The case for meanr-absolute deviation is incorporated in:

Corollary 4. Let (X,k·k) be a normed linear space. Ifx= (x1, ..., xn)∈Xⁿ and p= (p1, ..., pn)is a probability distribution with all pi nonzero, then for r≥1 we have

(3.9) Kr(p,x)≥ max

k∈{1,...,n}

( h

(1−pk)^1−rp^r_k+pk

i

xk−

n

X

l=1

plxl

r) .

Remark 6. Since the functionhr(t) := (1−t)^1−rt^r+t, r≥1, t∈[0,1)is strictly increasing on [0,1),then

k∈{1,...,n}min n

(1−pk)^1−rp^r_k+pk

o

=pm+ (1−pm)^1−rp^r_m, wherep_m:= min

k∈{1,...,n}p_k.By (3.9), we then obtain the following simpler inequality:

(3.10) Kr(p,x)≥h

pm+ (1−pm)^1−r·p^r_mi max

k∈{1,...,n}

xk−

n

X

l=1

plxl

p

,

which is perhaps more useful for applications(see also [8]).

4. Applications for f-Divergence Measures Given a convex functionf : [0,∞)→R, thef-divergence functional

(4.1) If(p,q) :=

n

X

i=1

qif p_i

qi

,

wherep= (p₁, . . . , p_n),q= (q₁, . . . , q_n) are positive sequences, was introduced by Csisz´ar in [1], as a generalized measure of information, a “distance function” on the set of probability distributionsPⁿ.As in [1], we interpret undefined expressions by

f(0) = lim

t→0+f(t), 0f 0

0

= 0, 0fa

0

= lim

q→0+qf a

q

=a lim

t→∞

f(t)

t , a >0.

The following results were essentially given by Csisz´ar and K¨orner [2]:

(i) If f is convex, then If(p,q)is jointly convex in pand q;

(ii) For every p,q∈Rⁿ₊,we have

(4.2) If(p,q)≥

n

X

j=1

qjf Pn

j=1p_j Pn

j=1qj

! . If f is strictly convex, equality holds in (4.2) iff

p₁ q1

=p₂ q2

=· · ·= p_n qn

.

If f is normalized, i.e.,f(1) = 0,then for every p,q∈Rⁿ₊ withPn i=1pi= Pn

i=1qi,we have the inequality

(4.3) If(p,q)≥0.

In particular, if p,q ∈Pⁿ, then (4.3) holds. This is the well-known positivity property of thef-divergence.

We endeavour to extend this concept for functions defined on a cone in a linear space as follows.

Firstly, we recall that the subsetKin a linear spaceX is acone if the following two conditions are satisfied:

(i) for anyx, y∈K we havex+y∈K;

(ii) for anyx∈Kand any α≥0 we haveαx∈K.

For a givenn-tuple of vectorsz= (z1, ..., zn)∈Kⁿand a probability distribution q∈Pⁿ with all values nonzero, we can define, for the convex function f :K→R, the followingf-divergence of zwith the distribution q

(4.4) If(z,q) :=

n

X

i=1

qif zi

q_i

.

It is obvious that ifX =R, K= [0,∞) andx=p∈Pⁿ then we obtain the usual concept of thef-divergence associated with a function f : [0,∞)→R.

Now, for a givenn-tuple of vectorsx= (x1, ..., xn)∈Kⁿ, a probability distribu- tionq∈Pⁿ with all values nonzero and for any nonempty subsetJ of{1, ..., n}we have

q_J:= Q_J,Q¯_J

∈P²

and

x_J := X_J,X¯_J

∈K² where, as above

X_J :=X

i∈J

x_i, and X¯_J :=XJ¯. It is obvious that

If(xJ,qJ) =QJf XJ

Q_J

+ ¯QJf X¯J

Q¯J

.

The following inequality for thef-divergence of ann-tuple of vectors in a linear space holds:

Theorem 3. Let f : K → R be a convex function on the cone K. Then for any n-tuple of vectors x= (x1, ..., xn)∈Kⁿ, a probability distributionq∈Pⁿ with all values nonzero and for any nonempty subsetJ of {1, ..., n}we have

If(x,q)≥ max

∅6=J⊂{1,...,n}If(xJ,qJ)≥If(xJ,qJ) (4.5)

≥ min

∅6=J⊂{1,...,n}I_f(x_J,q_J)≥f(X_n) whereX_n:=Pn

i=1x_i.

The proof follows by Theorem 1 and the details are omitted.

We observe that, for a givenn-tuple of vectorsx= (x1, ..., xn)∈Kⁿ,a sufficient condition for the positivity ofIf(x,q) for any probability distributionq∈Pⁿ with all values nonzero is that f(Xn)≥0. In the scalar case and ifx=p∈Pⁿ, then a sufficient condition for the positivity of thef-divergenceIf(p,q) is thatf(1)≥0.

The case of functions of a real variable that is of interest for applications is incorporated in:

Corollary 5. Let f : [0,∞)→R be a normalized convex function. Then for any p,q∈Pⁿ we have

(4.6) I_f(p,q)≥ max

∅6=J⊂{1,...,n}

Q_Jf

P_J QJ

+ (1−Q_J)f

1−P_J 1−QJ

(≥0). In what follows we provide some lower bounds for a number off-divergences that are used in various fields of Information Theory, Probability Theory and Statistics.

The total variation distance is defined by the convex function f(t) = |t−1|, t∈Rand given in:

(4.7) V(p, q) :=

n

X

j=1

q_j

p_j qj

−1

=

n

X

j=1

|p_j−q_j|.

The following improvement of the positivity inequality for the total variation distance can be stated as follows.

Proposition 1. For anyp,q∈Pⁿ, we have the inequality:

(4.8) V(p, q)≥2 max

∅6=J⊂{1,...,n}|PJ−QJ| (≥0).

The proof follows by the inequality (4.6) forf(t) =|t−1|, t∈R.

The K. Pearsonχ²-divergenceis obtained for the convex functionf(t) = (1−t)², t∈Rand given by

(4.9) χ²(p, q) :=

n

X

j=1

qj

pj

qj

−1 2

=

n

X

j=1

(pj−qj)² qj

.

Proposition 2. For anyp,q∈Pⁿ, χ²(p, q)≥ max

∅6=J⊂{1,...,n}

((PJ−QJ)² QJ(1−QJ)

) (4.10)

≥4 max

∅6=J⊂{1,...,n}(PJ−QJ)² (≥0).

Proof. On applying the inequality (4.6) for the functionf(t) = (1−t)², t∈R, we get

χ²(p, q)≥ max

∅6=J⊂{1,...,n}

(

(1−QJ)

1−P_J 1−QJ

−1 2

+QJ

P_J QJ

−1 2)

= max

∅6=J⊂{1,...,n}

((P_J−Q_J)² QJ(1−QJ)

) . Since

QJ(1−QJ)≤ 1

4[QJ+ (1−QJ)]²=1 4, then

(PJ−QJ)²

QJ(1−QJ) ≥4 (PJ−QJ)²

for eachJ⊂ {1, . . . , n},which proves the last part of (4.10).

The Kullback-Leibler divergence can be obtained for the convex function f : (0,∞)→R,f(t) =tlntand is defined by

(4.11) KL(p, q) :=

n

X

j=1

qj· pj

q_j ln pj

q_j

=

n

X

j=1

pjln pj

q_j

.

Proposition 3. For anyp,q∈Pⁿ, we have:

(4.12) KL(p, q)≥ln

"

max

∅6=J⊂{1,...,n}

(1−PJ

1−Q_J 1−PJ

· PJ

Q_J

^PJ)#

≥0.

Proof. The first inequality is obvious by Corollary 5. Utilising the inequality be- tween thegeometric mean and the harmonic mean,

x^αy^1−α≥ 1

α

x +^1−α_y , x, y >0, α∈[0,1]

we have forx= ^P_Q^J

J, y=_1−Q^1−PJ

J andα=PJ that 1−PJ

1−Q_J ^1−PJ

· PJ

Q_J ^PJ

≥1,

for anyJ ⊂ {1, . . . , n},which implies the second part of (4.12).

Another divergence measure that is of importance in Information Theory is the Jeffreys divergence

(4.13) J(p, q) :=

n

X

j=1

qj· pj

q_j −1

ln pj

q_j

=

n

X

j=1

(pj−qj) ln pj

q_j

, which is anf-divergence forf(t) = (t−1) lnt, t >0.

Proposition 4. For anyp,q∈Pⁿ, we have:

J(p, q)≥ln max

k∈{1,...,n}

((1−P_J)Q_J (1−QJ)PJ

(QJ−PJ))!

(4.14)

≥ max

k∈{1,...,n}

"

(QJ−PJ)² P_J+Q_J−2P_JQ_J

#

≥0.

Proof. On making use of the inequality (4.6) forf(t) = (t−1) lnt,we have J(p, q)

≥ max

k∈{1,...,n}

(1−QJ)

1−PJ

1−Q_J −1

ln

1−PJ

1−Q_J

+QJ

PJ

Q_J −1

ln PJ

Q_J

= max

k∈{1,...,n}

(Q_J−P_J) ln

1−PJ

1−Q_J

−(Q_J−P_J) ln PJ

Q_J

= max

k∈{1,...,n}

(Q_J−P_J) ln

(1−P_J)Q_J (1−QJ)PJ

, proving the first inequality in (4.14).

Utilising the elementary inequality for positive numbers, lnb−lna

b−a ≥ 2

a+b, a, b >0 we have

(QJ−PJ)

ln

1−PJ

1−Q_J

−ln PJ

Q_J

= (QJ−PJ)· ln

1−PJ

1−QJ

−ln

P_J QJ

1−PJ

1−QJ −_Q^PJ

J

·

1−PJ

1−QJ

− PJ

QJ

= (Q_J−P_J)² QJ(1−QJ)·

ln

1−PJ

1−Q_J

−ln

PJ

Q_J

1−P_J 1−QJ −_Q^PJ

J

≥ (Q_J−P_J)²

QJ(1−QJ)· 2

1−PJ

1−QJ +_Q^PJ

J

= 2 (Q_J−P_J)² PJ+QJ−2PJQJ

≥0,

for eachJ⊂ {1, . . . , n},giving the second inequality in (4.14).

References

[1] I. Csisz´ar, Information-type measures of differences of probability distributions and indirect observations,Studia Sci. Math. Hung.,2(1967), 299-318.

[2] I. Csisz´ar and J. K¨orner, Information Theory: Coding Theorems for Discrete Memoryless Systems, Academic Press, New York, 1981.

[3] S.S. Dragomir, An improvement of Jensen’s inequality, Bull. Math. Soc. Sci. Math.

Roumanie,34(82) (1990), No. 4, 291-296.

[4] S.S. Dragomir, Some refinements of Ky Fan’s inequality,J. Math. Anal. Appl.,163(2) (1992), 317-321.

[5] S.S. Dragomir, Some refinements of Jensen’s inequality,J. Math. Anal. Appl.,168(2) (1992), 518-522.

[6] S.S. Dragomir, A further improvement of Jensen’s inequality, Tamkang J. Math., 25(1) (1994), 29-36.

[7] S.S. Dragomir, A new improvement of Jensen’s inequality,Indian J. Pure and Appl. Math., 26(10) (1995), 959-968.

[8] S.S. Dragomir, A refinement of Jensen’s inequality with applications forf-divergence mea- sures, Preprint,Res. Rep. Coll.10(2007), Supp., Article 15.

[9] S.S. Dragomir, Refinements of the Jensen integral inequality,in preparation.

[10] S.S. Dragomir and N.M. Ionescu, Some converse of Jensen’s inequality and applications.Rev.

Anal. Num´er. Th´eor. Approx.23(1994), no. 1, 71–78. MR1325895 (96c:26012).

[11] S.S. Dragomir and C.J. Goh, A counterpart of Jensen’s discrete inequality for differential- ble convex mappings and applications in information theory, Mathl. Comput. Modelling, 24(1996), No. 2, 1-11.

[12] S.S. Dragomir, J. Peˇcari´c and L.E. Persson, Properties of some functionals related to Jensen’s inequality,Acta Math. Hung.,70(1-2) (1996), 129-143.

[13] J. Peˇcari´c and S.S. Dragomir, A refinements of Jensen inequality and applications, Studia Univ. Babe¸s-Bolyai, Mathematica,24(1) (1989), 15-19.

Research Group in Mathematical Inequalities & Applications, School of Engineering

& Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia.

E-mail address:sever.dragomir@vu.edu.au

URL:http://www.staff.vu.edu.au/rgmia/dragomir/