J. ofInequal. & Appl., 1997, Vol. 1, pp. 311-325 Reprints available directly from the publisher Photocopying permitted by license only
(C) 1997 OPA (Overseas Publishers Association) Amsterdam B.V. Published in The Netherlands under license by Gordon and Breach Science Publishers Printed in Malaysia
Some Counterpart Inequalities for a Functional Associated with Jensens Inequality
S.S. DRAGOMIRa,* and C.J. GOH b
Department of Mathematics, University of Transkei, Private Bag Xl, Unitra, Umtata 5100, South Africa b Department of Mathematics, University of Western Australia Nedlands, WA 6907, Australia

(Received November 1996)

We derive several inequalities for a functional connected with the well-known Jensen inequality on IR". Some applications for arithmetic and geometric means, and for the entropy mapping in information theory are also discussed.
Keywords: Jensens inequality; entropy function; superadditivity; monotonicity; arithmetic and geometric means.
AMS Subject Classification: 26D 15, 26D99.

INTRODUCTION

Let f C IR be a convex mapping defined on a convex set C in a linear space X. Define the functional:

.T(f, p, I, x)"

Pi f (Xi)
* SSD acknowledges the financial support from the University of Western Australia during his visit when this work was completed.
S.S. DRAGOMIR and C.J. GOH
(Pi)i6IN is a sequence of positive real numbers, I 79f(IN), i.e., I is a finite set of indices and x (x,i)iiU C X is a sequence of vectors in X. It is easy to see that, with the above assumptions,
(i).T(f, p, I, x) > 0, i.e., Jensens discrete inequality, and (ii) (otf +/3g, p, I, x) ot.T(f, p, I, x) +/3.T(g, p, I, x) > 0 for all or,/3 >_ 0 and f, g are convex mappings. It is instructive to examine the properties of this functional with respect to the second and the third arguments. The following results (Theorem 1.1 and 1.3) were established in 1 ]:
THEOREM 1.1 (Properties of with respect to the second argument) Let f C X IR be a convex mapping on the convex subset C of the linear space X and x (xi)iIU C C. Then: (i) For all p, q > 0 one has the inequality:

f is as above, p

one has:
.T(f, p + q, I, x) >.T(f, p, I, x) +.T(f, q, I, x) > 0,
where I is fixed in 7"gf(IN), i.e. the mapping f(f,., I, x) is superadditive. (ii) For all p q 0 (i.e., each component of p is greater or equal to the corresponding component in q, and each component of q is greater or equal to 0), one has the inequality;

"=

.T(f, p, I, x) > 9t(f, q, I, x) > 0

the second argument.

Consider the following subset of nonnegative sequences
for a fixed I 79f(IN), i.e. U(f,., I, x) is monotonically non-decreasing in

79r(I)

(Pi)ilN, Pi >_ O,

IN and

for a fixed I in 79f(IN). It is obvious that 79r(I) is a convex set as for p, q 6 79r (I) and or, fl > 0 and ot + fl 1 we have that tp + flq 6 7"9r (I). We shall first derive some convexity properties of the functional f" with respect to the second argument. The following corollary to Theorem 1.1 holds:
COROLLARY 1.2 The mapping f(f,., I, x) is concave on 79f(I)for every fixed I in 79f (IN) \ 0.

JENSENS INEQUALITY

Proof Let a,/3

> 0 with a +/

1. By (1) we have that
(f, up +/Sq, I, x) >_ (f, up, I, x) + (f,/q, I, x) for p, q
Since.T(f,., I, x) is homogeneous, we have

79r (I).

f(f, up, I, x)

ot(f, p, I, x)

f(f,/Sq, I, x) =/5.T(f, q, I, x+
and the proof is complete.
THEOREM 1.3 (Properties of.T" with respect to the third argument) Let f C c_ X --+ IR be a convex mapping on the convex subset C of the linear space X and x (Xi)iIN C C. Then: (i) For all J, K 79f (IN) with J A K 0, one has the inequality
.T(f, p, J t3 K, x) > (f, p, J, x) + (f, p, K, x) > 0,
where p > 0 is fixed, i.e., the mapping.T(f, p,., x) is superadditive as an index set mapping. (ii) For all J c_C_ K, I, K 7)f(IN), one has the inequality
)v(f, p, K, x) > (f, p, J, x) _> 0,

for a fixed p

> O, i.e.Y(f, p,., x) is monotonically non-decreasing as an index set function.
In this paper, we shall derive several counterpart inequalities for the functional of (1.1) for the case where the mapping f is defined on an open convex subset C of the linear vector space IR n and f is differentiable on C. In particular, we extend the results in Theorem 1.1 and Theorem 1.3 to come up with further counterpart inequalities with respect to the second and third arguments. This wil be taken up in the next two sections, where we will also discuss some applications of these results.
2 COUNTERPART INEQUALITIES WITH RESPECT TO THE

SECOND ARGUMENT

In this section, we show that the inequalities of Theorem 1.1 can be further refined if the first argument of the functional.T" is a differentiable function.
THEOREM 2.1 Let f: C c_ X IR be a differentiable convex mapping on the open convex set C. Thus for all p, q > 0 one has the inequality:

0 <.U(f, p + q, I, x)

U(f, p, I, x)

.Y(f, q, I, x)

eI -I- QI eI
where I 79f(IN) and x (Xi)iiN C C are fixed. Equality holds in both inequalities if and only if p q.
Since the mapping fC --+ IR is differentiable and convex in the open convex set C we have that:
f(x)- f(y) _> (Vf(y), x- y)
where x (Xl Xn), y (Yl Yn) C. Using the inequality (4) we can write for all x, y 6 C and t,/3 > 0 with ot +/3 > 0, that:

(Vf (x), y

(V f (y), y
and summing the obtained results, we obtain:
Now, if we multiply the inequality (2.3) by ot and the inequality (2.4) by/3

(t +/3) f

( )+/3

off (x) -/3f (y)

[(Vf(x)

(Vf(x)

(Vf(y), y

Vf(y), y

and thus we obtain
tx+fly)O<tf(x)+flf(Y)-(t+fl)f(

Vf(y), x

Now, it is easier to see that: 0 _< (f, p + q, I, x)

(f, p, I, x)

(f, q, I, x)

Epixi + Qlf

+ Qi) f
Ei i pi xi -t-- Q I PI + QI

PI + QI PI

(Vf(1) (-I EqiXi ) -I EpiXi

Ei I qi Xi

where in the last inequality we have used (2.5) with the choices: qkxk" or:P1 16--QI, x-Y
The following corollary is immediately obvious.
COROLLARY 2.2 Let f be as above. If I 79f(IN) \ 0 is fixed and p, q 79r (I), thus for all [0, 1] we have the inequality:

0 <_ (f, tp + (1

t)q, I, x)

tf(f, p, I, x)

t)(f, q, I, x)

<- t(1- t)

(E piXi)

v f (E qiXi)

E E qixi} "ice
3 COUNTERPART INEQUALITIES WITH RESPECT TO THE

THIRD ARGUMENT

Similarly, if the first argument of.T" is differentiable, then it is also possible to refine the result of Theorem 1.3 further.
THEOREM 3.1 Let f" C c_ IR n -. IR be a differentiable convex mapping defined on the open convex set C. Thenfor all J, K Pf (IN) with J flK 0, one has the inequality:

where p

0 < f(f, p, J

.T(f, p, J, x)

.T(f, p, K, x)

PjPK(v (-j iej

Vf(-K E pkXk),

0 is fixed and x

(xi)i IN C C.
Proof It is easy to see that

0 <.T(f, p, J 1

.Y(f, p, J, x)

f(f, p, K, x)

-fi-fj

-+- PK f

+ PK) f

-Jc" PK -p-1

eJ + PK

PjPK(vf(liej +

Vf(K kK pkXl), K E
and for the last inequality we used (2.4) with the choices"

fl-" PK

and y-
4 APPLICATIONS IN THE A-G MEAN INEQUALITY
We now present an application of the above results to the following wellknown Arithmetic mean- Geometric mean inequality (A-G Mean inequality, for short): An(p, x) >_ Gn(p, x), (4.1)

where x

(Xl, x2,

An (p, x)

Gn (p, x) :=

(Pl, P2,

where Pn _in=l Pi. If Pi 0 i 1, 2,. n, it is well-known that the holds in (2.6) if and only if X equality Xn. x2 For I 79f (IN), let us consider

AI(p, x) "---

GI (P, x)

where Xi, Pi > O, Vi

I. It is easy to see that for f (.)

In(.), we have:

(--ln(.), p,/,x)

AI(p x) >_ 0, G/(p,x)

which is a direct proof of (4.1) by Jensen inequality. A more sophisticated extension of the A-G inequality is as follows.

PROPOSITION 4.1

With the above assumptions, we have, for p, q

GI(p -1- q, x)

Ai(p, x)

Al(q, x)

_< exp

PI + QI

(AI(p,x)-AI(q,x)) AI(p, X)Al(q, x)

_,iI Pi

> 0 and Q I

Zi6I qi

> O.
Proof Applying Theorem 2.1

deduce that:

to the convex mapping f(.)

In(.), we

x)A- A)-l(q, PI+QI PI QI (Ai(p, x) AI(q, x)) e PI + QI AI(p, x)Ai(q, x)

(_A]_l(p,

x)). (A/(p, x)- Ai(q, x))
The conclusion of the Theorem is obtained by taking the exponential of each term in the above.
Similarly, applying Theorem 3.1 to the convex mapping following result:
PROPOSITION 4.2 With the above assumptions and for J, K K 0, we have the following inequality:

In(.) yields the

79f (IN), J N

GjjK(p, x)

< exp

PJPK PJK

> O,
Ajx) Aj(p, x) (AJ(px)-AK(px))2

aj(p, X)AK (p, x)

where Pj
YiK Pi > 0 and Pjt_JK iJt_JK Pi.
5 APPLICATIONS, FOR THE EXPONENTIAL MAPPING
In the A-G mean inequality, we use the convex mapping f (.) In(.). If use the other well-known convex mapping f (.) exp(.) instead, further we
inequalities on the arithmetic mean can be obtained directly from Theorem 2.1 and 3.1 as follows:
PROPOSITION 5.1 (i) For p, q O,
0 < (PI d- QI) exp[Ai(p + q, x)]
PI exp[Al(p, x)] QI exp[Ai(q, x)] PIQI < (exp[Ai(p, x)] -exp[Ai (q, x)])(Ai(p, x)-Ai(q, x)). (5.1) PI+QI

(ii) For J, K

79f(IN), J fq K

(exp[A/(p, x)]

Pj exp[Aj(p, x)]
0 < (PJtK) exp[AjuK(p, x)]

PK exp[AK(p, x)]

AK(p, x)) (5.2)
exp[AK (p, x)]) (Aj(p, x)
The above inequalities (5.1) d (5.2) can be used fuher to derive sil inequalities for geometric mes. One simply replaces all occurence of xi by In yi in (5.1) and (5.2) to yield the following:
PoosIIO 5.2 (i) For p, q O, and Yi 0 i I, 0 (PI + QI)G(p + q, y) PGt(p, y) QiGz(q, y) PIQI < (GI (p, y)-GI (q, y))(ln G(p, y)-ln Gz(q, y)). (5.3) PI+Q1 (ii) For J, K Pf (IN), J K

(Pj)Gj(p, y)

PjGj(p, y)

PG(p, y)

(Gs(o, y)
G(O, r))0n G(p, y)-ln G(p, y)). (5.4)
Note that the first inequality of (5.3) suggests some nd of concavity
propey of the geometric mean function G I with respect to the first vable p, d the first inequality of (5.4) suggests some nd of concavity propey of the geometric mean function GI as an index set function in I.
6 FURTHER INEQUALITIES FOR THE GEOMETRIC MEAN

Consider the mapping

f IR --+ IR+ defined by

f(.) := ln(1 + exp(.))

exp(.) with f(.) 1 + exp(.) exp(.) and f"(.) > 0. (1 + exp(.)) 2 Clearly f is convex in IR. Consider the following mapping:

where x, y

I(p, I, x, y) :--
e GI(p,x + Y) GI(p, x) + GI(p, y)
THEOREM 6.1 With the above assumptions, we have, (i)/fp, q 0, then
F(p+q, I,x,y) > F(p, I, x, y)F(q, I, x, y) >_ 1,
(ii) If p >_ q >_ O, then
i.e., the mapping F (., I, x, y) is supermultiplicative in the first argument.
F (p, I, x, y) >_ F (q, I, x, y) >_ 1,
i.e., the mapping F(., I, x, y) is monotone nondecreasing in the

argument.

xi Proof Let the vector z (zi)iI be such that zi ln i 6 I. For x and z In x Then, using the convenience, let y denotes the vector eI convex mapping f as defined in (6.1), we have,

0 _<.T(ln(1

+ exp(.)), I, p, z)

Pi In 1

lnii (1-+-/.)- PI In
=piln(xi+yi)-_pilnyi-PIln

In H(xi

l -+-exp Iln (/I (.)pi)

Gi(p,y)

--[- yi) pi

eI In (GI(p, x)

+GI(p, y)) -I- In (GI (p, y))Pl GI(p, x + y) In Gl(p, x) + Gi(p, y) In F (p, I, x, y).
Using the first inequality of Theorem 3.1, we have

In F(p + q, I, x, y)

(ln(1 + exp(.)), I, p + q, z) > (ln(1 + exp(.)), I, p, z) + (ln(1 ln[r (p, I, x, y)F (q, I, x, y)],

+ exp(.)), I, q, z)

from which (6.3) follows. Similarly, (6.4) follows from the direct application of the first inequality of Theorem 3.1.
The following corollary follows from Theorem 6.1 and the fact that 1-(tp, I, x, y) (F(p, I, x, y))t.
COROLLARY 6.2 With the above assumptions, we have:

1-(tp -t- (1

t)q, I, x, y) >_ [l"(p, I, x, y)]t [1- (q, I, x, y)]-t

for all

variable.
[0, 1], i.e., the mapping F(., I, x, y) is log-concave in the first
THEOREM 6.3 (i) If J, K Try(IN) with J

0, then

F (p, J t_J K, x, y) > F (p, J, x, y) F (p, K, x, y) > 1

i.e., the mapping (p,., x, y) is supermultiplicative as an index set mapping. (ii) If J c_C_ K, j =/: 0, then
1 < F (p, J, x, y) < F (p, K, x, y)

(iii) We have,

F (p, I, x, y)

(iv) We have,

sup F (p, J, x, y) > 1.

pi-l-Pj

F(p, I, x, y) > max

[(xi -1- yi)

(xj @ yj)PJ ] Pi

-[- Pj

[xix;J]Pi-]-PJ [yiPiy;J]PiJc-PJ
The proofs of the above follow directly from Theorem 3.1, the details of which are omitted.

THEOREM 6.4 Let p, q

O, then we have:
F(p + q, I, x, y) F(p, I, x, y)F(q, I, x, y) GI(p,x) GI(q,Y) < GI(p,y) Gi(q,x)
GI(p, x)Gi(q, y) Gi(q, x)Gi(p, y) (G ;;i + GI(p, y))(G,(q, x) iil; Y))
The first inequality is just (6.3). To derive the second inequality, we use the convex mapping f(.) ln(1 + exp(.)) in the second inequality of Theorem 2.1 to get

0 < U(f, p + q, I, z)

.T(f, p, I, z)

.Y(f, q, I, z)

(6.10)

PIQI < PI+QI

Using the fact that
[f(ai(p,z))- f(ai(q,z))] [ai(p,z) ai(q,z)]

exp(.) 1 + exp(.)

we have,

f(AI(p, z))

f(Al(q, z))
exp(Al (p, z)) + exp(Al (p, z))

Gi(q, )

exp(Ai (q, z)) + exp(Ai(q, z)) x

+ Gi(p,

I+GI q,
G(q, x) Gi(q, x) + GI(q, y)
GI(p, x) GI(p, x) + GI(p, y)
Thus, by (6.10) we deduce that
0 < In F (p + q, I, x, y)
In F (p, I, x, y) In F (q, I, x, y) PIQI GI(p,x)GI(q,y)-GI(q,x)GI(P,Y) < PI + QI (GI(p-) + GI(p, y))(Gi(q, x) -t- Gi(q, y)) GI(px) _ln GI(q,x GI(p, y) Gi(q, y)

which is equivalent to

0 _< In
GI(q,Y) z < ln G(p, y) G(q, x)
+ [ F(p )-y-i;q /, ] { GI(p,_x) }

q, /, x, y)

PIQI P+Q

[ (G(p) +

Gi(p,x)Gi(q, y) Gi(q,x)Gi(p, y) G(p, y))(Gi(q, x) + G(q, y))
The conclusion follows from taking the exponential of each term of the above.
Similarly, application of the second inequality of Theorem 3.2 leads to the following result, which we shall merely state without proof.

THEOREM 6.5 Let K, J

79f (IN) with K fq J

0 and K, J :/: 0, then,

F(p, J t_J K, x, y) F(p, J, x, y)F(fl, K, x, y) G.(P, x) GK(p, Y) z < Gy(p,y) GK(p,x)

(6.11)

Gy(p, x)G(p, y) G(p, x)Gy(p, y) -[ (G(p, x) -t- G,(p, y))(GK(p, x) + G(p, Y)) _!

7 APPLICATIONS IN INFORMATION THEORY
Another application of Theorem 3.1 can be found in Information Theory. Suppose X is a discrete random variable having range R {xi, Z} and 2" }. Let p be having a probability distribution {0 < pi Pr(X xi), the probability vector corresponding to the probability distribution of X. The b-Entropy of the random variable X is defined by [2]:

Hb (X)

Hb (p)

Pi lOgb

Several inequalities for the entropy function can be established merely by applying the Jensen ineqaulity, the following is one of them:

0 < Hb(X)

Now lets say we are interested in the entropy of two sub-probability vectors of p (upon appropriate normalization) and we wish to relate this to the entropy of the original probability vector as given in (7.1). For some index subset J,K c_C_ 27, J C K 27, J,K 0, J tO K 0, we define the new random variables Xj and X/c having range in Rj := {xi, J} and S/ := {xi, K} and respective probability distributions
P where P := ,jeJ PJ > 0and P; "= jer PJ > 0.Letpj {p/J, j 6 J} and PK {p.r j 6 K} be the probability vectors corresponding to the probability distribution of Xj and XK respectively. The entropies of the two
sub-probability vectors are defined in the usual manner:

"=-

> 0, i J}, and

> 0, i K},

Hb(X) Hb(XK)

THEOREM 7.1

0 _< log b
1 E p/ log __--y Pi E PiK lgb Pi1

K"

With the above assumptions, we have

I-I- Hb(X)

Pj[logb

IJI- Hb(Xj)]

PK[logb

IKI- Hb(XK)]

<PjPK(lnb 1)

Consequently, if ?,

1271- Hb(X)
IJI- Hb(Xj)] + PK[lOgb IKI- Hb(X)].
Proof In Theorem 1.3, let f (.)

1 lnb (.) and let Xi

/ Then,
log/) (.) which is convex with V f (.)

.T(f, p, 2-, x)

____pilOgbPi_l_lOgb(jKPi)+Pjij -J

PJ lgb

lgb Pi
logb IJ KI- Hb(X) P[logb P[log IKI- Hb(X)],

iK KK IJI-

PK log/)

(-K iK )

Hb(X)]
The conclusion thus follows from replacing the terms in Theorem 3.1 by (7.2) and (7.3).

References

S.S. Dragomir, J. Peari6 and L.E. Persson, Properties of some functionals related to Jensens inequality, A cta Math. Hungarica, 60 (1996), 129-143. [2] R.J. McEliece, The Theory of Information and Coding, Addison Wesley Publishing Company, Reading, 1977.