573
Internat. J. Math. & Math. Sci.
VOL. 20 NO. 3 (1997) 573-576
INEQUALITIES VIA
LAGRANGE MULTIPLIERS
W. T. SULAIMAN
Zarka Private University
P O. Box 150863
Zarka 13115, JORDAN
(Received August 8, 1993 and in revised form June 6, 1996)
ABSTRACT. An easy method is obtained to prove many inequalities using Lagrange mutipliers.
KEY WORDS AND PHRASES: Inequalities.
1991 AMS SUBJECT CLASSIFICATION CODES: 90C50, 980C39.
INTRODUCTION
Let us assume that d l, dn are unit perpendicular vectors in an n-dimensional space X. In
particular dl, ah, and d3 are the unit perpendicular vectors i, j, and k in the 3-dimensional space. Any
vector v in X is usually uniquely written in the form
1.
v
E Aida
for scalars A,. We define
V J(zl,
’A,(zl,...,z,)d,,
z,)
t=l
Kapur and Kumar (1986), have used the principle of dynamic programming to prove major inequalities
due to Shannon, genyi, and Holder, see [1]. In this note we give a new method using Lagrange
multipliers.
2.
SHANNON’S INEQUALITY
THEOREM 2.1. Given /9/- a,
qi
b, then
t=l
=1
aln(a/b) <
E p, ln(pi/qi),
p,, q,
The equality holds iffp q, for each i.
PROOF. Let the q,’s and a be fixed; set
f(1,
P, In(p,/q,)"
Pn)
p,, q,
t---I
we aim to minimize
f subject to the constraint
g(p
p,,)
p,
a
o.
_
>_ O.
t----1
O,
574
W T SULAIMAN
A 7 g because g is linear and f is convex, since its second
There is a minimum achieved where 7 f
order partials are all non-negative
i=1
=1
1 + In(pi/q.)
q
A
p
q,
a,
a
q2
b,
b
In(a/b)
p,
a ln(a/b),
Therefore
min
E p, ln(p,/q,)
or
<_
a ln(a/b)
If a
b
pi
ln(I&/q,).
1, we get Sharmon’s inequality
pi
ln(p,/qi) > 0 and
p, ln(pi/qi)
0 iff p,
q,
for each i.
i=1
3.
RENYI’S INEQUALITY
THEOREM 3.1. Given a,
1
b,
a,
b, then
,=1
t=l
1
(aab 1-a-a) < E
1
(pq_, p),
p,, q, >0, 0
<a#
)’ v,
, o
1.
The equality holds iffp, q/for each i.
PROOF. Let the qi’s and a be fixed and write
f (pl
1
pn
,=1
o
o(p
1
,)
t=l
by the convexity of f and linearity of g. Hence
1
aO b l-,
a-1
If a
b
,=1
a-1
1, we get Renyi’s inequality
a--1
4.
HOLDER’S INEQUALITY
THEOREM 4.1. Given a,P
t=l
A,
b, =B, abi=C, a,b,>_0, p,q>l,
,=1
,=1
+ =1,
INEQUALITIES VIA LAGRANGE MULTIPLIERS
575
C < AX/r’B 1/q.
(4 1)
then
PROOF. This follows from Renyi’s inequality, taking a
l/p,
q,q, or, we prove the
fi, b
ai
result directly as follows:
let the a’s and C be fixed and write
f(bl,
bn)
Aq/P
E bqt, g(bl
bn)
t=l
7f
bq-ld
A $7 9 =" q Aq/’
::, Aq/’b,q-1
=A
0
oa
A
=1
(4.2)
C
atb,
=1
:=1
(A/q)at
q/p
(A/q)C,
AqB
(A/q)A,
(4.2)
(4.3)
and
as p(q-
1)
(4.4)
q
Cq-1.
(4.3) & (4.4) =, k/q
Therefore, by the convexity of f and linearity of g,
rrfm(Aq/r’B)
Cq,
or
C < A1/’B 1/q.
5.
GENERALIZATIONS OF HOLDER’S INEQUALITY
THEOREM S.1. Given
A, b B, c" C, and
=1
?++;=1,
a
:=1
=1
a,b,c
D,
, b,, c >_ 0,
:=1
then
D < al/r’BUqC Ur.
PROOF. This follows by an easy application ofHolder’s inequality:
c
6.
MINKOWSKI’S INEQUALITY
THEOREM 6.1. Given a’ A,
i=l
Eb
=1
B, and E (a + ai)P
C<A+
PROOF. Let the b,’s and A be fixed and write
=1
C, q,, b, > 0, p > 1, then
576
W T SULAJMAN
f(1,
an)=
(, + b,) p, (
.al,...,a
t=l
::*.
a
A=O
t=l
ba
a
b,
--Co
an
Therefore,
(a, + ca,)’
max Ck
( + c)A
A7, +cA7,
AT,
+ BT,,
C7, < AT,
+ BT,.
or
-
7. ARITHMETIC-GEOMETRIC-MEAN INEQUALITY
THEOREM 7.1.
t
t=l
PROOF. Write
1
/"
Xtt=l
.q(Zl
,=n)
n
z,-C=O.
Let C be fixed, we have
t=l
n
Xt
n
t=l
=xt= -y
= C-- -yo
Therefore
maxy =-y=C,
#
or
X
t=l
_<
X.
t=l
ACKNOWLEDGMENT. The author is so grateful to the referee for his kind remarks, suggestions, and
improvements of this paper.
REFERENCES
KAPUR, J.N, KUMAR, V. and KUMAR, U., A measure of mutual divergence among a number of
probability distributions, ]nternat. J. Math. &Math. Sci., 10 (1987), 597-608.