Journal of Inequalities in Pure and
Applied Mathematics
INEQUALITIES RELATED TO REARRANGEMENTS OF
POWERS AND SYMMETRIC POLYNOMIALS
volume 4, issue 2, article 37,
2003.
CEZAR JOIŢA AND PANTELIMON STĂNICĂ
Department of Mathematics,
Lehigh University,
Bethlehem, PA 18015, USA.
E-Mail: cej3@lehigh.edu
URL: http://www.lehigh.edu/~cej3/cej3.html
Auburn University Montgomery,
Department of Mathematics,
Montgomery,
AL 36124-4023, USA.
EMail: pstanica@mail.aum.edu
URL: http://sciences.aum.edu/~stanica
Received 29 April, 2003;
accepted 19 May, 2003.
Communicated by: C. Niculescu
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2000
Victoria University
ISSN (electronic): 1443-5756
058-03
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Abstract
In [2] the second author proposed to find a description (or examples) of realvalued n-variable functions satisfying the following two inequalities:
if xi ≤ yi , i = 1, . . . , n, then F (x1 , . . . , xn ) ≤ F (y1 , . . . , yn ),
with strict inequality if there is an index i such that xi < yi ; and for 0 < x1 <
x2 < · · · < xn , then,
Inequalities Related to
Rearrangements of Powers and
Symmetric Polynomials
F (xx1 2 , xx2 3 , . . . , xxn1 ) ≤ F (xx1 1 , xx2 2 , · · · , xxnn ).
Cezar Joiţa and
Pantelimon Stănică
In this short note we extend in a direction a result of [2] and we prove a theorem that provides a large class of examples satisfying the two inequalities,
with F replaced by any symmetric polynomial with positive coefficients. Moreover, we find that the inequalities are not specific to expressions of the form xy ,
rather they hold for any function g(x, y) that satisfies some conditions. A simple
consequence of this result is a theorem of Hardy, Littlewood and Polya [1].
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2000 Mathematics Subject Classification: 05E05, 11C08, 26D05.
Key words: Symmetric Polynomials, Permutations, Inequalities.
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Both authors are associated with the Institute of Mathematics of Romanian Academy,
Bucharest – Romania.
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1
2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.
Introduction
In [2], the following problem was proposed: find examples of functions F :
Rn+ → R with the properties
(1.1)
if xi ≤ yi , i = 1, . . . , n, then F (x1 , . . . , xn ) ≤ F (y1 , . . . , yn ),
with strict inequality if there is an index i such that xi < yi ,
and
(1.2)
for 0 < x1 < x2 < · · · < xn , then,
x2
F (x1 , xx2 3 , . . . , xxn1 ) ≤ F (xx1 1 , xx2 2 , · · · , xxnn ).
Inequalities Related to
Rearrangements of Powers and
Symmetric Polynomials
Cezar Joiţa and
Pantelimon Stănică
In [2], the following result was proved.
Theorem 1.1. Assume that the permutation σ can be written as a product of
disjoint circular cycles C1 ×C2 ×· · ·×Cr , where each Ci is a cyclic permutation,
that is Ci (j) = j + ti , for some fixed ti . For any increasing sequence 0 < x1 <
· · · < xn , we have
n
X
(1.3)
i=1
n
Y
i=1
x
ai xi σ(i)
x
ai xi σ(i)
≤
≤
n
X
i=1
n
Y
ai xxi i , and
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i=1
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where ai ≥ 0 is increasing on the cycles Ci of σ.
J. Ineq. Pure and Appl. Math. 4(2) Art. 37, 2003
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(The condition on ai was inadvertently omitted in the final version of [2].)
In this short note we extend in a direction the previous result of [2] to any
permutation, not only the permutations which are products of circular cycles,
by proving (1.1) and (1.2) for symmetric polynomials with positive coefficients.
Finally, we prove that these inequalities are not specific only to rearrangements
of powers, that is, we find other classes of functions of 2-variables with real
values, say g(x, y), such that, for any σ ∈ Sn (the group of permutations), we
have
(1.4)
F (g(x1 , xσ(1) ), . . . , g(xn , xσ(n) )) ≤ F (g(x1 , x1 ), . . . , g(xn , xn )),
where F is any symmetric polynomial with positive coefficients.
Inequalities Related to
Rearrangements of Powers and
Symmetric Polynomials
Cezar Joiţa and
Pantelimon Stănică
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2.
The Results
Lemma 2.1. If f ∈ R[X1 , X2 ] is a symmetric polynomial with positive coefficients and (x1 , x2 ) ∈ R2+ and (y1 , y2 ) ∈ R2+ are such that x1 x2 ≤ y1 y2 and
xn1 + xn2 ≤ y1n + y2n , ∀ n ∈ N, then f (x1 , x2 ) ≤ f (y1 , y2 ).
Proof. We have
f (X1 , X2 ) =
X
aij X1i X2j =
X
X
aij X1i X2j + aji X1j X2i +
aii X1i X2i ,
i<j
where aij ∈ R+ . Since f is symmetric aij = aji , and therefore
X
X
f (X1 , X2 ) =
aij X1i X2j + X1j X2i +
aii X1i X2i
Inequalities Related to
Rearrangements of Powers and
Symmetric Polynomials
Cezar Joiţa and
Pantelimon Stănică
i<j
=
X
X
aij X1i X2i X1j−i + X2j−i +
aii X1i X2i .
i<j
It is clear now that the two conditions imposed on (x1 , x2 ) and (y1 , y2 ) imply
that f (x1 , x2 ) ≤ f (y1 , y2 ).
We will consider A ⊂ R and a function g : A × A → [0, ∞) with the
following property: for all x1 , x2 , y1 , y2 ∈ R such that x1 ≤ x2 and y1 ≤ y2 the
following two inequalities are satisfied:
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(2.1)
(2.2)
g(x1 , y2 )g(x2 , y1 ) ≤ g(x1 , y1 )g(x2 , y2 )
[g(x1 , y2 )]n + [g(x2 , y1 )]n ≤ [g(x1 , y1 )]n + [g(x2 , y2 )]n , ∀ n ∈ N.
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Theorem 2.2. Let F (X1 , X2 , . . . , Xn ) be a symmetric polynomial with positive
coefficients and g as above. Then for any σ ∈ Sn and any x1 , x2 , . . . , xn ∈ A
we have:
F (g(x1 , xσ(1) ), g(x2 , xσ(2) ), . . . , g(xn , xσ(n) ))
≤ F (g(x1 , x1 ), g(x2 , x2 ), . . . , g(xn , xn )).
Proof. Consider x1 , x2 , ..., xn ∈ A arbitrary and fixed. Without loss of generality we may assume that x1 ≤ x2 ≤ · · · ≤ xn . Let
m = max{F g(x1 , xσ(1) ), . . . , g(xn , xσ(n) ) | σ ∈ Sn }
Inequalities Related to
Rearrangements of Powers and
Symmetric Polynomials
Cezar Joiţa and
Pantelimon Stănică
and let
P = {σ ∈ Sn | F g(x1 , xσ(1) ), . . . , g(xn , xσ(n) ) = m}.
We would like to prove that e ∈ P where e is the identity. Let τ ∈ P the
permutation that has the minimum number of inversions among all elements of
P and suppose that τ 6= e. Since e is the only increasing permutation it follows
that there exists i ∈ {1, 2, . . . , n − 1} such that τ (i) > τ (i + 1). Without loss
of generality we may assume that i = 1. Consider τ 0 ∈ Sn defined as follows:
τ 0 (1) = τ (2), τ 0 (2) = τ (1) and τ 0 (j) = τ (j) if j ≥ 3. Then τ 0 has fewer
inversions than τ and therefore τ 0 6∈ P , which implies that:
(2.3) F g(x1 , xτ 0 (1) ), . . . , g(xn , xτ 0 (n) ) < F g(x1 , xτ (1) ), . . . , g(xn , xτ (n) ) .
Consider f (X1 , X2 ) = F (X1 , X2 , g(x3 , xτ (3) ), . . . , g(xn , xτ (n) )). It follows that
f is symmetric and has positive coefficients. If we set y1 = xτ 0 (1) = xτ (2) and
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y2 = xτ 0 (2) = xτ (1) it follows that y1 ≤ y2 . Using the two properties of g and
Lemma 2.1 we deduce that f (g(x1 , y2 ), g(x2 , y1 )) ≤ f (g(x1 , y1 ), g(x2 , y2 )) and
therefore
F g(x1 , xτ (1) ), . . . , g(xn , xτ (n) ) ≤ F g(x1 , xτ 0 (1) ), . . . , g(xn , xτ 0 (n) ) ,
which contradicts (2.3).
If g(x, y) = xy , then the conditions imposed on g are
xy12 xy21 ≤ xy11 xy22 ,
2
1
1
xny
+ xny
≤ xny
1
2
1
Inequalities Related to
Rearrangements of Powers and
Symmetric Polynomials
2
+ xny
2 ,
Cezar Joiţa and
Pantelimon Stănică
which are equivalent to
xy12 −y1 ≤ xy22 −y1 ,
n(y2 −y1 )
1
xny
1 (x1
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n(y2 −y1 )
1
− 1) ≤ xny
2 (x2
− 1).
The first inequality is certainly true as x1 ≤ x2 and y1 ≤ y2 . The second
inequality is true if 1 ≤ x1 ≤ x2 and y1 ≤ y2 . Therefore
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Corollary 2.3. The inequalities (1.1) and (1.2) are satisfied for all n-variable
symmetric polynomials with positive coefficients, defined on [1, ∞)n .
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If F (x1 , . . . , xn ) := x1 + · · · + xn , we can prove a result similar to the one
of Theorem 2.2 even if we significantly weaken the assumption on g.
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Theorem 2.4. Let A ⊂ R and g : A × A → R be a function such that ha,b (y) =
g(a, y) − g(b, y) (a > b) is increasing. Then for any x1 , x2 , . . . , xn ∈ A and any
σ ∈ Sn we have:
F g(x1 , xσ(1) ), g(x2 , xσ(2) ), . . . , g(xn , xσ(n) )
≤ F (g(x1 , x1 ), g(x2 , x2 ), . . . , g(xn , xn )) .
Proof. We follow the proof of Theorem 2.2 and the only thing we have to check
is that
F g(x1 , xτ (1) ), . . . , g(xn , xτ (n) ) ≤ F g(x1 , xτ 0 (1) ), . . . , g(xn , xτ 0 (n) ) .
But this inequality is equivalent to
Inequalities Related to
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Symmetric Polynomials
Cezar Joiţa and
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g(x1 , xτ (1) ) + g(x2 , xτ (2) ) ≤ g(x1 , xτ 0 (1) ) + g(x2 , xτ 0 (2) ).
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If we set y1 = xτ 0 (1) = xτ (2) and y2 = xτ 0 (2) = xτ (1) , it follows that y1 ≤ y2 and
the previous inequality can be written as
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g(x1 , y2 ) + g(x2 , y1 ) ≤ g(x1 , y1 ) + g(x2 , y2 ),
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which is equivalent to
hx2 ,x1 (y1 ) ≤ hx2 ,x1 (y2 ).
This inequality is satisfied because y1 ≤ y2 and hx2 ,x1 is increasing.
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Corollary 2.5. Let u, v be increasing functions on R with values in [1, ∞). The
following inequalities are true for all x1 , x2 , · · · , xn ∈ R
(2.4)
(2.5)
(2.6)
n
X
u(xi )v(xσ(i) ) ≤
i=1
n
X
i=1
n
Y
i=1
n
X
u(xi )v(xi ),
i=1
u(xi )v(xσ(i) ) ≤
n
X
u(xi )v(xi ) ,
i=1
u(xi )v(xσ(i) ) ≤
n
Y
u(xi )v(xi ) .
i=1
Proof. It suffices to prove that the following functions g(x, y) = u(x)v(y),
g(x, y) = u(x)v(y) , or g(x, y) = u(y)v(x) have the associated h’s increasing.
Let g(x, y) = u(x)v(y). Then h(y) = u(a)v(y)−u(b)v(y) = (u(a)−u(b))v(y)
which is increasing since u(a) ≥ u(b) and v(y) is increasing.
Let g(x, y) = u(x)v(y) . Then h(y) = u(a)v(y) − u(b)v(y) . Since u(a) ≥ u(b) ≥
1, and v(y) is increasing, by writing
!
v(y)
u(a)
−1 ,
h(y) = u(b)v(y)
u(b)
we see that h is increasing.
We remark that to prove (2.4) we only needed u, v to have positive values.
Using the previous remark, to show the last inequality, apply (2.4) with w =
log(u) and v (which are both increasing).
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Corollary 2.6. If the function h is decreasing on A, then all the inequalities are
reversed.
Remark 2.1. We see that Theorem 368 of [1] follows from (2.4) and Corollary
2.6.
Inequalities Related to
Rearrangements of Powers and
Symmetric Polynomials
Cezar Joiţa and
Pantelimon Stănică
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References
[1] G. HARDY, J.E. LITTLEWOOD AND G. PÓLYA, Inequalities, Cambridge
Univ. Press, 2001.
[2] P. STǍNICǍ, Inequalities on linear functions and circular powers, J.
Ineq. in Pure and Applied Math., 3(3) (2002), Art.43. [ONLINE: http:
//jipam.vu.edu.au/v3n3/015_02.html]
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Cezar Joiţa and
Pantelimon Stănică
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