Volume 9 (2008), Issue 4, Article 104, 5 pp.
A MULTINOMIAL EXTENSION OF AN INEQUALITY OF HABER
HACÈNE BELBACHIR
USTHB/ FACULTÉ DE M ATHÉMATIQUES
BP 32, E L A LIA , 16111 BAB E ZZOUAR
A LGER , A LGERIA .
hbelbachir@usthb.dz
Received 10 July, 2008; accepted 17 September, 2008
Communicated by L. Tóth
A BSTRACT. In this paper, we establish the following: Let a1 , a2 , . . . , am be non negative real
numbers, then for all n ≥ 0, we have
n
X
1
a1 + a2 + · · · + am
i 1 i2
im
a1 a2 · · · am ≥
.
n+m−1
m
m−1
i +i +···+i =n
1
2
m
The case m = 2 gives the Haber inequality. We apply the result to find lower bounds for the
sum of reciprocals of multinomial coefficients and for symmetric functions.
Key words and phrases: Haber inequality, multinomial coefficient, symmetric functions.
2000 Mathematics Subject Classification. 05A20, 05E05.
1. I NTRODUCTION
In 1978, S. Haber [3] proved the following inequality: Let a and b be non negative real
numbers, then for every n ≥ 0, we have
n
1
a+b
n
n−1
n−1
n
a + a b + · · · + ab
+b ≥
.
(1.1)
n+1
2
Another formulation of (1.1) is
f (x, y) ≥ f 12 , 12 for all non negative numbers x, y satisfying x + y = 1,
where
f (x, y) =
X
i+j=n
xi y j with x =
a
b
and y =
.
a+b
a+b
In 1983 [5], A. Mc.D. Mercer, using an analogous technique, gave an extension of Haber’s
inequality for convex sequences.
Special thanks to A. Chabour, S. Y. Raffed, and R. Souam for useful discussions. The proof given in the remark is due to A. Chabour.
201-08
2
H ACÈNE B ELBACHIR
Let (uk )0≤k≤n be a convex sequence, then the following inequality holds
n n
X
1 X
n
(1.2)
uk ≥
uk .
n + 1 k=0
k
k=0
In 1994 [1], using also the same tools, H. Alzer and J. Pečarić obtained a more general result
than the relation (1.2).
In 2004 [6], A. Mc.D. Mercer extended the result using an equivalent inequality of (1.1) as a
polynomial in x = ab , and deduced relations satisfying (1.2), see [1].
P
Let P (x) = nk=0 ak xk satisfy P (x) = (x − 1)2 Q (x) , where the coefficients of Q (x) are
real and non negative. Then if (uk )0≤k≤n is a convex sequence, we have
n
X
(1.3)
ak uk ≥ 0.
k=0
Our proposal is to establish an extension of the relation (1.1) to n real numbers.
2. M AIN R ESULT
In this section, we give an extension of the inequality given by the relation (1.1) for several
variables.
Theorem 2.1 (Generalized Haber inequality). Let a1 , a2 , . . . , am be non negative real numbers,
then for all n ≥ 0, one has
n
X
1
a1 + a2 + · · · + am
i1 i2
im
(2.1)
a1 a2 · · · am ≥
.
n+m−1
m
m−1
i +i +···+i =n
1
m
2
For another formulation of (2.1), let us consider the following homogeneous polynomial of
degree n
X
fm (x1 , x2 , . . . , xm ) =
xi11 xi22 · · · ximm
i1 +i2 +···+im =n
where x1 , x2 , . . . , xm are non negative real numbers satisfying the constraint x1 +x2 +· · ·+xm =
ai
1. By setting for all i = 1, . . . , m; xi = a1 +a2 +···+a
, the inequality given by (2.1) becomes
m
(2.2)
fm (x1 , x2 , . . . , xm ) ≥ fm m1 , m1 , . . . , m1
Proof. Let (y1 , y2 , . . . , ym ) be the values for which fm is minimal. It is well known that the
gradient of fm at (y1 , y2 , . . . , ym ) is parallel to that of the constraint which is (1, 1, . . . , 1) , one
then deduces
∂fm
∂fm
(x1 , . . . , xm )
=
(x1 , . . . , xm )
,
∂xα
∂xβ
xα =yα
xβ =yβ
for all α, β, 1 ≤ α 6= β ≤ m, which is equivalent to


m
X
X
 Y ij  iα −1 iβ
iα 
xj  yα yβ =
i1 +···+im =n
i1 +···+im =n
j=1
j6=α,β
i.e.

X
i1 +···+im =n


m
Y


m
 Y ij  iα iβ −1
iβ 
xj  yα yβ ,
j=1
j6=α,β

i −1
i 
xjj  yαiα −1 yββ (iα yβ − iβ yα ) = 0,
j=1
j6=α,β
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 104, 5 pp.
http://jipam.vu.edu.au/
M ULTINOMIAL I NEQUALITY OF H ABER
which one can write as


n
X
X
i −1


yαiα −1 yββ (iα yβ − iβ yα ) 
r=0
iα +iβ =r
3

X


i1 +···+im =n−r
ik 6=iα & ik 6=iβ

m
Y
i 
xjj  = 0.
j=1
j6=α,β
This last expression is a polynomial of several variables x1 , . . . , xj , . . . , xm (j 6= α, β) which
is null if all coefficients are zero. Then for yα = a and yβ = b, one obtains for every r = 0, . . . , n
X
ai−1 bj−1 (ib − ja) = 0.
i+j=r
By developing the sum and gathering the terms of the same power, one obtains
r−1
X
(2i + 1 − r) ai br−1−i = 0.
i=0
By gathering successively the extreme terms of the sum, we have
r+1
bX
2 c
(r − 2i − 1) ar−2i−1 − br−2i−1 a2i b2i = 0
i=0
which is equivalent to
(a − b)
r+1
bX
2 c r−2i−2
X
i=0
(r − 2i − 1) ak+2i br−k−2 = 0.
k=0
The double summation is positive, then one deduces that
a = b ⇐⇒ yα = yβ .
The symmetric group Sm acts naturally by permutations over R [x1 , x2 , . . . , xm ] and leaves
invariant fm (x1 , x2 , . . . , xm ) and x1 + x2 + · · · + xm = 1. Finally, one concludes that
1
y1 = y2 = · · · = ym = .
m
Remark 1. We can prove the above inequality using:
(1) induction over m exploiting Haber’s inequality and the well known relation
n
n − im
n
=
.
i1 , i2 , . . . , im
i1 , i2 , . . . , im−1
im
(2) the sectional method for the function
X
fm (x1 , x2 , . . . , xm ) =
xi11 xi22 · · · ximm
|i|=n
with the constraint x1 + x2 + · · · + xm = 1;
P
P
Let a1 , a2 , . . . , am and b1 , b2 , . . . , bm be real numbers such that i ai = 0 and i bi =
1, bi > 0, and consider the curve
X
Φ (t) =
(a1 t + b1 )i1 (a2 t + b2 )i2 · · · (am t + bm )im .
|i|=n
We prove that
b = (b1 , b2 , . . . , bm ) is a local minima for fm if and only if b1 = b2 =
1
· · · = bm = m .
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 104, 5 pp.
http://jipam.vu.edu.au/
4
H ACÈNE B ELBACHIR
Indeed, one has
X
i1 a 1 i2 a 2
i m am
i1 i2
im
∼
b1 b2 · · · bm
Φ (t) − Φ (0) =
+
+ ··· +
t + ···
b1
b2
bm
|i|=n
n+m−1
n+m−1
i1 a1
im am
(
)
∼
m−1
(b1 · · · bm )
+ ··· +
t + ··· .
=
m−1
b1
bm
∼ 2
If b1 = b2 = · · · = bm = m1 then Φ (t) − Φ (0)
P=aict + · · · , c > 0, . . .
If not, we can choose a1 , a2 , . . . , am such that i bi 6= 0, . . .
N.B. The possible nullity of some bi ’s is not a problem.
(3) the Popoviciu’s Theorem given in [7].
3. A PPLICATIONS
In this section we apply the previous result to find lower bounds for the sum of reciprocals of
multinomial coefficient and for two symmetric functions.
(1) Sum of reciprocals of multinomial coefficient.
Theorem 3.1. The following inequality holds
X
1
i1 +i2 +···+im =n
n
i1 ,...,im
n+m−1
m−1
m! · mn
≥
.
Proof. It suffices to integrate each side of the inequality given by the relation (2.2) :
fm (x1 , x2 , . . . , xm−1 , 1 − x1 − · · · − xm−1 ) ≥ fm
1 1
, , . . . , m1
m m
over the simplex
(
D=
xi , i = 1, . . . , m − 1 : xi ≥ 0,
m−1
X
)
xi ≤ 1 .
i=1
The left hand side gives under the sum the Dirichlet function (or the generalized beta
function) and is equal to the reciprocal of a multinomial coefficient. For the right hand
1
side we are led to compute the volume of the simplex D which is equal to m!
.
(2) An identity due to Sylvester in the 19th century, see [2, Thm 5], states that
Theorem 3.2. Let x1 , x2 , . . . , xm be independent variables. Then, one has in R[x1 , x2 ,
. . . , xm ]
m
X
X
xn+m−1
k1 k 2
i
km
Q
x1 x2 . . . xm =
.
j6=i (xi − xj )
i=1
k +···+k =n
1
m
As corollary of this theorem and Theorem 2.1, one obtains the following lower bound.
Corollary 3.3. Using the hypothesis of the above theorem, one has
n m
X
xn+m−1
x1 + x2 + · · · + xm
n+m−1
i
Q
≥
.
m
m−1
j6=i (xi − xj )
i=1
(3) The third application is about the symmetric polynomials. We need the following result:
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 104, 5 pp.
http://jipam.vu.edu.au/
M ULTINOMIAL I NEQUALITY OF H ABER
5
Theorem 3.4 ([2, Cor. 5] and [4, Th. 1]). Let x1 , x2 , . . . , xm be elements of unitary
commutative ring A with
X
Sk =
xi1 xi2 · · · xik ,
for 1 ≤ k ≤ m.
1≤i1 <i2 <···<ik ≤m
Then, for each positive integer n, one has
X
X k 1 + · · · + k m k1
km
x1 . . . xm =
(−1)n−k1 −···−km Sk11 . . . Skmm ,
k1 , . . . , k m
where the summations are taken over all m-tuples (k1 , k2 , . . . , km ) of integers kj ≥ 0
satisfying the relations k1 + k2 + · · · + km = n for the left hand side and k1 + 2k2 +
· · · + mkm = n for the right hand side.
This theorem and Theorem 2.1, give:
Corollary 3.5. Using the hypothesis of the last theorem, one has
S n
X k 1 + · · · + k m 1
1
n−k1 −···−km k1
km
(−1)
S
.
.
.
S
≥
.
1
m
n+m−1
k
,
.
.
.
,
k
m
1
m
m−1
where the summation is being taken over all m-tuples (k1 , k2 , . . . , km ) of integers kj ≥ 0
satisfying the relation k1 + 2k2 + · · · + mkm = n.
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Mat. [467], 11 (1994), 27–30.
[2] H. BELBACHIR AND F. BENCHERIF, Linear recurrent sequences and powers of a square matrix,
Electronic Journal of Combinatorial Number Theory, A12, 6 (2006).
[3] H. HABER, An elementary inequality, Internat. J. Math. and Math. Sci., 2(3) (1979), 531–535.
[4] J. MCLAUGHLIN AND B. SURY, Powers of matrix and combinatorial identities, Electronic Journal
of Combinatorial Number Theory, A13(5) (2005).
[5] A. McD. MERCER, A note on a paper by S. Haber, Internat. J. Math. and Math. Sci., 6(3) (1983),
609–611.
[6] A.McD. MERCER, Polynomials and convex sequence inequalities, J. Ineq. Pure Appl. Math., 6(1)
(2005), Art. 8. [ONLINE: http://jipam.vu.edu.au/article.php?sid=477].
[7] T. POPOVICIU, Notes sur les functions convexes d’ordre superieure (V), Bull. de la sci. Acad.
Roumaine, 22 (1940), 351–356.
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 104, 5 pp.
http://jipam.vu.edu.au/