A MULTINOMIAL EXTENSION OF AN INEQUALITY OF HABER

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Multinomial Inequality of Haber Hacène Belbachir vol. 9, iss. 4, art. 104, 2008

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A MULTINOMIAL EXTENSION OF AN INEQUALITY OF HABER

HACÈNE BELBACHIR

USTHB/ Faculté de Mathématiques BP 32, El Alia, 16111 Bab Ezzouar Alger, Algeria.

EMail:hbelbachir@usthb.dz

Received: 10 July, 2008

Accepted: 17 September, 2008

Communicated by: L. Tóth 2000 AMS Sub. Class.: 05A20, 05E05.

Key words: Haber inequality, multinomial coefficient, symmetric functions.

Abstract: In this paper, we establish the following: Leta1, a2, . . . , ambe non negative real numbers, then for alln≥0,we have

1

n+m−1 m−1

X

i₁+i₂+···+i_m=n

aⁱ₁¹aⁱ₂²· · ·aⁱm^m≥a1+a2+· · ·+am

m

ⁿ .

The casem = 2gives the Haber inequality. We apply the result to find lower bounds for the sum of reciprocals of multinomial coefficients and for symmetric functions.

Acknowledgements: 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.

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1. Introduction

In 1978, S. Haber [3] proved the following inequality: Letaandb be non negative real numbers, then for everyn ≥0,we have

(1.1) 1

n+ 1 aⁿ+aⁿ⁻¹b+· · ·+abⁿ⁻¹+bⁿ

≥

a+b 2

n

.

Another formulation of(1.1)is f(x, y)≥f ¹₂,¹₂

for all non negative numbersx, y satisfyingx+y = 1, where

f(x, y) = X

i+j=n

xⁱy^j with x= a

a+b andy = b a+b.

In 1983 [5], A. Mc.D. Mercer, using an analogous technique, gave an extension of Haber’s inequality for convex sequences.

Let(u_k)_0≤k≤nbe a convex sequence, then the following inequality holds

(1.2) 1

n+ 1

n

X

k=0

u_k ≥

n

X

k=0

n k

u_k.

In 1994 [1], using also the same tools, H. Alzer and J. Peˇcari´c 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 inx= ^a_b,and deduced relations satisfying(1.2), see [1].

LetP (x) = Pn

k=0a_kx^k satisfy P(x) = (x−1)²Q(x),where the coefficients ofQ(x)are real and non negative. Then if(u_k)_0≤k≤nis a convex sequence, we have

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(1.3)

n

X

k=0

a_ku_k≥0.

Our proposal is to establish an extension of the relation(1.1)tonreal numbers.

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2. Main Result

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). Leta₁, a₂, . . . , a_m be non negative real numbers, then for alln ≥0,one has

(2.1) 1

n+m−1 m−1

X

i1+i2+···+i_m=n

aⁱ₁¹aⁱ₂²· · ·aⁱ_m^m ≥

a₁ +a₂+· · ·+a_m m

n

.

For another formulation of (2.1), let us consider the following homogeneous polynomial of degreen

f_m(x₁, x₂, . . . , x_m) = X

i1+i2+···+im=n

xⁱ₁¹xⁱ₂²· · ·xⁱ_m^m

wherex₁, x₂, . . . , x_m are non negative real numbers satisfying the constraint x₁ + x₂+· · ·+x_m = 1.By setting for alli= 1, . . . , m;x_i = _a ^aⁱ

1+a2+···+am,the inequality given by(2.1)becomes

(2.2) f_m(x₁, x₂, . . . , x_m)≥f_m _m¹,_m¹, . . . ,_m¹

Proof. Let(y1, y2, . . . , ym)be the values for whichfmis minimal. It is well known that the gradient off_mat(y₁, y₂, . . . , y_m)is parallel to that of the constraint which is (1,1, . . . ,1),one then deduces

∂f_m

∂xα

(x₁, . . . , x_m) xα=yα

= ∂f_m

∂xβ

(x₁, . . . , x_m) xβ=yβ

,

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for allα, β,1≤α6=β ≤m,which is equivalent to

X

i1+···+im=n

i_α







m

Y

j=1 j6=α,β

xⁱ_j^j





yⁱ_α^α⁻¹y_βⁱ^β = X

i1+···+im=n

i_β







m

Y

j=1 j6=α,β

xⁱ_j^j





y_αⁱ^αy_βⁱ^β⁻¹,

i.e.

X

i1+···+im=n







m

Y

j=1 j6=α,β

xⁱ_j^j





y_αⁱ^α⁻¹yⁱ_β^β⁻¹(i_αy_β −i_βy_α) = 0, which one can write as

n

X

r=0



 X

iα+iβ=r

y_αⁱ^α⁻¹y_βⁱ^β⁻¹(i_αy_β−i_βy_α)











X

i1+···+im=n−r ik6=iα&ik6=iβ







m

Y

j=1 j6=α,β

xⁱ_j^j











= 0.

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 foryα =aandyβ =b,one obtains for everyr= 0, . . . , n

X

i+j=r

aⁱ⁻¹b^j−1(ib−ja) = 0.

By developing the sum and gathering the terms of the same power, one obtains

r−1

X

i=0

(2i+ 1−r)aⁱb^r−1−i = 0.

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By gathering successively the extreme terms of the sum, we have b^r+12 c

X

i=0

(r−2i−1) a^r−2i−1−b^r−2i−1

a²ⁱb²ⁱ = 0

which is equivalent to

(a−b) b^r+12 c

X

i=0

r−2i−2

X

k=0

(r−2i−1)a^k+2ib^r−k−2 = 0.

The double summation is positive, then one deduces that a=b⇐⇒y_α =y_β.

The symmetric groupSm acts naturally by permutations over R[x1, x2, . . . , xm] and leaves invariantf_m(x₁, x₂, . . . , x_m)andx₁ +x₂+· · ·+x_m = 1.Finally, one concludes that

y₁ =y₂ =· · ·=y_m = 1 m.

Remark 1. We can prove the above inequality using:

1. induction overmexploiting Haber’s inequality and the well known relation n

i₁, i₂, . . . , i_m

=

n−im

i₁, i₂, . . . , im−1

n i_m

.

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2. the sectional method for the function

f_m(x₁, x₂, . . . , x_m) = X

|i|=n

xⁱ₁¹xⁱ₂²· · ·xⁱ_m^m

with the constraintx₁+x₂+· · ·+x_m = 1;

Leta₁, a₂, . . . , a_m andb₁, b₂, . . . , b_m be real numbers such thatP

ia_i = 0and P

ib_i = 1, b_i >0,and consider the curve Φ (t) = X

|i|=n

(a₁t+b₁)ⁱ¹(a₂t+b₂)ⁱ²· · ·(a_mt+b_m)ⁱ^m.

We prove that b = (b₁, b₂, . . . , b_m) is a local minima for f_m if and only if b₁ =b₂ =· · ·=b_m = _m¹

. Indeed, one has

Φ (t)−Φ (0)∼= X

|i|=n

bⁱ₁¹bⁱ₂²· · ·bⁱ_m^m i₁a₁

b₁ + i₂a₂

b₂ +· · ·+i_ma_m b_m

t+· · ·

∼=

n+m−1 m−1

(b₁· · ·b_m)(

n+m−1 m−1 )

i1a1

b₁ +· · ·+ imam

b_m

t+· · · .

Ifb₁ =b₂ =· · ·=b_m = _m¹

thenΦ (t)−Φ (0)∼=ct²+· · · , c >0, . . . If not, we can choosea₁, a₂, . . . , a_msuch thatP

i ai

bi 6= 0, . . . N.B. The possible nullity of someb_i’s is not a problem.

3. the Popoviciu’s Theorem given in [7].

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3. Applications

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

i1+i2+···+im=n

1

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) :

f_m(x₁, x₂, . . . , xm−1,1−x₁− · · · −xm−1)≥f_m _m¹,_m¹, . . . ,_m¹ over the simplex

D=

(

x_i, i = 1, . . . , m−1 :x_i ≥0,

m−1

X

i=1

x_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 side we are led to compute the volume of the simplexDwhich is equal to _m!¹ .

2. An identity due to Sylvester in the 19th century, see [2, Thm 5], states that

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Theorem 3.2. Letx₁, x₂, . . . , x_m be independent variables. Then, one has in R[x₁, x₂, . . . , x_m]

X

k1+···+km=n

x^k₁¹x^k₂². . . x^k_m^m =

m

X

i=1

x^n+m−1_i Q

j6=i(x_i−x_j).

As corollary of this theorem and Theorem2.1, one obtains the following lower bound.

Corollary 3.3. Using the hypothesis of the above theorem, one has

m

X

i=1

x^n+m−1_i Q

j6=i(xi−xj) ≥

x₁+x₂+· · ·+x_m m

n

n+m−1 m−1

.

3. The third application is about the symmetric polynomials. We need the following result:

Theorem 3.4 ([2, Cor. 5] and [4, Th. 1]). Let x₁, x₂, . . . , x_m be elements of unitary commutative ringAwith

S_k = X

1≤i1<i2<···<i_k≤m

x_i₁x_i₂· · ·x_i_k, for1≤k≤m.

Then, for each positive integern, one has Xx^k₁¹. . . x^k_m^m =X

k₁+· · ·+k_m k1, . . . , km

(−1)^n−k¹^{−···−k}^mS^k₁¹. . .S^k_m^m, where the summations are taken over all m-tuples (k₁, k₂, . . . , k_m)of integers k_j ≥ 0satisfying the relations k₁+k₂+· · ·+k_m = n for the left hand side andk1+ 2k2+· · ·+mkm =nfor the right hand side.

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This theorem and Theorem2.1, give:

Corollary 3.5. Using the hypothesis of the last theorem, one has

1

n+m−1 m−1

X

k₁+· · ·+k_m k₁, . . . , k_m

(−1)^n−k¹^{−···−k}^mS^k₁¹. . .S^k_m^m ≥_S₁ m

ⁿ .

where the summation is being taken over all m-tuples(k₁, k₂, . . . , k_m)of inte- gersk_j ≥0satisfying the relationk₁ + 2k₂+· · ·+mk_m =n.

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References

[1] H. ALZERANDJ. PE ˇCARI ´C, On an inequality of A.M. Mercer, Rad Hrvatske Akad. Znan. Umj. Mat. [467], 11 (1994), 27–30.

[2] H. BELBACHIRANDF. 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 identi- ties, 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.