Two Discrete Inequalities of Grüss Type Via Pólya- Szegö and Shisha Results for Real Numbers

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Two Discrete Inequalities of Grüss Type Via Pólya- Szegö and Shisha Results for Real Numbers

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Dragomir, Sever S and Khan, Lutfar R (2002) Two Discrete Inequalities of Grüss Type Via Pólya-Szegö and Shisha Results for Real Numbers. RGMIA research report collection, 5 (3).

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TWO DISCRETE INEQUALITIES OF GR ÜSS TYPE VIA P ÓLYA-SZEG Ö AND SHISHA RESULTS FOR REAL NUMBERS

S.S. DRAGOMIR AND L. KHAN

Abstract. Some new Gr¨uss type discrete inequalities for nonnegative real numbers and applications for the moments of guessing mappings are given.

1. Introduction

In 1950, Biernacki, Pidek and Ryll-Nardzewski [1] proved the following Gr¨uss type discrete inequality.

If ¯a = (a1, . . . , an) and ¯b = (b1, . . . , bn) are such that there exists the real numbersa, A, b, B with

(1.1) a≤ai≤A, b≤bi≤B, i∈ {1, . . . , n}

then

Cn ¯a,¯b

≤ 1 n

hn 2

i 1− 1

n hn

2 i

(A−a) (B−b) (1.2)

= 1

n² n²

4

(A−a) (B−b)

≤ 1

4(A−a) (B−b) where

Cn ¯a,¯b := 1

n

X

i=1

aibi−1 n

n

X

i=1

ai· 1 n

n

X

i=1

bi.

A weighted version of the above result has been obtained in 1988 by Andrica and Badea [2].

Let¯a,¯bsatisfy (1.1) and¯p= (p1, . . . , pn) be ann−tuple of nonnegative numbers withP_n>0.IfS is a subset of {1, . . . , n}that minimises the expression

(1.3)

X

i∈S

pi−1 2Pn

,

then

Cn p,¯ ¯a,b¯

≤PS

Pn

1−PS

Pn

(A−a) (B−b) (1.4)

≤1

4(A−a) (B−b),

Date: May 24, 2002.

1991Mathematics Subject Classification. Primary 26D15; Secondary 94A05.

Key words and phrases. Integral Inequalities, Grüss Type Discrete Inequalities, Pólya-Szegö Inequality, Shisha Inequality.

1

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wherePS :=P

i∈Spi where Cn ¯p,¯a,¯b

:= 1 P_n

n

X

i=1

piaibi− 1 P_n

n

X

i=1

piai· 1 P_n

n

X

i=1

pibi.

Recently, Dragomir and Booth [3] obtained the following result.

If¯a,b¯are realn−tuples and¯pis nonnegative withP_n>0, then

(1.5)

C_n ¯p,¯a,¯b

≤ max

1≤j≤n−1|∆a_j| max

1≤j≤n−1|∆b_j|C_n(¯p,¯e,¯e)

where ¯e = (1,2, . . . , n) and ∆aj := aj+1−aj is the forward difference, andj = 1, . . . , n−1. Note that

(1.6) C_n(¯p,¯e,¯e) = 1 P_n²

n

X

i=1

i²p_i− 1 Pn

n

X

i=1

ip_i

!2

.

In particular, we have

(1.7)

Cn ¯a,¯b ≤ 1

12 n²−1

1≤j≤n−1max |∆aj| max

1≤j≤n−1|∆bj|. The constant ₁₂¹ is best possible.

In 2002, Dragomir [4] extended the above result for the p−norm. Namely, he proved that

(1.8)

C_n ¯p,¯a,¯b ≤ 1

P_n² X

1≤j<i≤n

(i−j)

n−1

X

k=1

|∆a_k|^p

!¹_p _n−1 X

k=1

|∆b_k|^q

!¹_q

wherep >1, ¹_p+¹_q = 1.

(1.9)

Cn ¯a,¯b ≤ 1

6· n²−1 n

n−1

X

k=1

|∆ak|^p

!¹p n−1

X

k=1

|∆bk|^q

!¹q

.

The constant ¹₆ is best possible.

The case of one-norm [5], can be stated as follows:

(1.10)

C_n p,¯ ¯a,¯b ≤ 1

2· 1 P_n²

n

X

i=1

p_i(P_n−p_i)

n−1

X

k=1

|∆a_k|

n−1

X

k=1

|∆b_k|.

(1.11)

C_n ¯a,b¯ ≤1

2

1−1 n

ⁿ⁻¹ X

k=1

|∆ak|

n−1

X

k=1

|∆bk|. The constant ¹₂ is sharp.

Another direction was considered by Cerone and Dragomir in [8].

If¯a,¯bare realn−tuples and¯pis a positiven−tuple and there existsm, M ∈R such that

(1.12) m≤a_i≤M,

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INTEGRAL INEQUALITIES OF GR ¨USS TYPE 3

then one has the inequality Cn ¯p,¯a,b¯

≤1

2(M−m) 1 P_n

n

X

i=1

pi

bi− 1 P_n

n

X

j=1

pjbj

.

The constant ¹₂ is best possible. In particular, we have C_n ¯a,b¯

≤1

2(M−m)· 1 n

n

X

i=1

b_i−1 n

n

X

j=1

b_j .

The constant ¹₂ is best possible.

In this paper we obtain different Gr¨uss type discrete inequalities for nonnegative real numbers by the use of some counterpart results for the Cauchy-Buniakowsky- Schwarz inequality. Application for the moments of guessing mapping are also given.

2. Discrete Inequalities The following Gr¨uss type inequality holds.

Theorem 1. Let ¯a= (a1, . . . , an)and¯b= (b1, . . . , bn)be two sequences of positive real numbers with

(2.1) 0< a≤ai≤A <∞and0< b≤bi≤B <∞for each i∈ {1, . . . , n}. Then one has the inequality

(2.2)

Cn ¯a,b¯ ≤1

4 ·(A−a) (B−b)

√aAbB 1 n

n

X

i=1

ai· 1 n

n

X

i=1

bi.

The constant ¹₄ is best possible in (2.2) in the sense that it cannot be replaced by a smaller constant.

Proof. We have, by the Cauchy-Buniakowski-Schwarz inequality for double sums, the inequality

Cn ¯a,¯b (2.3)

=

1 2n²

n

X

i,j=1

(ai−aj) (bi−bj)

≤ 1 2n²

n

X

i,j=1

|(ai−aj) (bi−bj)|

≤ 1 2n²





n

X

i,j=1

(a_i−a_j)²

n

X

i,j=1

(b_i−b_j)²





1 2

= 1

2n²



4



n

n

X

i=1

a²_i −

n

X

i=1

a_i

!²





n

n

X

i=1

b²_i −

n

X

i=1

b_i

!²







1 2

=



 1 n

n

X

i=1

a²_i − 1 n

n

X

i=1

a_i

!2



1 2

 1 n

n

X

i=1

b²_i − 1 n

n

X

i=1

b_i

!2



1 2

.

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Utilising the P´olya-Szeg¨o inequality [19]

(2.4) 1≤

Pn

i=1z_i²Pn i=1u²_i (Pn

i=1ziui)² ≤1 4

rM₁M₂ m1m2

+

rm₁m₂ M1M2

!² ,

provided 0< m₁≤z_i≤M₁<∞,0< m₂≤u_i≤M₂<∞,i∈ {1, . . . , n},we may state that

nPn i=1a²_i (Pn

i=1ai)²

≤ 1 4

rA a +

ra A

!2

= 1

4·(A+a)² aA giving

nPn

i=1a²_i −(Pn i=1a_i)² (Pn

i=1ai)² ≤ 1

4·(A+a)²

aA −1 = (A−a)² 4aA , that is,

(2.5) n

n

X

i=1

a²_i −

n

X

i=1

ai

!²

≤ (A−a)² 4aA

n

X

i=1

ai

!² .

In a similar fashion, we obtain

(2.6) n

n

X

i=1

b²_i −

n

X

i=1

b_i

!2

≤(B−b)² 4bB

n

X

i=1

b_i

!2

. Using (2.3), (2.5) and (2.6), we deduce the desired inequality (2.2).

Now, assume that the inequality in (2.2) holds with a constantc >0,i.e.,

(2.7)

Cn ¯a,¯b

≤c(A−a) (B−b)

√ aAbB

1 n

n

X

i=1

ai· 1 n

n

X

i=1

bi.

If we choosen= 2, a1=b1, a2=b2, a1=a, a2=A,then from (2.7) we get 1

2 a²+A²

−1

4(a+A)²≤c(A−a)²

aA ·(a+A)² 4 giving

1

4(A−a)²≤c(A−a)²

aA ·(a+A)² 4 from where we get

(2.8) aA≤c(a+A)² for any 0< a < A <∞.

Leta= 1−ε, A= 1 +ε, withε∈(0,1). Then from (2.8) we get 1−ε²≤4c for anyε∈(0,1),which shows thatc≥ ¹₄.

Remark 1. We will now compare the inequality (2.2) with the Gr¨uss inequality

(2.9)

C_n ¯a,b¯ ≤1

4(A−a) (B−b), provideda≤ai≤Aandb≤bi≤B, i∈ {1, . . . , n}.

We consider, for a, b >0,the quantity U := 1

√

aAbB · 1 n

n

X

i=1

a_i· 1 n

n

X

i=1

b_i

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and we will assume thata=b, A=B, ai=bi, i∈ {1, . . . , n}. Thus U = (Pn

i=1ai)² n²aA .

Choosen= 3, a1=a2= 1, a3=x.ThusAa=xand we have U(x) = (x+ 2)²

9x . We observe that

U(x)−1 = x²−9x+ 4

9x =(x−1) (x−4)

9x ,

showing that if x∈(0,1]∪[4,∞), U(x)≥1 while forx∈(1,4), U(x)<1.

In conclusion, the bound provided by (2.2) is sometimes better, and at other times, worse than the bound provided by the Gr¨uss inequality.

The second result of Gr¨uss type is embodied in the following theorem.

Theorem 2. Let ¯a= (a1, . . . , an)and¯b= (b1, . . . , bn)be two sequences of positive real numbers satisfying (2.2). Then one has the inequality

(2.10)

C_n ¯a,b¯ ≤√

A−√ a √

B−√

br Pn i=1ai

n ·

Pn i=1bi

n .

The constant c= 1is the best possible in the sense that it cannot be replaced by a smaller constant.

Proof. We shall use Shisha’s inequality [20]

(2.11)

Pn i=1z_i² Pn

i=1ziyi

− Pn

i=1z_iy_i Pn

i=1y_i² ≤

rM1

m2

+ rm1

M2

!² , provided 0< m₁≤z_i≤M₁<∞and 0< m₂≤y_i≤M₂<∞.

If in (2.11) we choosez_i=a_i, y_i= 1, then we get

(2.12) 0≤ 1

n

X

i=1

a²_i − 1 n

n

X

i=1

ai

!2

≤ Pn

i=1ai

n

√ A−√

a2

.

Similarly

(2.13) 0≤ 1

n

X

i=1

b²_i − 1 n

n

X

i=1

bi

!2

≤ Pn

i=1bi

n

√ B−√

b² .

Now, making use of (2.3), (2.12) and (2.13), we obtain the desired inequality (2.10).

To prove the sharpness of the constant, assume that (2.10) holds with a constant c >0,i.e.,

(2.14) 1 n

n

X

i=1

aibi− 1 n

n

X

i=1

ai· 1 n

n

X

i=1

bi

≤c√ A−√

a √ B−√

br Pn i=1ai

n ·

Pn i=1bi

n .

If we choosen= 2, a₁=b₁, a₂=b₂, a₁=a, a₂=A,then from (2.14) we get 1

4(A−a)²≤c√ A−√

a2

·a+A 2 ,

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that is, 1 4

√ A−√

a²√ A+√

a²

≤c√ A−√

a²

·a+A 2 , giving for any 0< a < A <∞that

(2.15)

√ A+√

a²

≤2c(a+A).

If in (2.15) we choosea= 1−ε, A= 1 +ε, ε∈(0,1),we get √

1−ε+√ 1 +ε²

≤ 4c.Lettingε→0+,we deducec≥1,and the theorem is proved.

Remark 2. We shall show that at some times, the Gr¨uss inequality (2.8) is better, and at other times, the inequality (2.10) is better.

If we choosea_i=b_i, i= 1, n, a=b, A=B,we have to compare I1:= 1

4(A−a)² with

I2:=√ A−√

a2Pn i=1ai

n .

If we assume that a= 0, A= 1, then I₁= 1

4, I₂= Pn

i=1ai

n i= 1, n showing that for 0 ≤ a_i ≤ 1 with

Pn i=1a_i

n < ¹₄, (2.10) is better than the Gr¨uss inequality while for

Pn i=1a_i

n >¹₄, the Gr¨uss inequality provides a better bound.

Remark 3. We will show now the fact that the bounds provided by (2.2) and (2.5) cannot generally be compared.

Assume thata_i=b_i, (i= 1, . . . , n), a=b, A=b and consider J1:= 1

4

(A−a)² aA

1 n

n

X

i=1

ai

!2

J₂:=√ A−√

a²Pn i=1a_i

n .

If we choosea= 1, A= 4,we get J1= 9

16x², J2=xwherex:=

Pn i=1ai

n ∈[1,4]. We observe thatJ1−J2=^x(9x−16)₁₆ showing that forx∈

1,¹⁶₉

the bound provided by (2.10) is better than the bound provided by (2.10) while for x ∈ ¹⁶₉,4

, the conclusion is the other way around.

3. Applications for Moments of Guessing Mappings

In 1994, J.L. Massey [14] considered the problem of guessing the value taken on by a discrete random variable X in one trial of a random experiment by asking questions of the form

(3.1) “Did X take on itsi^th possible value?”

until the answer is

(3.2) “Yes!”.

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This problem arises for instance when a cryptologist must try out possible secret keys one at a time afterminimising the possibilities by some cryptoanalysis.

Consider a random variableX with finite rangeX={x1, . . . , xn}and distribu- tionP_X(x_k) =p_k fork= 1,2, . . . , n.

A one-to-one functionG:χ→ {1, . . . , n}is a guessing function forX.Thus

(3.3) E(G^m) :=

n

X

k=1

k^mpk

is the m^th moment of this function, provided we renumber thexi such thatxk is always thek^th guess.

In [14], Massey observed that, E(G), the average number of guesses, is min- imised by a guessing strategy that guesses the possible values of X in decreasing order of probability.

In the same paper [14], Massey proved that

(3.4) E(G)≥ 1

42^H(X)+ 1 provided H(X)≥2 bits, for an optimal guessing strategy, whereH(X) is the Shannon entropy

(3.5) H(X) =−

n

X

i=1

p_ilog₂(p_i).

He also has shown thatE(G) may be arbitrarily large whenH(X) is an arbitrarily small positive number such that there is no interesting upper bound on E(G) in terms ofH(X).

In 1996, Arikan [15] has proved that any guessing algorithm for X obeys the lower bound

(3.6) E(G^ρ)≥

Pn

k=1p

1 1+ρ

k

^1+ρ

[1 + lnn]^ρ , ρ≥0 while an optimal guessing algorithm forX satisfies

(3.7) E(G^ρ)≤

" _n X

k=1

p

1 1+ρ

k

#^1+ρ

, ρ≥0.

In 1997, Bozta¸s [16] proved that form≥1,integer (3.8) E(G^m)≤ 1

m+ 1

" _n X

k=1

p

1 1+m

k

#1+m

+ 1

m+ 1

m+ 1 2

E G^m−1

−

m+ 1 3

E G^m−2

+· · ·+ (−1)^m+1

provided the guessing strategy satisfies:

(3.9) p

1 1+m

k+1 ≤ 1 k

p

1 1+m

1 +· · ·+p

1 1+m

k

, k= 1, . . . , n−1.

In 1997, Dragomir and Bozta¸s [17] obtained for any guessing sequence:

(3.10)

E(G)−n+ 1 2

≤ (n−1) (n+ 1)

6 max

1≤i<j≤n|p_i−p_j|,

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(3.11)

E(G)−n+ 1 2

≤ v u u

t(n−1) (n+ 1)

nkpk²₂−1

12 ,

wherekpk²₂=Pn

i=1p²_i and (3.12)

E(G)−n+ 1 2

≤ n+ 1

2 n−

n+ 1 2

1≤k≤nmax

pk−1 n , where [x] is the integer part ofx.

For other results on E(G^p), p > 0 see also [18]. We mention only, by making use of Gr¨uss inequality, one has for p, q >0 that

(3.13)

E G^p+q

−E(G^p)E(G^q) ≤1

4(n^q−1) (n^p−1).

The above result may be complemented in the following way (see for example [11]).

Theorem 3. With the above assumptions, we have the inequality

(3.14)

E G^p+q

−1 +n^q

2 E(G^p)−1 +n^p

2 E(G^q) +1 +n^q

2 ·1 +n^p 2

≤ 1

4(n^q−1) (n^p−1). for any p, q >0.

Applications for different particular instances of p, q >0 may be provided, but we omit the details.

The following result also holds [9].

Theorem 4. AssumeS_n(p), p >0denotes the sum ofp-power of the firstnnatural numbers, that is

Sn(p) :=

n

X

k=1

i^p.

If

p_i≤(≥)1

n fori≤

S_n(p) n

1/p

and

pi≥(≤)1

n fori≥

Sn(p) n

^1/p + 1

wherebxcdenotes the integer part of x, then we have the inequality E(G^p)≥(≤)1

nSn(p).

We are able now to state the first reasult for the momments of guessing mapping that may be obtained by the use of the inequality (2.2).

Theorem 5. If the probability distribution(p1, ..., pn)satisfies the assumption (3.15) 0< p_m≤p_i ≤p_M for anyi∈ {1, ..., n},

then one has the inequality

E(G^p)− 1 nSn(p)

≤1 4

(pM−pm)

n^p/2+1 · n^p−1

√p_mp_M ·Sn(p).

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In particular, forp= 1,we have the inequality

E(G)−n+ 1 2

≤1 8

(pM−pm)

√n · n²−1

√pmpM

.

If one uses the other Gr¨uss type inequality (2.10), then one may state the following result as well.

Theorem 6. If the probability distribution (p₁, ..., p_n) satisfies the assumption (3.15), then one has the inequality

E(G^p)− 1 nSn(p)

≤(√

pM −√ pm)√

n^p−1 p Sn(p).

In particular, forp= 1,we have the inequality

E(G)−n+ 1 2

≤(√

pM−√ pm) √

n−1

rn(n+ 1)

2 .

References

[1] Biernacki, M., Pidek, H. and Ryll-Nardzewski, C. (1950), Sur une inégalité entre des intégrales definies,Ann. Univ. Mariae Curie-Skolodowska,A4, 1-4.

[2] Andrica, D. and Badea, C. (1988), Gr¨uss’ inequality for positive linear functionals,Periodica Math. Hungarica,19(2), 155-167.

[3] Dragomir, S.S. and Booth, G.L. (2000), On a Gr¨uss-Lupa¸s type inequality and its application for the estimation ofp−moments of guessing mappings,Math. Comm.,5, 117-126.

[4] Dragomir, S.S. (2002), Another Gr¨uss type inequality for sequences of vectors in normed linear spaces and applications,J. Comp. Analysis & Appl.,4(2), 157-172.

[5] Dragomir, S.S. (2002), A Gr¨uss type inequality for sequences of vectors in normed linear spaces, (Preprint) RGMIA Res. Rep. Coll., 5(2), Article 9. (ONLINE:

http://rgmia.vu.edu.au/v5n2.html)

[6] Dragomir, S. S. (2001), Integral Gr¨uss inequality for mappings with values in Hilbert spaces and applications.J. Korean Math. Soc.38, no. 6, 1261–1273.

[7] Dragomir, S. S. (1999), A generalization of Gr¨uss’s inequality in inner product spaces and applications.J. Math. Anal. Appl.237, no. 1, 74–82

[8] Cerone, P. and Dragomir, S.S. (2002), A refinement of Gr¨uss’ inequality and applications, RGMIA Res. Rep. Coll.,5(2002), No. 2, Article 15.

[9] Cerone, P. and Dragomir, S.S. (2002), New inequalities for ˇCebyˇsev functional involving two n-tuples of real numbers and applications, (Preprint)RGMIA Res. Rep. Coll.,5(3), Article 4. (ONLINE:http://rgmia.vu.edu.au/v5n3.html)

[10] Dragomir, S.S. and Peˇcari´c, J.(1989), Refinements of some inequalities for isotonic functionals, Anal. Num. Theor. Approx.,18, 61-65.

[11] Dragomir, S.S. (2002), A companion of the Gr¨uss inequality and applications,RGMIA Res.

Rep. Coll.,5, Supplement, Article 13.

[12] Fink, A. M. (1999), A treatise on Gr¨uss’ inequality.Analytic and geometric inequalities and applications, 93–113, Math. Appl., 478, Kluwer Acad. Publ., Dordrecht,

[13] Peˇcaric, J. (1980), On some inequalities analogous to Gr¨uss inequality. Mat. Vesnik 4(17)(32), no. 2, 197–202.

[14] Massey J.L. (1994), Guessing and entropy,Proc. 1994 IEEE Int. Symp. on Inf. Th.,(Trond- heim, Norway, 1994), p. 204.

[15] Arikan E. (1996), An inequality on guessing and its application to sequential decoding,IEEE Tran. Inf. Th.,42(1), 99-105.

[16] Bozta¸s S. (1997), Comments on “An Inequality of Guessing and Its Applications to Sequential Decoding”,IEEE Tran. Inf. Th.,43(6), 2062-2063.

[17] Dragomir S.S. and Bozta¸s S. (1997), Some estimates of the average number of guesses to determine a random variable, Proc. 1997 IEEE Int. Symp. on Inf. Th., (Ulm, Germany, 1997), p. 159.

[18] Dragomir S.S. and Bozta¸s S. (1998), Estimation of arithmetic means and their applications in guessing theory,Math. Comput. Modelling,28(10) (1998), 31-43.

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[19] P´olya, G. and Szeg¨o (1925),Aufgaben und Lehrsatze aus der Analysis, Vol. 1, Berlin 1925, pp. 57 and 213-214.

[20] Shisha, O. (1967),Inequalities I, New York, London, 293-308.

School of Communications and Informatics, Victoria University of Technology, PO Box 14428, MCMC 8001, Victoria, Australia.

E-mail address:[email protected]

URL:http://rgmia.vu.edu.au/SSDragomirWeb.html