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volume 5, issue 3, article 58, 2004.

Received 09 January, 2004;

accepted 22 April, 2004.

Communicated by:P.S. Bullen

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Journal of Inequalities in Pure and Applied Mathematics

A DISCRETE EULER IDENTITY

A. AGLI Ć ALJINOVI Ć AND J. PE ˇCARI Ć

Department of Applied Mathematics

Faculty of Electrical Engineering and Computing Unska 3, 10 000 Zagreb, Croatia.

EMail:andrea@zpm.fer.hr Faculty of Textile Technology University of Zagreb Pierottijeva 6, 10000 Zagreb Croatia.

EMail:pecaric@mahazu.hazu.hr

c

2000Victoria University ISSN (electronic): 1443-5756 086-04

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A Discrete Euler Identity A. Aglić Aljinović and J. Peˇcarić

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J. Ineq. Pure and Appl. Math. 5(3) Art. 58, 2004

http://jipam.vu.edu.au

Abstract

A discrete analogue of the weighted Montgomery identity (i.e. Euler identity) for finite sequences of vectors in normed linear space is given as well as a discrete analogue of Ostrowski type inequalities and estimates of difference of two arithmetic means.

2000 Mathematics Subject Classification:26D15

Key words: Discrete Montgomery identity, Discrete Ostrowski inequality.

1. Introduction

The following Ostrowski inequality is well known [10]:

f(x)− 1 b−a

Z b a

f(t)dt

≤

"

1

4+ x− ^a+b₂ 2

(b−a)²

#

(b−a)kf⁰k_∞.

It holds for everyx∈[a, b]wheneverf : [a, b]→Ris continuous on[a, b]and differentiable on(a, b)with derivativef⁰ : (a, b)→Rbounded on(a, b)i.e.

kf⁰k_∞= sup

t∈(a,b)

|f⁰(t)|<+∞.

Let f : [a, b] → R be differentiable on [a, b], f⁰ : [a, b] → R integrable on [a, b]andw : [a, b] → [0,∞)some probability density function, i.e. integrable function satisfying Rb

a w(t)dt = 1; defineW(t) = Rt

aw(x)dx for t ∈ [a, b], W(t) = 0fort < aandW(t) = 1fort > b. The following identity, given by Peˇcari´c in [11], is the weighted Montgomery identity

f(x) = Z b

a

w(t)f(t)dt+ Z b

a

P_w(x, t)f⁰(t)dt, where the weighted Peano kernel is

P_w(x, t) =

( W(t), a≤t≤x, W(t)−1 x < t≤b.

All results in this paper are discrete analogues of results from [1]. The aim of this paper is to prove the discrete analogue of the weighted Euler identity for

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finite sequences of vectors in normed linear spaces and to use it to obtain some new discrete Ostrowski type inequalities as well as estimates of differences between two (weighted) arithmetic means. In Section2, a discrete weighted Mont- gomery (i.e. Euler) identity is presented. In Section 3, Ostrowski’s inequality and its generalization are proved. These are the discrete analogues of some results from [6]. In Section4, estimates of differences between two (weighted) arithmetic means are given and these are the discrete analogues of some results from [2], [3], [4], [5] and [12].

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2. Discrete Weighted Euler Identity

Let x₁, x₂, . . . , x_n be a finite sequence of vectors in the normed linear space (X,k·k) and w1, w2, . . . , wn finite sequence of positive real numbers. If, for 1≤k ≤n,

W_k =

k

X

i=1

w_i, W_k=

n

X

i=k+1

w_i =W_n−W_k,

then we have, see [9],

(2.1)

n

X

i=1

wixi

=x_kW_n+

k−1

X

i=1

W_i(x_i−x_i+1) +

n−1

X

i=k

W_i(x_i+1−x_i), 1≤k ≤n.

The difference operator∆is defined by

(2.2) ∆x_i =x_i+1−x_i.

So using formula (2.1), we get the discrete analogue of weighted Mont- gomery identity

(2.3) x_k = 1

W_n

n

X

i=1

w_ix_i+

n−1

X

i=1

D_w(k, i) ∆x_i,

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where the discrete Peano kernel is defined by

(2.4) D_w(k, i) = 1 W_n ·

( W_i, 1≤i≤k−1,

−Wi

, k ≤i≤n.

If we takew_i = 1, i = 1, . . . , n, thenW_i = i andW_i = n −i, and (2.3) reduces to the discrete Montgomery identity

(2.5) x_k = 1

n

X

i=1

x_i+

n−1

X

i=1

D_n(k, i) ∆x_i,

where

D_n(k, i) = ( i

n, 1≤i≤k−1,

i

n−1, k≤i≤n.

Ifn∈N,∆ⁿis inductively defined by

∆ⁿx_i = ∆ⁿ⁻¹(∆x_i).

It is then easy to prove, by induction or directly using the elementary theory of operators,see [8], that

∆ⁿx_i =

n

X

k=0

n k

(−1)^n−kx_i+k.

In the next theorem we give the generalization of the identity (2.3).

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Theorem 2.1. Let(X,k·k)be a normed linear space,x₁, x₂, . . . , x_na finite se- quence of vectors inX, w1, w2, . . . , wnfinite sequence of positive real numbers.

Then for allm ∈ {2,3, . . . , n−1}andk ∈ {1,2, . . . , n}the following identity is valid:

(2.6) x_k= 1 W_n

n

X

i=1

w_ix_i+

m−1

X

r=1

1 n−r

n−r

X

i=1

∆^rx_i

!

×

n−1

X

i1=1 n−2

X

i2=1

· · ·

n−r

X

ir=1

D_w(k, i₁)D_n−1(i₁, i₂)· · ·D_n−r+1(i_r−1, i_r)

!

+

n−1

X

i1=1 n−2

X

i2=1

· · ·

n−m

X

im=1

D_w(k, i₁)D_n−1(i₁, i₂)· · ·D_n−m+1(i_m−1, i_m) ∆^mx_i_m.

Proof. We prove our assertion by induction with respect to m. For m = 2we have to prove the identity

xk = 1 W_n

n

X

i=1

wixi+ 1 n−1

n−1

X

i=1

∆xi

! _n−1 X

i=1

Dw(k, i)

!

+

n−1

X

i=1 n−2

X

j=1

D_w(k, i)Dn−1(i, j) ∆²x_j.

Applying the identity (2.5) for the finite sequence of vectors∆xi,i= 1,2, . . . , n−

1,we obtain

∆x_i = 1 n−1

n−1

X

i=1

∆x_i+

n−2

X

j=1

Dn−1(i, j) ∆²x_j

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so, again using (2.3), we have

x_k = 1 W_n

n

X

i=1

w_ix_i+

n−1

X

i=1

D_w(k, i) 1 n−1

n−1

X

i=1

∆x_i+

n−2

X

j=1

Dn−1(i, j) ∆²x_j

!

= 1 W_n

n

X

i=1

w_ix_i+ 1 n−1

n−1

X

i=1

∆x_i

! _n−1 X

i=1

D_w(k, i)

!

+

n−1

X

i=1 n−2

X

j=1

D_w(k, i)Dn−1(i, j) ∆²x_j.

Hence the identity (2.6) holds form = 2.

Now, we assume that it holds for a natural number m ∈ {2,3, . . . , n−2}.

Applying the identity (2.5) for the∆^mxim

∆^mx_i_m = 1 n−m

n−m

X

i=1

∆^mx_i+

n−m−1

X

im+1=1

Dn−m(i_m, i_m+1) ∆^m+1x_i_m+1

and using the induction hypothesis, we get

x_k = 1 W_n

n

X

i=1

w_ix_i+

m−1

X

r=1

1 n−r

n−r

X

i=1

∆^rx_i

!

×

n−1

X

i1=1 n−2

X

i2=1

· · ·

n−r

X

ir=1

D_w(k, i₁)Dn−1(i₁, i₂)· · ·Dn−r+1(ir−1, i_r)

!

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+

n−1

X

i1=1 n−2

X

i2=1

· · ·

n−m

X

im=1

D_w(k, i₁)Dn−1(i₁, i₂)· · ·Dn−m+1(im−1, i_m)

×



 1 n−m

n−m

X

i=1

∆^mx_i+

n−m−1

X

im+1=1

Dn−m(i_m, i_m+1) ∆^m+1x_i_m+1









= 1 W_n

n

X

i=1

w_ix_i+

m

X

r=1

1 n−r

n−r

X

i=1

∆^rx_i

!

×

n−1

X

i1=1 n−2

X

i2=1

· · ·

n−r

X

ir=1

D_w(k, i₁)D_n−1(i₁, i₂)· · ·D_n−r+1(i_r−1, i_r)

!

+

n−1

X

i1=1 n−2

X

i2=1

· · ·

n−(m+1)

X

im+1=1

D_w(k, i₁)Dn−1(i₁, i₂)· · ·Dn−m(i_m, i_m+1) ∆^m+1x_i_m+1.

We see that (2.6) is valid form+ 1and our assertion is proved.

Remark 2.1. Form=n−1(2.6) becomes

x_k = 1 W_n

n

X

i=1

w_ix_i+

n−2

X

r=1

1 n−r

n−r

X

i=1

∆^rx_i

!

×

n−1

X

i1=1 n−2

X

i2=1

· · ·

n−r

X

ir=1

!

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+

n−1

X

i1=1 n−2

X

i2=1

· · ·

1

X

in−1=1

D_w(k, i₁)Dn−1(i₁, i₂)· · ·D₂(in−2, in−1) ∆ⁿ⁻¹x_i_n−1.

Corollary 2.2. Let(X,k·k)be a normed linear space,x₁, x₂, . . . , x_na finite se- quence of vectors inX. Then for allm∈ {2,3, . . . , n−1}andk ∈ {1,2, . . . , n}

the following identity is valid:

x_k = 1 n

n

X

i=1

x_i+

m−1

X

r=1

1 n−r

n−r

X

i=1

∆^rx_i

!

×

n−1

X

i1=1 n−2

X

i2=1

· · ·

n−r

X

ir=1

D_n(k, i₁)D_n−1(i₁, i₂)· · ·D_n−r+1(i_r−1, i_r)

!

+

n−1

X

i1=1 n−2

X

i2=1

· · ·

n−m

X

im=1

Dn(k, i1)Dn−1(i1, i2)· · ·Dn−m+1(im−1, im) ∆^mxim.

Proof. Apply Theorem2.1withw_i = 1,i= 1, . . . , n.

Remark 2.2. If we apply (2.6) withn = 2l−1andk =lwe get

x_l = 1 W_2l−1

2l−1

X

i=1

w_ix_i+

m−1

X

r=1

1 2l−1−r

2l−1−r

X

i=1

∆^rx_i

!

×

2l−2

X

i1=1 2l−3

X

i2=1

· · ·

2l−1−r

X

ir=1

D_w(l, i₁)D2l−2(i₁, i₂)· · ·D2l−r(ir−1, i_r)

!

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+

2l−2

X

i1=1 2l−3

X

i2=1

· · ·

2l−1−m

X

im=1

D_w(l, i₁)D_2l−2(i₁, i₂)· · ·D_2l−m(i_m−1, i_m) ∆^mx_i_m.

We may regard this identity as a generalized midpoint identity since for m= 1 it reduces to

(2.7) x_l = 1

W_2l−1

2l−1

X

i=1

w_ix_i+

2l−2

X

i=1

D_w(l, i) ∆x_i

and further forwi = 1, i= 1,2, . . . ,2l−1to

(2.8) x_l = 1

2l−1

X

i=1

x_i+ 1 2l−1

l−1

X

i=1

i(∆x_i−∆x2l−1−i).

Similarly, if we apply (2.6) withk = 1and then withk =n, then sum these two equalities and divide them by2, we get

(2.9) x1 +xn

2 = 1

W_n

n

X

i=1

w_ix_i+

m−1

X

r=1

1 n−r

n−r

X

i=1

∆^rx_i

!

×

n−1

X

i1=1

· · ·

n−r

X

ir=1

D_w(1, i₁) +D_w(n, i₁)

2 Dn−1(i₁, i₂)· · ·Dn−r+1(ir−1, i_r)

!

+

n−1

X

i1=1

· · ·

n−m

X

im=1

D_w(1, i₁) +D_w(n, i₁)

2 Dn−1(i₁, i₂)· · ·Dn−m+1(im−1, i_m) ∆^mx_i_m.

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We may regard this identity as a generalized trapezoid identity since form = 1 it reduces to

(2.10) x₁+x_n

2 = 1

W_n

n

X

i=1

w_ix_i+

n−1

X

i=1

D_w(1, i) +D_w(n, i) 2 ∆x_i, and further forw_i = 1, i= 1,2, . . . , nto

(2.11) x₁+x_n

2 = 1 n

n

X

i=1

x_i+ 1 n

n−1

X

i=1

i−n

2

∆x_i.

(2.8) and (2.11) were obtained by Dragomir in [7].

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3. Discrete Ostrowski Type Inequalities

The Bernoulli numbersB_i,i≥0, are defined by the implicit recurrence relation

m

X

i=0

m+ 1 i

B_i =

( 1, if m = 0, 0, if m 6= 0.

If, forn∈Nandm∈R,we write

Sm(n) = 1^m+ 2^m+ 3^m+· · ·+ (n−1)^m, it is well known, see [8], that ifm∈N

S_m(n) = 1 m+ 1

m

X

i=0

m+ 1 i

B_i n^m+1−i.

Theorem 3.1. Let(X,k·k)be a normed linear space,x₁, x₂, . . . , x_na finite se- quence of vectors inX, w₁, w₂, . . . , w_nfinite sequence of positive real numbers.

Let also (p, q)be a pair of conjugate exponents¹, m ∈ {2,3, . . . , n−1} and k ∈ {1,2, . . . , n}the following inequality holds:

(3.1)

xk− 1 W_n

n

X

i=1

wixi−

m−1

X

r=1

1 n−r

n−r

X

i=1

∆^rxi

!

×

n−1

X

i1=1 n−2

X

i2=1

· · ·

n−r

X

ir=1

!

1That is:1< p, q <∞,¹_p+¹_q = 1

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≤

n−1

X

i1=1 n−2

X

i2=1

· · ·

n−m+1

X

im−1=1

D_w(k, i₁)Dn−1(i₁, i₂)· · ·Dn−m+1(im−1,·) q

k∆^mxk_p,

where

k∆^mxk_p =









 _n−m

P

i=1

k∆^mx_ik^p ¹_p

, if1≤p < ∞,

1≤i≤n−mmax k∆^mx_ik ifp=∞.

Proof. By using the (2.6) and the Hölder inequality.

Corollary 3.2. Let (X,k·k) be a normed linear space, x₁, x₂, . . . , x_n a finite sequence of vectors in X, w₁, w₂, . . . , w_n a finite sequence of positive real numbers. Let also (p, q) be a pair of conjugate exponents. Then for all k ∈ {1,2, . . . , n}the following inequalities hold:

x_k− 1 W_n

n

X

i=1

w_ix_i

≤









 1 W_n

_n P

i=1

|k−i|wi

· k∆xk_∞,

1 W_n

k−1

P

i=1 i

P

j=1

w_j

!q

+

n−1

P

i=k n

P

j=i+1

w_i

!q!¹_q

· k∆xk_p, 1

W_nmax{Wk−1, W_n−W_k} · k∆xk₁.

Proof. By using the discrete analogue of the weighted Montgomery identity

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(2.3) and applying the Hölder inequality we get

x_k− 1 W_n

n

X

i=1

w_ix_i

≤ kD_w(k,·)k_qk∆xk_p.

We have

kD_w(k,·)k₁ = 1 W_n

k−1

X

i=1

|W_i|+

n−1

X

i=k

−W_i

!

= 1 Wn

k−1

X

i=1

(k−i)w_i+

n−k

X

i=1

iw_k+i

!

= 1 Wn

n

X

i=1

|k−i|w_i

and the first inequality is proved.

Since

kD_w(k,·)k_q = 1 W_n

k−1

X

i=1

|W_i|^q+

n−1

X

i=k

−W_i

q

!¹_q

= 1 W_n

k−1

X

i=1 i

X

j=1

w_j

!q

+

n−1

X

i=k n

X

j=i+1

w_i

!q!¹_q

the second inequality is proved.

Finally, for the third

kD_w(k,·)k_∞= 1

W_n max{Wk−1, W_n−W_k},

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which completes the proof.

The first and the third inequality from Corollary3.2 and also the following corollary was proved by Dragomir in [7].

Corollary 3.3. Let (X,k·k) be a normed linear space, x1, x2, . . . , xn a finite sequence of vectors in X, w₁, w₂, . . . , w_n finite sequence of positive real num- bers, and also let (p, q) be a pair of conjugate exponents. Then for all k ∈ {1,2, . . . , n}the following inequalities hold:

(3.2)

x_k− 1 n

n

X

i=1

x_i

≤











1 n

n²−1

4 + k− ⁿ⁺¹₂ 2

· k∆xk_∞,

1

n(S_q(k) +S_q(n−k+ 1))¹^q · k∆xk_p,

1

nmax{k−1, n−k} · k∆xk₁.

Proof. If we apply Corollary3.2withw_i = 1,i= 1,2, . . . , n, or use the discrete Montgomery identity (2.5), we have

xk− 1 n

n

X

i=1

xi

=

n−1

X

i=1

Dn(k, i) ∆xi

≤

n−1

X

i=1

|D_n(k, i)|^q

!¹_q _n−1 X

i=1

k∆x_ik^p

!¹_p .

Since forq = 1

n−1

X

i=1

|D_n(k, i)|= 1 n

n²−1

4 +

k− n+ 1 2

2! ,

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the first inequality follows.

For the second let1< q <∞

n−1

X

i=1

|D_n(k, i)|^q = 1 n^q

k−1

X

i=1

i^q+

n−1

X

i=k

(n−i)^q

!

= 1

n^q (S_q(k) +S_q(n−k+ 1)) the second inequality follows.

Finally forq =∞and

1≤i≤n−1max {|D(k, i)|}= 1

n max{k−1, n−k}

implies the last inequality.

Corollary 3.4. Assume that all assumptions from Theorem 3.1hold. Then the following inequality holds

x_l− 1 W2l−1

2l−1

X

i=1

w_ix_i−

m−1

X

r=1

1 2l−1−r

2l−1−r

X

i=1

∆^rx_i

!

×

2l−2

X

i1=1 2l−3

X

i2=1

· · ·

2l−1−r

X

ir=1

D_w(l, i₁)D2l−2(i₁, i₂)· · ·D2l−r(ir−1, i_r)

!

≤

2l−2

X

i1=1 2l−3

X

i2=1

· · ·

2l−m

X

im−1=1

D_w(l, i₁)D2l−2(i₁, i₂)· · ·D2l−m(im−1,·) q

k∆^mxk_p;

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it may be regarded as a generalized midpoint inequality since for m = 1 it reduces to

x_l− 1 W2l−1

2l−1

X

i=1

w_ix_i

≤









 1 W_2l−1

_2l−1 P

i=1

|l−i|w_i

· k∆xk_∞,

1 W2l−1

l−1

P

i=1 i

P

j=1

w_j

!q

+

2l−2

P

i=l n

P

j=i+1

w_i

!q!¹_q

· k∆xk_p, 1

W2l−1

max{W_l−1, W_2l−1−W_l} · k∆xk₁; if in additionw_i = 1, i= 1,2, . . . ,2l−1it further reduces to

(3.3)

x_l− 1 2l−1

2l−1

X

i=1

x_i

≤











l(l−1)

2l−1 · k∆xk_∞, 1

2l−1(2S_q(l))¹^q · k∆xk_p, l−1

2l−1 · k∆xk₁.

Proof. Apply (3.1) withn= 2l−1andk=lto get the first inequality.

For the second, takingm= 1, or applying Hölder’s inequality to (2.7), gives

x_l− 1 W2l−1

2l−1

X

i=1

w_ix_i

=

2l−2

X

i=1

D_w(l, i) ∆x_i

≤ kD_w(l,·)k_qk∆xk_p.

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Now

kD_w(l,·)k₁ = 1 W2l−1

l−1

X

i=1

|W_i|+

2l−1

X

i=l

−W_i

!

= 1

W2l−1 2l−1

X

i=1

|l−i|w_i

! ,

kD_w(l,·)k_q= 1 W2l−1

l−1

X

i=1

|W_i|^q+

2l−1

X

i=l

−W_i

q

!¹_q

= 1

W2l−1 l−1

X

i=1 i

X

j=1

w_j

!q

+

2l−2

X

i=l n

X

j=i+1

w_i

!q!¹_q ,

kD_w(l,·)k_∞ = 1

W_2l−1 max{Wl−1, W2l−1−W_l} and the second inequality is proved.

Now if we take w_i = 1, i = 1,2, . . . ,2l − 1, or apply inequality (3.2) with n = 2l−1andk =l,

kD2l−1(l,·)k₁ = 1 2l−1

2l−1

X

i=1

|l−i|= l(l−1) 2l−1 ,

kD2l−1(l,·)k_q= 1 2l−1

2l−1

X

i=1

|l−i|^q

!¹_q

= 1

2l−1(2S_q(l))¹^q , kD_2l−1(l,·)k_∞ = 1

2l−1max{l−1,2l−1−l}= l−1 2l−1, and thus the third inequality is proved.

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Corollary 3.5. Let all the assumptions from Theorem 3.1 hold. Then the fol- lowing inequality holds:

x₁+x_n

2 − 1

W_n

n

X

i=1

w_ix_i−

m−1

X

r=1

1 n−r

n−r

X

i=1

∆^rx_i

!

×

n−1

X

i1=1

· · ·

n−r

X

ir=1

D_w(1, i₁) +D_w(n, i₁)

2 Dn−1(i1, i2)· · ·Dn−r+1(ir−1, ir)

!

≤

n−1

X

i1=1

· · ·

n−m+1

X

im−1=1

D_w(1, i₁) +D_w(n, i₁)

2 Dn−1(i1, i2)· · ·Dn−m+1(im−1,·) q

× k∆^mxk_p; this may regarded as a generalized trapezoid inequality since for m = 1 it reduces to

x1+xn

2 − 1

W_n

n

X

i=1

w_ix_i

≤









 Pn−1

i=1

Wi

Wn − ¹₂

· k∆xk_∞, Pn−1

i=1

Wi

Wn − ¹₂

q¹_q

· k∆xk_p,

maxn

w1

Wn − ¹₂ ,

wn

Wn − ¹₂

o· k∆xk₁.

and if in addition,w_i = 1, i= 1,2, . . . , nit further reduces to

(3.4)

x₁ +x_n 2 − 1

n

X

i=1

x_i

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≤











1

n n−1−_n

2

_n

2

· k∆xk_∞,











1

n 2S_q ⁿ₂¹_q

· k∆xk_p, ifnis even,

1 n

Sq(n−1)

2^q−1 −2S_q ⁿ⁻¹₂ ¹_q

· k∆xk_p, ifnis odd,

n−2

2n · k∆xk₁.

Proof. To obtain the first inequality take (2.9) and apply Hölder’s inequality.

For the second we takem= 1or apply Hölder’s inequality to (2.10),

x₁+x_n

2 − 1

W_n

n

X

i=1

wixi

=

n−1

X

i=1

D_w(1, i) +D_w(n, i)

2 ∆xi

≤

D_w(1,·) +D_w(n,·) 2

q

k∆xk_p.

Now

D_w(1,·) +D_w(n,·) 2

1

=

n−1

X

i=1

W_i−W_i 2W_n

=

n−1

X

i=1

W_i W_n − 1

2 ,

Dw(1,·) +Dw(n,·) 2

q

=

n−1

X

i=1

Wi

W_n − 1 2

q!¹_q ,

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D_w(1,·) +D_w(n,·) 2

_∞

= max

1≤i≤n−1

W_i W_n −1

2

= max

W1

W_n − 1 2 ,

Wn−1

W_n − 1 2

= max

w1

W_n − 1 2 ,

wn

W_n −1 2

and the second inequality is proved.

Now if we take w_i = 1, i = 1,2, . . . , n, or use (2.11) and apply Hölder’s inequality, we get

x₁+x_n 2 − 1

n

X

i=1

x_i

≤ i n − 1

2 q

k∆xk_p.

Forq= 1 i n −1

2 1

= 1 n

n−1

X

i=1

i−n

2 = 1

n

n−1−jn 2

k jn 2

k

;

for1< q <∞ i n −1

2 q

= 1 n

n−1

X

i=1

i− n

2

q!¹_q

=











1

n 2S_q ⁿ₂¹_q

, ifnis even,

1 n

_S

q(n−1)

2^q−1 −2S_q ⁿ⁻¹₂ ¹_q

, ifnis odd;

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and forq =∞ i n − 1

2 _∞

= max

1≤i≤n−1

i n −1

2

= n−2 2n .

Remark 3.1. The first inequality from (3.3) was obtained by Dragomir in [7]

and also an incorrect version of the first inequality from (3.4), viz.:

x₁+x_n 2 − 1

n

X

i=1

x_i

≤







k−1

2 k∆xk_∞, ifn = 2k,

2k²+2k+1

2(2k+1) k∆xk_∞, ifn = 2k+ 1.

The second coefficient ^2k_2(2k+1)²^+2k+1 should be _2k+1^k² since

1 2k+ 1

(2k+ 1)−1−

2k+ 1 2

= k² 2k+ 1.

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4. Estimates of the Differences Between Two Weighted Arithmetic Means

In this section we will give the estimates of the differences between two weighted arithmetic means using the discrete weighted Montgomery (Euler) identity. We suppose l, m, n ∈ N. The first method is by subtracting two weighted Mont- gomery identities. The second is by summing the discrete weighted Mont- gomery identity. Both methods are possible for both the case1 ≤l ≤ m ≤n, i.e. [l, m]⊆[1, n]and the case1≤l≤n ≤m, i.e.[1, n]∩[l, m] = [l, n].

Theorem 4.1. Let(X,k·k)be a normed linear space,x₁, x₂, . . . , xmax{m,n}a fi- nite sequence of vectors inX,l, m, n ∈N,w1, w2, . . . , wnandul, ul+1, . . . , um, two finite sequences of positive real numbers. Let also W = Pn

i=1w_i, U = Pm

i=lu_i and fork∈N

W_k =







k

P

i=1

wi, 1≤k ≤n, W, k > n,

(4.1) U_k =











0, k < l,

k

P

i=l

u_i l≤k ≤m, U, k > m.

If[1, n]∩[l, m]6=∅, then, for both cases[l, m]⊆[1, n]and[1, n]∩[l, m] = [l, n],

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the next formula is valid

(4.2) 1

W

n

X

i=1

wixi− 1 U

m

X

i=l

uixi =

max{m,n}

X

i=1

K(i) ∆xi,

where

K(i) = U_i U − W_i

W, 1≤i≤max{m, n}. Proof. Fork ∈([1, n]∩[l, m])∩N, we subtract the identities

x_k = 1 W

n

X

i=1

w_ix_i+

n−1

X

i=1

D_w(k, i) ∆x_i,

and

x_k = 1 U

m

X

i=l

u_ix_i+

m−1

X

i=l

D_u(k, i) ∆x_i.

Then put

K(k, i) =D_u(k, i)−D_w(k, i). AsK(k, i)does not depend onkwe write simplyK(i):

(4.3) K(i) =











−^W_Wⁱ, 1≤i≤l−1,

Ui

U − ^W_Wⁱ, l≤i≤m, 1−^W_Wⁱ, m+ 1 ≤i≤n,

if [l, m]⊆[1, n],

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(4.4) K(i) =











−^W_Wⁱ, 1≤i≤l−1,

Ui

U − ^W_Wⁱ, l ≤i≤n,

Ui

U −1, n+ 1≤i≤m,

if [1, n]∩[l, m] = [l, n].

Theorem 4.2. Let all assumptions from Theorem4.1 hold and(p, q)be a pair of conjugate exponents. Then we have

1 W

n

X

i=1

w_ix_i− 1 U

m

X

i=l

u_ix_i

≤ kKk_qk∆xk_p.

The constantkKk_q is sharp for1≤p≤ ∞.

Proof. We use the identity (4.2) and apply the Hölder inequality to obtain

1 W

n

X

i=1

w_ix_i− 1 U

m

X

i=l

u_ix_i

=

max{m,n}

X

i=1

K(i) ∆x_i

≤ kKk_qk∆xk_p.

For the proof of the sharpness of the constant kKk_q, we will find x, a finite sequence of vectors inX such that

max{m,n}

X

i=1

K(i) ∆x_i

=





max{m,n}

X

i=1

|K(i)|^q





1 q

k∆xk_p.

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For1< p <∞takexto be such that

∆x_i = sgnK(i)· |K(i)|^p−1¹ . Forp=∞take

∆x_i = sgnK(i).

Forp= 1we will find a finite sequence of vectorsxsuch that

max{m,n}

X

i=1

K(i) ∆x_i

= max

1≤i≤max{m,n}|K(i)|





max{m,n}

X

i=1

|∆x_i|



.

Suppose that |K(i)|attains its maximum ati0 ∈ ([1, n]∪[l, m])∩N. First we assume thatK(i₀)>0. Definexsuch that∆x_i₀ = 1and∆x_i = 0,i6=i₀,i.e.

xi =

( 0, 1≤i≤i₀,

1, i₀+ 1< i≤max{m, n}. Then,

max{m,n}

X

i=1

K(i) ∆x_i

=|K(i₀)|= max

1≤i≤max{m,n}|K(i)|





max{m,n}

X

i=1

|∆x_i|



,

and the statement follows. In the caseK(i₀) <0, we takexsuch that∆x_i₀ =