A discrete analogue of the weighted Montgomery identity (i.e

http://jipam.vu.edu.au/

Volume 5, Issue 3, Article 58, 2004

A DISCRETE EULER IDENTITY

A. AGLI ´C ALJINOVI ´C AND J. PE ˇCARI ´C DEPARTMENT OFAPPLIEDMATHEMATICS

FACULTY OFELECTRICALENGINEERING ANDCOMPUTING

UNSKA3, 10 000 ZAGREB, CROATIA. andrea@zpm.fer.hr FACULTY OFTEXTILETECHNOLOGY

UNIVERSITY OFZAGREB

PIEROTTIJEVA6, 10000 ZAGREB

CROATIA.

pecaric@mahazu.hazu.hr

Received 09 January, 2004; accepted 22 April, 2004 Communicated by P.S. Bullen

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.

Key words and phrases: Discrete Montgomery identity, Discrete Ostrowski inequality.

2000 Mathematics Subject Classification. 26D15.

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 every x ∈ [a, b]whenever f : [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)|<+∞.

Letf : [a, b]→Rbe differentiable on[a, b],f⁰ : [a, b]→Rintegrable on[a, b]andw: [a, b]→ [0,∞) some probability density function, i.e. integrable function satisfying Rb

aw(t)dt = 1;

ISSN (electronic): 1443-5756

c 2004 Victoria University. All rights reserved.

086-04

defineW(t) = Rt

aw(x)dxfor 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 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 Section 2, a discrete weighted Montgomery (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 Section 4, 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].

2. DISCRETEWEIGHTED EULERIDENTITY

Let x₁, x₂, . . . , x_n be a finite sequence of vectors in the normed linear space (X,k·k) and w₁, w₂, . . . , w_nfinite sequence of positive real numbers. If, for1≤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

w_ix_i =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 Montgomery 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,

where the discrete Peano kernel is defined by

(2.4) Dw(k, i) = 1

W_n ·

( W_i, 1≤i≤k−1,

−W_i

, k ≤i≤n.

If we take w_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

n

X

i=1

x_i+

n−1

X

i=1

D_n(k, i) ∆x_i,

where

Dn(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).

Theorem 2.1. Let(X,k·k)be a normed linear space,x₁, x₂, . . . , x_na finite sequence of vectors inX, w₁, w₂, . . . , w_nfinite 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)Dn−1(i₁, i₂)· · ·Dn−r+1(ir−1, i_r)

!

+

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) ∆^mx_im.

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

x_k= 1 Wn

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)D_n−1(i, j) ∆²x_j.

Applying the identity (2.5) for the finite sequence of vectors∆x_i,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

so, again using (2.3), we have x_k = 1

Wn 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 numberm∈ {2,3, . . . , n−2}. Applying the identity (2.5) for the∆^mx_im

∆^mxim = 1 n−m

n−m

X

i=1

∆^mxi+

n−m−1

X

im+1=1

Dn−m(im, im+1) ∆^m+1xim+1

and using the induction hypothesis, we get 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

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

!

+

n−1

X

i1=1 n−2

X

i2=1

· · ·

n−m

X

im=1

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

×



 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_im+1









= 1 Wn

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_im+1.

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

Remark 2.2. 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

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

· · ·

1

X

in−1=1

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

Corollary 2.3. Let(X,k·k)be a normed linear space,x₁, x₂, . . . , x_na finite sequence 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)Dn−1(i₁, i₂)· · ·Dn−r+1(ir−1, i_r)

!

+

n−1

X

i1=1 n−2

X

i2=1

· · ·

n−m

X

im=1

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

Proof. Apply Theorem 2.1 withwi = 1,i= 1, . . . , n.

Remark 2.4. If we apply (2.6) withn= 2l−1andk =lwe get 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

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

!

+

2l−2

X

i1=1 2l−3

X

i2=1

· · ·

2l−1−m

X

im=1

D_w(l, i₁)D2l−2(i₁, i₂)· · ·D2l−m(im−1, i_m) ∆^mx_im.

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

(2.7) xl = 1

W2l−1 2l−1

X

i=1

wixi+

2l−2

X

i=1

Dw(l, i) ∆xi

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

(2.8) x_l= 1

2l−1

2l−1

X

i=1

x_i+ 1 2l−1

l−1

X

i=1

i(∆x_i−∆x_2l−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+x_n

2 = 1

Wn 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 D_n−1(i₁, i₂)· · ·D_n−r+1(i_r−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_im. We may regard this identity as a generalized trapezoid identity since form= 1it reduces to

(2.10) x₁+x_n

2 = 1

Wn 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].

3. DISCRETEOSTROWSKI TYPEINEQUALITIES

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

m

X

i=0

m+ 1 i

Bi =

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

If, forn ∈Nandm∈R,we write

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

Sm(n) = 1 m+ 1

m

X

i=0

m+ 1 i

Bin^m+1−i.

Theorem 3.1. Let(X,k·k)be a normed linear space,x₁, x₂, . . . , x_na finite sequence of vectors in X, w₁, w₂, . . . , w_n finite 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)

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)

!

≤

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 _p¹

, 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_na 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 allk∈ {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|w_i

· 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.

1That is:1< p, q <∞,¹_p+¹_q = 1

Proof. By using the discrete analogue of the weighted Montgomery identity (2.3) and applying the Hölder inequality we get

x_k− 1 Wn

n

X

i=1

w_ix_i

≤ kD_w(k,·)k_qk∆xk_p.

We have

kD_w(k,·)k₁ = 1 Wn

k−1

X

i=1

|W_i|+

n−1

X

i=k

−W_i

!

= 1 W_n

k−1

X

i=1

(k−i)w_i+

n−k

X

i=1

iw_k+i

!

= 1 W_n

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 Wn

max{Wk−1, W_n−W_k},

which completes the proof.

The first and the third inequality from Corollary 3.2 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, . . . , x_na finite sequence of vectors in X, w₁, w₂, . . . , w_n finite sequence of positive real numbers, and also let(p, q) be a pair of conjugate exponents. Then for allk∈ {1,2, . . . , n}the following inequalities hold:

(3.2)

xk− 1 n

n

X

i=1

xi

≤











1 n

n²−1

4 + k− ⁿ⁺¹₂ 2

· k∆xk_∞,

1

n(Sq(k) +Sq(n−k+ 1))^1q · k∆xk_p,

1

nmax{k−1, n−k} · k∆xk₁.

Proof. If we apply Corollary 3.2 withwi = 1,i = 1,2, . . . , n, or use the discrete Montgomery identity (2.5), we have

x_k− 1 n

n

X

i=1

x_i

=

n−1

X

i=1

D_n(k, i) ∆x_i

≤

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! ,

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 3.1 hold. Then the following in- equality 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

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

!

≤

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;

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

x_l− 1 W2l−1

2l−1

X

i=1

w_ix_i

≤





















 1 W2l−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{Wl−1, W2l−1−Wl} · 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))^1q · 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.

Now

kD_w(l,·)k₁ = 1 W_2l−1

l−1

X

i=1

|W_i|+

2l−1

X

i=l

−W_i

!

= 1

W_2l−1

2l−1

X

i=1

|l−i|w_i

! ,

kDw(l,·)k_q = 1 W2l−1

l−1

X

i=1

|Wi|^q+

2l−1

X

i=l

−Wi

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 W2l−1

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

Now if we takew_i = 1, i= 1,2, . . . ,2l−1, or apply inequality (3.2) withn = 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))^1q,

kD2l−1(l,·)k_∞ = 1

2l−1max{l−1,2l−1−l}= l−1 2l−1,

and thus the third inequality is proved.

Corollary 3.5. Let all the assumptions from Theorem 3.1 hold. Then the following 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, i_r)

!

≤

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 form = 1it reduces to

x₁+x_n

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

n

X

i=1

x_i

≤























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 −2Sq n−1 2

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

Wn n

X

i=1

w_ix_i

=

n−1

X

i=1

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

2 ∆x_i

≤

Dw(1,·) +Dw(n,·) 2

q

k∆xk_p.

Now

D_w(1,·) +D_w(n,·) 2

1

=

n−1

X

i=1

W_i−W_i 2Wn

=

n−1

X

i=1

W_i Wn

− 1 2 ,

D_w(1,·) +D_w(n,·) 2

q

=

n−1

X

i=1

W_i Wn

− 1 2

q!¹_q ,

Dw(1,·) +Dw(n,·) 2

_∞

= max

1≤i≤n−1

Wi

W_n − 1 2

= max

W₁ Wn

− 1 2 ,

Wn−1

Wn

− 1 2

= max

w₁ W_n − 1

2 ,

w_n W_n − 1

2

and the second inequality is proved.

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

x₁+x_n 2 − 1

n

n

X

i=1

xi

≤ 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;

and forq=∞

i n −1

2 _∞

= max

1≤i≤n−1

i n −1

2

= n−2 2n .

Remark 3.6. The first inequality from (3.3) was obtained by Dragomir in [7] and also an in- correct version of the first inequality from (3.4), viz.:

x₁+x_n 2 − 1

n

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

2k+ 1 2

= k² 2k+ 1.

4. ESTIMATES OF THEDIFFERENCESBETWEEN 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 Montgomery identities. The second is by summing the discrete weighted Montgomery 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 finite sequence of vectors inX,l, m, n∈N,w1, w2, . . . , wnandul, ul+1, . . . , um,two finite sequences of positive real numbers. Let alsoW =Pn

i=1w_i,U =Pm

i=lu_iand fork ∈N

W_k =







k

P

i=1

w_i, 1≤k≤n, W, k > n,

(4.1) U_k =











0, k < l,

k

P

i=l

ui 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], the next formula is valid

(4.2) 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,

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],

(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 Theorem 4.1 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_qis sharp for1≤p≤ ∞.

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

1 W

n

X

i=1

wixi− 1 U

m

X

i=l

uixi

=

max{m,n}

X

i=1

K(i) ∆xi

≤ kKk_qk∆xk_p.

For the proof of the sharpness of the constantkKk_q, we will findx, 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.

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 at i₀ ∈ ([1, n]∪[l, m])∩N. First we assume that K(i₀)>0. Definexsuch that∆x_i0 = 1and∆x_i = 0,i6=i₀,i.e.

x_i =

( 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 =−1and∆x_i = 0, i6=i₀,i.e.

x_i =

( 1, 1≤i≤i0,

0, i₀+ 1≤i≤max{m, n},

and the rest of proof is the same as above.

Corollary 4.3. Assume all assumptions from the Theorem 4.2 hold and additionally assume 1≤l < m≤n. Then we have

1 W

n

X

i=1

w_ix_i− 1 U

m

X

i=l

u_ix_i

≤































"_l−1 X

i=1

W_i W

+

m

X

i=l

U i U −W_i

W

+

n

X

i=m+1

1−W_i W

#

k∆xk_∞,

"_l−1 X

i=1

W_i W

q

+

m

X

i=l

U i U − W_i

W

q

+

n

X

i=m+1

1− W_i W

q#¹_q

k∆xk_p,

max

Wl−1

W ,1− W_m+1 W , max

l≤i≤m

U i U −W_i

W

k∆xk₁,

and for1≤l < n≤m

1 W

n

X

i=1

w_ix_i− 1 U

m

X

i=l

u_ix_i

≤































"_l−1 X

i=1

W_i W

+

n

X

i=l

U i U −W_i

W

+

m

X

i=n+1

U_i U −1

#

k∆xk_∞,

"_l−1 X

i=1

W_i W

q

+

n

X

i=l

U i U −W_i

W

q

+

m

X

i=n+1

U_i U −1

q#¹_q

k∆xk_p,

max

Wl−1

W ,1− U_n+1 U ,max

l≤i≤n

U i U − W_i

W

k∆xk₁.

Proof. Directly from the Theorem 4.2.

Remark 4.4. If we suppose n = m in both of the cases1 ≤ l < m ≤ n and 1 ≤ l < n ≤ m,then the analogous results coincides.

Remark 4.5. By settingl = m = k and u_k = 1in the first inequality from Corollary 4.3 we get the weighted Ostrowski inequality from Corollary 3.2.

Corollary 4.6. If all assumptions from Theorem 4.2 hold and, in addition, assume1≤k ≤m, then we have

1 W

k

X

i=1

w_ix_i− 1 U

m

X

i=k

u_ix_i

≤































"_k−1 X

i=1

W_i W

+

U_k U − W_k

W

+

m

X

i=k+1

U_i U −1

#

k∆xk_∞,

"_k−1 X

i=1

Wi

W

q

+

Uk

U −Wk

W

q

+

m

X

i=k+1

Ui

U −1

q#¹_q

k∆xk_p,

max

Wk−1

W ,1− Uk+1

U ,

Uk

U − Wk

W

k∆xk₁.

Proof. By settingn =l=kin the second inequality from the Corollary 4.3.

The second method of giving an estimate of the difference between the two weighted arith- metic means is by summing the weighted Montgomery identity. In this way we also get formula (4.2). For the case1≤l ≤m ≤nletj ∈ {l, l+ 1, . . . , m},from (2.3) we have

u_jx_j =u_j 1 W

n

X

i=1

w_ix_i+u_j

n−1

X

i=1

D_w(j, i) ∆x_i, so

1 U

m

X

j=l

u_jx_j = 1 U

m

X

j=l

u_j

! 1 W

n

X

i=1

w_ix_i+ 1 U

m

X

j=l

u_j

n−1

X

i=1

D_w(j, i) ∆x_i.

By interchange of the order of summation we get 1

U

m

X

j=l

u_jx_j − 1 W

n

X

i=1

w_ix_i

= 1 U

m

X

j=l

u_j

j−1

X

i=1

Wi

W ∆x_i+ 1 U

m

X

j=l

u_j

n−1

X

i=j

Wi

W −1

∆x_i

= 1 U

l−1

X

i=1 m

X

j=l

u_jW_i

W ∆x_i+ 1 U

m−1

X

i=l m

X

j=i+1

u_jW_i W∆x_i

+ 1 U

m−1

X

i=l i

X

j=l

u_j W_i

W −1

∆x_i+ 1 U

n−1

X

i=m m

X

j=l

u_j W_i

W −1

∆x_i

=

l−1

X

i=1

W_i W∆x_i+

m−1

X

i=l

1− U_i

U W_i

W∆x_i

+

m−1

X

i=l

U_i U

W_i W −1

∆x_i+

n−1

X

i=m

W_i W −1

∆x_i

=

l−1

X

i=1

W_i W∆xi+

m

X

i=l

W_i W − U_i

U

∆xi+

n−1

X

i=m+1

W_i W −1

∆xi.

This identity is equivalent to (4.2) with (4.3).

For case1≤l ≤n≤m, letj ∈ {l, l+ 1, . . . , n},and again from (2.3) we have u_jx_j =u_j 1

W

n

X

i=1

w_ix_i+u_j

n−1

X

i=1

D_w(j, i) ∆x_i, so

n

X

j=l

u_jx_j =

n

X

j=l

u_j 1 W

n

X

i=1

w_ix_i+

n

X

j=l

u_j

n−1

X

i=1

D_w(j, i) ∆x_i and

1 U

m

X

j=l

u_jx_j− 1 U

m

X

j=n+1

u_jx_j

= 1 U

m

X

j=l

u_j

! 1 W

n

X

i=1

w_ix_i− 1 U

m

X

j=n+1

u_j

! 1 W

n

X

i=1

w_ix_i

+ 1 U

n

X

j=l

uj

!_n−1 X

i=1

Dw(j, i) ∆xi.

Thus 1 U

m

X

j=l

ujxj− 1 W

n

X

i=1

wixi

= 1 U

m

X

j=n+1

uj xj − 1 W

n

X

i=1

wixi

! + 1

U

n

X

j=l

uj n−1

X

i=1

Dw(j, i) ∆xi.