Generalization of weighted Ostrowski inequality with applications in numerical integration

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Adv. Inequal. Appl. 2019, 2019:7 https://doi.org/10.28919/aia/3998 ISSN: 2050-7461

GENERALIZATION OF WEIGHTED OSTROWSKI INEQUALITY WITH APPLICATIONS IN NUMERICAL INTEGRATION

NAZIA IRSHAD, ASIF R. KHAN, MUHAMMAD AWAIS SHAIKH∗

Department of Mathematics, University of Karachi, Karachi-75270, Pakistan

Copyright c2019 the authors. This is an open access article distributed under the Creative Commons Attribution License, which permits

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract. An integral inequality of Ostrowski type with weights is established for differentiable functions up to second order, whose second derivatives are bounded and first derivatives are absolutely continuous. The inequality of weighted integral is then functional to weighted composite quadrature rules.

Keywords:weighted Ostrowski’s inequality; numerical integration. 2010 AMS Subject Classification:65D30.

1. Introduction

It is a known that Ostrowski type inequalities can be used to estimate the absolute deviation

from its integral mean. These inequalities can be used to provide explicit error bounds for

nu-merical quadrature formulae. The weighted version of Ostrowski inequality was first presented

in 1983 by J. E. Peˇcari´c and B. Savi´c [10]. Keeping in view the importance of this inequality, for

last few decades, the researchers are in continuous effort to obtain sharp bounds of Ostrowski’s

∗_{Corresponding author}

E-mail address: [email protected] Received January 15, 2019

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inequality in terms of weight.

We have introduced a weighted inequality which has its applications in Numerical quadrature

rules and Probability theory. The probability density function, defined by us, has produced the

results of Ostrowski’s inequality in a more organized way.

In 1938, a Ukrainian Mathematician Alexandar Markowich Ostrowski discovered an inequality

called Ostrowski inequality, which states that.

Lemma 1.1. [15] Let ϕ: I⊆R→Rbe a differentiable function on I0(I0is the interior of I)

and leta₀,b₁∈I0witha₀<b₁. Ifϕ0:(a0,b1)→Ris bounded on(a0,b1), i.e.,

||ϕ0||∞= sup

t∈(a0,b1)

|ϕ0(t)|<∞,

Then the inequality is

ϕ(v)− 1

b1−a0

Z b₁

a0

ϕ(t)dt

≤ "

1

4+

(v−a0+b1

2 ) 2

(b1−a0)2 #

(b₁−a₀)||ϕ0||∞

for allv∈[a0,b1].The number 1₄ is sharp which cannot be replaced by a smaller one.

G. V. Milovanovi´c and J. E. Pe˘cari´c proved a generalised Ostrowski’s inequality in 1976 forn

-times differentiable functions [9]. In which he pointed out the speciality of twice differentiable

functions [8, p. 470].

Lemma 1.2. [15] Letϕ: I ⊆_R→_Rbe a differentiable function on I0(I0 is the interior of I) and leta₀,b₁∈I0witha₀<b₁. Ifϕ0:(a0,b1)→Ris bounded on(a0,b1), i.e.,

||ϕ0||∞= sup

t∈(a0,b1)

|ϕ0(t)|<∞,

Then the inequality is

ϕ(v)− 1

b₁−a₀

Z b₁

a0

ϕ(t)dt

≤ "

1

4+

(v−a0+b1

2 )2

(b₁−a₀)2 #

(b1−a0)||ϕ0||∞

for allv∈[a₀,b₁].The number 1₄ is sharp which cannot be replaced by a smaller one.

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Ris bounded on(a0,b1). Then the following inequality holds for all v∈[a0,b1].

1 2

ϕ(v) +(v−a0)ϕ(a0) + (b1−v)ϕ(b1)

b₁−a₀

− 1 b₁−a₀

Z b₁

a0

ϕ(t)dt

≤ kϕ 00_k

∞

4 (b1−a0)

2 "

1

12+

(v−a0+b1

2 ) 2

(b₁−a₀)2 #

In 1999, the following inequality is proved by Cerone et.al. in [1].

Lemma 1.4. [15]Let the following assumption(1.2)be valid for all v∈[a0,b1]

ϕ(v)−

v−a0+b1

2

ϕ0(v)− 1

b₁−a₀

Z b

a0

1ϕ(t)dt

≤ "

(b₁−a₀)2

24 +

1 2

v−a0+b1

2

2#

kϕ00||∞≤

(b₁−a₀)2

6 kϕ

00_||

∞

In the same year, the following result is stated by Dragomir and Barnett in [2].

Lemma 1.5. [15]Let the following assumption(1.2)be valid for all v∈[a0,b1]

ϕ(v)−ϕ(b1)−ϕ(a0)

b₁−a₀

v−a0+b1

2

− 1 b₁−a₀

Z b

a0

1ϕ(t)dt

≤ (b1−a0) 2

2

  

  



v−a0+b1

2 b₁−a₀

!2

+1

4



 2

+ 1

12

  

 

kϕ00||∞≤

(b1−a0)2

6 kϕ

00_||

∞.

An Ostrowski type integral inequality for functions whose second derivatives are bounded

is established by P. Cerone, S. S. Dragomir and J. Roumeliotis in [1]. S. S. Dragomir and N.

S. Barnett in [2] have also established a similar inequality. in [4], an Ostrowski type integral

inequality is similar in sense as that of [1] or [2] pointed out by S. S. Dragomir and A. Sofo.

The following integral inequality in [4] is proved by S. S. Dragomir and A. Sofo.

Lemma 1.6. [15]Let the functionϕ :[a0,b1]→ Ris considered having first derivative is ab-solutely continuous on [a0,b1] and the second derivative ϕ00 ∈ L∞ [a0,b1]. Then we get the

following inequality for all v∈[a₀,b₁]

Z b₁

a0

ϕ(t)−(b1−a0) 2

ϕ(v) +ϕ(a0) +ϕ(b1) 2

+b1−a0

2

v−a0+b1

2

ϕ0(v)

≤ kϕ00k∞ 1 3

v−a0+b1

2

3

+(b1−a0)

3

48

!

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In present paper, we give weighted version of Ostrowski-type inequality (1) using different

kernel and will also state its application in Numerical quadrature rules, probability theory and

special means. This paper is arranged in the following approach. The first part is based on

preliminaries whereas in second part we are using weights in inequality (1) from which we get

main result, which is in fact probability density functions. In the last part we would state about

the application in terms of composite quadrature rules.

3. Main results

We use here a weighted form of integral inequality which is proved by S. S. Dragomir and

A. Sofo in [4], and also we are using it in numerical integration.

Theorem 3.1.

Z b₁

a0

ϕ(t)w(t)dt−1 2

ϕ(v) +b1−a0

2 [ϕ(a0)w(a0) +ϕ(b1)w(b1)]

−

Z b₁

a0

ϕ(t)

t− a0+b1

2

w0(t)dt −

Z b₁

a0

w(t)

v−a0+b1

2

ϕ0(v)dt

≤ ϕ00

∞

Z b₁

a0

w(u)du _v2

4 −

a₀+b₁

4 v

+

Z b₁

a0

_a 0+b1

4 t−

t2

4

w(t)dt

(2)

where w:[a₀,b₁]→[0,∞)is a PDF.

Proof.Let us begin with the following integral identity [8] (see also [6]).

ψ(v) =

Z b₁

a0

w(t)ψ(t)dt+

Z b₁

a0

P_w(v,t)ψ0(t)dt (3)

for all v∈[a₀,b₁], given that f is absolutely continuous on [a₀,b₁] and the weighted kernel

P_w:[a₀,b₁]×[a₀,b₁]→_Ris given by:

Pw(v,t) =

          

         

Z t

a0

w(u)du, if t∈[a₀,v],

Z t

a₀+b₁

2

w(u)du, if t∈(v,a₀+b₁−v],

Z t

b1

(5)

wheret∈[a0,b1]. Let us consider,ψ(v) =

v−a0+b1

2

ϕ0(v). Then(3)implies

v−a0+b1

2

ϕ0(v) =

Z b₁

a0

w(t)

t−a0+b1

2

ϕ0(t)dt+

Z b₁

a0

Pw(v,t)

ϕ0(t)

+

t−a0+b1

2

ϕ00(t)dt.

(4)

Using integration by parts, we get

Z b₁

a0

t−a0+b1

2

w(t)ϕ0(t)dt = b1−a0

2 [ϕ(a0)w(a0) +ϕ(b1)w(b1)]

−

Z b₁

a0

ϕ(t)w(t)dt−

Z b₁

a0

ϕ(t)

t−a0+b1

2

w0(t)dt,

(5)

also

Z b₁

a0

Pw(v,t)ϕ0(t)dt=ϕ(v)−

Z b₁

a0

ϕ(t)w(t)dt. (6)

Now using equations(5)and(6)in(4), we get

v−a0+b1

2

ϕ0(v) =b1−a0

2 [ϕ(a0)w(a0) +ϕ(b1)w(b1)]

+ϕ(v)−2

Z b₁

a0

ϕ(t)w(t)dt−

Z b₁

a0

ϕ(t)

t−a0+b1

2

w0(t)dt

+

Z b₁

a0

P_w(v,t)

t−a0+b1

2

ϕ00(t)dt.

or

Z b₁

a0

ϕ(t)w(t)dt =b1−a0

4 [ϕ(a0)w(a0) +ϕ(b1)w(b1)] + 1 2ϕ(v)

−1

2

v−a0+b1

2

ϕ0(v)−1 2

Z b₁

a0

ϕ(t)

t−a0+b1

2

w0(t)dt

+1

2

Z b₁

a0

Pw(v,t)

t−a0+b1

2

(6)

for allv∈[a₀,b₁], which gives us

Z b₁

a0

ϕ(t)w(t)dt−1 2

ϕ(v) +b1−a0

2 [ϕ(a0)w(a0) +ϕ(b1)w(b1)]

−

Z b₁

a0

ϕ(t)

t−a0+b1

2

w0(t)dt−

v−a0+b1

2

ϕ0(v)

= 1 2

Z b₁

a0

P_w(v,t)

t−a0+b1

2

ϕ00(t)dt

≤ 1 2

Z b₁

a0

|P_w(v,t)|

t−a0+b1

2 ϕ00(t)

dt

Then we have

Z b₁

a0

|Pw(v,t)|

t−a0+b1

2 ϕ00(t)

dt

≤ kϕ00k∞

Z b₁

a0

|P_w(v,t)|

t−a0+b1

2 dt, (7) Also I =

Z b₁

a0

|P_w(v,t)|

t−a0+b1

2 dt or I= Z v a0 Z t a0

w(u)du

t−a0+b1

2 dt+

Z a₀+b₁−v

v Z t

a₀+b₁

2

w(u)du ×

t−a0+b1

2 dt+

Z b₁

a0+b1−v

Z t b1

w(u)du

t−a0+b1

2 dt. (8)

Now, we have a case:

forv∈ha₀,a0+b1

2 i

, we find

I=

Z v

a0

Z t

a0

w(u)du a0+b1

2 −t

dt+

Z a0

+b₁

2

v

Z b₁

t

w(u)du

×

a₀+b₁

2 −t

dt+

Z b₁

a₀+b₁

2

Z b₁

t

w(u)du t−a0+b1

2

(7)

After solving the equations,we find

I=

Z v

a0

w(u)du

v2

2 −

a₀+b₁

2 v − Z v a0 t2 2 −

a₀+b₁

2 t

w(t)dt

+

Z a0+₂b1

v

w(u)du

v2

2 −

a₀+b₁

2 v

+

Z a0+₂b1

v

a₀+b₁

2 t−

t2

2

w(t)dt

−

Z a₀+b₁−v

a₀+b₁

2

w(u)du _a

0+b1

2 v−

v2

2

+

Z a₀+b₁−v

a₀+b₁

2

_a 0+b1

2 −

t2

2

w(t)dt

−

Z b₁

a0+b1−v

w(u)du

a0+b1

2 v−

v2

2

+

Z b₁

a0+b1−v

a0+b1

2 t−

t2

2

w(t)dt

=

Z v

a0

w(u)du

v2

2 −

a₀+b₁

2 v + Z v a0

a₀+b₁

2 t−

t2

2

w(t)dt

+

Z a0+b1

2

v

w(u)du

v2

2 −

a₀+b₁

2 v

+

Z a0+b1

2

v

a₀+b₁

2 t−

t2

2

w(t)dt

+

Z a₀+b₁−v

a₀+b₁

2

w(u)du

v2

2 −

a0+b1

2 v

+

Z a₀+b₁−v

a₀+b₁

2

a0+b1

2 −

t2

2

w(t)dt

+

Z b₁

a0+b1−v

w(u)du

v2

2 −

a₀+b₁

2 v

+

Z b₁

a0+b1−v

a₀+b₁

2 t−

t2

2

w(t)dt

(9)

and finally

I=

Z b₁

a0

w(u)du

v2

2 −

a0+b1

2 v

+

Z b₁

a0

a0+b1

2 t−

t2

2

w(t)dt

(10)

Using equations (7), (8), and (9),we find

Z b₁

a0

ϕ(t)w(t)dt−1

2

ϕ(v) +b1−a0

2 [ϕ(a0)w(a0) +ϕ(b1)w(b1)]

−

Z b₁

a0

ϕ(t)

t−a0+b1

2

w0(t)dt−

v−a0+b1

2

ϕ0(v)

≤ kϕ 00_k

∞ 2

Z b₁

a0

w(u)du

v2

2 −

a₀+b₁

2 v

+

Z b₁

a0

a₀+b₁

2 t−

t2

2

w(t)dt

= ϕ00

∞

Z b₁

a0

w(u)du

v2

4 −

a₀+b₁

4 v

+

Z b₁

a0

a₀+b₁

4 t−

t2

4

w(t)dt

(8)

which is our required result.

remarks 3.2.If we put w(t)≡ 1

b1−a0 in (2), then we will get

Z b₁

a0

ϕ(t)−(b1−a0) 2

ϕ(v) +ϕ(a0) +ϕ(b1) 2

+b1−a0

2

v−a0+b1

2

ϕ0(v)

≤ kϕ00k∞

(b₁−a₀)

4

x−a0+b1

2

+(a0−b1)

3

48 (11)

.

remarks 3.3.If we put the midpoint v= a0+b1

2 in (2), then we will get

Z b₁

a0

ϕ(t)w(t)dt−1 2

ϕ

a₀+b₁

2

+b1−a0

2 (ϕ(a0)w(a0) +ϕ(b1)w(b1))

−

Z b₁

a0

ϕ(t)

t−a0+b1

2

w0(t)dt

≤ ϕ00

∞

Z b₁

a0

a₀+b₁

4 t−

t2

4

w(t)dt −

Z b₁

a0

w(u)du

a₀+b₁

4

2

(12)

.

remarks 3.4.If we put w(t)≡ _b 1

1−a0 in (12), then we will get

Z b₁

a0

ϕ(t)dt−(b1−a0) 2

ϕ

a0+b1

2

+b1−a0

2 (ϕ(a0)w(a0) +ϕ(b1)w(b1))

−(b₁−a₀)

Z b₁

a0

ϕ(t)

t−a0+b1

2

w0(t)dt ≤ ϕ00 ∞

Z b₁

a0

a0+b1

4 t−

t2

4

dt −

Z b₁

a0

du

a0+b1

4

2

(13)

.

remarks 3.5.If we examine an approximation for the end point v=a0in (2), then we will get

Z b₁

a0

ϕ(t)w(t)dt−1

2[ϕ(a0) +

b1−a0

2 (ϕ(a0)w(a0) +ϕ(b1)w(b1))

−

Z b₁

a0

ϕ(t)

t−a0+b1

2

w0(t)dt + (b₁−a₀)

Z b₁

a0

w(t)ϕ0(a0)dt]|

≤ ϕ00

∞

Z b₁

a0

_a 0+b1

4 t−

t2

4

w(t)dt −

Z b₁

a0

w(u)du _a

0b1

4

(9)

remarks 3.6.If we examine an approximation for the end point v=b₁in (2), we will get

Z b₁

a0

ϕ(t)w(t)dt−1 2

ϕ(b1) +

b1−a0

2 (ϕ(a0)w(a0) +ϕ(b1)w(b1))

−

Z b₁

a0

ϕ(t)

t−a0+b1

2

w0(t)dt −

Z b₁

a0

w(t)(b₁−a₀)ϕ0(b1)dt

≤ ϕ00

∞

Z b₁

a0

a0+b1

4 t −

t2

4

w(t)dt−

Z b₁

a0

w(u)du

a0b1

4

.

3.Application in Numerical Integration

Theorem 3.1. Let I_n:a₀=v₀<v₁< . . . <v_n₋₁<v_n=b₁ be a partition of interval [a₀,b₁], ∆ζ =vi+1−vi,, a0i≤ξi≤b1i, where a0i=vi, b1i=vi+1, i=0, . . . ,n−1then

Z v_i₊₁

vi

ϕ(t)w(t)dt =S(ϕ,ϕ0,In,v,w) +R(ϕ,ϕ0,In,v,w)

where

|S(ϕ,ϕ0,In,v,w)|

= 1

2

n−1

∑

i=0

ϕ(ξi) +

∆ζ

2 [ϕ(vi)w(vi) +ϕ(vi+1)w(vi+1)]

−

Z v_i₊₁

vi

ξ−

v_i+v_i₊₁

2

ϕ0(ξi)dt

−

Z v_i₊₁

vi

ϕ(t)

t−vi+vi+1

2

w0(t)dt

∆ζ

and

R(ϕ,ϕ0,I_n,v,w) ≤

ϕ00

∞

n−1

∑

i=0

Z v_i₊₁

vi

(v_i+v_i₊₁)t

4 −

t2

4

w(t)dt

+

Z v_i₊₁

vi

ξ2

4 −

(v_i+v_i₊₁)ξ 4

.

(14)

.

Proof. Applying inequality (2) on ξi ∈[xi,xi+1] and summing over i from to n−1 and using

(10)

b1−a0 in (2) then we will get

|S(ϕ,ϕ0,In,v,w)|=

(15)

(v_i₊₁−v_i)

2

n−1

∑

i=0

ϕ(v) +ϕ(vi) +ϕ(vi+1)

2 +

v_i₊₁−v_i

2

v−vi+vi+1

2

ϕ0(v)

and

ϕ00

∞

(v_i₊₁−v_i)

4

n−1

∑

i=0

ξi−

v_i+v_i₊₁

2

+(vi−vi+1)

3

48 (16)

.

for allξi∈[vi,vi+1]

remarks 3.7.forξi= vi+₂vi+1(i=0,· · ·n−1),in (2) then the following quadrature rule is:

S(ϕ,ϕ0,In,v,w) =

1 2

n−1

∑

i=0

ϕ(ξi) +

∆ζ

2 [ϕ(vi)w(vi) +ϕ(vi+1)w(vi+1)]

−

Z v_i₊₁

vi

ϕ(t)

t−vi+vi+1

2

w0(t)dt

∆ζ (17)

and

ϕ00

∞

Z v_i₊₁

vi

w(t)dt

(v_i+v_i+1)t

2 −

t2

2

+

Z v_i₊₁

vi

vi+vi+1

4

2

(18)

.

vi+1−vi in (12) then we will get

|S(ϕ,ϕ0,In,v,w)|= (v_i₊₁−v_i)

2

n−1

∑

i=0

ϕ

v_i+v_i₊₁

2

+b1−vi

2 (ϕ(vi)w(vi) +ϕ(vi+1)w(vi+1))

−(v_i+1−vi)

Z v_i₊₁

vi

ϕ(t)

t−vi+vi+1

2

w0(t)dt

(11)

and

R(ϕ,ϕ0,I_n,v,w)

≤ϕ00

∞

n−1

∑

i=0

Z v_i₊₁

vi

v_i+v_i+1

4 t−

t2

4

dt −

Z v_i₊₁

vi

du

v_i+v_i+1

4

2

(20)

.

remarks 3.9.If we examine an approximation for the end point v=v_iin (2), we will get

|S(ϕ,ϕ0,In,v,w)|=

1 2

n−1

∑

i=0

ϕ(vi) +

v_i+1−vi

−

Z v_i+1

vi

ϕ(t)

t−vi+vi+1

2

w0(t)dt+ (vi+1−vi)

Z v_i+1

vi

w(t)ϕ0(vi)dt

(21)

and

n−1

∑

i=0

ϕ00

∞

Z v_i₊₁

vi

vi+vi+1

4 t−

t2

4

w(t)dt −

Z v_i₊₁

vi

w(u)du v_iv

i+1

4

.

remarks 3.10.If we examine an approximation for the end point v=b₁in (2), we will get:

|S(ϕ,ϕ0,In,v,w)|=

1 2

n−1

∑

i=0

ϕ(vi+1) +

v_i+1−vi

−

Z v_i₊₁

vi

ϕ(t)

t−vi+vi+1

2

w0(t)dt−

Z v_i₊₁

vi

w(t)(v_i₊₁−v_i)ϕ0(vi+1)dt

(22)

and

ϕ00

∞

n−1

∑

i=0

Z b₁

vi

v_i+v_i₊₁

4 t −

t2

4

w(t)dt−

Z v_i₊₁

vi

w(u)duvivi+1

4

(12)

.

4.Application for Probability Density Function

Let X be a continuous random variable having the probability density function

Ψ:[a0,b1]→R+ and the cumulative distribution function

Ψ:[a0,b1]→[0,1],i.e,

Ψ(ξ) =

Z ξ

a0

ψ(t)dt, ξ ∈[α,β]⊂[a0,b1],

is the expectation of the random variable X on the interval [a0;b1] and weighted expectation

would be

Ew(X) =

Z b₁

a0

tψ(t)w(t)dt,

t−ξ+b1−a0

2 ≤ξ ≤t−ξ−

b1−a0

2

is the expected of the random variable X on the interval [a₀,b₁]. so, we get the following theorem.

Theorem 4.Let the suppositions of Theorem2.1be valid if probability density function belongs to L2[a0,b1]space, then we get the following inequality

b₁w(b₁)−E_w(X)−

Z b₁

a0

tΨ(t)w0(t)dt

−1

2

Ψ(v) +b1−a0

2 [Ψ(a0)w(a0) ]Ψ(b1)w(b1)−

Z b₁

a0

Ψ(t)

t− a0+b1

2

×w0(t)dt−

Z b₁

a0

w(t)

v−a0+b1

2

Ψ0(v)dt

≤ Ψ00

∞

v2

4 −

a₀+b₁

4 v

+

Z b₁

a0

a₀+b₁

4 t−

t2

4

w(t)dt

(24)

(13)

Proof. Putψ =Ψin(2)we get(24), by using this identities

Z b₁

a0

Ψ(t)w(t)dt =b1w(b1)−Ew(X)−

Z b₁

a0

tΨ(t)w0(t)dt

wherew:[a₀,b₁]→[0,∞)is a probability density function

Proof. Putψ =Ψin(2)we get(24), by using this identities

Z b₁

a0

Ψ(t)w(t)dt =b1w(b1)−Ew(X)−

Z b₁

a0

tΨ(t)w0(t)dt

wherew:[a0,b1]→[0,∞)is a probability density function.

Conclusion

Weighted Peano Kernels are used for Ostrowski type inequalities, depending on the second

derivatives, are stated in this paper. In the research paper [5] and [11], Weighted Ostrowski

type inequalities for the second derivative of the functions are addressed.A generalization and

extension of the inequalities is presented in [5] and[11]. we have presented a generalization

(2)of the inequality(1)found in[5]for twice differentiable function whose first derivatives are

absolutely continuous and 2nd derivative belong toL_∞−(a₀,b₁)by presenting a parameterλ ∈ [0,1]. This generalization also results in finding a inequality for a specific value ofλ as stated

in remarks . The inequality thus found has a better bound than the inequalities presented in[5]

and [11]. Remarks also shows that the perturbed trapezoid inequality that can be found from

(2)is better than the perturbed inequalities presented in[5]and[11]of perturbed trapezoid type.

The inequality is then functional for a partition of the interval[a₀,b₁] to find some composite quadrature rules and also functional to special means.

Conflict of Interests

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