A new integral version of generalized Ostrowski

A new integral version of generalized Ostrowski- Grüss type inequality with applications

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Dragomir, Sever S, Khan, Asif R, Khan, Maria, Mehmood, Faraz and Shaikh, Muhammad Awais (2022) A new integral version of generalized Ostrowski- Grüss type inequality with applications. Journal of King Saud University - Science, 34 (5). ISSN 1018-3647

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Original article

A new integral version of generalized Ostrowski-Grüss type inequality with applications

Sever S. Dragomir

^a

, Asif R. Khan

^b

, Maria Khan

^b,c

, Faraz Mehmood

^c,⇑

, Muhammad Awais Shaikh

^d

aDepartment of Mathematics, College of Engineering & Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia

bDepartment of Mathematics, University of Karachi, University Road, Karachi 75270, Pakistan

cDepartment of Mathematics, Dawood University of Engineering and Technology, New M. A Jinnah Road, Karachi 74800, Pakistan

dDepartment of Mathematics, Nabi Bagh Z. M. Govt. Science College, Saddar, Karachi 74400, Pakistan

a r t i c l e i n f o

Article history:

Received 25 July 2021 Revised 10 April 2022 Accepted 22 April 2022 Available online 28 April 2022

2010 Mathematics Subject Classifications:

26D15 26D20 26D99

Keywords:

Ostrowski-Grüss inequality Cˇ ebyšev functional Korkine’s identity Cauchy-Schwartz inequality numerical integration Special means

a b s t r a c t

Our aim is to improve and further generalize the result of integral Ostrowski-Grüss type inequalities involving differentiable functions and then apply these obtained inequalities to probability theory, spe- cial means and numerical integration.

Ó2022 Published by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

InOstrowski (1938), Ostrowski presented an inequality which is now known as ‘‘Ostrowski’s inequality” stated below:

fðzÞ 1 nm

Z n m

fð

s

Þd

s

6 1

4þðz^mþn₂ Þ² ðnmÞ²

" #

ðnmÞM; z2 ½m;n ð1:1Þ

wheref:½m;n !Ris a differentiable function such thatjf⁰ðzÞj6M, for everyz2 ½m;n.

In present era, a large number of papers has been written about generalizations of Ostrowski’s inequality see for example (Anastassiou, 1997; Cheng, 2001; Dragomir and Wang, 1997;

Irshad and Khan, 2017; Liu, 2008; Matic´ et al., 2000; Milovanovic and Pecaric, 1976; Shaikh et al., 2021; Zafar and Mir, 2010).

Ostrowski’s inequality has proven to be an important tool for improvement of various branches of mathematical sciences. Very well said (Zafar, 2010) ‘‘Inequalities involving integrals that create bounds in the physical quantities are of great significance in the sense that these kinds of inequalities are not only used in approx- imation theory, operator theory, nonlinear analysis, numerical integration, stochastic analysis, information theory, statistics and probability theory but we may also see their uses in the various fields of biological sciences, engineering and physics”.

In the history, an important inequality that ‘‘estimate for the difference between the product of the integral of two functionals and the integral of their product” is known as ‘‘Grüss inequality”.

https://doi.org/10.1016/j.jksus.2022.102057

1018-3647/Ó2022 Published by Elsevier B.V. on behalf of King Saud University.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

⇑Corresponding author.

E-mail addresses:sever.dragomir@vu.edu.au(S.S. Dragomir),asifrk@uok.edu.pk (A.R. Khan),maria.khan@duet.edu.pk(M. Khan),faraz.mehmood@duet.edu.pk(F.

Mehmood).

Peer review under responsibility of King Saud University.

Production and hosting by Elsevier

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Journal of King Saud University – Science

j o u r n a l h o m e p a g e : w w w . s c i e n c e d i r e c t . c o m

This celebrated integral inequality was proved byGrüss (1935)in 1935, is stated below (see alsoMitrinovic´ et al. (1994)[p. 296]),

nm1

Rn

mfðzÞ

g

ðzÞdz _nm¹ Rn mfðzÞdz

₁

nm

Rn m

g

ðzÞdz

6¹₄ðM1m1ÞðN1n1Þ ð1:2Þ

provided thatfand

g

are integrable functions on½m;nsuch that m16fðzÞ6M1; n16

g

ðzÞ6N1;

8z2 ½m;n, wherem1;M1;n1;N1are real constants.

By using Grüss inequality, Dragomir and Wang proved an inequality, in the year 1997, which we would refer as

‘‘Ostrowski-Grüss inequality” (Dragomir and Wang, 1997) which is stated as follows:

Proposition 1.1. Supposef:I!Rbe a function differentiable in the interior I^o of I, where I #R, and let m;n2I^o and n>m. If

c

₆f⁰ðzÞ6C;z2 ½m;nfor real constants

c

_;C, then

fðzÞ 1 nm

Z n m

fð

s

Þd

s

fðnÞ fðmÞ

nm zmþn 2

61

4ðnmÞðC

c

Þ ð1:3Þ

holds,8z2 ½m;n.

Above inequality gives a relationship between Ostrowski inequality(1.1)and Grüss inequality(1.2).

Iffandgbelong toL2½m;n, then the Cˇ ebyšev functionalTðf;

g

Þis defined as

Tðf;

g

Þ ¼ 1 nm

Z n m

fðzÞ

g

ðzÞdz

1

nm Z n

m

fðzÞdz

1 nm

Z n m

g

ðzÞdz

:

FromMatic´ et al. (2000)pre-Grüss inequality is given below.

Proposition 1.2. Let f;

g

_:½m;n !R be integrable such that f

g

2Lðm;nÞ. If

c

6

g

ðzÞ6C ^{for z}2 ½m;n;

then jTðf;

g

Þj61

2ðC

c

Þ ffiffiffiffiffiffiffiffiffiffiffiffiffi Tðf;fÞ

p :

In the article (Matic´ et al., 2000) of year 2000, Matic´, Pecaric´ and Ujevic´ improved inequality (1.1), by using pre-Grüss inequality, which is as follows:

Proposition 1.3. Supposef:I!Rbe a function differentiable in the interior I^o of I, where I#R, and let m:n2I^o and n>m. If

c

6f⁰ðzÞ6C;z2 ½m;nfor real constants

c

;C^{, then}

fðzÞ 1 nm

Z n m

fð

s

Þd

s

fðnÞ fðmÞ

nm zmþn 2

6 1

4 ffiffiffi

p ðn3 mÞðC

c

Þ holds,8z2 ½m;n.

In the article (Barnett et al., 2000), by using Cˇ ebyšev functional, improved the Matic´-Pecˇaric´-Ujevic´ result(1.3)in terms of ‘‘Eucli- dean norm” as under:

Proposition 1.4. Let functionf:½m;n !Rbe an absolutely contin- uous and derivativef⁰2L2½m;n. If

c

6f⁰ð

s

Þ6Calmost everywhere for

s

2 ½m;n, then8z2 ½m;n

fðzÞ _nm¹ Rn

mfð

s

Þd

s

^fðnÞfðmÞ_nm z^mþn₂

6^nm₂^pffiffi_3nm¹ kf⁰k²₂^fðnÞfðmÞ_nm 2¹

2

6₄¹ffiffi

p3ðnmÞðC

c

Þ

ð1:4Þ

holds.

This article is divided into six sections: the 1st section totally based on introduction and preliminaries. In the 2nd section, we would give our main result about generalization of integral Ostrowski-Grüss type inequalities and would discuss its different special cases. In the 3rd, 4th and 5th sections, using the obtained result we would give some applications to probability theory, spe- cial means and numerical integration respectively and the 6th con- cludes the article.

2. New generalization of integral Ostrowski-Grüss type inequality

Our main theorem of this section is given in the following:

Theorem 2.1. Letf:½m;n !Rbe a differentiable function whose 1st derivative belongs to L2ðm;nÞ. If

c

6f⁰ð

s

Þ6Calmost everywhere for

s

2 ½m;n, then8z2 ½mþk^nm₂ ;^mþn2 andk2 ½0;1

ð1kÞfðzÞþfðmþnzÞ

2 þk^{fðmÞþfðnÞ}_{2 nm}¹ Rn mfð

s

Þd

s

6 ^ðnmÞ₁₂ ²ð3k²3kþ1Þ þz^mþn₂ 2

ð1kÞ h

þ^ðnmÞð1kÞ₂ ²z^mþn₂ i¹2 1

nmkf⁰k²₂^fðnÞfðmÞ_nm 2

¹

2

6¹2ðC

c

Þ ^ðnmÞ₁₂ ²ð3k²3kþ1Þ þz^mþn₂ 2

ð1kÞ h

þ^ðnmÞð1kÞ₂ ²z^mþn₂ i¹2

ð2:1Þ

holds.

Proof. We begin the proof of this theorem by defining the piece- wise continuous functionK:½m;n²!Rfork2 ½0;1as:

Kðz;

s

;kÞ ¼

s

mk^ðnmÞ₂ ; if

s

2 ½m;z;

s

^mþn₂ ; if

s

2 ðz;mþnz;

s

nþk^ðnmÞ₂ ; if

s

2 ðmþnz;n;

8>

>>

>>

><

>>

>>

>>

:

by Korkine’s identity Tðf;gÞ:¼ 1

2ðnmÞ² Z n

m

Z n m

ðfð

s

Þ fðsÞÞðgð

s

Þ gðsÞÞd

s

ds; ð2:2Þ we obtain

1 nm

Z n m

Kðz;

s

;kÞf⁰ð

s

Þd

s

1 nm

Z n m

Kðz;

s

;kÞdt Z n

m

f⁰ð

s

Þd

s

¼ 1

2ðnmÞ² Z n

m

Z n m

ðKðz;

s

;kÞ Kðz;s;kÞÞðf⁰ð

s

Þ f⁰ðsÞÞd

s

ds:

ð2:3Þ

S.S. Dragomir, A.R. Khan, M. Khan et al. Journal of King Saud University – Science 34 (2022) 102057

2

Since

nm1

Rn

mKðz;

s

;kÞf⁰ð

s

Þd

s

¼ ð1kÞfðzÞþfðmþnzÞ

2 þk^{fðmÞþfðnÞ}_{2 nm}¹ Rn

mfð

s

Þd

s

; Rn

mKðz;

s

;kÞdt¼0;

then by(2.3)we get the following identity ð1kÞfðzÞþfðmþnzÞ

2 þk^{fðmÞþfðnÞ}_{2 nm}¹ Rn mfð

s

Þd

s

¼_2ðnmÞ¹ 2

Rn m

Rn

mðKðz;

s

;kÞ Kðz;s;kÞÞðf⁰ð

s

Þ f⁰ðsÞÞd

s

ds; ð2:4Þ

8z2 ½mþk^nm₂ ;^mþn2 andk2 ½0;1.

By applying Cauchy-Schwartz inequality for double integrals, we can write

1 2ðnmÞ²

Rn m

Rn

mðKðz;

s

;kÞ Kðz;s;kÞÞðf⁰ð

s

Þ f⁰ðsÞÞd

s

ds

6 _2ðnmÞ¹ 2

Rn m

Rn

mðKðz;

s

;kÞ Kðz;s;kÞÞ²d

s

ds

¹₂

_2ðnmÞ¹ 2

Rn m

Rn

mðf⁰ð

s

Þ f⁰ðsÞÞ²d

s

ds

¹₂

:

ð2:5Þ

However

1 2ðnmÞ²

Rn m

Rn

mðKðz;

s

;kÞ Kðz;s;kÞÞ²d

s

ds

¼_ðnmÞ¹ Rn

mK²ðz;

s

;kÞdt _nm¹ Rn

mKðz;

s

;kÞd

s

2

¼_ðnmÞ^{1 2}₃ zmk^nm₂ 3

z^mþn₂ 3

þ^k3ðnmÞ₁₂ ³

h i

:

ð2:6Þ

Consider above terms in the following and simplifying:

zmk^nm₂

3

z^mþn₂ 3

¼^ðnmÞ₈ ³ð1kÞ³þ³₂z^mþn₂ 2

ðnmÞð1kÞ þ³₄ðnmÞ²ð1kÞ²z^mþn₂

;

ð2:7Þ

and 1 2ðnmÞ²

Z _n

m

Z _n

m

ðf⁰ð

s

Þ f⁰ðsÞÞ²d

s

ds¼ 1

ðnmÞk kf^{0 2}₂ fðnÞ fðmÞ nm

2

: ð2:8Þ Using(2.4), (2.6), (2.7) and (2.8), we get the 1st inequality of (2.1). Since

c

₆f⁰ð

s

Þ6Calmost everywhere for

s

2 ½m;n, by apply- ing Grüss inequality(1.2)we get

06 1 nm

Zn m

ðf⁰ð

s

ÞÞ²dt 1 nm

Z n m

f⁰ð

s

Þd

s

2

61

4ðC

c

Þ²; ð2:9Þ which completes the proof of last inequality of(2.1). h

Following remark (Remark 1 of Barnett et al. (2000)) is also valid for our main result.

Remark 2.2. SinceL1½m;n L2½m;n(and the inclusion is strict), then we remark that the inequality(2.1)can be applied also for the mappings f whose derivatives are unbounded on ðm;nÞ, but f⁰2L2½m;n.

Remark 2.3. Since 3k²3kþ161;8 k2 ½0;1and this is mini- mum whenk¼¹₂. Therefore,(2.1) captures various special cases of main result which is obtained by authors of article (Barnett et al., 2000) as can be seen in remark given below.

Remark 2.4. We can get different special cases of(2.1)by using several values of k by fixing z¼^mþn₂ . Under the assumptions of Theorem 2.1following results (special cases) are valid:

Special Case I:Fork¼1(2.1)gives trapezoid inequality

fðmÞþfðnÞ 2 _nm¹ Rn

mfð

s

Þd

s

6₂¹ffiffi

p3hðnmÞkf⁰k²₂ðfðnÞ fðmÞÞ²i¹₂ 6₄¹ffiffi

p3ðC

c

ÞðnmÞ;

which is Remark 3.2ðiÞofZafar (2010).

Special Case II:Fork¼0(2.1)gives mid-point ineqality f^mþn₂

_nm¹ Rn mfð

s

Þd

s

6₂^p1ffiffi₃hðnmÞkf⁰k²₂ðfðnÞ fðmÞÞ²i¹

2

6₄¹ffiffi

p3ðC

c

ÞðnmÞ:

which is Corollary 1 ofBarnett et al. (2000)and Remark 3.2ðiiÞof Zafar (2010).

Special Case III:Fork¼¹₂(2.1)gives averaged mid-point and trapezoid inequality

fðmÞþ2fð Þ^mþn₂ ^þfðnÞ

4 _nm¹ Rn

mfð

s

Þd

s

6₄¹ffiffi

p3hðnmÞkf⁰k²₂ðfðnÞ fðmÞÞ²i¹₂ 6₈¹ffiffi

p3ðC

c

ÞðnmÞ:

which is Remark 3.2ðiiiÞofZafar (2010).

Special Case IV: For k¼¹₃ (2.1) gives a variant of Simpson’s inequality for differentiable functionf

fðmÞþ4fð Þ^mþn₂ ^þfðnÞ

6 _nm¹ Rn

mfð

s

Þd

s

6¹₆hðnmÞkf⁰k²₂ðfðnÞ fðmÞÞ²i¹₂ 6₁₂¹ðC

c

ÞðnmÞ:

which is Remark 3.2ði

v

^ÞofZafar (2010).

3. Application to probability theory

Suppose random variable ‘Z’ be continuous with PDF f:½m;n !R^þand CDFU:½m;n ! ½0;1is defined as

U^{ðzÞ ¼Z z}

m

fð

s

Þd

s

; z2 mþknm 2 ;mþn

2

h i

; and

EðZÞ ¼ Z n

m

s

fð

s

Þd

s

;

is expectation of random variable ‘Z’ on½m;n. Then we have follow- ing result:

Theorem 3.1. Let the suppositions ofTheorem2.1be valid and if PDF f2L2½m;n, then

ð1kÞU^{ðzÞ þ}U^ðmþnzÞ

2 þ^k₂^nEðZÞ_nm

6 1

nm

ðnmÞ²

12 ð3k²3kþ1Þ þ zmþn 2

2

ð1kÞ

"

þðnmÞð1kÞ²

2 zmþn

2

#¹²

ðnmÞkU^0k221

h i¹₂

6ðHhÞ 2

ðnmÞ²

12 ð3k²3kþ1Þ þ zmþn 2

2

ð1kÞ

"

þðnmÞð1kÞ²

2 zmþn

2

#¹²

;

ð3:1Þ

where h6U^0ð

s

Þ6H,8

s

2 ½m;n.

Proof.Putf¼U^{in(2.1)we obtain}(3.2), by applying the identity Z n

m U^ð

s

Þd

s

¼nEðZÞ where U^{ðmÞ ¼}0; U^{ðnÞ ¼}1:

h

Corollary 3.2.Under the assumptions as stated inTheorem3.1, if we put z¼^mþn₂ , then

ð1kÞU ^mþn2

þ₂^knEðZÞ_nm

6₂^p1ffiffi₃ð3k²3kþ1Þ¹²hðnmÞkU^0k221i¹₂ 6^ðnmÞ₄ ffiffi

p3 ð3k²3kþ1Þ¹²ðHhÞ hold for h6U^0ð

s

Þ6H8

s

2 ½m;n.

Remark 3.3.The Corollary 3.2 is in fact Corollary 3.1 of Zafar (2010).

4. Application to special means

Before we proceed further we need here some definitions of special means.

Special Means:These means can be found inZafar (2010).

(a) Arithmetic Mean A¼mþn

2 ; m;nP0:

(b) Geometric Mean G¼Gðm;nÞ ¼ ffiffiffiffiffiffiffi

pmn

; m;nP0:

(c) Harmonic Mean H¼Hðm;nÞ ¼ 2

1

mþ¹_n; m;n>0:

(d) Logarithmic Mean

L¼Lðm;nÞ ¼

m; if m¼n

lnnlnnmm; if m–n;

8>

<

>: m;n>0:

(e) Identric Mean

I¼Iðm;nÞ ¼

m; if m¼n ln ð Þ^{mmnn nm1}

e

!

; if m–n;

8>

><

>>

: m;n>0:

(f)p-Logarithmic Mean

Lp¼Lpðm;nÞ ¼

m; if m¼n

n^pþ1m^pþ1 ðpþ1ÞðnmÞ

¹_p

; if m–n;

8>

><

>>

:

wherep2Rn f1;0g;m;n>0. It is known that ‘‘Lp is monotoni- cally increasing overp2R”, ‘‘L0¼I” and ‘‘L1¼L”.

Example 4.1. ConsiderfðzÞ ¼z^p;p2Rn f1;0g;then for n>m 1

nm Z n

m

fð

s

Þd

s

¼L^p_pðm;nÞ;

fðnÞ fðmÞ

nm ¼pL^p1_p1ðm;nÞ;

fðmÞ þfðnÞ

2 ¼Aðm^p;n^pÞ;

mþn 2 ¼A

and 1

nmkf⁰k²₂¼ 1 nm

Z n m

jf⁰ð

s

Þj²dt

¼p²L^2ðp1Þ_2ðp1Þ; wherez2 ½mþk^nm₂ ;^mþn₂ .

Therefore,(2.1)becomes

ð1kÞ^{zpþðmþnzÞ}₂ ^pþkAðm^p;n^pÞ L^p_p

jpjh^ðnmÞ₁₂²ð3k²3kþ1Þ þ ðzAÞ²ð1kÞ þ^ðnmÞð1kÞ₂ ²ðzAÞi¹₂

L^2ðp1Þ_2ðp1ÞL^2ðp1Þ_ðp1Þ h i¹₂

: ð4:1Þ

Choosez¼Ain(4.1), get ð1kÞA^pþkAðm^p;n^pÞ L^p_p

6jpj^nm₂ ffiffi

p3ð3k²3kþ1Þ¹²hL^2ðp1Þ_2ðp1ÞL^2ðp1Þ_ðp1Þi¹₂

;

which is minimum fork¼¹₂. Moreover fork¼1 Aðm^p;n^pÞ L^p_p

6nm

2 ffiffiffi

p jpj3 hL^2ðp1Þ_2ðp1ÞL^2ðp1Þ_ðp1Þ i¹₂ :

Example 4.2. ConsiderfðzÞ ¼¹_z; z–0, then

nm1

Z n m

fð

s

Þd

s

¼ L¹ðm;nÞ;

fðnÞfðmÞ

nm ¼ _G¹2;

fðmÞþfðnÞ

2 ¼ ^A

G²;

1 nm

Z n m

jf⁰ð

s

Þj²dt¼ ^m2_3m^þmnþn3n^{3 2}

and _nm¹ Z n

m

jf⁰ð

s

Þj²dt fðnÞ fðmÞ nm

2

¼ ^ðnmÞ_3m3n³²¼^ðnmÞ2

3G⁶ ; wherez2 ½mþk^nm₂ ;^mþn₂ ð0;1Þ.

S.S. Dragomir, A.R. Khan, M. Khan et al. Journal of King Saud University – Science 34 (2022) 102057

4

Therefore,(2.1)becomes

ð1kÞ

2 1

zþ_ðmþnzÞ¹

þk_G^A2¹_L

6h^ðnmÞ₁₂²ð3k²3kþ1Þ þ ðzAÞ²ð1kÞ

þ^ðnmÞð1kÞ₂ ²ðzAÞi¹_2ðnmÞ ffiffi3

pG³: ð4:2Þ

If we choose z = A in(4.2), we get ð1kÞ1

AþkA G²1

L

6ðnmÞ²

6G³ ð3k²3kþ1Þ¹²: Fork¼1

A G²1

L

6ðnmÞ² 6G³ :

Example 4.3.ConsiderfðzÞ ¼lnz;z>0, then

1 nm

Z n m

fð

s

Þd

s

¼ lnðIðm;nÞÞ;

fðnÞfðmÞ nm ¼ ¹_L;

fðmÞþfðnÞ

2 ¼ lnG;

nm1

Z n m

jf⁰ð

s

Þj²dt¼ _G¹2 and

1 nm

Zn m

jf⁰ð

s

Þj²dt fðnÞ fðmÞ nm

2

¼ ^L2G2

L²G² ; wherez2 ½mþk^nm₂ ;^mþn₂ ð0;1Þ.

Therefore,(2.1)becomes ln ^{ðzðmþnzÞÞ}

ð1kÞ 2 G^k I

6h^ðnmÞ₁₂²ð3k²3kþ1Þ þ ðzAÞ²ð1kÞ þ^ðnmÞð1kÞ₂ ²ðzAÞi¹

2 ðL²G²Þ¹² LG : Forz¼A

ln A^ð1kÞG^k I !

6ðnmÞ 2 ffiffiffi

p3

LGð3k²3kþ1ÞðL²G²Þ¹

2:

Fork¼1 ln G

I

6ðnmÞ 2 ffiffiffi

p3

LGðL²G²Þ¹²:

5. Application to numerical integration

To get the composite quadrature rules, we have to let Ij:m¼z0<z1<. . .<zj1<zj¼nbe the partision of the interval

½m;n;hj¼zjþ1zj; k2 ½0;1;zjþk^h₂^j6

g

j6^zjþz2^jþ1;j2 f0;. . .; i1g, then the following results hold:

Theorem 5.1. If

c

6 f⁰ð

s

Þ 6 C ^{almost everywhere for}

s

2 ½zjþk^h₂^j;zjþ1(j2 f0;. . .;i1g), then under the assumptions of

Theorem2.1the following quadrature formula holds Z n

m

fð

s

Þd

s

¼Qðf;f⁰;Ij;

g

;kÞ þRðf;f⁰;Ij;

g

;kÞ; ð5:1Þ where

Qðf;f⁰;Ij;

g

;kÞ ¼Pⁱ¹

j¼0hjhð1kÞ^{fðgjÞþfðzj}₂^þzjþ1gjÞþk^{fðzjÞþfðz}₂ ^jþ1Þi ð5:2Þ

and remainder R satisfies the estimate

jRðf;f⁰;Ij;g;kÞj6Pⁱ¹

j¼0 h²_j

12ð3k²3kþ1Þ þgj^zjþz₂^jþ12

ð1kÞ

þ^hjð1kÞ₂ ²g_j^zjþz₂^jþ1i¹₂

hjkf⁰k²₂fðzjþ1Þ fðzjÞ2

h i¹₂

6¹₂ðCcÞPⁱ¹

j¼0

hj h²_j

12ð3k²3kþ1Þ þg_j^zjþz₂^jþ12

ð1kÞ

þ^hjð1kÞ₂ ²g_j^zjþz₂^jþ1i¹₂ :

ð5:3Þ

Proof. By using inequalities (2.4), (2.5) and (2.9) on z_jþk^h₂^j6

g

_j6^zjþz₂^jþ1and summing overjfrom 0 toi1, then we get required result. h

By putting several values ofkand by fixing

g

j¼^zjþz₂^jþ1, under the assumptions ofTheorem 5.1 following results (special cases) are valid.

Special Case I:Putk¼1 in(5.2) and (5.3), we have Q f;f⁰;Ij;zjþzjþ1

2 ;1

¼1 2

Xⁱ¹

j¼0

hjðfðzjÞ þfðzjþ1ÞÞ

and

jRf;f⁰;Ij;^zjþz₂^jþ1;1 j 6₂¹ffiffi

p3Xⁱ¹

j¼0

hjhhjkf⁰k²₂fðzjþ1Þ fðzjÞ2i¹₂ 6₄¹ffiffi

p3ðC

c

ÞXⁱ¹

j¼0

h²_j:

Special Case II:Putk¼0 in(5.2) and (5.3), we have Q f;f⁰;Ij;zjþzjþ1

2

¼Xⁱ¹

j¼0

hjfðzjþzjþ1

2 Þ and

jRf;f⁰;Ij;^zjþz₂^jþ1 j 6₂¹ffiffi

p3Pⁱ¹

j¼0

hjhhjkf⁰k²₂fðzjþ1Þ fðzjÞ2i¹₂ 6₄¹ffiffi

p3ðC

c

ÞPⁱ¹

j¼0

h²_j:

Special Case III:Putk¼¹₂in(5.2) and (5.3), we have Q f;f⁰;Ij;zjþzjþ1

2 ;1 2

¼1 4

Xⁱ¹

j¼0

hj fðzjÞ þ2fðzjþzjþ1

2 Þ þfðzjþ1Þ

and

jRf;f⁰;Ij;^zjþz₂^jþ1;¹₂ j 6₄^p1ffiffi₃Pⁱ¹

j¼0

hjhhjkf⁰k²₂fðz_jþ1Þ fðzjÞ2i¹

26₈^p1ffiffi₃ðC

c

ÞPⁱ¹

j¼0

h²_j:

Special Case IV:Putk¼¹₃in(5.2) and (5.3), we have Q f;f⁰;Ij;zjþzjþ1

2 ;1 3

¼1 6

Xⁱ¹

j¼0

hj fðzjÞ þ4fðzjþzjþ1

2 Þ þfðzjþ1Þ

and

jRf;f⁰;Ij;^zjþz₂^jþ1;¹₃ j 6¹₆Pⁱ¹

j¼0hjhhjkf⁰k²₂fðzjþ1Þ fðzjÞ2i¹₂

6₁₂¹ðC

c

ÞPⁱ¹

j¼0h²_j:

6. Conclusion

Using three step kernel, we have obtained new generalized Ostrowski-Grüss type inequalities(2.1)which is a variant of(1.4) which was obtained in article (Barnett et al., 2000). By fixing

z¼^mþn₂ and by choosing different values of parameterkwe cap- tured many results stated in Barnett et al. (2000) and Zafar (2010). We also got different important results from our main results as special cases such as trapezoidal inequality, mid-point inequality, averaged mid-point and trapezoidal inequality and Simpson’s inequality. Moreover, applications are deduced for prob- ability theory, special means and numerical integration.

Conflict of Interest:Authors declared: No conflict of interest.

Acknowledgment

The research of the Second Author is financially supported by the Dean’s Science Research Grant 2021-2022, University of Karachi, with grant number, KURP Award Letter No. 161/2021-22.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, in the online version, athttps://doi.org/10.1016/j.jksus.2022.102057.

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