649–664 DOI: 10.18514/MMN.2018.1661 SIMPSON TYPE QUANTUM INTEGRAL INEQUALITIES FOR CONVEX FUNCTIONS M

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Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 19 (2018), No. 1, pp. 649–664 DOI: 10.18514/MMN.2018.1661

SIMPSON TYPE QUANTUM INTEGRAL INEQUALITIES FOR CONVEX FUNCTIONS

M. TUNÇ , E. G ÖV, AND S. BALGEÇ TI Received 07 May, 2015

Abstract. In this paper we establish some new Simpson type quantum integral inequalities for convex functions. Moreover, we obtain some inequalities for special means.

2010Mathematics Subject Classification: 26D10; 26D15; 34A08

Keywords: integral inequalities, q-calculus, convex functions, Simpson’s inequality

1. INTRODUCTION

A functionf WJ R!Ris said to be convex onJ if the inequality

f .t uC.1 t / v/tf .u/C.1 t / f .v/ (1.1) holds for allu; v2J andt2Œ0; 1. We say thatf is concave if. f /is convex.

Convex functions play an important role in mathematical inequalities. The most famous inequality have been used with convex functions is Hermite-Hadamard, which is stated as follows:

Let f WJ R!Rbe a convex function and u; v 2J with u < v. Then the following double inequalities hold:

f

uCv 2

1

v u

Z v u

f .x/ dxf .u/Cf .v/

2 : (1.2)

In recent years quantum calculus has been actively studied. There are numer- ous applications in many mathematical areas like special functions, integral trans- forms, quantum mechanics, information technology and mathematical inequalities.

At present q analogous of many inequalities have been established. In the view of these developments q-convexity and convexity of q analogous of the inequalities has also been considered, see [3–5,7,9–11].

The inequality given below is well known in the literature as Simpson’s inequality:

1 b a

Z b a

f .x/ dx 1 3

f .a/Cf .b/

2 C2f

aCb 2

1

2880 f^.4/

1.b a/⁴ (1.3)

c 2018 Miskolc University Press

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where the mappingf WŒa; b!Ris assumed to be four times continuously differen- tiable on the interval andf^.4/to be bounded on.a; b/, that is,

f^.4/

1D sup

t2.a;b/

ˇ ˇ ˇf^.4/

ˇ ˇ ˇ<1:

Inequality (1.3) have been studied by many authors. For more details see [1,2,6,8].

The aim of this paper is to establish q analogues of Simpson type inequalities based on convexity. The consequences of Simpson type inequalities for convex functions are given as special cases whenq!1.

2. PRELIMINARIES

In this section, we recall some previously known concepts and basic results.

Let J DŒa; bR be an interval and 0 < q < 1 be a constant. We define q- derivative of a functionf WJ !Rat a pointx2J onŒa; bas follows.

Definition 1. Assumef WJ !Ris a continuous function and letx2J. Then the expression

aDqf .x/Df .x/ f .qxC.1 q/ a/

.1 q/ .x a/ ; x¤a; aDqf .a/ D lim

x!aaDqf .x/ ; (2.1) is called theq-derivative onJ of functionf atx.

Also ifaD0in (2.1), then 0Dqf .a/ DDqf , whereDq is theq-derivative of the functionf .x/defined by

Dqf .x/Df .x/ f .qx/

.1 q/ x : For more details, see [7].

Lemma 1([10]). Let˛2R, then we have

aDq.x a/^˛ D

1 q^˛ 1 q

.x a/^{˛ 1}: (2.2)

Definition 2. Letf WJ R!Rbe a continuous function. Thenq-integral onJ is defined as

Z x a

f .t /adqt D.1 q/ .x a/

1

X

nD0

qⁿf qⁿxC 1 qⁿ a

(2.3) forx2J. IfaD0in (2.2), then we have the classicalq-integral [7].

Moreover, ifv2.a; x/then the definiteq-integral onJ is defined by Z x

v

f .t /adqt D Z x

a

f .t /adqt Z v

a

f .t /adqt

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D.1 q/ .x a/

1

X

nD0

qⁿf qⁿxC 1 qⁿ a

.1 q/ .v a/

1

X

nD0

qⁿf qⁿvC 1 qⁿ a

:

Lemma 2([11]). For˛2Rn f 1g, the following formula holds:

Z x a

.t a/^˛adqt D

1 q 1 q^˛^C¹

.x a/^˛^C¹: (2.4)

3. RESULTS

We begin with the following lemma.

Lemma 3. Letf WJ !Rbe a continuous function and0 < q < 1. IfaDqf is an integrable function onJ^ı(the interior ofJ), then the following inequality holds:

1 6

f .a/C4f

aCb 2

Cf .b/

1

b a Z b

a

f .x/adqx

D.b a/

Z 1 0

p .t /aDqf ..1 t / aCt b/ 0dqt (3.1) where

p .t /D

qt ¹₆ ; t 2 0;¹₂ qt ⁵₆ ; t 2₁

2; 1 : Proof. From Definition1and Definition2, we have Z ¹₂

0

qt 1

6

aD_qf ..1 t / aCt b/ ₀d_qt D

Z ¹₂

0

qt aDqf ..1 t / aCt b/ 0dqt 1 6

Z ¹₂

0

aDqf ..1 t / aCt b/ 0dqt D

Z ¹₂

0

qf ..1 t / aCt b/ f ..1 qt / aCqt b/

.1 q/ .b a/ ⁰dqt

1 6

Z ¹₂

0

f ..1 t / aCt b/ f ..1 qt / aCqt b/

.1 q/ .b a/ t ⁰dqt D1

2

1

X

nD0

qⁿ^C¹f 1 ¹₂qⁿ

aC¹2qⁿb .b a/

1 2

1

X

nD0

qⁿ^C¹f 1 ¹₂qⁿ^C¹

aC¹2qⁿ^C¹b .b a/

1 6

1

X

nD0

f 1 ¹₂qⁿ

aC¹₂qⁿb

.b a/ C1

6

1

X

nD0

f 1 ¹₂qⁿ^C¹

aC¹₂qⁿ^C¹b .b a/

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D1 2q

1

X

nD0

qⁿf 1 ¹₂qⁿ

aC¹₂qⁿb .b a/

1 2

1

X

nD1

qⁿf 1 ¹₂qⁿ

aC¹₂qⁿb .b a/

1 6

1 b a

1

X

nD0

f

1 1 2qⁿ

aC1

2qⁿb

C1 6

1 b a

1

X

nD1

f

1 1 2qⁿ

aC1

2qⁿb

D 1

b a.1 q/1 2

1

X

nD0

qⁿf

1 1 2qⁿ

aC1

2qⁿb

C1 2

1 b af

aCb 2

1 6

1 b a

f

aCb 2

f .a/

D 1 b a

Z ¹₂

0

f ..1 t / aCt b/0dqtC 1 3 .b a/f

aCb 2

C 1

6 .b a/f .a/ : For the second part of the integral, we have

Z 1

1 2

qt 5

6

aDqf ..1 t / aCt b/ 0dqt D

Z 1 0

qt 5

6

aDqf ..1 t / aCt b/ 0dqt Z ¹₂

0

qt 5

6

aDqf ..1 t / aCt b/ 0dqt and similarly we obtain

Z 1 0

qt 5

6

aDqf ..1 t / aCt b/ 0dqt D 1

b a Z 1

0

f ..1 t / aCt b/0dqtC 1

6 .b a/f .b/C 5

6 .b a/f .a/

and Z ¹₂

0

qt 5

6

aDqf ..1 t / aCt b/ 0dqt D 1

b a Z ¹₂

0

f ..1 t / aCt b/0dqt 1 3 .b a/f

aCb 2

C 5

6 .b a/f .a/ : Thus, we have

Z 1 0

p .t /aDqf ..1 t / aCt b/ 0dqt

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D 1 b a

Z 1 0

f ..1 t / aCt b/0dqt C 2 3 .b a/f

aCb 2

C 1

6 .b a/ff .a/Cf .b/g D 1

.b a/² Z b

a

f .x/adqxC 2 3 .b a/f

aCb 2

C 1

6 .b a/ff .a/Cf .b/g

We complete the proof.

Remark1. Ifq!1, then (3.1) reduces to 1

6

f .a/C4f

aCb 2

Cf .b/

1

b a Z b

a

f .x/ dx

D.b a/

Z 1 0

p .t / f⁰.t bC.1 t / a/ dt;

where

p .t /D

t ¹₆; t2 0;¹₂ t ⁵₆ t2₁

2; 1 : See also [1, Lemma 1].

Lemma 4. Let0 < q < 1be a constant. Then, Z ¹₂

0

.1 t / ˇ ˇ ˇ ˇ

qt 1 6 ˇ ˇ ˇ ˇ

0dqt D 1 216

36q³C12q²C12qC1

q³C2q²C2qC1 : (3.2) Proof. By computing directly and using (2.4), we have

Z ¹₂

0

.1 t / ˇ ˇ ˇ ˇ

qt 1 6 ˇ ˇ ˇ ˇ

0dqt D

Z ¹₂

0

ˇ ˇ ˇ ˇ

qt 1 6 ˇ ˇ ˇ ˇ

0dqt Z ¹₂

0

t ˇ ˇ ˇ ˇ

qt 1 6 ˇ ˇ ˇ ˇ

0dqt D

Z _6q¹

0

1 6 qt

0dqt C Z ¹₂

1 6q

qt 1

6

0dqt Z _6q¹

0

t 1

6 qt

0dqtC Z ¹₂

1 6q

t

qt 1 6

0dqt

!

D Z _6q¹

0

1 6 qt

0dqt C Z ¹₂

0

qt 1

6

0dqt Z _6q¹

0

qt 1

6

0dqt Z _6q¹

0

t 1

6 qt

0dqtC Z ¹₂

0

t

qt 1 6

0dqt

Z _6q¹

0

t

qt 1 6

0dqt

!

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D 1 36

6q 1 qC1

1 216

18q²C18q 7 q³C2q²C2qC1 D 1

216

36q³C12q²C12qC1 q³C2q²C2qC1 :

Lemma 5. Let0 < q < 1be a constant. Then,

Z 1

1 2

.1 t / ˇ ˇ ˇ ˇ

qt 5 6 ˇ ˇ ˇ ˇ

0dqt D 1 216

12q²C12qC5

q³C2q²C2qC1: (3.3) Proof. By computing directly and using (2.4), we have

Z 1

1 2

.1 t / ˇ ˇ ˇ ˇ

qt 5 6 ˇ ˇ ˇ ˇ

0dqt D

Z 1

1 2

ˇ ˇ ˇ ˇ

qt 5 6 ˇ ˇ ˇ ˇ

0dqt Z 1

1 2

t ˇ ˇ ˇ ˇ

qt 5 6 ˇ ˇ ˇ ˇ

0dqt

D Z _6q⁵

1 2

5 6 qt

0dqtC Z 1

5 6q

qt 5

6

0dqt Z _6q⁵

1 2

t 5

6 qt

0dqt C Z 1

5 6q

t

qt 5 6

0dqt

!

D 5 36 .qC1/

1 216

18q²C18qC25 q³C2q²C2qC1 D 1

216

12q²C12qC5 q³C2q²C2qC1:

Theorem 1. Letf WJ !Rbe a continuous function and0 < q < 1:Ifˇ

ˇ_aDqfˇ ˇis convex and integrable function onJ^ı, then the following inequality holds:

1 6

f .a/C4f

aCb 2

Cf .b/

1

b a Z b

a

f .x/adqx (3.4)

.b a/

12

2q²C2qC1 q³C2q²C2qC1

ˇ

ˇ_aDqf .b/ˇ ˇC1

3

6q³C4q²C4qC1 q³C2q²C2qC1

ˇ

ˇ_aDqf .a/ˇ ˇ

: Proof. Using Lemma3and the convexity ofˇ

ˇ_aDqfˇ

ˇonJ^ı, we have ˇ

ˇ ˇ ˇ ˇ 1 6

f .a/C4f aCb

2

Cf .b/

1

b a Z b

a

f .x/adqx ˇ ˇ ˇ ˇ ˇ

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D.b a/

ˇ ˇ ˇ ˇ ˇ

Z ¹₂

0

qt 1

6

aDqf .t bC.1 t / a/0dqt

C Z 1

1 2

qt 5

6

aDqf .t bC.1 t / a/0dqt ˇ ˇ ˇ ˇ ˇ .b a/

Z ¹₂

0

ˇ ˇ ˇ ˇ

qt 1 6 ˇ ˇ ˇ ˇ ˇ

ˇaDqf .t bC.1 t / a/ˇ ˇ0dqt

C Z 1

1 2

ˇ ˇ ˇ ˇ

ˇ_aDqf .t bC.1 t / a/ˇ ˇ₀dqt

!

.b a/

Z ¹₂

0

ˇ ˇ ˇ ˇ

qt 1 6 ˇ ˇ ˇ ˇ

tˇ

ˇaDqf .b/ˇ

ˇC.1 t /ˇ

ˇaDqf .a/ˇ ˇ

0dqt

C.b a/

Z 1 1 2

ˇ ˇ ˇ ˇ

qt 5 6 ˇ ˇ ˇ ˇ

tˇ

ˇ_aDqf .b/ˇ

ˇC.1 t /ˇ

ˇ_aDqf .a/ˇ ˇ

0dqt

Dˇ

ˇaDqf .b/ˇ ˇ.b a/

Z ¹₂

0

t ˇ ˇ ˇ ˇ

qt 1 6 ˇ ˇ ˇ ˇ

0dqt Cˇ

ˇaDqf .a/ˇ ˇ.b a/

Z ¹₂

0

.1 t / ˇ ˇ ˇ ˇ

qt 1 6 ˇ ˇ ˇ ˇ

0dqt

Cˇ

ˇ_aDqf .b/ˇ ˇ.b a/

Z 1 1 2

t ˇ ˇ ˇ ˇ

qt 5 6 ˇ ˇ ˇ ˇ⁰

dqt Cˇ

ˇ_aDqf .a/ˇ ˇ.b a/

Z 1 1 2

.1 t / ˇ ˇ ˇ ˇ

dqt

Applying Lemma4and Lemma5, we have ˇ

ˇ ˇ ˇ ˇ 1 6

f .a/C4f aCb

2

Cf .b/

1

b a Z b

a

f .x/adqx ˇ ˇ ˇ ˇ ˇ .b a/

216

18q²C18q 7 q³C2q²C2qC1

ˇ

ˇ_aDqf .b/ˇ

ˇC.b a/

216

ˇ

ˇ_aDqf .a/ˇ ˇ

C.b a/

216

ˇ

ˇ_aDqf .b/ˇ

ˇC.b a/

216

ˇ

ˇ_aDqf .a/ˇ ˇ

D.b a/

12

ˇ

ˇaDqf .b/ˇ

ˇC.b a/

36

ˇ

ˇaDqf .a/ˇ ˇ:

The proof is complete.

Remark2. Ifq!1, then (3.4) reduces to ˇ

ˇ ˇ ˇ ˇ 1 6

f .a/C4f aCb

2

Cf .b/

1

b a Z b

a

f .x/ dx ˇ ˇ ˇ ˇ ˇ

5 .b a/

72

f⁰.a/Cf⁰.b/