After this, a new fractional Hermite-Hadamard inequality which is not a result of Hermite-Hadamard- Fej´er inequality and better than given in [9] by Sarıkaya et al

Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 19 (2018), No. 2, pp. 1007–1017 DOI: 10.18514/MMN.2018.2441

IMPROVEMENT OF FRACTIONAL HERMITE-HADAMARD TYPE INEQUALITY FOR CONVEX FUNCTIONS

MEHMET KUNT, ˙IMDAT ˙IS¸CAN, SERCAN TURHAN, AND D ¨UNYA KARAPINAR Received 02 November, 2017

Abstract. In this paper, it is proved that fractional Hermite-Hadamard inequality and fractional Hermite-Hadamard-Fej´er inequality are just results of Hermite-Hadamard-Fej´er inequality. After this, a new fractional Hermite-Hadamard inequality which is not a result of Hermite-Hadamard- Fej´er inequality and better than given in [9] by Sarıkaya et al. is obtained. Also, a new equality is proved and some new fractional midpoint type inequalities are given. Our results generalizes the results given in [5] by Kırmacı.

2010Mathematics Subject Classification: 26A51; 26A33; 26D10

Keywords: convex functions, Hermite-Hadamard inequalities, Riemann-Liouville fractional in- tegrals, midpoint type inequalities

1. INTRODUCTION

Letf WI R!Rbe a convex function defined on the intervalIof real numbers anda; b2I witha < b. The inequality

f

aCb 2

1

b a

Z b a

f .x/dx f .a/Cf .b/

2 (1.1)

is well known in the literature as Hermite-Hadamard’s inequality [2,3].

The most well-known inequalities related to the integral mean of a convex func- tion f are the Hermite Hadamard inequality or its weighted versions, the so-called Hermite-Hadamard-Fej´er inequality.

In [1], Fej´er established the following Fej´er inequality which is the weighted gen- eralization of Hermite-Hadamard inequality (1.1):

Theorem 1. Letf WŒa; b!Rbe convex function. Then, the inequality f

aCb 2

Z b a

g.x/dx Z b

a

f .x/g.x/dxf .a/Cf .b/

2

Z b a

g.x/dx (1.2) holds, where g WŒa; b!R is nonnegative, integrable and symmetric to ^aC₂^b (i.e.

g .x/Dg .aCb x/for allx2Œa; b).

c 2018 Miskolc University Press

In [5], Kırmacı used the following equality to obtain midpoint type inequalities and some applications:

Lemma 1. Leta; b2I witha < b andf WI^ı!Ris a differentiable mapping (I^ıthe interior ofI). Iff⁰2L Œa; b, then we have

1

b a

Z b a

f .u/ du f

aCb 2

D.b a/

Z 1=2 0

tf⁰.t aC.1 t / b/ dtC Z 1

1=2

.t 1/ f⁰.t aC.1 t / b/ dt:

(1.3)

Following definitions of the left and right side Riemann-Liouville fractional integ- rals are well known in the literature.

Definition 1. Leta; b2Rwitha < bandf 2L Œa; b. The left and right Riemann- Liouville fractional integralsR_aCf andR_b f of order > 0are defined by

R_aCf .x/D 1 . /

Z x a

.x t / ¹f .t /dt; x > a

and

R_b f .x/D 1 . /

Z b x

.t x/ ¹f .t /dt; x < b

respectively, where . /is the Gamma function defined by . /D R1 0

e ^tt ¹dt (see [6, page 69] and [10, page 4]).

In [9], Sarıkaya et al. proved the following fractional Hermite-Hadamard type inequality:

Theorem 2. Letf WŒa; b!Rbe a positive function with 0a < b andf 2 LŒa; b. Iff is a convex function onŒa; b, then the following inequalities for frac- tional integrals holds:

f

aCb 2

.C1/

2 .b a/

h

R_aCf .b/CR_b f .a/

i

f .a/Cf .b/

2 (1.4)

with > 0.

Remark1. In Theorem2, it is not necessary supposing thatf be a positive func- tion anda; bare positive real numbers. From the Definition1, it is clear thata; bare any real numbers such asa < b.

In [4], ˙Is¸can proved the following fractional Hermite-Hadamard-Fej´er type in- equality:

Theorem 3. Letf WŒa; b!Rbe a convex function witha < bandf 2LŒa; b.

IfgWŒa; b!Ris nonnegative, integrable and symmetric to ^aC₂^b, then the following inequality for fractional integrals holds:

f

aCb 2

h

R_aCg.b/CR_b g.a/i h

R_aC.fg/ .b/CR_b .fg/ .a/i

f .a/Cf .b/

2 h

R_aCg.b/CR_b g.a/i (1.5)

with > 0.

In [7], Kunt et al. proved the following left Riemann-Liouville fractional Hermite- Hadamard type inequality and next equality:

Theorem 4. Leta; b2Rwitha < bandf WŒa; b!Rbe a convex function. If f 2L Œa; b, then the following inequality for the left Riemann-Liouville fractional integral holds:

f

aCb C1

.C1/

.b a/ R_aCf .b/ f .a/Cf .b/

C1 (1.6)

with > 0.

Lemma 2. Leta; b2Rwitha < bandf WŒa; b!Rbe a differentiable function on.a; b/. Iff⁰2LŒa; b, then the following equality for the left Riemann-Liouville fractional integrals holds:

.C1/

.b a/ R_aCf .b/ f

aCb C1

D.b a/

2 4

R

C1

0 tf⁰.t aC.1 t / b/ dt CR1

C1

t 1

f⁰.t aC.1 t / b/ dt 3 5

(1.7)

with > 0.

In [8], Kunt et al. proved the following right Riemann-Liouville fractional Hermite- Hadamard type inequality and next equality:

Theorem 5. Leta; b2Rwitha < bandf WŒa; b!Rbe a convex function. If f 2L Œa; b, then the following inequality for the right Riemann-Liouville fractional integral holds:

f

aCb C1

.C1/

.b a/ R_b f .a/f .a/Cf .b/

C1 (1.8)

with > 0.

Lemma 3. Leta; b2Rwitha < bandf WŒa; b!Rbe a differentiable function on.a; b/. Iff⁰2LŒa; b, then the following equality for the right Riemann-Liouville fractional integrals holds:

.C1/

.b a/ R_b f .a/ f

aCb C1

D.b a/

2 4

R

C1

0 tf⁰.t bC.1 t / a/ dt CR1

C1

1 t

f⁰.t bC.1 t / a/ dt 3 5

(1.9)

with > 0.

In our studies we noticed that fractional Hermite-Hadamard type inequality given in Theorem2and fractional Hermite-Hadamard-Fej´er type inequality given in The- orem3are just result of Hermite-Hadamard-Fej´er inequality (given in Theorem1), with a special selection of the weighted function. This show how strong the Hermite- Hadamard-Fej´er inequality is. However, we will prove new fractional Hermite-Ha- damard type inequality which is not a result of Theorem1. Also, we will have new fractional midpoint type inequalities.

2. RESULTS OFHERMITE-HADAMARD-FEJER INEQUALITY´ Proposition 1. Theorem2is a result of Theorem1.

Proof. In Theorem1, let we chooseg .x/D.x a/ ¹C.b x/ ¹ for > 0, a; b2RandgWŒa; b!R(It is clearg .x/nonnegative, integrable and symmetric to

aCb

2 ). Computing the following integrals, we have Z b

a

g.x/dxD Z b

a

.x a/ ¹C.b x/ ¹dxD2 .b a/

; (2.1)

Z b a

f .x/g.x/dxD Z b

a

h

.x a/ ¹C.b x/ ¹ i

f .x/dx (2.2)

D Z b

a

.x a/ ¹f .x/dxC Z b

a

.b x/ ¹f .x/dx

D . /h

R_aCf .b/CR_b f .a/i :

Combining.1:2/,.2:1/and.2:2/we have.1:4/. This completes the proof.

Proposition 2. Theorem3is a result of Theorem1.

Proof. In Theorem1, let we choosew .x/Dh

.x a/ ¹C.b x/ ¹i

g .x/for > 0,a; b2R, gWŒa; b!Randg .x/nonnegative, integrable and symmetric to

aCb

2 (It is clearw .x/ nonnegative, integrable and symmetric to ^aC₂^b). Computing the following integrals, we have

Z b a

w .x/ dxD Z b

a

h

.x a/ ¹C.b x/ ¹ i

g .x/ dx (2.3)

D Z b

a

.x a/ ¹g .x/ dxC Z b

a

.b x/ ¹g .x/ dx

D . /h

R_aCg.b/CR_b g.a/i

;

Z b a

f .x/w.x/dxD Z b

a

h

.x a/ ¹C.b x/ ¹ i

f .x/g .x/ dx (2.4)

D Z b

a

.x a/ ¹f .x/g .x/ dxC Z b

a

.b x/ ¹f .x/g .x/ dx

D . /h

R_aC.fg/ .b/CR_b .fg/ .a/i :

Combining.1:2/,.2:3/and.2:4/we have.1:5/. This completes the proof.

Remark2. Theorem4and Theorem5are not results of Theorem1.

3. IMPROVEMENT OF FRACTIONALHERMITE-HADAMARD TYPE INEQUALITY

We will use Theorem4and Theorem5to have new fractional Hermite-Hadamard type inequality better than.1:4/.

Theorem 6. Leta; b2Rwitha < bandf WŒa; b!Rbe a convex function. If f 2L Œa; b, then the following inequality for fractional integral holds:

f

aCb C1

Cf

aCb C1

2 .C1/

2 .b a/

h

R_aCf .b/CR_b f .a/i

f .a/Cf .b/

2

(3.1) with > 0.

Proof. If .1:6/ and .1:8/ gather side by side and dividing into 2, it is hold the

desired result.

Remark 3. Since, f is a convex function on Œa; b, it is clear f aCb

2

f aCb

C1

Cf

aCb C1

2 for > 0. It means that (1) Theorem6is better than Theorem2,

(2) In Theorem6if one takesD1, one has.1:1/, (3) Theorem6is not a result of Theorem1.

4. NEW FRACTIONAL MIDPOINT TYPE INEQUALITIES

We will now prove an equality to have new fractional midpoint type inequalities.

Lemma 4. Leta; b2Rwitha < bandf WŒa; b!Rbe a differentiable function on .a; b/. If f⁰ 2LŒa; b, then the following equality for the fractional integrals holds:

.C1/

2 .b a/

h

R_aCf .b/CR_b f .a/

i f

aCb C1

Cf

aCb C1

2 (4.1)

Db a 2

2 6 6 6 6 6 6 4

R

C1

0 tf⁰.t aC.1 t / b/ dt CR1

C1

t 1

f⁰.t aC.1 t / b/ dt CR

C1

0 tf⁰.t bC.1 t / a/ dt CR1

C1

1 t

f⁰.t bC.1 t / a/ dt 3 7 7 7 7 7 7 5

Proof. If .1:7/ and .1:9/ gather side by side and dividing into 2, it is hold the

desired result.

Corollary 1. In Lemma4, if one takesD1, one has Lemma1.

Theorem 7. Leta; b2Rwitha < bandf WŒa; b!Rbe a differentiable function on .a; b/. If jf⁰j is convex on Œa; b, then the following fractional midpoint type inequality holds:

ˇ ˇ ˇ ˇ ˇ ˇ

.C1/

2 .b a/

h

R_aCf .b/CR_b f .a/i f

aCb C1

Cf

aCb C1

2

ˇ ˇ ˇ ˇ ˇ ˇ

(4.2)

.b a/ ^C1 .C1/^C2

ˇˇf⁰.a/ˇ ˇCˇ

ˇf⁰.b/ˇ ˇ

with > 0.

Proof. Using Lemma4and the convexity ofjf⁰j, we have ˇ

ˇ ˇ ˇ ˇ ˇ

.C1/

2 .b a/

h

R_aCf .b/CR_b f .a/

i f

aCb C1

Cf

aCb C1

2

ˇ ˇ ˇ ˇ ˇ ˇ

b a

2 2 6 6 6 6 6 6 4

R

C1

0 tjf⁰.t aC.1 t / b/jdt CR1

C1

1 t

jf⁰.t aC.1 t / b/jdt CR

C1

0 tjf⁰.t bC.1 t / a/jdt CR1

C1

1 t

jf⁰.t bC.1 t / a/jdt 3 7 7 7 7 7 7 5

b a 2

2 6 6 6 6 6 6 4

R

C1

0 tŒtjf⁰.a/j C.1 t /jf⁰.b/j dt CR1

C1

1 t

Œtjf⁰.a/j C.1 t /jf⁰.b/j dt CR

C1

0 tŒtjf⁰.b/j C.1 t /jf⁰.a/j dt CR1

C1

1 t

Œtjf⁰.b/j C.1 t /jf⁰.a/j dt 3 7 7 7 7 7 7 5

Db a 2

2 4

R

C1

0 tŒjf⁰.a/j C jf⁰.b/j dt CR1

C1

1 t

Œjf⁰.a/j C jf⁰.b/j dt 3 5

Db a 2

"

Z _C1

0

tdtC Z 1

C1

1 t

dt

#

ˇˇf⁰.a/ˇ ˇCˇ

ˇf⁰.b/ˇ ˇ

D.b a/ ^C1 .C1/^C2

ˇˇf⁰.a/ˇ ˇCˇ

ˇf⁰.b/ˇ ˇ :

This completes the proof.

Corollary 2. In Theorem7, if one takesD1, one has [5, Theorem 2.2].

Theorem 8. Leta; b2Rwitha < bandf WŒa; b!Rbe a differentiable function on.a; b/. Ifjf⁰j^qis convex onŒa; bforq1, then the following fractional midpoint type inequality holds:

ˇ ˇ ˇ ˇ ˇ ˇ

.C1/

2 .b a/

h

R_aCf .b/CR_b f .a/i f

aCb C1

Cf

aCb C1

2

ˇ ˇ ˇ ˇ ˇ ˇ

(4.3)

b a 2

^C1 .C1/^C2

2 6 6 6 6 6 6 6 6 6 4

C2jf⁰.a/j^qC_C²₂jf⁰.b/j^q_q¹ C_.

C1/C2^C1

2.C2/ jf⁰.a/j^qC⁴₂.^.C^C2/^1/jf⁰.b/j^q1_q C

C2jf⁰.b/j^qC_C²₂jf⁰.a/j^q_q¹ C_.

C1/C2^C1

2.C2/ jf⁰.b/j^qC⁴₂.^.C^C2/^1/jf⁰.a/j^q1_q 3 7 7 7 7 7 7 7 7 7 5

with > 0.

Proof. Using Lemma 4, power mean inequality and the convexity ofjf⁰j^q, we have

ˇ ˇ ˇ ˇ ˇ ˇ

.C1/

2 .b a/

h

R_aCf .b/CR_b f .a/i f

aCb C1

Cf

aCb C1

2

ˇ ˇ ˇ ˇ ˇ ˇ

b a 2

2 6 6 6 6 6 6 4

R

C1 0 tˇ

ˇf⁰.t aC.1 t / b/ˇ ˇdt CR1

C1

1 tˇ

ˇf⁰.t aC.1 t / b/ˇ ˇdt CR

C1 0 tˇ

ˇf⁰.t bC.1 t / a/ˇ ˇdt CR1

C1

1 tˇ

ˇf⁰.t bC.1 t / a/ˇ ˇdt

3 7 7 7 7 7 7 5

b a 2

2 6 6 6 6 6 6 6 6 6 6 6 6 6 4

R

C1 0 tdt

1 ¹_q R

C1 0 tˇ

ˇf⁰.t aC.1 t / b/ˇ ˇ

qdt ¹_q

C

R1 C1

1 t

dt 1 _q¹

R1 C1

1 t

ˇˇf⁰.t aC.1 t / b/ˇ ˇ

qdt ¹_q

C

R

C1 0 tdt

1 _q¹ R

C1 0 tˇ

ˇf⁰.t bC.1 t / a/ˇ ˇ

qdt

1 q

C

R1 C1

1 t

dt 1 _q¹

R1 C1

1 tˇ

ˇf⁰.t bC.1 t / a/ˇ ˇ

qdt ¹_q

3 7 7 7 7 7 7 7 7 7 7 7 7 7 5

b a 2

^C1 .C1/^C2

!1 ¹_q

2 6 6 6 6 6 6 6 6 6 6 6 6 6 4

R

C1 0 th

tˇ ˇf⁰.a/ˇ

ˇ

qC.1 t /ˇ ˇf⁰.b/ˇ

ˇ

qi dt

1 q

C

R1 C1

1 t h

tˇ ˇf⁰.a/ˇ

ˇ

qC.1 t /ˇ ˇf⁰.b/ˇ

ˇ

qi dt

¹_q

C

R

C1 0 th

tˇ ˇf⁰.b/ˇ

ˇ

qC.1 t /ˇ ˇf⁰.a/ˇ

ˇ

qi dt

¹_q

C

R1 C1

1 t h

tˇ ˇf⁰.b/ˇ

ˇ

qC.1 t /ˇ ˇf⁰.a/ˇ

ˇ

qi dt

¹_q 3 7 7 7 7 7 7 7 7 7 7 7 7 7 5

b a 2

^C1 .C1/^C2

2 6 6 6 6 6 6 6 6 6 6 6 4

C2

ˇˇf⁰.a/ˇ ˇ

qC_C²₂ˇ ˇf⁰.b/ˇ

ˇ

q¹_q

C

.C1/C2^C1 2.C2/

ˇˇf⁰.a/ˇ ˇ

qC⁴₂.^.CC2/^1/

ˇˇf⁰.b/ˇ ˇ

q¹_q

C

C2

ˇ ˇf⁰.b/ˇ

ˇ

qC_C²₂ˇ ˇf⁰.a/ˇ

ˇ

q¹_q

C

.C1/C2^C1 2.C2/

ˇˇf⁰.b/ˇ ˇ

qC^{4 .C1/}

2.C2/

ˇˇf⁰.a/ˇ ˇ

q¹_q 3 7 7 7 7 7 7 7 7 7 7 7 5 :

This completes the proof.

Corollary 3. In Theorem8, if one takes D1, one has the following midpoint type inequality,

ˇ ˇ ˇ ˇ ˇ

1

b a

Z b a

f .u/ du f

aCb 2

ˇ ˇ ˇ ˇ ˇ

b a 8

2 4

1 3

ˇˇf⁰.a/ˇ ˇ

qC²₃ˇ ˇf⁰.b/ˇ

ˇ

q_q¹

C

2 3

ˇ ˇf⁰.a/ˇ

ˇ

qC¹₃ˇ ˇf⁰.b/ˇ

ˇ

q¹_q

3 5:

(4.4) Theorem 9. Leta; b2Rwitha < bandf WŒa; b!Rbe a differentiable function on.a; b/. Ifjf⁰j^qis convex onŒa; bforq > 1, then the following fractional midpoint

type inequality holds:

ˇ ˇ ˇ ˇ ˇ ˇ

.C1/

2 .b a/

h

R_aCf .b/CR_b f .a/i f

aCb C1

Cf

aCb C1

2

ˇ ˇ ˇ ˇ ˇ ˇ

(4.5)

b a 2

2 6 6 6 6 6 6 6 6 6 6 6 4

R

C1 0 t^pdt

p¹

²

2.C1/²jf⁰.a/j^qC_2.^2C2

C1/²jf⁰.b/j^q1q

C R1

C1 1 t^p dt_p¹

2C1

2.C1/²jf⁰.a/j^qC_2.¹

C1/²jf⁰.b/j^q1_q C

R

C1 0 t^pdt

¹_p

²

2.C1/²jf⁰.b/j^qC_2.^2C2

C1/²jf⁰.a/j^q1_q CR1

C1 1 tp

dtp¹

2C1

2.C1/²jf⁰.b/j^qC_2.¹

C1/²jf⁰.a/j^q1q

3 7 7 7 7 7 7 7 7 7 7 7 5

with _p¹C¹_q D1and > 0.

Proof. Using Lemma4, Holder inequality and the convexity ofjf⁰j^q, we have ˇ

ˇ ˇ ˇ ˇ ˇ

.C1/

2 .b a/

h

R_aCf .b/CR_b f .a/i f

aCb C1

Cf

aCb C1

2

ˇ ˇ ˇ ˇ ˇ ˇ

b a 2

2 6 6 6 6 6 4

R

C1

0 tjf⁰.t aC.1 t / b/jdt CR1

C1 1 t

jf⁰.t aC.1 t / b/jdt CR

C1

0 tjf⁰.t bC.1 t / a/jdt CR1

C1 1 t

jf⁰.t bC.1 t / a/jdt 3 7 7 7 7 7 5

b a 2

2 6 6 6 6 6 6 6 6 6 6 6 4

R

C1 0 t^pdt

p¹ R

C1

0 jf⁰.t aC.1 t / b/j^qdt ¹q

C R1

C1 1 t^p dt_p¹

R1

C1jf⁰.t aC.1 t / b/j^qdt¹_q C

R

C1 0 t^pdt

p¹ R

C1

0 jf⁰.t bC.1 t / a/j^qdt ¹q

CR1

C1 1 tp

dtp¹R1

C1jf⁰.t bC.1 t / a/j^qdt¹q

3 7 7 7 7 7 7 7 7 7 7 7 5

b a 2

2 6 6 6 6 6 6 6 6 6 6 6 4

R

C1 0 t^pdt

p¹ R

C1 0

tjf⁰.a/j^qC.1 t /jf⁰.b/j^q dt

¹q

CR1

C1 1 t^p

dtp¹ R1 C1

tjf⁰.a/j^qC.1 t /jf⁰.b/j^q dt¹q

C

R

C1 0 t^pdt

_p¹ R

C1 0

tjf⁰.b/j^qC.1 t /jf⁰.a/j^q dt

¹_q

C R1

C1 1 t^p dt_p¹

R1 C1

tjf⁰.b/j^qC.1 t /jf⁰.a/j^q dt¹_q

3 7 7 7 7 7 7 7 7 7 7 7 5

b a 2

2 6 6 6 6 6 6 6 6 6 6 6 4

R

C1 0 t^pdt

¹_p ²

2.C1/²jf⁰.a/j^qC_2.^2C2

C1/²jf⁰.b/j^q1q

C R1

C1 1 t^p dt_p¹

2C1

2.C1/²jf⁰.a/j^qC_2.¹

C1/²jf⁰.b/j^q_q¹ C

R

C1 0 t^pdt

p¹

²

2.C1/²jf⁰.b/j^qC_2.^2C2

C1/²jf⁰.a/j^q1_q CR1

C1 1 tp

dtp¹

2C1

2.C1/²jf⁰.b/j^qC_2.¹

C1/²jf⁰.a/j^qq¹

3 7 7 7 7 7 7 7 7 7 7 7 5 :

This completes the proof.

Corollary 4. In Theorem9, if one takesD1, one has [5, Theorem 2.3].

5. COMPETING INTERESTS

The authors declare that they have no competing interests.

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Authors’ addresses

Mehmet Kunt

Karadeniz Technical University, Department of Mathematics, 61080 Trabzon, Turkey E-mail address:mkunt@ktu.edu.tr

˙Imdat ˙Is¸can

Giresun University, Department of Mathematics, 28200 Giresun, Turkey E-mail address:imdati@yahoo.com;imdat.iscan@giresun.edu.tr

Sercan Turhan

Giresun University, Department of Mathematics, 28200 Giresun, Turkey E-mail address:sercan.turhan@giresun.edu.tr

D ¨unya Karapınar

Karadeniz Technical University, Department of Mathematics, 61080 Trabzon, Turkey E-mail address:dunyakarapinar@ktu.edu.tr