On generalized fractional integral inequalities for twice differentiable convex functions

(1)

Contents lists available atScienceDirect

Journal of Computational and Applied

Mathematics

journal homepage:www.elsevier.com/locate/cam

On generalized fractional integral inequalities for twice

differentiable convex functions

Pshtiwan Othman Mohammed

a,∗

, Mehmet Zeki Sarikaya

b

a_{Department of Mathematics, College of Education, University of Sulaimani, Sulaimani, Kurdistan Region, Iraq} b_{Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey}

a r t i c l e i n f o

Article history:

Received 11 January 2019

Received in revised form 17 September 2019 MSC: 26D07 26D10 26D15 26A33 60E15 Keywords:

Generalized fractional integral Convex function

Integral inequalities Random variable

a b s t r a c t

In this article, some new generalized fractional integral inequalities of midpoint and trapezoid type for twice differentiable convex functions are obtained. In view of this, we obtain new integral inequalities of midpoint and trapezoid type for twice differentiable convex functions in a form classical integral and Riemann–Liouville fractional integrals. Finally, we apply our new inequalities to construct inequalities involving moments of a continuous random variable.

1. Introduction

A function g

:

I

⊆

_R

→

_{R is said to be convex on the interval I, if the inequality}

g(

ζ

x

+

(1

−

ζ

)y)

≤

ζ

g(x)

+

(1

−

ζ

)g(y) (1)

holds for all x

,

y

∈

Iand

ζ ∈ [

0

,

1]. We say that g is concave if

−

g is convex.

For differentiable convex functions, many equalities or inequalities have been established by many authors; for example Hermite–Hadamard type inequality [1], Ostrowski type inequality [2], Hardy type inequality [3], Olsen type inequality [4] and Gagliardo–Nirenberg type inequality [5]. But for twice differentiable convex functions, the most important inequalities are midpoint type inequality [6] and trapezoidal type inequality [7] which are defined by:

⏐

∫

v u g(x)dx

−

(

v −

u)g

(

u

+

v

2

)⏐

⏐

≤

(

v −

u) 3 24



g ′′



_∞

,

(2)

⏐

∫

v u g(x)dx

−

v −

u 2

[

g(u)

+

g(

v

)]

⏐

≤

(

v −

u) 3 12



g ′′



_∞

,

(3)

respectively, where g

: [

u

, v] ⊂

R

→

R is assumed to be twice differentiable function on an open interval (u, v) with the second derivative bounded on (u

, v

). That is,



g′′



∞

:=

supx∈(u,v)

|

g ′′

(x)|

< ∞

. For more details and applications on inequalities(2)and(3), see [6–9].

∗

Corresponding author.

E-mail addresses: pshtiwansangawi@gmail.com(P.O. Mohammed),sarikayamz@gmail.com(M.Z. Sarikaya). https://doi.org/10.1016/j.cam.2020.112740

(2)

Now, we recall some necessary definitions and mathematical preliminaries of the generalized fractional integrals which are defined by Sarikaya and Ertugral [10].

Let

ϑ : [

0

, ∞

)

→ [0

, ∞

) which satisfies the following conditions:

∫

1

0

ϑ

(

ζ

)

ζ

d

ζ < ∞.

We define the following left-sided and right-sided generalized fractional integral operators, respectively, as follows: u+Iϑg(x)

=

∫

x u

ϑ

(x

−

ζ

) x

−

ζ

g(

ζ

)d

ζ,

x

>

u

,

(4) v−I_ϑg(x)

=

∫

v x

ϑ

(

ζ −

x)

ζ −

x g(

ζ

)d

ζ,

x

< v.

(5)

The most important feature of generalized fractional integrals is that they generalize some types of fractional integrals such as Riemann–Liouville fractional integral [1,11,12], k-Riemann–Liouville fractional integral [13], Katugampola fractional integrals [14,15], conformable fractional integral [16], Hadamard fractional integrals [6], and so on. These important special cases of the integral operators(4)and(5)are mentioned below:

(a) if

ϑ

(

ζ

)

=

ζ

, the operators(4)and(5)reduce to the Riemann integral:

Iu+g(x)

=

∫

x u g(

ζ

)d

ζ,

x

>

u

,

I_v−g(x)

=

∫

v x g(

ζ

)d

ζ,

x

< v.

(b) if

ϑ

(

ζ

)

=

_Γζα

(α), the operators(4)and(5)reduce to the Riemann–Liouville fractional integral [6]: Iu+g(x)

=

1 Γ(

α

)

∫

x u (x

−

ζ

)α−1g(

ζ

)d

ζ,

x

>

u

,

I_v−g(x)

=

1 Γ(

α

)

∫

v x (

ζ −

x)α−1g(

ζ

)d

ζ,

x

< v.

(c) if

ϑ

(

ζ

)

=

ζ α k

kΓk(α), the operators(4)and(5)reduce to the k-Riemann–Liouville fractional integral:

Iu+,kg(x)

=

1 kΓk(

α

)

∫

x u (x

−

ζ

)αk−1_g(

ζ

_)d

ζ,

_x

>

_u

,

I_v−_,_kg(x)

=

1 kΓk(

α

)

∫

v x (

ζ −

x)αk−1_g(

ζ

_)d

ζ,

_x

< v,

where Γk(

α

)

=

∫

∞ 0

ζ

α−1_e−ζk k _d

ζ,

R(α₎

>

₀

,

and Γk(

α

)

=

k α k−1Γ

(

α

k

) ,

R(α)

>

0;k

>

0

which are given by Mubeen and Habibullah in [17].

Recently, in [10], Sarikayal and Ertugra established the following generalized fractional integral inequalities of trapezoid type for differentiable convex function:

Theorem 1.1. Let g

: [

u

, v] ⊂

_R

→

R be a differentiable function on (u, v) with u

< v

. If

|

g′

_|

is convex on

[

u

, v]

. Then

⏐

g(u)

+

g(

v

) 2

−

1 2Λ(1)[u+Iϑg(

v

)

+

v−Iϑg(u)]

⏐

≤

v −

u Λ(1)

∫

1 0

ζ|

Λ(1

−

ζ

)

−

Λ(

ζ

)|d

ζ

( |

g ′ (u)| + |g′ (

v

)| 2

)

,

where Λ(

ζ

)

=

∫

ζ 0

ϑ

((

v −

u)z) z dz

< ∞.

(3)

: [

u

, v] ⊂

_R

→

< v

. If

|

g′

|

qis convex on

[

u

, v]

. Then

⏐

g(u)

+

g(

v

) 2

−

1 2Λ(1)[u+Iϑg(

v

)

+

v −I_ϑg(u)]

⏐

≤

v −

u 2Λ(1)

(∫

1 0

|

Λ(1

−

ζ

)

−

Λ(

ζ

)|pd

ζ

)

1 p

_{( |}

_g′ (u)|q

+ |

g′ (

v

)|q 2

)

1q

,

where 1_p

+

1 q

=

1

,

q

>

1.

Furthermore, in [18], Ertugral and Sarikaya established the following generalized fractional integral inequalities of trapezoid type for differentiable convex function:

: [

u

, v] ⊂

_R

→

_{R be an absolutely continuous function on}

[

u

, v]

such that g′

_∈

L1([u

, v]

), where u

, v ∈ [

u

, v]

with u

< v

. If the function

|

g′

_|

is convex on

[

u

, v]

. Then

⏐

∇(0)g(

v

)

+

∆(0)g(u)

v −

u

−

1

v −

u[u+Iϑg(

v

)

+

v−Iϑg(u)]

⏐

≤

v −

x

v −

u

|

g ′ (x)|

∫

1 0

|∇(

ζ

)|

ζ

d

ζ +

x

−

u

v −

u

|

g ′ (x)|

∫

1 0

|

∆(

ζ

)|

ζ

d

ζ

+

v −

x

v −

u

|

g ′ (

v

)|

∫

1 0

|∇(

ζ

)|(1

−

ζ

)d

ζ +

x

−

u

v −

u

|

g ′ (u)|

∫

1 0

|

∆(

ζ

)|(1

−

ζ

)d

ζ,

where ∆(

ζ

)

=

∫

1 ζ

ϑ

((x

−

u)z) z dz

< ∞, ∇

(

ζ

)

=

∫

1 ζ

ϑ

((

v −

x)z) z dz

< ∞.

: [

u

, v] ⊂

R

→

< v

. If the function

|

g′

|

qis convex on

[

u

, v]

. Then

⏐

g(u)

+

g(

v

) 2

−

1 2Λ(1)[u+Iϑg(

v

)

+

v−Iϑg(u)]

⏐

≤

v −

x

v −

u

(∫

1 0

|∇(

ζ

)|pd

ζ

)

1p

_{( |}

_g′ (u)|q

+ |

g′ (

v

)|q 2

)

1q

+

v −

x

v −

u

(∫

1 0

|

∆(

ζ

)|pd

ζ

)

1 p

_{( |}

g′ (u)|q

+ |

g′ (

v

)|q 2

)

1q

,

where 1_p

+

1_q

=

1

,

q

>

1.

Tomar et al. [11] established the following Hermite–Hadamard type inequalities via Riemann–Liouville fractional integrals for twice differentiable functions:

Lemma 1.1. Let g

:

I

⊂

_R

→

_{R be twice differentiable function on the interior I}o_{of an interval I such that g}′′

_∈

L(

[

u

, v]

),

where u

, v ∈

Iwith u

< v

. Then for each h

∈

(0

,

1) and

α >

0 the following equality holds:

Hg(g;

α,

u

, v

)

=

(

v −

u)2 8

×

[

∫

1 0

ζ

α+1_g′′

(

ζ

2u

+

2

−

ζ

2

v

)

d

ζ +

∫

1 0

ζ

α+1_g′′

(

2

−

ζ

2 u

+

ζ

2

v

)

d

ζ

],

(6) where Hg(g;

α,

u

, v

)

=

2α−1_Γ₍

_{α +}

₂₎ (

v −

u)α

[

Iα

(

u+_v 2

)

+g(

v

)

+

Iα

(

u+_v 2

)

−g(u)

]

−

(

α +

1)g

(

u

+

v

2

)

.

Theorem 1.5. With the similar assumptions ofLemma 1.1, if

|

g′′

_|

is convex on

[

u

, v]

, then for each h

∈

(0

,

1). Then

⏐

H_g(g;

α,

u

, v

)

⏐

⏐ ≤

(

v −

u)2 8(

α +

2)

(|

g ′′ (u)| + |g′′(

v

)|

)

.

(7)

(4)

|

g′′

|

qis convex on

[

u

, v]

. Then

⏐

H_g(g;

α,

u

, v

)

⏐

⏐ ≤

(

v −

u)2 8(

α +

2)

(

2 (

α +

1)p

+

1

)

1p

(|

g′′(u)| + |g′′(

v

)|

)

.

(8) where 1_p

+

1 q

=

1

,

q

>

1.

|

g′′

_|

q

is convex on

[

u

, v]

for q

>

1. Then

⏐

Hg(g;

α,

u

, v

)

⏐

⏐ ≤

(

v −

u)2 8

(

1

α +

2

)

1−1_q

[

(

1 2(

α +

3)

|

g ′′ (u)|q

+

(

1

α +

2

−

1 2(

α +

3)

)

|

g′′(

v

)|q

)

1_q

+

((

1

α +

2

−

1 2(

α +

3)

)

|

g′′(u)|q

+

1 2(

α +

3)

|

g ′′ (

v

)|q

)

1_q

]

.

There are many papers which study and give significant results for twice differentiable functions via fractional integrals; for more details see [12,19–23].

In this paper, we established some new midpoint and trapezoid type inequalities for twice differential convex functions via generalized fractional integrals.

2. Midpoint type inequalities for twice differentiable functions with generalized fractional integrals

Here we give midpoint type inequalities for twice differential convex functions via generalized fractional integral operator. To obtain these results, we need the following lemma:

Lemma 2.1. Let g

:

I

⊆

_R

→

_{R be twice differentiable function on I}o_{such that g}′′

_∈

L(

[

u

, v]

), where u

, v ∈

Iowith u

< v

. Then

[

(

u+v 2

)

+I_ϑg(

v

)

+(

u+v 2

)

−I_ϑg(u)

]

−

2Υ(1)g

(

u

+

v

2

)

=

(

v −

u) 2 4

[∫

1 0 Ω(

ζ

)g′′

(

ζ

2u

+

2

−

ζ

2

v

)

d

ζ +

∫

1 0 Ω(

ζ

)g′′

(

2

−

ζ

2 u

+

ζ

2

v

)

d

ζ

]

,

(9) where Ω(

ζ

)

=

∫

ζ 0 Υ(s)ds

,

Υ(s)

=

∫

s 0

ϑ ((

v−u 2

)

z

)

z dz

< ∞.

(10)

Proof. First, we let

¯

h1

+ ¯

h2

:=

∫

1 0 Ω(

ζ

)g′′

(

ζ

2u

+

2

−

ζ

2

v

)

d

ζ +

∫

1 0 Ω(

ζ

)g′′

(

2

−

ζ

2 u

+

ζ

2

v

)

d

ζ .

(11)

On integrating by parts twice one can have

¯

h1

=

∫

1 0 Ω(

ζ

)g′′

(

ζ

2u

+

2

−

ζ

2

v

)

d

ζ

=

2

v −

u

[

−

Ω(

ζ

)g′

(

ζ

2u

+

2

−

ζ

2

v

)⏐

⏐

1 0

+

∫

1 0 Υ(

ζ

)g′

(

ζ

2u

+

2

−

ζ

2

v

)

d

ζ

]

=

2

v −

u

[

−

Ω(1)g′

(

u

+

v

2

)

+

∫

1 0 Υ(

ζ

)g′

(

ζ

2u

+

2

−

ζ

2

v

)

d

ζ

]

=

4 (

v −

u)2

[

−

v −

u 2 Ω(1)g ′

(

u

+

v

2

)

−

Υ(1)g

(

u

+

v

2

)

+

∫

1 0

ϑ ((

v−u 2

)

t

)

ζ

g

(

ζ

2u

+

2

−

ζ

2

v

)

d

ζ

]

.

(5)

By changing the variable x by x

=

ζ₂u

+

2−₂ζ

v

, we get

¯

h1

=

4 (

v −

u)2

[

−

v −

u 2 Ω(1)g ′

(

u

+

v

2

)

−

Υ(1)g

(

u

+

v

2

)

+

∫

v u+_v 2

ϑ

(

v −

x)

v −

x g(x)dx

]

=

4 (

v −

u)2

[

−

v −

u 2 Ω(1)g ′

(

u

+

v

2

)

−

Υ(1)g

(

u

+

v

2

)

+(

u+v 2

)

+I_ϑg(

v

)

]

.

Analogously, we have

¯

h2

=

∫

1 0 Ω(

ζ

)g′′

(

ζ

2u

+

2

−

ζ

2

v

)

d

ζ

=

2

v −

u

[

Ω(1)g′

(

u

+

v

2

)

−

∫

1 0 Υ(

ζ

)g′

(

2

−

ζ

2 u

+

ζ

2

v

)

d

ζ

]

=

4 (

v −

u)2

[ v −

u 2 Ω(1)g ′

(

u

+

v

2

)

−

Υ(1)g

(

u

+

v

2

)

+

∫

1 0

ϑ ((

v−u 2

)

t

)

ζ

g

(

2

−

ζ

2 u

+

ζ

2

v

)

d

ζ

]

=

4 (

v −

u)2

[

v −

u 2 Ω(1)g ′

(

u

+

v

2

)

−

Υ(1)g

(

u

+

v

2

)

+(

u+_v 2

)

−I_ϑg(u)

]

.

Usingh

¯

1andh

¯

2in(11)and multiplying the result by (v−u)

2

4 , we get equality(9). □

Remark 2.1. With the similar assumptions ofLemma 2.1, if

ϑ

(

ζ

)

=

ζ

, then identity(9)reduces to the following identity: 1

v −

u

∫

v u g(x)dx

−

g

(

u

+

v

2

)

=

(

v −

u) 2 16

[∫

1 0

ζ

2_g′′

(

ζ

2u

+

2

−

ζ

2

v

)

d

ζ +

∫

1 0

ζ

2_g′′

(

2

−

ζ

2 u

+

ζ

2

v

)

d

ζ

]

which is proved by Sarikaya and Kiris in [24].

Remark 2.2. With the similar assumptions ofLemma 2.1, if

ϑ

(

ζ

)

=

_Γζ₍α_α₎, then identity(9)reduces to identity(6).

Corollary 2.1. With the similar assumptions ofLemma 2.1, if

ϑ

(

ζ

)

=

ζ

α

k

kΓk(

α

)

, then we get the following identity:

(

v −

u)2 8

[∫

1 0

ζ

αk+1_g′′

(

ζ

2u

+

2

−

ζ

2

v

)

d

ζ +

∫

1 0

ζ

αk+1_g′′

(

2

−

ζ

2 u

+

ζ

2

v

)

d

ζ

]

=

2 α k−1Γ_k₍

α +

_2k) (

v −

u)αk

[

Iα

(

u+v 2

)

+ ,kg(

v

)

+

I α

(

u+v 2

)

− ,kg(u)

]

−

(

α +

k)g

(

u

+

v

2

)

.

Theorem 2.1. Let g

:

I

⊆

_R

→

_{R be twice differentiable function on I}osuch that g′′

_∈

L(

[

u

, v]

), where u

, v ∈

Iowith u

< v

. If the function

|

g′′

_|

is convex on

[

u

, v]

. Then

⏐

[

(

u+_v 2

)

+I_ϑg(

v

)

+(

u+_v 2

)

−I_ϑg(u)

]

−

2Υ(1)g

(

u

+

v

2

)⏐

⏐

≤

(

v −

u) 2 4

(

|

g′′ (u)| + |g′′ (

v

)|

)

∫

1 0

|

Ω(

ζ

)|d

ζ.

(12)

Proof. Using equality(9)and the convexity of

|

g′′

_|

on

[

u

, v]

, we have

⏐

[

(

u+_v 2

)

+I_ϑg(

v

)

+(

u+_v 2

)

−I_ϑg(u)

]

−

2Υ(1)g

(

u

+

v

2

)⏐

⏐

≤

(

v −

u) 2 4

⏐

∫

1 0 Ω(

ζ

)g′′

(

ζ

2u

+

2

−

ζ

2

v

)

d

ζ +

∫

1 0 Ω(

ζ

)g′′

(

2

−

ζ

2 u

+

ζ

2

v

)

d

ζ

⏐

≤

(

v −

u) 2 4

[∫

1 0

|

Ω(

ζ

)|

⏐

g′′

(

ζ

2u

+

2

−

ζ

2

v

)⏐

⏐

d

ζ +

∫

1 0

|

Ω(

ζ

)|

⏐

g′′

(

2

−

ζ

2 u

+

ζ

2

v

)⏐

⏐

d

ζ

]

(6)

≤

(

v −

u) 2 4

[∫

1 0

|

Ω(

ζ

)|

(

ζ

2

|

g ′′ (u)| +2

−

ζ

2

|

g ′′ (

v

)|

)

d

ζ +

∫

1 0

|

Ω(

ζ

)|

(

2

−

ζ

2

|

g ′′ (u)| +

ζ

2

|

g ′′ (

v

)|

)

d

ζ

]

≤

(

v −

u) 2 8

[

|

g′′(u)|

∫

1 0

|

Ω(

ζ

)|t d

ζ + |

g′′(

v

)|

∫

1 0

|

Ω(

ζ

)|(2

−

ζ

)d

ζ

+ |

g′′(u)|

∫

1 0

|

Ω(

ζ

)|(2

−

ζ

)d

ζ + |

g′′(

v

)|

∫

1 0

|

Ω(

ζ

)|t d

ζ

]

=

(

v −

u) 2 4

(

|

g′′(u)| + |g′′(

v

)|

)

∫

1 0

|

Ω(

ζ

)|d

ζ.

Hence the proof is completed. □

Remark 2.3. With the similar assumptions ofTheorem 2.1, if

ϑ

(

ζ

)

=

ζ

, then inequality(12)reduces to the following inequality:

⏐

1

v −

u

∫

v u g(x)dxf

−

(

u

+

v

2

)⏐

⏐

≤

(

v −

u) 2 24

[ |

g′′ (u)| + |g′′ (

v

)| 2

]

which is proved by Sarikaya and Kiris in [24].

ϑ

(

ζ

)

=

ζ

α

Γ(

α

), then inequality(12)reduces to the inequality

(7).

Corollary 2.2. With the similar assumptions ofTheorem 2.1, if

ϑ

(

ζ

)

=

ζ

α k kΓk(

α

) . Then

⏐

2αk−1Γ_k₍

α +

_2k) (

v −

u)αk

[

Iα

(

u+_v 2

)

+ ,kg(

v

)

+

I α

(

u+_v 2

)

− ,kg(u)

]

−

(

α +

k)g

(

u

+

v

2

)

⏐

_⏐

⏐

≤

(

v −

u) 2 8(

α +

2k)

(

|

g′′(u)| + |g′′(

v

)|

).

:

I

⊆

_R

→

_∈

L(

[

u

, v]

), where u

, v ∈

Io_{with u}

_{< v}

_.

If the function

|

g′′

|

qis convex on

[

u

, v]

. Then

⏐

[

(

u+v 2

)

+I_ϑg(

v

)

+(

u+v 2

)

−I_ϑg(u)

]

−

2Υ(1)g

(

u

+

v

2

)⏐

⏐

≤

(

v −

u) 2 4

(∫

1 0

|

Ω(

ζ

)|pd

ζ

)

1p

{

_{( |}

g′′ (u)|q

+

3|g′′ (

v

)|q 4

)

1_q

+

(

3|g′′ (u)|q

+ |

g′′ (

v

)|q 4

)

1_q

}

≤

(

v −

u) 2 22q

(∫

1 0

|

Ω(

ζ

)|pd

ζ

)

1p

₍

|

g′′ (u)| + |g′′ (

v

)|

),

(13) where 1_p

+

1 q

=

1

,

q

>

1.

Proof. Using equality(9), the convexity of

|

g′′

_|

q_on

_[

_u

_{, v]}

_{and Hölder’s inequality, we have}

⏐

[

(

u+_v 2

)

+I_ϑg(

v

)

+(

u+_v 2

)

−I_ϑg(u)

]

−

2Υ(1)g

(

u

+

v

2

)⏐

⏐

≤

(

v −

u) 2 4

{

(∫

1 0

|

Ω(

ζ

)|pd

ζ

)

1p

(∫

1 0

⏐

g′′

(

ζ

2u

+

2

−

ζ

2

v

)⏐

⏐

q d

ζ

)

1q

+

(∫

1 0

|

Ω(

ζ

)|pd

ζ

)

1 p

(∫

1 0

⏐

g′′

(

2

−

ζ

2 u

+

ζ

2

v

)⏐

⏐

q d

ζ

)

1 q

}

≤

(

v −

u) 2 4

(∫

1 0

|

Ω(

ζ

)|pd

ζ

)

1 p

{

(∫

1 0

(

ζ

2

|

g ′′ (u)|q

+

2

−

ζ

2

|

g ′′ (

v

)|q

)

d

ζ

)

1 q

(7)

+

(∫

1 0

(

2

−

ζ

2

|

g ′′ (u)|q

+

ζ

2

|

g ′′ (

v

)|q

)

d

ζ

)

1 q

}

≤

(

v −

u) 2 4

(∫

1 0

|

Ω(

ζ

)|pd

ζ

)

1p

{

_{( |}

g′′(u)|q

+

3|g′′(

v

)|q 4

)

1q

+

(

3|g′′(u)|q

+ |

g′′(

v

)|q 4

)

1q

}

.

Let c1

= |

g ′′ (u)|q

,

c2

=

3|g ′′ (u)|q

,

d1

=

3|g ′′ (

v

)|q

,

d2

= |

g ′′

(

v

)|q, then using the fact that n

∑

i=1

(

ci

+

di

)

m

≤

n

∑

i=1 cm i

+

n

∑

i=1

+

dm i

,

0

≤

m

<

1

,

and 1

+

31q

_≤

_{4, we have}

⏐

[

(

u+_v 2

)

+I_ϑg(

v

)

+(

u+_v 2

)

−I_ϑg(u)

]

−

2Υ(1)g

(

u

+

v

2

)⏐

⏐

≤

(

v −

u) 2 22q

(∫

1 0

|

Ω(

ζ

)|pd

ζ

)

1p

₍

|

g′′ (u)| + |g′′ (

v

)|

).

This completes the proof of(13). □

ϑ

(

ζ

)

=

ζ

, then inequalities(13)reduce to the following inequalities:

⏐

1

v −

u

∫

v u g(x)dxg

(

u

+

v

2

)⏐

⏐

≤

(

v −

u) 2 16(2p

+

1)1p

{

( |

g′′ (u)|q

+

3|g′′ (

v

)|q 4

)

1_q

+

(

3|g′′ (u)|q

+ |

g′′ (

v

)|q 4

)

1_q

}

≤

(

v −

u) 2 22+1q_(2p

₊

₁₎ 1 p

(

|

g′′(u)| + |g′′(

v

)|

)

which are proved by Sarikaya and Kiris in [24].

ϑ

(

ζ

)

=

_Γζα

(α), then inequalities(13)reduce to the inequalities

(8).

Corollary 2.3. With the similar assumptions ofTheorem 2.2, if

ϑ

(

ζ

)

=

ζ

α k kΓk(

α

) . Then

⏐

2αk−1Γ_k₍

α +

_2k) (

v −

u)αk

[

Iα

(

u+_v 2

)

+ ,kg(

v

)

+

I α

(

u+_v 2

)

− ,kg(u)

]

−

(

α +

k)g

(

u

+

v

2

)

⏐

_⏐

⏐

≤

(

v −

u) 2 8

(

k (

α +

k)p

+

k

)

1_p

{

_{( |}

g′′ (u)|q

+

3|g′′ (

v

)|q 4

)

1q

+

(

3|g′′ (u)|q

+ |

g′′ (

v

)|q 4

)

1q

}

≤

(

v −

u) 2 23q

(∫

1 0

|

Ω(

ζ

)|pd

ζ

)

1p

₍

|

g′′(u)| + |g′′(

v

)|

).

3. Trapezoid type inequalities for twice differentiable functions with generalized fractional integrals

Here we give trapezoid type inequalities for twice differential convex functions via generalized fractional integral operator. To obtain these results, we need the following lemma:

Lemma 3.1. Let g

:

I

⊆

_R

→

_∈

L(

[

u

, v]

), where u

, v ∈

Io_{with u}

_{< v}

_. Then g(u)

+

g(

v

) 2

−

1 2Φ(0)

[

(

u+_v 2

)

+I_ϑg(

v

)

+ (

u+_v 2

)

−I_ϑg(u)

]

(14)

=

(

v −

u) 2 8Φ(0)

[∫

1 0 Ω

(ζ)

g′′

(

ζ

2u

+

2

−

ζ

2

v

)

d

ζ +

∫

1 0 Ω

(ζ )

g′′

(

2

−

ζ

2 u

+

ζ

2

v

)

d

ζ

]

.

(8)

where the functionsΩ

(ζ)

andΦ(

ζ

) are defined by Ω

(ζ ) =

∫

ζ 0 Φ(s)ds

,

Φ(s)

=

∫

1 s

ϑ (

v−u 2 z

)

z dz

< ∞.

(15)

Proof. On integrating by parts twice one can have J1

=

∫

1 0 Ω

(ζ )

g′′

(

ζ

2u

+

2

−

ζ

2

v

)

d

ζ

= −

2

v −

uΩ

(ζ)

g ′

(

ζ

2a

+

2

−

ζ

2

v

)⏐

⏐

1 0

+

2

v −

u

∫

1 0 Φ(

ζ

)g′

(

ζ

2u

+

2

−

ζ

2

v

)

d

ζ

= −

2

v −

uΩ

(

1

)

g ′

(

u

+

v

2

)

+

2

v −

u

×

[

−

2

v −

uΦ(

ζ

)g

(

ζ

2u

+

2

−

ζ

2

v

)⏐

⏐

1 0

−

2

v −

u

∫

1 0

ϑ (

v−u 2 t

)

ζ

g

(

ζ

2u

+

2

−

ζ

2

v

)

d

ζ

]

= −

2

v −

uΩ

(

1

)

g ′

(

u

+

v

2

)

+

4Φ(0)

(v −

u

)

2g(

v

)

−

4

(v −

u

)

2

∫

1 0

ϑ (

v−u 2 t

)

ζ

g

(

ζ

2u

+

2

−

ζ

2

v

)

d

ζ.

By changing the variable x by x

=

ζ₂u

+

2−₂ζ

v

, we get

J₁

= −

2

v −

uΩ

(

1

)

g ′

(

u

+

v

2

)

+

4Φ(0)

(v −

u

)

2g(

v

)

−

4

(v −

u

)

2

∫

v u+v 2

ϑ

(

v −

x)

v −

x g(x)dx

= −

2

v −

uΩ

(

1

)

g ′

(

u

+

v

2

)

+

4Φ(0)

(v −

u

)

2g(

v

)

−

4

(v −

u

)

2

(

u+2v

)

+I_ϑg(

v

)

.

Analogously, we have J2

=

∫

1 0 Ω

(ζ )

g′′

(

2

−

ζ

2 u

+

ζ

2

v

)

d

ζ

=

2

v −

uΩ

(

1

)

g ′

(

u

+

v

2

)

+

4Φ(0)

(v −

u

)

2g(

v

)

−

4

(v −

u

)

2

(

u+2v

)

−I_ϑg(u)

.

FromJ1andJ2, we get

(v −

u

)

2 8Φ(0)

(

J1

+

J2

) =

g(u)

+

g(

v

) 2

−

1 2Φ(0)

[

(

u+_v 2

)

+I_ϑg(

v

)

+ (

u+_v 2

)

−I_ϑg(u)

] ,

which completes the proof of(14). □

Corollary 3.1. With the similar assumptions ofLemma 3.1if

ϑ (ζ) = ζ

, we have the following important identity: g(u)

+

g(

v

) 2

−

1

v −

u

∫

v u

(

ζ −

1 2

ζ

2

)

g(

ζ

)d

ζ

=

(

v −

u) 2 8

[∫

1 0

(

ζ −

1 2

ζ

2

)

g′′

(

ζ

2u

+

2

−

ζ

2

v

)

d

ζ +

∫

1 0

(

ζ −

1 2

ζ

2

)

g′′

(

2

−

ζ

2 u

+

ζ

2

v

)

d

ζ

]

.

ϑ (ζ) =

_Γζ₍α_α₎, then for x

=

u+₂v, we get g(u)

+

g(

v

) 2

−

2α−1Γ(

α +

1)

(v −

u

)

α

[

Iα

(

u+v 2

)

+g(

v

)

+

Iα

(

u+v 2

)

−g(u)

]

=

(

v −

u) 2 8

[

∫

1 0

(

ζ −

1

α +

1

ζ

α+1

)

g′′

(

ζ

2u

+

2

−

ζ

2

v

)

d

ζ

+

∫

1 0

(

ζ −

1

α +

1

ζ

α+1

)

g′′

(

2

−

ζ

2 u

+

ζ

2

v

)

d

ζ

]

.

(9)

ϑ (ζ) =

ζ α k kΓk(α), then g(u)

+

g(

v

) 2

−

2αk−1Γ_k₍

α +

_k)

(v −

u

)

αk

[

Iα,k

(

u+_v 2

)

+g(

v

)

+

Iα, k

(

u+_v 2

)

−g(u)

]

=

(

v −

u) 2 8

[

∫

1 0

(

ζ −

k

α +

k

ζ

α k+1

)

g′′

(

ζ

2u

+

2

−

ζ

2

v

)

d

ζ

+

∫

1 0

(

ζ −

k

α +

k

ζ

α k+1

)

g′′

(

2

−

ζ

2 u

+

ζ

2

v

)

d

ζ

]

.

:

I

⊆

_R

→

∈

L(

[

u

, v]

), where u

, v ∈

Io with u

< v

. If the function

|

g′′

|

is convex on

[

u

, v]

, then

⏐

g(u)

+

g(

v

) 2

−

1 2Φ(0)

[

(

u+_v 2

)

+I_ϑg(

v

)

+ (

u+_v 2

)

−I_ϑg(u)

]

⏐

≤

(

v −

u) 2 8Φ(0)

(∫

1 0

|

Ω(

ζ

)|d

ζ

)

(

|

g′′ (u)| + |g′′ (

v

)|

).

(16)

Proof. Using equality(14)and the convexity of

|

g′′

|

on

[

u

, v]

, we have

⏐

g(u)

+

g(

v

) 2

−

1 2Φ(0)

[

(

u+_v 2

)

+I_ϑg(

v

)

+ (

u+_v 2

)

−I_ϑg(u)

]

⏐

≤

(

v −

u) 2 8Φ(0)

⏐

[∫

1 0 Ω

(ζ )

g′′

(

ζ

2u

+

2

−

ζ

2

v

)

d

ζ +

∫

1 0 Ω

(ζ )

g′′

(

2

−

ζ

2 u

+

ζ

2

v

)

d

ζ

]⏐

⏐

≤

(

v −

u) 2 8Φ(0)

[∫

1 0

|

Ω(

ζ

)|

⏐

g′′

(

ζ

2u

+

2

−

ζ

2

v

)⏐

⏐

d

ζ +

∫

1 0

|

Ω(

ζ

)|

⏐

g′′

(

2

−

ζ

2 u

+

ζ

2

v

)⏐

⏐

d

ζ

]

≤

(

v −

u) 2 8Φ(0)

[∫

1 0

|

Ω(

ζ

)|

(

ζ

2

|

g ′′ (u)| +2

−

ζ

2

|

g ′′ (

v

)|

)

d

ζ +

∫

1 0

|

Ω(

ζ

)|

(

2

−

ζ

2

|

g ′′ (u)| +

ζ

2

|

g ′′ (

v

)|

)

d

ζ

]

=

(

v −

u) 2 8Φ(0)

(∫

1 0

|

Ω(

ζ

)|d

ζ

)

(

|

g′′(u)| + |g′′(

v

)|

),

which completes the proof of(16). □

Corollary 3.4. With the similar assumptions ofTheorem 3.1if

ϑ (ζ) = ζ

, we have

⏐

g(u)

+

g(

v

) 2

−

1

v −

u

∫

v u g(

ζ

)d

ζ

⏐

≤

(

v −

u) 2 24

(

|

g′′(u)| + |g′′(

v

)|

).

ϑ (ζ) =

=

u+v

2 , we have

⏐

g(u)

+

g(

v

) 2

−

2α−1_Γ₍

_{α +}

₁₎

(v −

u

)

α

[

Iα

(

u+_v 2

)

+g(

v

)

+

Iα

(

u+_v 2

)

−g(u)

]⏐

⏐

≤

(

v −

u) 2 8

(

α

(

α +

3) 2(

α +

1)(

α +

2)

)

(

|

g′′(u)| + |g′′(

v

)|

).

ϑ (ζ) =

ζ α k kΓk(α), we have

⏐

g(u)

+

g(

v

) 2

−

2αk−1Γ_k₍

α +

_k)

(v −

u

)

αk

[

Iα,k

(

u+_v 2

)

+g(

v

)

+

Iα, k

(

u+_v 2

)

−g(u)

]

⏐

≤

(

v −

u) 2 8

(

α

(

α +

3k) 2(

α +

k)(

α +

2k)

)

(

|

g′′(u)| + |g′′(

v

)|

).

(10)

:

I

⊆

_R

→

∈

L(

[

u

, v]

), where u

, v ∈

Iowith u

< v

. If the function

|

g′′

_|

q is convex on

[

u

, v]

, then

⏐

g(u)

+

g(

v

) 2

−

1 2Φ(0)

[

(

u+_v 2

)

+I_ϑg(

v

)

+ (

u+_v 2

)

−I_ϑg(u)

]

⏐

≤

(

v −

u) 2 8Φ(0)

(∫

1 0

|

Ω(

ζ

)|p_d

_ζ

)

1p

{

_{( |}

g′′(u)|q

+

3|g′′(

v

)|q 4

)

1q

+

(

3|g′′(u)|q

+ |

g′′(

v

)|q 4

)

1q

}

≤

(

v −

u) 2 23qΦ₍₀₎

(∫

1 0

|

Ω(

ζ

)|pd

ζ

)

1 p

₍

|

g′′(u)| + |g′′(

v

)|

),

(17) where 1 p

+

1 q

=

1

,

q

>

1.

Proof. Using equality(14), the convexity of

|

g′′

_|

q

_,

q

>

1 and Hölder’s inequality, we get

⏐

g(u)

+

g(

v

) 2

−

1 2Φ(0)

[

(

u+v 2

)

+I_ϑg(

v

)

+ (

u+v 2

)

−I_ϑg(u)

]

⏐

≤

(

v −

u) 2 8Φ(0)

{

(∫

1 0

|

Ω(

ζ

)|pd

ζ

)

1 p

(∫

1 0

⏐

g′′

(

ζ

2u

+

2

−

ζ

2

v

)⏐

⏐

q d

ζ

)

1 q

+

(∫

1 0

|

Ω(

ζ

)|pd

ζ

)

1p

(∫

1 0

⏐

g′′

(

2

−

ζ

2 u

+

ζ

2

v

)⏐

⏐

q d

ζ

)

1q

}

≤

(

v −

u) 2 8Φ(0)

(∫

1 0

|

Ω(

ζ

)|pd

ζ

)

1 p

{

(∫

1 0

(

ζ

2

|

g ′′ (u)|q

+

2

−

ζ

2

|

g ′′ (

v

)|q

)

d

ζ

)

1 q

+

(∫

1 0

(

2

−

ζ

2

|

g ′′ (u)|q

+

ζ

2

|

g ′′ (

v

)|q

)

d

ζ

)

1 q

}

≤

(

v −

u) 2 8Φ(0)

(∫

1 0

|

Ω(

ζ

)|pd

ζ

)

1p

{

_{( |}

g′′ (u)|q

+

3|g′′ (

v

)|q 4

)

1_q

+

(

3|g′′ (u)|q

+ |

g′′ (

v

)|q 4

)

1_q

}

.

Let c1

= |

g′′(u)|q

,

c2

=

3|g′′(u)|q

,

d1

=

3|g′′(

v

)|q

,

d2

= |

g′′(

v

)|q, then using the fact that

n

∑

i=1

(

ci

+

di

)

m

≤

n

∑

i=1 c_im

+

n

∑

i=1

+

dm_i

,

0

≤

m

<

1

,

and 1

+

31q

_≤

_{4, we get}

⏐

g(u)

+

g(

v

) 2

−

1 2Φ(0)

[

(

u+_v 2

)

+I_ϑg(

v

)

+ (

u+_v 2

)

−I_ϑg(u)

]

⏐

_⏐

⏐

≤

(

v −

u) 2 23qΦ₍₀₎

(∫

1 0

|

Ω(

ζ

)|pd

ζ

)

1 p

₍

|

g′′(u)| + |g′′(

v

)|

).

This ends the proof of(17). □

ϑ (ζ) = ζ

, we get the following inequality:

⏐

g(u)

+

g(

v

) 2

−

1

v −

u

∫

v u

(

ζ −

1 2

ζ

2

)

g(

ζ

)d

ζ

⏐

≤

(

v −

u) 2 8

(

1 p

+

1

−

1 3

(

1 2

)

p

)

1p

{

_{( |}

_g′′ (u)|q

+

3|g′′ (

v

)|q 4

)

1q

+

(

3|g′′ (u)|q

+ |

g′′ (

v

)|q 4

)

1q

}

≤

(

v −

u) 2 23q

(

1 p

+

1

−

1 3

(

1 2

)

p

)

1p

(

|

g′′(u)|q

+ |

g′′(

v

)|q

).

(11)

Corollary 3.8. Under assumption ofTheorem 3.2if

ϑ (ζ) =

=

u+₂v, we get

⏐

g(u)

+

g(

v

) 2

−

2α−1_Γ₍

_{α +}

₁₎

(v −

u

)

α

[

Iα

(

u+_v 2

)

+g(

v

)

+

Iα

(

u+_v 2

)

−g(u)

]⏐

⏐

≤

(

v −

u) 2 8

(

1 p

+

1

−

(

1

α +

1

)

p

(

1 (

α +

1)p

+

1

))

1p

×

{

( |

g′′ (u)|q

+

3|g′′ (

v

)|q 4

)

1q

+

(

3|g′′ (u)|q

+ |

g′′ (

v

)|q 4

)

1q

}

≤

(

v −

u) 2 23q

(

1 p

+

1

−

(

1

α +

1

)

p

(

1 (

α +

1)p

+

1

))

1p

₍

|

g′′(u)|q

+ |

g′′(

v

)|q

).

ϑ (ζ) =

ζ α k kΓk(α), then we get

⏐

g(u)

+

g(

v

) 2

−

2αk−1Γ_k₍

α +

_k)

(v −

u

)

αk

[

Iα,k

(

u+_v 2

)

+g(

v

)

+

Iα, k

(

u+_v 2

)

−g(u)

]

⏐

≤

(

v −

u) 2 8

(

1 p

+

1

−

(

k

α +

k

)

p

(

k (

α +

k)p

+

1

))

1p

×

{

( |

g′′ (u)|q

+

3|g′′ (

v

)|q 4

)

1q

+

(

3|g′′ (u)|q

+ |

g′′ (

v

)|q 4

)

1q

}

≤

(

v −

u) 2 23q

(

1 p

+

1

−

(

k

α +

k

)

p

(

k (

α +

k)p

+

1

))

1p

₍

|

g′′(u)|q

+ |

g′′(

v

)|q

).

4. Application: Moments of the random variable

Distribution functions and density functions provide complete descriptions of the distribution of probability for a given random variable. However, they do not allow us to easily make comparisons between two different distributions. The set of moments that uniquely characterizes the distribution under reasonable conditions are useful in making comparisons. Knowing the probability function, we can determine moments if they exist. Applying the mathematical inequalities, some estimations for the moments of random variables were recently studied by many authors, for more details see [25–32].

Let

χ

be a random variable whose probability function is h

:

I

⊂

_R

→

_R+, where h is a convex function on the interval

of real numbers I such that u

, v ∈

Iwith u

< v

. Denote by Mr(x) the rth moment about any arbitrary point x of the random variable

χ,

r

≥

0, defined as

Mr(x)

=

∫

v

u

(

z

−

x

)

rh(z)dz

,

r

=

0

,

1

,

2

, . . .

(18)

In view of(18), we obtain the following propositions:

Proposition 4.1. Let

χ

:

I

⊂

_R

→

_R+, where h is a convex function on the interval of real numbers I such that u

, v ∈

Iwith u

< v

. Then

[

(

u+_v 2

)

+I_ϑM_r(

v

)

+(

u+_v 2

)

−I_ϑM_r(u)

]

−

2Υ(1)Mr

(

u

+

v

2

)

=

r(r

−

1)(

v −

u) 4

[∫

1 0 Ω(

ζ

)Mr−2

(

ζ

2a

+

2

−

ζ

2

v

)

d

ζ +

∫

1 0 Ω(

ζ

)Mr−2

(

2

−

ζ

2 u

+

ζ

2b

)

d

ζ

]

=

r(r

−

1)(

v −

u) 4

{

∫

1 0 Ω(

ζ

)

(

∫

v u

(

z

−

(

ζ

2a

+

2

−

ζ

2

v

))

r−2 h(z)dz

)

d

ζ

+

∫

1 0 Ω(

ζ

)

(

∫

v u

(

z

−

(

2

−

ζ

2 u

+

ζ

2b

))

r−2 h(z)dz

)

d

ζ

}

(19) for r

≥

2k

,

k

=

1

,

2

, . . .

.

(12)

χ

:

I

⊂

_R

→

, v ∈

Iwith u

< v

. If the function

|

h

|

is bounded, then

⏐

[

(

u+v 2

)

+I_ϑM_r(

v

)

+(

u+v 2

)

−I_ϑM_r(u)

]

−

2Υ(1)Mr

(

u

+

v

2

)⏐

⏐

≤

r

(

1

+

(−1) r

₎

₍

_{v −}

_u)r

_∥

_h

_∥

∞ 2r+1

∫

1 0 Ω(

ζ

)

(

ζ

r−1

−

(

ζ −

2)r−1

)

d

ζ

for r

≥

2k

,

k

=

1

,

2

, . . .

.

Proof. By usingProposition 4.1and since

|

h

|

is a bounded function, we have

⏐

[

(

u+_v 2

)

+I_ϑM_r(

v

)

+(

u+_v 2

)

−I_ϑM_r(u)

]

−

2Υ(1)Mr

(

u

+

v

2

)⏐

⏐

≤

r(r

−

1)(

v −

u) 4

{

∫

1 0 Ω(

ζ

)

(

∫

v u

(

z

−

(

ζ

2a

+

2

−

ζ

2

v

))

r−2

|

h(z)

|

dz

)

d

ζ

+

∫

1 0 Ω(

ζ

)

(

∫

v u

(

z

−

(

2

−

ζ

2 u

+

ζ

2b

))

r−2

|

h(z)

|

dz

)

d

ζ

}

≤

r

(

1

+

(−1) r

₎

₍

_{v −}

_u)r

_∥

_h

_∥

∞ 2r+1

∫

1 0 Ω(

ζ

)

(

ζ

r−1

−

(

ζ −

2)r−1

)

d

ζ .

This completes the proof. □

Corollary 4.1. Under the similar assumptions ofProposition 4.2, if 1.

ϑ (ζ) = ζ

, then

⏐

1

v −

u

∫

v u g(

ζ

)d

ζ −

f

(

u

+

v

2

)⏐

⏐

≤

r

(

1

+

(−1) r

₎

₍

_{v −}

_u)r+1

_∥

_h

_∥

∞ 2r+4

(

1 r

+

2

+

(−1) r₂r+2

_β

1 2 (3

,

r)

)

.

2.

ϑ (ζ) =

=

u+v 2 , we have

⏐

2α−1_Γ₍

_{α +}

₁₎

(v −

u

)

α

[

Iα

(

u+_v 2

)

+g(

v

)

+

Iα

(

u+_v 2

)

−g(u)

]

−

(

α +

1)f

(

u

+

v

2

)⏐

⏐

≤

r

(

1

+

(−1) r

₎

₍

_{v −}

_u)α+r

_∥

_h

_∥

∞ 2α+r+2Γ₍

α +

₂₎

(

1

α +

r

+

1

+

(−1) r₂α+r+1

_β

1 2(

α +

2

,

r)

)

.

χ

:

I

⊂

_R

→

, v ∈

Iwith u

< v

and h

∈

Lp

[

u

, v],

p

>

1. If the function

|

h

|

is bounded, then

⏐

[

(

u+_v 2

)

+I_ϑM_r(

v

)

+(

u+_v 2

)

−I_ϑM_r(u)

]

−

2Υ(1)Mr

(

u

+

v

2

)⏐

⏐

≤

r(r

−

1)

(

1

+

(−1) (r−2)q+1 q

)

∥

h

∥

_p

×

[

(

v −

u)(r−1)q+1 2rq+1_((r

₋

_2)q

₊

₁₎

]

1q

∫

1 0 Ω(

ζ

)

(

ζ

(r−2)q+1

₋

₍

_{ζ −}

₂₎(r−2)q+1

)

1q_d

_ζ

for r

≥

2k

,

k

=

1

,

2

, . . .

and 1_p

+

1 q

=

1.

Proof. Since h

∈

Lp

[

u

, v]

, then by Holder’s integral inequality we have

∫

v u

(

z

−

(

ζ

2a

+

2

−

ζ

2

v

))

r−2 h(z)dz

≤

(∫

v u gp(z)dz

)

1_p

(

∫

v u

(

z

−

(

ζ

2a

+

2

−

ζ

2

v

))

(r−2)q dz

)

1q

= ∥

h

∥

_p

[

(

v −

u)(r−2)q+1

(

ζ

(r−2)q+1

₋

₍

_{ζ −}

₂₎(r−2)q+1

)

2(r−2)q+1_((r

₋

_2)q

₊

₁₎

]

1_q

.

(20)

(13)

Analogously,

∫

v u

(

z

−

(

2

−

ζ

2 u

+

ζ

2b

))

r−2 h(z)dz

≤

(∫

v u gp(z)dz

)

1p

(

∫

v u

(

z

−

(

2

−

ζ

2 u

+

ζ

2b

))

(r−2)q dz

)

1q

= ∥

h

∥

p

[

(−1)(r−2)q+2(

v −

u)(r−2)q+1

(

ζ

(r−2)q+1

−

(

ζ −

2)(r−2)q+1

)

2(r−2)q+1_((r

₋

_2)q

₊

₁₎

]

1_q

.

(21)

Using(20)and(21)in(19), we get the desired result. □

χ

:

I

⊂

_R

→

, v ∈

Iwith u

< v

. Then

Mr(u)

+

Mr(

v

) 2

−

1 2Φ(0)

[

(

u+v 2

)

+I_ϑM_r(

v

)

+(

u+v 2

)

−I_ϑM_r(u)

]

=

r(r

−

1)(

v −

u) 8Φ(0)

[∫

1 0 Ω(

ζ

)Mr−2

(

ζ

2a

+

2

−

ζ

2

v

)

d

ζ −

∫

1 0 Ω(

ζ

)Mr−2

(

2

−

ζ

2 u

+

ζ

2b

)

d

ζ

]

=

r(r

−

1)(

v −

u) 8Φ(0)

{

∫

1 0 Ω(

ζ

)

(

∫

v u

(

z

−

(

ζ

2a

+

2

−

ζ

2

v

))

r−2 h(z)dz

)

d

ζ

−

∫

1 0 Ω(

ζ

)

(

∫

v u

(

z

−

(

2

−

ζ

2 u

+

ζ

2b

))

r−2 h(z)dz

)

d

ζ

}

(22) for r

≥

2k

,

k

=

1

,

2

, . . .

.

Proof. Proof of this proposition follows from setting g(x)

=

Mr(x) in the equality(14). □

χ

:

I

⊂

_R

→

, v ∈

Iwith u

< v

. If the function

|

h

|

is bounded, then

⏐

Mr(u)

+

Mr(

v

) 2

−

1 2Φ(0)

[

(

u+v 2

)

+I_ϑM_r(

v

)

+(

u+v 2

)

−I_ϑM_r(u)

]

⏐

_⏐

⏐

≤

r

(

1

+

(−1) r

₎

₍

_{v −}

_u)r

_∥

_h

_∥

∞ 2r+2Φ₍₀₎

∫

1 0 Ω(

ζ

)

(

ζ

r−1

−

(

ζ −

2)r−1

)

d

ζ

for r

≥

2k

,

k

=

1

,

2

, . . .

.

Proof. By usingProposition 4.4and since

|

h

|

is a bounded function, we have

⏐

Mr(u)

+

Mr(

v

) 2

−

1 2Φ(0)

[

(

u+_v 2

)

+I_ϑM_r(

v

)

+(

u+_v 2

)

−I_ϑM_r(u)

]

⏐

≤

r(r

−

1)(

v −

u) 8Φ(0)

{

∫

1 0 Ω(

ζ

)

(

∫

v u

(

z

−

(

ζ

2a

+

2

−

ζ

2

v

))

r−2

|

h(z)

|

dz

)

d

ζ

+

∫

1 0 Ω(

ζ

)

(

∫

v u

(

z

−

(

2

−

ζ

2 u

+

ζ

2b

))

r−2

|

h(z)

|

dz

)

d

ζ

}

≤

r

(

1

+

(−1) r

₎

₍

_{v −}

_u)r

_∥

_h

_∥

∞ 2r+2_Φ₍₀₎

∫

1 0 Ω(

ζ

)

(

ζ

r−1

₋

₍

_{ζ −}

₂₎r−1

)

d

ζ.