International Journal of Analysis and Applications

Volume 17, Number 3 (2019), 464-478

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-17-2019-464

FRACTIONAL EXPONENTIALLY m-CONVEX FUNCTIONS AND INEQUALITIES

SAIMA RASHID1,2,∗, MUHAMMAD ASLAM NOOR2 AND KHALIDA INYAT NOOR2

1Government College University, Faisalabad, Pakistan

2COMSATS University Islamabad, Islamabad, Pakistan

∗Corresponding author: saimarashid@gcuf.edu.pk

Abstract. In this article, we introduce a new class of convex functions involving m ∈ [0, 1], which is called

exponentially m-convex function. Some new Hermite-Hadamard inequalities for exponentially m-convex

functions via Reimann-Liouville fractional integral are deduced. Several special cases are discussed. Results

proved in this paper may stimulate further research in different areas of pure and applied sciences.

1. Introduction

Convex functions and their variant forms are being used to study a wide class of problems which arises in

various branches of pure and applied sciences. This theory provides us a natural, unified and general frame-

work to study a wide class of unrelated problems. For recent applications, generalizations and other aspects

of convex functions and their variant forms, see [3–13, 15, 18–21, 24–27, 29–31] and the references therein.

An important class of convex functions, which is called exponential convex functions, was introduced and

studied by Antczak [2], Dragomir et al [10] and Noor et al [19]. Alirezai and Mathar [1] have investi-

gated their basic properties along with their potential applications in statistics and information theory,

Received 2019-02-19; accepted 2019-03-25; published 2019-05-01.

2000 Mathematics Subject Classification. 26D15, 26D10, 90C23.

Key words and phrases. convex function; exponential convex function; Reimann-Liouville Fractional integral inequalit;

Holder’s inequality and power-mean inequality.

c©2019 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

464

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-464


Int. J. Anal. Appl. 17 (3) (2019) 465

see [1, 2, 14]. Awan et al [3] and Pecaric and Jaksetic [26] defined another kind of exponential convex func-

tions and have shown that the class of exponential convex functions unifies various unrelated concepts. It

has been shown [17]that the minimum of the differentiable exponentially convex functions on the convex sets

can be characterized by an inequality, which is called the exponentially variational inequality. Exponentially

variational inequalities can be viewed a natural generalization of the variational inequalities, see [32]. For

the applications and numerical methods of variational inequalities, see Noor [16].

Toader [31] defined the m-convexity, an intermediate between the usual convexity and star shaped prop-

erty. If m = 0, we have the concept of star shaped functions on [a,b].

We would like to emphasize that exponentially convex functions and m-convex functions are two distinct

classes of convex functions. It is natural to introduce a new class of convex functions, which unifies these

concepts. Motivated by these facts, we introduce a new class of convex functions, which is called exponen-

tially m-convex functions.

The advantages of fractional calculus have been described and pointed out in the last few decades by many

authors. Fractional calculus is based on derivatives and integrals of fractional order, fractional differential

equations and methods of their solution. The most celebrated inequality has been studied extensively since

it is established, is the Hermite-Hadamard inequality not only established for classical integrals but also for

fractional integrals, see [18, 20, 27, 29].

In this paper, we obtain some new Hermite-Hadamard type inequality for exponentially m-convex func-

tions via Riemann-Liouville fractional integrals. Some special cases are also discussed which can be obtained

from results. The ideas and techniques of this paper may motivate further research in this field.

2. Preliminaries

First of all, we recall the following basic concepts.

Definition 2.1. [9, 31]. A set K ⊂ R is said to be a m-convex set with respect to a fixed constant m ∈ [0, 1],

if

(1 − t)a + mtb ∈ K, ∀a,b ∈ K,t ∈ [0, 1].

The m-convex set contains the line segment between points a and mb for every pair of points a and b of K.

Toader [31] defined the notion of m-convex functions as follows.



Int. J. Anal. Appl. 17 (3) (2019) 466

Definition 2.2. [31]. A function f : K ⊂ R → R is said to be a m-convex function, where m ∈ [0, 1], if

f((1 − t)a + mtb) ≤ (1 − t)f(a) + mtf(b), ∀a,b ∈ K,t ∈ [0, 1].

Remark 2.1. Clearly, a 1-convex function is a convex function in the ordinary sense. The 0-convex function

are the starshaped functions. If we take m = 1, then we recapture the concept of convex functions and if we

take t = 1, then

f(mb) ≤ mf(b) ∀b ∈ K.

This shows that the function f is sub-homogenous.

We now consider class of exponentially convex function, which are mainly due to Antczak [2], Dragomir [8]

and Noor et al [20], respectively.

Definition 2.3. [2, 8, 20]. A function f : K ⊂ R −→ R is said to be exponentially convex function,if

ef((1−t)a+tb) ≤ [(1 − t)ef(a) + tef(b)], a,b ∈ K, t ∈ [0, 1], (2.1)

where f is positive.

For the applications of exponentially convex functions in different field of statistics, information theory

and mathematical sciences, see [1–3, 17] and the references therein.

We would like to point out that u ∈ K is the minimum of the differentiable exponentially convex function

f, if and only if, u ∈ K satisfies the inequality

〈f′(u)ef(u),v −u〉≥ 0, ∀v ∈ K,

which is called the exponentially variational inequalities. For the more details, see Noor and Noor [17].

We now introduce a new concept of exponentially m-convex function.

Definition 2.4. Let f : K ⊂ R −→ R is said to be an exponentially m-convex function, where m ∈ (0, 1], if

ef((1−t)a+mtb) ≤ [(1 − t)ef(a) + mtef(b)], a,b ∈ K, t ∈ [0, 1]. (2.2)

For t = 1
2
, we have

ef(
a+mb

2
) ≤

ef(a) + mef(b)

2
, ∀a,b ∈ K. (2.3)

The function f is called exponentially Jensen m-convex function.

We now give the Definition of the fractional integral, which is mainly due to [27].



Int. J. Anal. Appl. 17 (3) (2019) 467

Definition 2.5. [27]. Let α > 0 with n− 1 < α ≤ n, n ∈ N, and 1 < x < v. The left- and right-hand side

Riemann-Liouville fractional integrals of order α of function f are given by

Jαu+f(x) =
1

Γ(α)

x∫
u

(x− t)α−1f(t)dt,

and

Jαv−f(x) =
1

Γ(α)

v∫
x

(t−x)α−1f(t)dt,

where Γ(α) is the gamma function.

We also made the convention J0
u+

= J0
v−

= f(x).

We recall the special functions which are known as Gamma function,

Γ(x) =

∞∫
0

e−ttx−1dt.

For appropriate and suitable choice of m, one can obtain several new and known classes of exponentially

convex functions as special cases. This shows that the concept of exponentially m-convex function is quite

general and unifying one.

3. Main Results

“In this section, we obtain Hermite-Hadamard type inequalities for exponentially m-convex function via

Reimann-Liouville fractional integral.

Throughout this section, let I = [a,mb] be an interval in real line. From now onward, we take I = [a,mb],

unless otherwise specified.”

Theorem 3.1. Let f : I ⊂ R → R be an exponentially convex function, where m ∈ (0, 1]. If f ∈ L[a,mb],

then “

ef(
a+mb

2
) ≤

Γ(α + 1)

2(mb−a)α

{
Jα(a)+e

f(mb) + mα+1Jα(b)−e
f( a

m
)

}
≤
α[ef(a) + m2ef(

b
m

)] + [mef(b) + mef(
a
m

)]

α(α + 1)
. (3.1)

”



Int. J. Anal. Appl. 17 (3) (2019) 468

Proof. “Let f be an exponentially m-convex function, from the inequality (2.2). Then, we have” “

ef(
x+my

2
) ≤

ef(x) + mef(y)

2
, ∀x,y ∈ [a,mb].

” “Substituting x = at + m(1 − t)b and y = (1 − t) a
m

+ mt b
m

for t ∈ [0, 1]. Then” “

2ef(
a+mb

2
) ≤ [ef(at+m(1−t)b) + mef((1−t)

a
m

+mt b
m

)].

” “Multiplying both sides of the above inequality with tα−1, and integrating over [0, 1], we have” “

2

α
ef(

a+mb
2

) ≤
1∫

0

tα−1[ef(at+m(1−t)b) + mef((1−t)
a
m

+mt b
m

)]dt

=
1

(mb−a)α

{ mb∫
a

(mb−u)α−1ef(u)du + mα+1
b∫

a
m

(v −
a

m
)α−1ef(v)dv

}

=
Γ(α)

(mb−a)α

{
Jα(a)+e

f(mb) + mα+1Jα(b)−e
f( a

m
)

}
,

” from which, one has “

ef(
a+mb

2
) ≤

Γ(α + 1)

2(mb−a)α

{
Jα(a)+e

f(mb) + mα+1Jα(b)−e
f( a

m
)

}
. (3.2)

” On the other hand exponentially m-convexity of f gives “

ef(at+m(1−t)b) + mef((1−t)
a
m

+mt b
m

) ≤ t[ef(a) + m2ef(
b
m

) + (1 − t)[mef(b) + mef(
a
m

)].

”. Multiplying both sides of the above inequality with tα−1, and integrating over [0, 1], we have “

1∫
0

tα−1ef(at+m(1−t)b)dt + m

1∫
0

tα−1ef((1−t)
a
m

+mt b
m

)dt

≤
1∫

0

tα[ef(a) + m2ef(
b
m

)]dt +

1∫
0

(tα−1 − tα)[mef(b) + mef(
a
m

)]dt.

”

“

Γ(α)

(mb−a)α

{
Jα(a)+e

f(mb) + mα+1Jα(b)−e
f( a

m
)

}
≤
ef(a) + m2ef(

b
m

)

α + 1
+
mef(b) + mef(

a
m

)

α(α + 1)
,

” from which one has “

Γ(α + 1)

2(mb−a)α

{
Jα(a)+e

f(mb) + mα+1Jα(b)−e
f( a

m
)

}
≤
α[ef(a) + m2ef(

b
m

)] + [mef(b) + mef(
a
m

)]

α(α + 1)
. (3.3)

” “Combining inequality (3.2) and inequality (3.3), we get (3.4).” �



Int. J. Anal. Appl. 17 (3) (2019) 469

“

Corollary 3.1. If we choose m = 1 in Theorem 3.1, then we have a new result

ef(
a+b
2

) ≤
Γ(α + 1)

2(b−a)α

{
Jα(a)+e

f(b) + Jα(b)−e
f(a)

}
≤

[ef(a) + ef(b)]

α
.

” “

Corollary 3.2. If we choose m = 1 and α = 1 in Theorem 3.1, then we have a new result

2ef(
a+b
2

) ≤
1

b−a

b∫
a

ef(x)dx ≤ 2[ef(a) + ef(b)].

”

“

Theorem 3.2. Let f,g : I ⊂ R → R be an exponentially m-convex function, where m ∈ (0, 1]. If fg ∈

L[a,mb], then

Γ(α + 1)

(mb−a)α

{
Jαmb−e

f(a) + Jαa+e
g(mb)

}
≤
ef(a) + meg(b) + α(mef(b) + eg(a))

α(α + 1)
.

”

Proof. Let f,g : I ⊂ R → R be an exponentially m-convex function. Then “

ef(a(1−t)+mtb) ≤ (1 − t)ef(a) + tmef(b), a,b ∈ [a,mb], t ∈ [0, 1],

eg(at+m(1−t)b) ≤ teg(a) + m(1 − t)eg(b), a,b ∈ [a,mb], t ∈ [0, 1].

” Adding both sides of the above inequalities, we have “

ef(a(1−t)+mtb) + eg(at+m(1−t)b) ≤ (1 − t)[ef(a) + meg(b)] + t[mef(b) + eg(a)].

” “Multiplying both sides of the above inequality with tα−1, and integrating over [0, 1], we have” “

1∫
0

tα−1[ef(a(1−t)+mtb) + eg(at+m(1−t)b)]dt

≤
1∫

0

(tα−1 − tα)[ef(a) + meg(b)]dt +
1∫

0

tα[mef(b) + eg(a)]dt.

” “

Γ(α)

(mb−a)α

{ mb∫
a

(u−a)α−1ef(u)du +
mb∫
a

(mb−v)α−1eg(v)dv
}

≤
ef(a) + meg(b)

α(α + 1)
+
α(mef(b) + eg(a))

α(α + 1)
,



Int. J. Anal. Appl. 17 (3) (2019) 470

” from which, we have “

Γ(α + 1)

(mb−a)α

{
Jαmb−e

f(a) + Jαa+e
g(mb)

}
≤
ef(a) + meg(b) + α(mef(b) + eg(a))

α(α + 1)
,

” which is the required result. �

“

Corollary 3.3. If we choose m = 1 in Theorem 3.2, then we have a new result

Γ(α + 1)

(b−a)α

{
Jαb−e

f(a) + Jαa+e
g(b)

}
≤
ef(a) + eg(b) + α(ef(b) + eg(a))

α(α + 1)
.

” “

Corollary 3.4. If we choose m = 1 and α = 1 in Theorem 3.2, then we have a new result

1

b−a

b∫
a

[eg(x) + ef(x)]dx ≤
ef(a) + ef(b) + eg(a) + eg(b)

2
.

”

“We now derive Hermite-Hadamard type inequalities for m-convex functions using Reimann-Liouville

fractional integral.”

Theorem 3.3. “Let α > 0 and f : I ⊂ R → R be an exponentially convex function, where m ∈ (0, 1]. If

f ∈ L[a,mb], then” “

ef(
a+mb

2
) ≤

2α−1Γ(α + 1)

(mb−a)α
[
Jα
( a+mb

2
)+
ef(mb) + mα+1Jα

( a+mb
2m

)−
ef(

a
m

)
]

≤
α

4(α + 1)

[
ef(a) −m2ef(

a
m2

)
]

+
m

2

[
ef(b) + me

f( a
m2

)
]
.

”

Proof. “Let f be an exponentially m-convex function. Then, from the inequality (2.3), we have” “

ef(
x+my

2
) ≤

ef(x) + mef(y)

2
, x,y ∈ I.

” “Substituting x = t
2
a + m2−t

2
b, y = 2−t

2m
a + t

2
b for t ∈ [0, 1]. Then” “

2ef(
a+mb

2
) ≤ ef(

t
2
a+m 2−t

2
b) + mef(

2−t
2m

a+ t
2
b).



Int. J. Anal. Appl. 17 (3) (2019) 471

” “Multiplying both sides of the above inequality with tα−1, and integrating over [0, 1], we have” “

2

α
ef(

a+mb
2

) ≤
1∫

0

tα−1ef(
t
2
a+m 2−t

2
b)dt + m

1∫
0

tα−1ef(
2−t
2m

a+ t
2
b)dt

=
2

a−mb

a+mb
2∫

mb

(
2(mb−u)
mb−a

)α−1
ef(u)du +

2m2

mb−a

a+mb
2m∫
a
m

(
2(v − a

m
)

b− a
m

)α−1
ef(v)dv

=
2αΓ(α)

(mb−a)α
[
Jα
( a+mb

2
)+
ef(mb) + mα+1Jα

( a+mb
2m

)−
ef(

a
m

)
]
.

” Thus “

ef(
a+mb

2
) ≤

2α−1Γ(α + 1)

(mb−a)α
[
Jα
( a+mb

2
)+
ef(mb) + mα+1Jα

( a+mb
2m

)−
ef(

a
m

)
]

=
α

2

1∫
0

tα−1
{
ef(

t
2
a+m 2−t

2
b) + mef(

2−t
2m

a+ t
2
b)

}
dt

≤
α

2

1∫
0

tα−1
{[ t

2
ef(a) +

m(2 − t)
2

ef(b)
]

+ m
[
m

2 − t
2

e
f( a

m2
)

+
t

2
ef(b)

]}
dt

=
α

4

[
ef(a) −m2ef(

a
m2

)
] 1∫
0

tαdt +
mα

2

[
ef(b) + me

f( a
m2

)

1∫
0

tα−1dt
]

=
α

4(α + 1)

[
ef(a) −m2ef(

a
m2

)
]

+
m

2

[
ef(b) + me

f( a
m2

)
]
,

” which is the required result. �

Lemma 3.1. “ Let α > 0 and f : I ⊂ R → R be a differentiable exponentially m-convex function on the

interior I◦ of I, where m ∈ (0, 1]. If |f′| ∈ L[[a,mb], is a m-convex function, then” “

2α−1Γ(α + 1)

(mb−a)α
[
Jα
( a+mb

2
)+
ef(mb) + mα+1Jα

( a+mb
2m

)−
ef(

a
m

)
]
−

1

2

[
ef(

a+mb
2

) + mef((
a+mb
2m

)

]

=
mb−a

4

[ 1∫
0

tαef(
t
2
a+m 2−t

2
b)f′(

t

2
a + m

2 − t
2

b)dt

−
1∫

0

tαef(
2−t
2m

a+ t
2
b)f′(

2 − t
2m

a +
t

2
b)dt

]
. (3.4)

”



Int. J. Anal. Appl. 17 (3) (2019) 472

Proof. It suffices to note that “

mb−a
4

[ 1∫
0

tαef(
t
2
a+m 2−t

2
b)f′(

t

2
a + m

2 − t
2

b)dt

]

=
mb−a

4

[
2

mb−a
ef(

a+mb
2

) −
2α

(a−mb)

a+mb
2∫

mb

(
2(mb−x)
mb−a

)α−1
2ef(x)dx

a−mb

]

=
mb−a

4

[
−

2

mb−a
ef(

a+mb
2

) +
2α+1Γ(α + 1)

(mb−a)α+1
Jα
( a+mb

2
)−
ef(mb)

]
. (3.5)

” Similarly, one can have “

−
mb−a

4

[ 1∫
0

tαef(
2−t
2m

a+ t
2
b)f′(

2 − t
2m

a +
t

2
b)dt

]

= −
mb−a

4

[
2m

mb−a
ef(

a+mb
2m

) −
2α+1Γ(α + 1)

(mb−a)α+1
Jα
( a+mb

2m
)+
ef(

a
m

)

]
. (3.6)

” Adding (3.5) and (3.6), gives (3.4). �

Theorem 3.4. “Let α > 0 and f : I ⊂ R → R be a differentiable exponentially m-convex function on the

interior I◦ of I, where m ∈ (0, 1]. If |f′| ∈ L[[a,mb], is a m-convex function, then” “

∣∣∣∣2α−1Γ(α + 1)(mb−a)α [Jα( a+mb2 )+ef(mb) + mα+1Jα( a+mb2m )−ef( am )]
−

1

2

[
ef(

a+mb
2

) + mef((
a+mb
2m

)

]∣∣∣∣
≤
mb−a

4

{
1

4(α + 3)

{∣∣ef(a)f′(a)∣∣ + ∣∣ef(b)f′(b)∣∣} + m2(α2 + 7α + 14)
4(α + 1)(α + 2)(α + 3){∣∣ef(b)f′(b)∣∣ + ∣∣ef( am2 )f′( a

m2
)
∣∣} + m(α + 4)

((α + 2)(α + 3))

{
∆1(a,b) + ∆2(

a

m2
,b)
}}

,

” where “

∆1(a,b) =
∣∣ef(a)f′(b)∣∣ + ∣∣ef(b)f′(a)∣∣,

and

∆2(
a

m2
,b) =

∣∣ef( am2 )f′(b)∣∣ + ∣∣ef(b)f′( a
m2

)
∣∣.

”



Int. J. Anal. Appl. 17 (3) (2019) 473

Proof. “Using Lemma 3.1and exponentially m-convexity function of f, we have” “∣∣∣∣2α−1Γ(α + 1)(mb−a)α [Jα( a+mb2 )+ef(mb) + mα+1Jα( a+mb2m )−ef( am )]
−

1

2

[
ef(

a+mb
2

) + mef((
a+mb
2m

)

]∣∣∣∣
≤
mb−a

4

[ 1∫
0

tα
{∣∣∣∣ef( t2 a+m 2−t2 b)f′( t2a + m2 − t2 b)

∣∣∣∣
+

∣∣∣∣ef( 2−t2m a+ t2 b)f′( 2 − t2m a + t2b)
∣∣∣∣
}
dt

]
. (3.7)

” Using m-convexity of f, we have “∣∣∣∣∣ef( t2 a+m 2−t2 b)f′( t2a + m2 − t2 b)
∣∣∣∣∣ +
∣∣∣∣∣ef( 2−t2m a+ t2 b)f′( 2 − t2m a + t2b)

∣∣∣∣∣
≤
{[ t

2
|ef(a)| +

m(2 − t)
2

|ef(b)|
][ t

2
|f′(a)| +

m(2 − t)
2

|f′(b)|
]}

{[
m

2 − t
2
|ef(

a
m2

)| +
t

2
|ef(b)|

][m(2 − t)
2

|f′
a

m2
| +

t

2
|f′(b)|

]}
=
t2

4

∣∣ef(a)f′(a)∣∣ + m2(2 − t)2
4

∣∣ef(b)f′(b)∣∣ + m(2 − t)t
4

[∣∣ef(a)f′(b)∣∣ + ∣∣ef(b)f′(a)∣∣]
+
m2(2 − t)2

4

∣∣ef( am2 )f′( a
m2

)
∣∣ + t2

4

∣∣ef(b)f′(b)∣∣ + m(2 − t)t
4

[∣∣ef( am2 )f′(b)∣∣
+
∣∣ef(b)f′( a

m2
)
∣∣]

=
t2

4

{∣∣ef(a)f′(a)∣∣ + ∣∣ef(b)f′(b)∣∣} + m2(2 − t)2
4

{∣∣ef(b)f′(b)∣∣ + ∣∣ef( am2 )f′( a
m2

)
∣∣}

+
m(2 − t)t

4

{
∆1(a,b) + ∆2(

a

m2
,b)
}
. (3.8)

” Thus “ ∣∣∣∣2α−1Γ(α + 1)(mb−a)α [Jα( a+mb2 )+ef(mb) + mα+1Jα( a+mb2m )−ef( am )]
−

1

2

[
ef(

a+mb
2

) + mef((
a+mb
2m

)

]∣∣∣∣
≤
mb−a

4

[ 1∫
0

tα
{
t2

4

{∣∣ef(a)f′(a)∣∣ + ∣∣ef(b)f′(b)∣∣} + m2(2 − t)2
4

{∣∣ef(b)f′(b)∣∣
+
∣∣ef( am2 )f′( a

m2
)
∣∣} + m(2 − t)t

4

{
∆1(a,b) + ∆2(

a

m2
,b)
}}

dt

]
=
mb−a

4

{
1

4(α + 3)

{∣∣ef(a)f′(a)∣∣ + ∣∣ef(b)f′(b)∣∣} + m2(α2 + 7α + 14)
4(α + 1)(α + 2)(α + 3){∣∣ef(b)f′(b)∣∣ + ∣∣ef( am2 )f′( a

m2
)
∣∣} + m(α + 4)

((α + 2)(α + 3))

{
∆1(a,b) + ∆2(

a

m2
,b)
}}

,

” which is the required result. �



Int. J. Anal. Appl. 17 (3) (2019) 474

Corollary 3.5. [21]. “If we choose m = 1 and α = 1, in Theorem 3.4, then” “

∣∣∣∣ef( a+b2 ) − 1b−a
b∫
a

ef(x)dx

∣∣∣∣
≤
b−a

4

[
7{
∣∣ef(a)f′(a)∣∣ + ∣∣ef(b)f′(b)∣∣} + 10[∆1(a,b) + ∆2(a,b)]

24

]
.

”

Theorem 3.5. “Let f : I ⊂ R → R be differentiable exponentially m-convex function on the interior I◦ of

I, where m ∈ (0, 1]. If |f′| ∈ L[[a,mb], is a m-convex function on I and p−1 + q−1 = 1, where q > 1, then

we have” “

∣∣∣∣2α−1Γ(α + 1)(mb−a)α [Jα( a+mb2 )+ef(mb) + mα+1Jα( a+mb2m )−ef( am )]
−

1

2

[
ef(

a+mb
2

) + mef((
a+mb
2m

)

]∣∣∣∣
≤

mb−a
4(α + 1)

1
p

[{
1

4(α + 3)
|ef(a)f′(a)|q +

m2(α2 + 7α + 14)

4(α + 1)(α + 2)(α + 3)
|ef(b)f′(b)|q

+
m(α + 4)

(α + 2)(α + 3)
∆3(a,b)

}1
q

+

{
m2(α2 + 7α + 14)

4(α + 1)(α + 2)(α + 3)
|ef(

a
m2

)
f′(

a

m2
)|q

+
1

4(α + 3)
|ef(b)f′(b)|q +

m(α + 4)

(α + 2)(α + 3)
∆4(

a

m2
,b)

}1
q
]
,

” “where

∆3(a,b) = |ef(a)f′(a)|q + |ef(b)f′(b)|q

and

∆4(
a

m2
,b) = |ef(

a
m2

)
f′(

a

m2
)|q + |ef(b)f′(b)|q.

”

Proof. Using Lemma 3.1 and the power mean inequality, we have “

∣∣∣∣2α−1Γ(α + 1)(mb−a)α [Jα( a+mb2 )+ef(mb) + mα+1Jα( a+mb2m )−ef( am )]
−

1

2

[
ef(

a+mb
2

) + mef((
a+mb
2m

)

]∣∣∣∣



Int. J. Anal. Appl. 17 (3) (2019) 475

” “

≤
mb−a

4

[ 1∫
0

tα
{∣∣∣∣ef( t2 a+m 2−t2 b)f′( t2a + m2 − t2 b)

∣∣∣∣
+

∣∣∣∣ef( 2−t2m a+ t2 b)f′( 2 − t2m a + t2b)
∣∣∣∣
}
dt

]

=
mb−a

4

(
1

(α + 1)

)1
p
[{ 1∫

0

tα
{∣∣∣∣ef( t2 a+m 2−t2 b)f′( t2a + m2 − t2 b)

∣∣∣∣qdt
}1

q

+

{ 1∫
0

tα
∣∣∣∣ef( 2−t2m a+ t2 b)f′( 2 − t2m a + t2b)

∣∣∣∣qdt
}1

q
]

=
mb−a

4

(
1

(α + 1)

)1
p
[{ 1∫

0

tα
[t2

4
|ef(a)f′(a)|q +

m2(2 − t)2

4
|ef(b)f′(b)|q

+
mt(2 − t)

4
∆3(a,b)

]}1q
+

{ 1∫
0

tα
[t2

4
|ef(b)f′(b)|q +

m2(2 − t)2

4
|ef(

a
m2

)
f′(

a

m2
)|q

+
mt(2 − t)

4
∆4(

a

m2
,b)
]}1q ]

=
mb−a

4(α + 1)
1
p

[{
1

4(α + 3)
|ef(a)f′(a)|q +

m2(α2 + 7α + 14)

4(α + 1)(α + 2)(α + 3)
|ef(b)f′(b)|q

+
m(α + 4)

(α + 2)(α + 3)
∆3(a,b)

}1
q

+

{
m2(α2 + 7α + 14)

4(α + 1)(α + 2)(α + 3)
|ef(

a
m2

)
f′(

a

m2
)|q

+
1

4(α + 3)
|ef(b)f′(b)|q +

m(α + 4)

(α + 2)(α + 3)
∆4(

a

m2
,b)

}1
q
]
,

” which completes the proof. �

Corollary 3.6. [21]. If we choose m = 1 and α = 1, in Theorem 3.5, then

∣∣∣∣ef( a+b2 ) − 1b−a
b∫
a

ef(x)dx

∣∣∣∣
≤

b−a
4(2)

1
p

[{
3|ef(a)f′(a)|q + 11|ef(b)f′(b)|q + 20∆3(a,b)

48

}1
q

+

{
11|ef(a)f′(a)|q + 3|ef(b)f′(b)|q + 20∆4(a,b)

48

}1
q
]
,

Theorem 3.6. Let f : I ⊂ R → R be differentiable exponentially m-convex function on the interior I◦ of I,

where m ∈ (0, 1]. If |f′| ∈ L[[a,mb], is a m-convex function on I and p−1 + q−1 = 1, where q ≥ 1, then we



Int. J. Anal. Appl. 17 (3) (2019) 476

have “ ∣∣∣∣2α−1Γ(α + 1)(mb−a)α [Jα( a+mb2 )+ef(mb) + mα+1Jα( a+mb2m )−ef( am )]
−

1

2

[
ef(

a+mb
2

) + mef((
a+mb
2m

)

]∣∣∣∣
≤

mb−a
4(pα + 1)

1
p

[{
|ef(a)f′(a)|q + 7m2|ef(b)f′(b)|q + 2m∆3(a,b)

12

}1
q

+

{
7m2|ef(

a
m2

)
f′( a

m2
)|q + |ef(b)f′(b)|q + 2m∆4( am2 ,b)

12

}1
q
]
.

”

Proof. Using Lemma3.1 and the Holder’s inequality, we have “∣∣∣∣2α−1Γ(α + 1)(mb−a)α [Jα( a+mb2 )+ef(mb) + mα+1Jα( a+mb2m )−ef( am )]
−

1

2

[
ef(

a+mb
2

) + mef((
a+mb
2m

)

]∣∣∣∣
≤
mb−a

4

[( 1∫
0

tpαdt

)1
p
{ 1∫

0

∣∣∣∣ef( t2 a+m 2−t2 b)f′( t2a + m2 − t2 b)
∣∣∣∣qdt

}1
q

+

( 1∫
0

tpαdt

)1
p
{ 1∫

0

∣∣∣∣ef( 2−t2m a+ t2 b)f′( 2 − t2m a + t2b)
∣∣∣∣qdt

}1
q
]

” “

≤
mb−a

4(pα + 1)
1
p

[{ 1∫
0

[t2
4
|ef(a)f′(a)|q +

m2(2 − t)2

4
|ef(b)f′(b)|q +

mt(2 − t)
4

∆3(a,b)
]
dt

}1
q

+

{[m2(2 − t)2
4

|ef(
a

m2
)
f′(

a

m2
)|q +

t2

4
|ef(b)f′(b)|q +

mt(2 − t)
4

∆4(
a

m2
,b)
]}1q ]

=
mb−a

4(pα + 1)
1
p

[{
|ef(a)f′(a)|q + 7m2|ef(b)f′(b)|q + 2m∆3(a,b)

12

}1
q

+

{
7m2|ef(

a
m2

)
f′( a

m2
)|q + |ef(b)f′(b)|q + 2m∆4( am2 ,b)

12

}1
q
]
,

” which completes the proof. �

Corollary 3.7. [21]. If we choose m = 1 and α = 1, in Theorem 3.6, then “

∣∣∣∣ef( a+b2 ) − 1b−a
b∫

a

e
f(x)

dx

∣∣∣∣ ≤ b−a
4(p + 1)

1
p

[{
|ef(a)f′(a)|q + 7|ef(b)f′(b)|q + 2∆3(a, b)

12

}1
q

+

{
7|ef(a)f′(a)|q + |ef(b)f′(b)|q + 2∆4(a, b)

12

}1
q
]
.

”



Int. J. Anal. Appl. 17 (3) (2019) 477

4. Conclusions

In this paper, we have introduced and studied a new class of exponentially convex functions involving the

parameter m. We have obtained several new Hermite-Hadamard inequalities via Riemann-Liouville frac-

tional integrals. It is shown that previously known results can be obtained as special cases from our results.

It is shown that the class of exponentially m-convex functions is quite general, flexible and unifying one.

Acknowledgments: The authors would like to thank the Rector, COMSATS University Islamabad, Isla-

maabd, Pakistan for providing excellent research and academic environments.

References

[1] G. Alirezaei and R. Mathar, On exponentially concave functions and their impact in information theory, Information

Theory and Applications Workshop, San Diego, California, USA, 2018.

[2] T. Antczak, (p,r)-invex sets and functions, J. Math. Anal. Appl. 263(2001), 355-379.

[3] M. U. Awan, M. A. Noor and K. I. Noor, Hermite-Hadamard inequalities for exponentiaaly convex functions, Appl. Math.

Inform. Sci. 2(12)(2018), 405-409.

[4] M. K. Bakula, M. E. Ozdemir and J. Pecaric, Hadamard type inequalities for m-convex and (α; m)-convex functions, J.

Inequal. Pure Appl. Math. 9(4)(2008), Art. ID 96.

[5] I. A. Baloch and I. Iscan, Some Hermite-Hadamard type inequalities for harmonically (s; m)-convex functions in second

sense, arXiv:1604.08445v1 [math.CA], 2016.

[6] M. K. Bacul, J. Pecaric and M. Ribicic, Companion inequalities to Jensen’s inequality for m-convex and (α,m) convex

functions, J. Inequal. Pure. Appl. Math. 7(5)(2006), Art. ID 194.

[7] M. Braccamonte, J. Gimenez, N. Merentes and M. Vivas, Fejer type inequalities for m-convex functions, Publicaciones en

Ciencias y Tecnoloǵıa, 10(1)(2016), 7-11.

[8] S. S. Dragomir and I. Gomm, Some Hermite-Hadamard type inequalities for functions whose exponentials are convex,

Stud. Univ. Babes-Bolyai Math. 60(4)(2015), 527-534.

[9] S. S. Dragomir and G. Toader, Some inequalities for m-convex functions, Stud. Univ. Babes-Bolyai Math. 38 (1993), 21-28.

[10] S. S. Dragomir, On some new inequalities of Hermite-Hadamard type for m - convex functions, Tamkang J. Math.

33(1)(2002), 45-55.

[11] L. Fejéer, Uber die fourierreihen, II, Math Naturwise. Anz Ungar. Akad. Wiss. (24)(1906), 369-390.

[12] J. Hadamard, Etude sur les proprietes des fonctions entieres e.t en particulier dune fonction consideree par Riemann. J.

Math. Pure Appl. (58)(1893), 171-215.

[13] C. Y. He, Y. Wang, B. Y.Xi and F. Qi, Hermite-Hadamard type inequalities for (α; m)-HA and strongly (α; m)-HA convex

functions, J. Nonlinear Sci. Appl. (10)(2017), 205-214.

[14] M. Mahdavi, Exploiting Smoothness in Statistical Learning, Sequential Prediction, and Stochastic Optimization. East

Lansing, MI, USA: Michigan State University, (2014).

[15] C. P. Niculescu and L. E. Persson, Convex Functions and Their Applications. Springer-Verlag, New York, (2018).

[16] M. A. Noor, Some developments in general variational inequalities, Appl. Math. Comput. 152(2004), 199-277.

[17] M. A. Noor and K. I. Noor, Exponentially convex functions, Preprint.



Int. J. Anal. Appl. 17 (3) (2019) 478

[18] M. A. Noor, K. I. Noor and M. U. Awan, Fractional Hermite-Hadamard inequalities for convex functions and applications,

Tbilisi J. Math. 8(2)(2015), 103-113.

[19] M. A. Noor, K. I. Noor, and S. Rashid, Exponential r-convex functions and inequalities, Preprint.

[20] M. A. Noor, K. I. Noor and S. Rashid, Fractal exponential convex functions and inequalities, Preprint.

[21] S. Rashid, M. A. Noor and K. I. Noor, Modified exponential convex functions and inequalities, Open Acess J. Math. Theor.

Phy. 2(2)(2019), 45-51.

[22] M. E. Ozdemir, M. Avci. and H. Kavurmaci, Hermite-Hadamard type inequalities via (α,m)-convexity, J. Comput. Math.

Appl. 61(2011), 2614-2620.

[23] S. Pal and T. K. L. Wong, Exponentially concave functions and a new information geometry, Ann. Probab. 46(2)(2018),

1070-1113.

[24] J. Park, New Ostrowski-like type inequalities for differentable (s,m)-convex mappings, Int. J. Pure Appl. Math. 78 (8)

(2012), 1077-1089.

[25] J. E. Pecaric, F. Proschan and Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academics

Press, New York, (1992).

[26] J. Pecaric and J. Jaksetic, Exponential onvexity, Euler-Radau expansions and stolarsky means, Rad Hrvat. Matematicke

Znanosti, 515(2013), 81-94

[27] I. Podlubny, Fractional Differential Equations: Mathematics in Science and Engineering, Academic Press, San Diego,

(1999).

[28] M. Rostamian, S. S. Dragomir and M. D. L. Sen, Estimation type results related to Fejér inequality with applications, J.

Inequal. Appl, 2018 (2018), Art. ID 85.

[29] E. Set, A. O. Akdemir and I. Mumcu, The Hermite-Hadamard type inequality and its extensions for conformable fractional

integrals of any order α > 0, Preprint.

[30] G. Toader, Some generalizations of the convexity, Proc. Colloq. Approx. Opt. Cluj-Napoca (Romania), University of

Cluj-Napoca, 1984, 329-338.

[31] G. Toader, The order of starlike convex function, Bull. Appl. Comp. Math. 85(1998), 347-350.

[32] G. Stampacchia, Formes bilineaires coercivvities sur les ensembles convexes, C. R. Acad. Sci. Paris, 258(1964), 4413-4416.


	1. Introduction
	2. Preliminaries
	3. Main Results
	4. Conclusions
	References