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 involvingm∈[0,1],which is called exponentially m-convex function. Some new Hermite-Hadamard inequalities for exponentiallym-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. 2000Mathematics 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.
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 funcfunc-tions 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. Ifm= 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-tiallym-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 setK⊂Ris said to be am-convex set with respect to a fixed constantm∈[0,1], if
(1−t)a+mtb∈K, ∀a, b∈K, t∈[0,1].
The m-convex set contains the line segment between pointsaandmbfor every pair of points aandb ofK.
Definition 2.2. [31]. A functionf :K⊂R→Ris said to be a m-convex function, wherem∈[0,1], if
f((1−t)a+mtb)≤(1−t)f(a) +mtf(b), ∀a, b∈K, t∈[0,1].
Remark 2.1. Clearly, a1-convex function is a convex function in the ordinary sense. The0-convex function are the starshaped functions. If we takem= 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 functionf 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 functionf :K⊂R−→Ris 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)
wheref 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 thatu∈K is the minimum of the differentiable exponentially convex function
f,if and only if, u∈K satisfies the inequality
hf0(u)ef(u), v−ui ≥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 exponentiallym-convex function.
Definition 2.4. Let f :K⊂R−→Ris said to be an exponentially m-convex function, wherem∈(0,1], if
ef((1−t)a+mtb)≤[(1−t)ef(a)+mtef(b)], a, b∈K, t∈[0,1]. (2.2)
Fort= 12,we have
ef(a+mb2 )≤e
f(a)+mef(b)
2 , ∀a, b∈K. (2.3)
The functionf is called exponentially Jensenm-convex function.
Definition 2.5. [27]. Letα >0 with n−1< α≤n, n∈N, and1< x < v. The left- and right-hand side Riemann-Liouville fractional integrals of orderαof functionf are given by
Juα+f(x) = 1 Γ(α)
x Z
u
(x−t)α−1f(t)dt,
and
Jvα−f(x) = 1 Γ(α)
v Z
x
(t−x)α−1f(t)dt,
whereΓ(α)is the gamma function.
We also made the conventionJu0+ =J 0
v− =f(x).
We recall the special functions which are known as Gamma function,
Γ(x) =
∞
Z
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 exponentiallym-convex function via
Reimann-Liouville fractional integral.
Throughout this section, letI = [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+mb2 ) ≤ Γ(α+ 1)
2(mb−a)α
J(αa) +e
f(mb)+mα+1Jα
(b)−e f(a
m)
≤ α[e
f(a)+m2ef(b
m)] + [mef(b)+mef( a m)]
α(α+ 1) . (3.1)
Proof. “Letf be an exponentiallym-convex function, from the inequality (2.2). Then, we have” “
ef(x+my2 )≤e
f(x)+mef(y)
2 , ∀x, y∈[a, mb].
” “Substitutingx=at+m(1−t)b andy= (1−t)ma +mtmb fort∈[0,1].Then” “
2ef(a+mb2 ) ≤[ef(at+m(1−t)b)+mef((1−t) a m+mt
b m)].
” “Multiplying both sides of the above inequality withtα−1,and integrating over [0,1],we have” “
2
αe
f(a+mb
2 ) ≤
1
Z
0
tα−1[ef(at+m(1−t)b)+mef((1−t)ma+mt b m)]dt
= 1
(mb−a)α Zmb
a
(mb−u)α−1ef(u)du+mα+1
b Z
a m
(v− a
m)
α−1ef(v)dv
= Γ(α) (mb−a)α
J(αa)+ef(mb)+mα+1J(αb)−ef(ma)
,
”from which, one has “
ef(a+mb2 )≤ Γ(α+ 1)
2(mb−a)α
J(αa)+ef(mb)+mα+1J(αb)−ef(ma)
. (3.2)
”On the other hand exponentially m-convexity off gives“
ef(at+m(1−t)b)+mef((1−t)ma+mt b
m)≤t[ef(a)+m2ef( b
m)+ (1−t)[mef(b)+mef( a m)].
”. Multiplying both sides of the above inequality withtα−1, and integrating over [0,1],we have“
1
Z
0
tα−1ef(at+m(1−t)b)dt+m 1
Z
0
tα−1ef((1−t)ma+mt b m)dt
≤
1
Z
0
tα[ef(a)+m2ef(mb)]dt+ 1
Z
0
(tα−1−tα)[mef(b)+mef(ma)]dt.
”
“
Γ(α) (mb−a)α
J(αa)+ef(mb)+mα+1J(αb)−ef(ma)
≤ e
f(a)+m2ef(b m)
α+ 1 +
mef(b)+mef(ma) α(α+ 1) ,
”from which one has “
Γ(α+ 1) 2(mb−a)α
J(αa)+ef(mb)+mα+1J(αb)−ef(ma)
≤α[e
f(a)+m2ef(b
m)] + [mef(b)+mef( a m)]
α(α+ 1) . (3.3)
“
Corollary 3.1. If we choosem= 1 in Theorem3.1, then we have a new result
ef(a+b2 )≤ Γ(α+ 1)
2(b−a)α
J(αa) +e
f(b)+Jα
(b)−e f(a)
≤ [e
f(a)+ef(b)]
α .
” “
Corollary 3.2. If we choosem= 1 andα= 1in Theorem 3.1, then we have a new result
2ef(a+b2 )≤ 1 b−a
b Z
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 f g ∈
L[a, mb],then
Γ(α+ 1) (mb−a)α
Jmbα −ef(a)+Jaα+eg(mb)
≤e
f(a)+meg(b)+α(mef(b)+eg(a))
α(α+ 1) .
”
Proof. Letf, g:I⊂R→Rbe an exponentiallym-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 withtα−1,and integrating over [0,1],we have” “
1
Z
0
tα−1[ef(a(1−t)+mtb)+eg(at+m(1−t)b)]dt
≤
1
Z
0
(tα−1−tα)[ef(a)+meg(b)]dt+
1
Z
0
tα[mef(b)+eg(a)]dt.
” “
Γ(α) (mb−a)α
Zmb
a
(u−a)α−1ef(u)du+ mb Z
a
(mb−v)α−1eg(v)dv
≤e
f(a)+meg(b)
α(α+ 1) +
”from which, we have“
Γ(α+ 1) (mb−a)α
Jmbα −ef(a)+Jaα+eg(mb)
≤e
f(a)+meg(b)+α(mef(b)+eg(a))
α(α+ 1) ,
”which is the required result.
“
Corollary 3.3. If we choosem= 1 in Theorem3.2, then we have a new result
Γ(α+ 1) (b−a)α
Jbα−ef(a)+Jaα +e
g(b)
≤e
f(a)+eg(b)+α(ef(b)+eg(a))
α(α+ 1) .
” “
Corollary 3.4. If we choosem= 1 andα= 1in Theorem 3.2, then we have a new result
1
b−a
b Z
a
[eg(x)+ef(x)]dx≤e
f(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+mb2 )≤ 2
α−1Γ(α+ 1)
(mb−a)α
J(αa+mb 2 )+
ef(mb)+mα+1J(αa+mb 2m )−
ef(ma)
≤ α
4(α+ 1)
ef(a)−m2ef(ma2)+m
2
ef(b)+mef(ma2).
”
Proof. “Letf be an exponentiallym-convex function. Then, from the inequality (2.3), we have” “
ef(x+my2 )≤ e
f(x)+mef(y)
2 , x, y∈I.
” “Substitutingx= t
2a+m 2−t
2 b, y= 2−t
2ma+ t
2bfort∈[0,1]. Then” “
2ef(a+mb2 )≤ef( t 2a+m
2−t
2 b)+mef( 2−t 2ma+
” “Multiplying both sides of the above inequality withtα−1,and integrating over [0,1],we have” “
2
αe
f(a+mb2 )≤ 1
Z
0
tα−1ef(2ta+m 2−t
2 b)dt+m 1
Z
0
tα−1ef(22m−ta+t2b)dt
= 2
a−mb a+mb
2 Z
mb
2(mb−u)
mb−a
α−1
ef(u)du+ 2m
2
mb−a a+mb
2m Z
a m
2(v− a m)
b− a m
α−1 ef(v)dv
= 2 αΓ(α)
(mb−a)α
Jα (a+mb2 )+e
f(mb)+mα+1Jα
(a+mb2m )−e f(a
m).
”Thus“
ef(a+mb2 ) ≤2
α−1Γ(α+ 1)
(mb−a)α
J(αa+mb 2 )+
ef(mb)+mα+1J(αa+mb 2m )−
ef(ma)
=α 2
1
Z
0 tα−1
ef(2ta+m 2−t
2 b)+mef( 2−t 2ma+
t 2b)
dt ≤α 2 1 Z 0 tα−1
t
2e
f(a)+m(2−t)
2 e f(b)
+m
m2−t
2 e f( a
m2)+ t
2e f(b)
dt
=α 4
ef(a)−m2ef(ma2) 1
Z
0
tαdt+mα 2
ef(b)+mef(ma2) 1
Z
0
tα−1dt
= α
4(α+ 1)
ef(a)−m2ef(ma2)+ m
2
ef(b)+mef(ma2),
”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 interiorI◦ of I,wherem∈(0,1].If |f0| ∈L[[a, mb], is am-convex function, then” “
2α−1Γ(α+ 1) (mb−a)α
J(αa+mb 2 )+
ef(mb)+mα+1J(αa+mb 2m )−
ef(ma)−1
2
ef(a+mb2 )+mef(( a+mb
2m )
=mb−a 4
Z1
0
tαef(t2a+m 2−t
2 b)f0(t
2a+m 2−t
2 b)dt
−
1
Z
0
tαef(22m−ta+ t
2b)f0(2−t
2m a+ t
2b)dt
. (3.4)
Proof. It suffices to note that“
mb−a
4 Z1
0
tαef(2ta+m 2−t
2 b)f0(t
2a+m 2−t
2 b)dt
= mb−a 4
2
mb−ae
f(a+mb
2 )− 2α
(a−mb)
a+mb 2 Z
mb
2(mb−x)
mb−a
α−12ef(x)dx
a−mb
= mb−a 4
− 2
mb−ae
f(a+mb 2 )+2
α+1Γ(α+ 1)
(mb−a)α+1 J
α
(a+mb 2 )−
ef(mb)
. (3.5)
”Similarly, one can have“
−mb−a 4
Z1
0
tαef(22m−ta+ t
2b)f0(2−t
2m a+ t
2b)dt
=−mb−a 4
2m
mb−ae
f(a+mb 2m )−2
α+1Γ(α+ 1)
(mb−a)α+1 J
α
(a+mb2m )+e f(a
m)
. (3.6)
”Adding (3.5) and (3.6), gives (3.4).
Theorem 3.4. “Let α >0 andf : I⊂R→Rbe a differentiable exponentially m-convex function on the interiorI◦ of I,wherem∈(0,1].If |f0| ∈L[[a, mb], is am-convex function, then” “
2α−1Γ(α+ 1)
(mb−a)α
Jα (a+mb2 )+e
f(mb)+mα+1Jα
(a+mb2m )−e f(a
m)
−1 2
ef(a+mb2 )+mef(( a+mb
2m )
≤mb−a 4
1
4(α+ 3)
ef(a)f0(a) +
ef(b)f0(b)
+
m2(α2+ 7α+ 14)
4(α+ 1)(α+ 2)(α+ 3)
ef(b)f0(b) +
e
f( a m2)f0( a
m2)
+
m(α+ 4) ((α+ 2)(α+ 3))
∆1(a, b) + ∆2( a m2, b)
,
”where“
∆1(a, b) =
ef(a)f0(b) +
ef(b)f0(a) ,
and
∆2( a m2, b) =
ef(
a
m2)f0(b)+ef(b)f0( a m2)
.
Proof. “Using Lemma 3.1and exponentiallym-convexity function off,we have” “
2α−1Γ(α+ 1)
(mb−a)α
J(αa+mb 2 )+
ef(mb)+mα+1J(αa+mb 2m )−
ef(ma)
−1 2
ef(a+mb2 )+mef(( a+mb 2m )
≤mb−a 4 Z1 0 tα
ef(2ta+m 2−t
2 b)f0(t
2a+m 2−t
2 b) +
ef(22m−ta+ t
2b)f0(2−t
2m a+ t
2b) dt . (3.7)
”Usingm-convexity off, we have“
ef(t2a+m 2−t
2 b)f0(t
2a+m 2−t
2 b) +
ef(22m−ta+t2b)f0(2−t
2m a+ t
2b) ≤ t 2|e
f(a)|+m(2−t)
2 |e f(b)|t
2|f
0(a)|+m(2−t)
2 |f
0(b)|
m2−t
2 |e f( a
m2)|+t
2|e
f(b)|m(2−t) 2 |f
0 a
m2|+ t
2|f
0(b)|
= t
2
4
ef(a)f0(a) +
m2(2−t)2
4
ef(b)f0(b) +
m(2−t)t
4
ef(a)f0(b) +
ef(b)f0(a)
+m
2(2−t)2
4 e
f( a m2)f0( a
m2)
+ t2 4 e
f(b)
f0(b)+
m(2−t)t
4
e f( a
m2)f0(b)
+ef(b)f0(
a m2)
= t 2 4
ef(a)f0(a) +
ef(b)f0(b)
+
m2(2−t)2
4
ef(b)f0(b) +
e
f( a m2)f0( 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 )+e
f(mb)+mα+1Jα
(a+mb2m )−e f(a
m)
−1 2
ef(a+mb2 )+mef(( a+mb 2m )
≤mb−a 4 Z1 0 tα t2 4
ef(a)f0(a) +
ef(b)f0(b)
+
m2(2−t)2
4
ef(b)f0(b)
+ef( a m2)f0( a
m2)
+
m(2−t)t
4
∆1(a, b) + ∆2( a m2, b)
dt
=mb−a 4
1
4(α+ 3)
ef(a)f0(a) +
ef(b)f0(b)
+
m2(α2+ 7α+ 14)
4(α+ 1)(α+ 2)(α+ 3)
ef(b)f0(b) +
e
f( a m2)f0(
a m2)
+
m(α+ 4) ((α+ 2)(α+ 3))
∆1(a, b) + ∆2( a m2, b)
,
Corollary 3.5. [21]. “If we choosem= 1 andα= 1,in Theorem 3.4, then” “
ef(a+b2 )− 1 b−a
b Z
a
ef(x)dx
≤b−a 4
7{
ef(a)f0(a) +
ef(b)f0(b)
+ 10[∆1(a, b) + ∆2(a, b)]
24
.
”
Theorem 3.5. “Let f :I⊂R→Rbe differentiable exponentially m-convex function on the interior I◦ of
I, where m∈(0,1]. If |f0| ∈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
J(αa+mb 2 )+
ef(mb)+mα+1J(αa+mb 2m )−
ef(ma)
−1 2
ef(a+mb2 )+mef(( a+mb
2m )
≤ mb−a 4(α+ 1)1p
1 4(α+ 3)|e
f(a)f0(a)|q+ m
2(α2+ 7α+ 14)
4(α+ 1)(α+ 2)(α+ 3)|e
f(b)f0(b)|q
+ m(α+ 4)
(α+ 2)(α+ 3)∆3(a, b) 1q
+
m2(α2+ 7α+ 14) 4(α+ 1)(α+ 2)(α+ 3)|e
f( a m2)f0( a
m2)|
q
+ 1 4(α+ 3)|e
f(b)f0(b)|q+ m(α+ 4) (α+ 2)(α+ 3)∆4(
a m2, b)
1q
,
” “where
∆3(a, b) =|ef(a)f0(a)|q+|ef(b)f0(b)|q
and
∆4( a
m2, b) =|e
f( a m2)f0( a
m2)|
q+|ef(b)f0(b)|q.
”
Proof. Using Lemma3.1and the power mean inequality, we have“
2α−1Γ(α+ 1) (mb−a)α
J(αa+mb 2 )+
ef(mb)+mα+1J(αa+mb 2m )−
ef(ma)
−1 2
ef(a+mb2 )+mef(( a+mb
2m )
” “
≤mb−a 4 Z1 0 tα
ef(t2a+m 2−t
2 b)f0(t
2a+m 2−t
2 b) +
ef(22m−ta+ t
2b)f0(2−t
2m a+ t
2b) dt
=mb−a 4
1
(α+ 1)
1pZ1
0 tα
ef(t2a+m 2−t
2 b)f0(t
2a+m 2−t
2 b) q dt 1q + Z1 0 tα
ef(22m−ta+ t
2b)f0(2−t
2m a+ t
2b) q dt 1q
=mb−a 4
1
(α+ 1)
1pZ1
0 tαt
2
4|e
f(a)f0(a)|q+m2(2−t)2 4 |e
f(b)f0(b)|q
+mt(2−t)
4 ∆3(a, b)
1q
+ Z1
0 tαt
2
4|e
f(b)f0(b)|q+m
2(2−t)2
4 |e f( a
m2)f0( a m2)|
q
+mt(2−t) 4 ∆4(
a m2, b)
1q
= mb−a 4(α+ 1)p1
1
4(α+ 3)|e
f(a)f0(a)|q+ m2(α2+ 7α+ 14) 4(α+ 1)(α+ 2)(α+ 3)|e
f(b)f0(b)|q
+ m(α+ 4)
(α+ 2)(α+ 3)∆3(a, b) 1q
+
m2(α2+ 7α+ 14)
4(α+ 1)(α+ 2)(α+ 3)|e f( a
m2)f0( a m2)|
q
+ 1 4(α+ 3)|e
f(b)f0(b)|q+ m(α+ 4) (α+ 2)(α+ 3)∆4(
a m2, b)
1q
,
”which completes the proof.
Corollary 3.6. [21]. If we choosem= 1andα= 1, in Theorem3.5, then
ef(a+b2 )− 1 b−a
b Z
a
ef(x)dx
≤ b−a 4(2)1p
3|ef(a)f0(a)|q+ 11|ef(b)f0(b)|q+ 20∆
3(a, b)
48
1q
+
11|ef(a)f0(a)|q+ 3|ef(b)f0(b)|q+ 20∆
4(a, b)
48
1q
,
have“
2α−1Γ(α+ 1)
(mb−a)α
J(αa+mb 2 )+
ef(mb)+mα+1J(αa+mb 2m )−
ef(ma)
−1 2
ef(a+mb2 )+mef(( a+mb 2m )
≤ mb−a 4(pα+ 1)1p
|ef(a)f0(a)|q+ 7m2|ef(b)f0(b)|q+ 2m∆
3(a, b)
12
1q
+
7m2|ef( a m2)f0( a
m2)|
q+|ef(b)f0(b)|q+ 2m∆
4(ma2, b)
12
1q
.
”
Proof. Using Lemma3.1and the Holder’s inequality, we have“
2α−1Γ(α+ 1)
(mb−a)α
J(αa+mb 2 )+
ef(mb)+mα+1J(αa+mb 2m )−
ef(ma)
−1 2
ef(a+mb2 )+mef(( a+mb 2m )
≤ mb−a 4
Z1
0 tpαdt
p1Z1
0
ef(t2a+m 2−t
2 b)f0(t
2a+m 2−t
2 b) q dt 1q + Z1 0 tpαdt
1pZ1
0
ef(22m−ta+2tb)f0(2−t
2m a+ t
2b) q dt 1q ” “
≤ mb−a 4(pα+ 1)p1
Z1
0
t
2
4|e
f(a)f0(a)|q+m
2(2−t)2
4 |e
f(b)f0(b)|q+mt(2−t)
4 ∆3(a, b) dt 1q + m
2(2−t)2
4 |e f( a
m2)f0( a m2)|
q+t
2
4|e
f(b)f0(b)|q+mt(2−t) 4 ∆4(
a m2, b)
1q
= mb−a 4(pα+ 1)p1
|ef(a)f0(a)|q+ 7m2|ef(b)f0(b)|q+ 2m∆
3(a, b)
12
1q
+
7m2|ef( a m2)f0( a
m2)|
q+|ef(b)f0(b)|q+ 2m∆
4(ma2, b)
12
q1
,
”which completes the proof.
Corollary 3.7. [21]. If we choosem= 1andα= 1, in Theorem3.6, then“
ef(a+b2 )− 1 b−a
b
Z
a
ef(x)dx
≤ b−a
4(p+ 1)1p
|ef(a)f0
(a)|q+ 7|ef(b)f0
(b)|q+ 2∆ 3(a, b)
12
1q
+
7|ef(a) f0(a)|q
+|ef(b) f0(b)|q
+ 2∆4(a, b)
12
1q
.
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 exponentiallym-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.
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