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Baleanu et al. Advances in Difference Equations (2020) 2020:374
https://doi.org/10.1186/s13662-020-02837-0
R E S E A R C H Open Access
Some modifications in conformablefractional integral
inequalitiesDumitru Baleanu1,2,3, Pshtiwan Othman Mohammed4* ,
Miguel Vivas-Cortez5 andYenny Rangel-Oliveros5
*Correspondence:[email protected] of
Mathematics,College of Education, University ofSulaimani,
Sulaimani, KurdistanRegion, IraqFull list of author information
isavailable at the end of the article
AbstractThe prevalence of the use of integral inequalities has
dramatically influenced theevolution of mathematical analysis. The
use of these useful tools leads to fasteradvances in the
presentation of fractional calculus. This article investigates
theHermite–Hadamard integral inequalities via the notion
of�-convexity. After that, weintroduce the notion of�μ-convexity in
the context of conformable operators. Inview of this, we establish
some Hermite–Hadamard integral inequalities (bothtrapezoidal and
midpoint types) and some special case of those inequalities as
well.Finally, we present some examples on special means of real
numbers. Furthermore,we offer three plot illustrations to clarify
the results.
MSC: Primary 26D07; secondary 26D10; 26D15; 26A33
Keywords: Integral inequality; Conformable operator; Convex
functions
1 IntroductionFor any v1, v2 ∈ [a, b] and � ∈ [0, 1], the
real-valued function g on an interval [a, b] is calleda convex
function if the following holds:
g(�v1 + (1 – �)v2
) ≤ �g(v1) + (1 – �)g(v2). (1.1)
The theory and application of convexity has a close relationship
with theory and ap-plication of inequalities or integral
inequalities. The convex function (1.1) has been ex-tended and
generalized in several directions, such as pseudo-convex [1],
quasi-convex [2],strongly convex [3], �-convex [4], s-convex [5],
h-convex [6, 7], (α, m)-convex [8, 9], invexand preinvex [10–12],
and other kinds of convex functions by a number of mathemati-cians;
see [13–21] for more details.
Integral inequalities form an essential field of study among the
field of mathematicalanalysis. They have been vital in providing
bounds to solve some boundary value prob-lems in fractional
calculus, and in establishing the existence and uniqueness of
solutionsfor certain fractional differential equations. Convexity
plays an important role in the fieldof integral inequality due to
the behavior of its definition. Also, there is a strong
connection
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https://doi.org/10.1186/s13662-020-02837-0http://crossmark.crossref.org/dialog/?doi=10.1186/s13662-020-02837-0&domain=pdfhttp://orcid.org/0000-0001-6837-8075mailto:[email protected]
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between convexity and integral inequality. For this reason, many
known integral inequal-ities have been established in the
literature. The Hermite–Hadamard (HH) inequality isthe most well
known one: for an L1 convex function g : I ⊆ R→ R with v1, v2 ∈ I ,
v1 < v2,the HH inequality is defined as follows:
g(
v1 + v22
)≤ 1
v2 – v1
∫ v2
v1g(x) dx ≤ g(v1) + g(v2)
2. (1.2)
A huge number of researchers in the field of applied mathematics
have dedicated theirinterest to generalize, improve, refine,
counterpart, and extend HH inequality (1.2) forvarious types of
convex functions; see e.g. [22–30].
Recently, Samet [31] introduced a new notion of convexity for
certain functions thatdepends on some axioms. This often
generalizes various types of convexity e.g. �-convexfunctions,
α-convex functions, h-convex functions, and so on. Also, for
further details,visit [16, 32].
Throughout our study, we suppose that I ⊆ R (R the set of real
numbers), V ={(η1,η2);η� ∈ [v1, v2],� = 1, 2} and R̄ =
{(η1,η2,η3);ηi ∈ R,� = 1, 2, 3}. Then the family ofF of functions �
: R̄× [0, 1] → R satisfies the major axioms [31]:
(Λ1) If y� ∈ L1(0, 1), � = 1, 2, 3, then for every γ ∈ [0, 1] we
have∫ 1
0�
(y1(η), y2(η), y3(η),γ
)dη
= �(∫ 1
0y1(η) dη,
∫ 1
0y2(η) dη,
∫ 1
0y3(η) dη,γ
).
(Λ2) For every u ∈ L1(0, 1), w ∈ L∞(0, 1), and (z1, z2) ∈ R2, we
have∫ 1
0�
(w(η)u(η), w(η)z1, w(η)z2,η
)dη = T�,w
(∫ 1
0w(η)u(η) dη, z1, z2
),
where T�,w : R̄→ R is a function depending on (F , w). Moreover,
it is a nondecreas-ing function according to the first
variable.
(Λ3) For any (w, y1, y2, y3) ∈ R4, y4 ∈ [0, 1], we have
w�(y1, y2, y3, y4) = �(wy1, wy2, wy3, y4) + Lw,
where Lw ∈ R is a constant (depending on w).
Definition 1.1 Let g : [v1, v2] ⊆ R→ R with v1 < v2 be a
function, then we say that g is aconvex function according to � ∈F
(or briefly �-convex function) iff
�̄(g(ηx + (1 – η)y
), g(x), g(y),η
) ≤ 0, (x, y,η) ∈ V × [0, 1].
Remark 1.1 Suppose that (v1, v2) ∈ R2 with v1 < v2,(i) if g :
[v1, v2] ⊂ R→ R is an ε-convex function, or equivalently [26]
g(ηx + (1 – η)y
) ≤ ηg(x) + (1 – η)g(y) + ε, (x, y,η) ∈ V × [0, 1],
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then we define the functions � : R̄× [0, 1] → R as follows:
�(y1, y2, y3, y4) = y1 – y4y2 – (1 – y4)y3 – ε, (1.3)
and T�,w : R̄× [0, 1] → R as
T�,w(y1, y2, y3) = y1 –(∫ 1
0tw(η) dη
)y2 –
(∫ 1
0(1 – η)w(η) dη
)y3 – ε. (1.4)
For
Lw = (1 – w)ε, (1.5)
we can observe that � ∈F and
�(g(ηx + (1 – η)y
), g(x), g(y),η
)= g
(ηx + (1 – η)y
)– ηg(x) – (1 – η)g(y) – ε ≤ 0,
and this tells us g is an �-convex function. In a particular
case, we take ε = 0 to show thatg is an �-convex function according
to � when g is assumed to be a convex function.
(ii) If g : [v1, v2] ⊂ R→ R is a μ-convex function with μ ∈ (0,
1], or equivalently
g(ηx + (1 – η)y
) ≤ ημg(x) + (1 – ημ)g(y), (x, y,η) ∈ V × [0, 1].
Then we define the function � : R̄× [0, 1] → R as follows:
�(y1, y2, y3, y4) = y1 – yμ4 y2 –(1 – yμ4
)y3, (1.6)
and T�,w : R̄× [0, 1] → R as
T�,w(y1, y2, y3) = y1 –(∫ 1
0ημw(η) dη
)y2 –
(∫ 1
0
(1 – ημ
)w(η) dη
)y3. (1.7)
For Lw = 0, we can observe that � ∈F and
�(g(ηx + (1 – η)y
), g(x), g(y),η
)= g
(ηx + (1 – η)y
)– ημg(x) –
(1 – ημ
)g(y) – ε ≤ 0,
or g is an �-convex function.(iii) If h : I → R is a function
and it is not identically 0, where (0, 1) ⊆ I . Also, suppose
that g : [v1, v2] ⊂ I → [0,∞) is an h-convex function, that
is,
g(ηx + (1 – η)y
) ≤ h(η)g(x) + h(1 – η)g(y), (x, y,η) ∈ V × [0, 1].
Then we define the functions � : R̄× [0, 1] → R as follows:
�(y1, y2, y3, y4) = y1 – h(y4)y3 – h(1 – y4)y2, (1.8)
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and T�,w : R̄× [0, 1] → R as
T�,w(y1, y2, y3) = y1 –(∫ 1
0h(η)w(η) dη
)y2 –
(∫ 1
0h(1 – η)w(η) dη
)y3. (1.9)
For Lw = (1 – w)ε, we can observe that � ∈F and
�(g(ηx + (1 – η)y
), g(x), g(y),η
)= g
(ηx + (1 – η)y
)– h(η)g(x) – h(1 – η)g(y) – ε ≤ 0,
or we can say that g is an �-convex function.
In recent years, many possible inequalities have been proposed
in the context of frac-tional calculus including the midpoint and
trapezoidal formula inequalities and inequal-ities for ε-convexity,
α-convexity, (α, m)-convexity, and h-convexity; see [26, 31, 33]
formore details.
2 Conformable fractional operators and �μ-convexityIn the last
fifteen years, the definition of fractional calculus has been more
appropriateto describe historical dependence processes than the
local limit definitions of integer or-dinary differential equations
(ODEs) or partial differential equations (PDEs), and has re-ceived
more and more attention in many mathematical and physical fields,
see for de-tails [34–44]. Differential equations of fractional
order are more accurate than differentialequations of integer order
in describing the nature of things and objective laws. In
1695,Leibnitz discovered fractional derivatives, and after that
more and more researchers havededicated themselves to the study of
fractional calculus. The most commonly used frac-tional calculus
definitions are Riemann–Liouville definition, Caputo definition,
and con-formable fractional definition in basic mathematical and
engineering application research.In the present paper, we deal with
the conformable fractional definition [45–47] in orderto obtain our
results.
In this section, we recall some preliminaries and properties on
conformable fractionalcalculus. For further details and
applications, see the previously published articles [33,
45–54].
Definition 2.1 ([47]) Let g : [0,∞) → R, then the μth order
conformable derivative of gat η is defined by
Dμ(g)(η) = lim�→0
g(η + �η1–μ) – g(η)�
, μ ∈ (0, 1),η > 0. (2.1)
For μ-differentiable function g in some (0,μ),μ > 0, limt→0+
g(μ)(η) exist, define
g(μ)(0) = limt→0+
g(μ)(η).
Furthermore, if g is differentiable, then we have
Dμ(g)(η) = η1–μg ′(η), where g ′(η) = lim�→0
g(η + �) – g(η)�
. (2.2)
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Observe that we can write g(μ)(η) for dμdμη (g(η)) or simply
Dμ(g)(η) to denote a μth orderconformable derivative of g at η.
Furthermore, if the μth order conformable derivative ofg exists,
then we can simply say g is μ-differentiable.
Theorem 2.1 ([48]) Assume that μ ∈ (0, 1] and f , g are two
μ-differentiable functions ata point η > 0. Then we have:
1. Dμ(v1f + v2g) = v1Dμ(f ) + v2Dμ(g) for all v1, v2 ∈ R,2.
Dμ(fg) = fDμ(g) + gDμ(f ),3. Dμ( fg ) =
gDμ(f )–fDμ(g)g2 ,
4. Dμ(c) = 0 for each constant function, namely g(η) = c,5.
Dμ(1) = 0,6. Dμ( 1μη
μ) = 1.
Some basic properties of conformable operator are now stated,
which are useful in whatfollows.
Definition 2.2 ([47]) Assume that μ ∈ (0, 1], 0 ≤ v1 ≤ v2, and g
: [v1, v2] ⊂ R → R, thenwe say that a function g is μ-fractional
integrable on the interval [v1, v2] if the followingintegral
∫ v2
v1g(η) dμη =
∫ v2
v1g(η)ημ–1 dη (2.3)
exists and is finite.
Remark 2.1(a) We indicate by L1μ([v1, v2]) all μ-fractional
integrable functions on an interval
[v1, v2].(b) The usual Riemann improper integral has the
form
Iv1μ (g)(η) = Iv11
(ημ–1g
)=
∫ t
v1xμ–1g(x) dx, μ ∈ (0, 1]. (2.4)
Theorem 2.2 ([47, 48]) Let g : (v1, v2) → R be differentiable
and μ ∈ (0, 1]. Then, for allη > v1, we have
Iv1μ Dv1μ (g)(η) = g(η) – g(v1).
Theorem 2.3 ([51]) Suppose that g : [v1,∞) → R such that g(n) is
continuous. Then, foreach η > v1, we have
Dv1μ Iv1μ (g)(η) = g(η), μ ∈ (n, n + 1],
which is called the inverse property.
Theorem 2.4 ([47, 48]) Let g : [v1, v2] ⊂ R→ R be two functions
with fg is differentiable.Then
∫ v2
v1g(x)Dv1μ (h)(x) dμx = gh|v2v1 –
∫ v2
v1h(x)Dv1μ (g)(x) dμx.
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Theorem 2.5 ([47, 48]) Let f , g : [v1, v2] ⊂ R→ R be a
continuous function on [v1, v2] andwith 0 ≤ v1 ≤ v2. Then
∣∣Iv1μ (g)(η)∣∣ ≤ Iv1μ |f |(η), μ ∈ (0, 1].
It is time to define the concept of �μ-convexity on conformable
integrals, namely thefamily of �μ.
The family of �μ of functions �μ : R̄× [0, 1] → R satisfies the
major axioms:(Λ̄1) If y� ∈ L1(0, 1), � = 1, 2, 3, then for every γ
∈ [0, 1] we have
∫ 1
0�μ
(y1(η), y2(η), y3(η),γ
)dη
= �μ(∫ 1
0y1(η) dη,
∫ 1
0y2(η) dη,
∫ 1
0y3(η) dη,γ
).
(Λ̄2) For every u ∈ L1(0, 1), w ∈ L∞(0, 1), and (z1, z2) ∈ R2,
we have∫ 1
0�μ
(w(η)u(η), w(η)z1, w(η)z2,η
)dη = T�μ ,w
(∫ 1
0w(η)u(η) dη, z1, z2
),
where T�μ ,w : R̄ → R is a nondecreasing function according to
the first variablewhich depends on (�μ, w).
(Λ̄3) For any (w, y1, y2, y3) ∈ R4, y4 ∈ [0, 1], we have
w�μ(y1, y2, y3, y4) = �μ(wy1, wy2, wy3, y4) + Lw,
where Lw ∈ R is a constant (depending on w).
Definition 2.3 Let μ ∈ (0, 1] and g : [v1, v2] ⊂ R→ R with v1
< v2 be a function, then wesay g is a conformable convex
function according to �μ ∈ F (or briefly �μ-conformableconvex
function) if
�μ
(g(ημxμ +
(1 – ημ
)yμ
), g
(xμ
), g
(yμ
),ημ
) ≤ 0, (x, y,η) ∈ V × [0, 1].
Remark 2.2 Suppose that (v1, v2) ∈ R2 with v1 < v2.(i) Let g
: [v1, v2] ⊂ R→ R be an ε-conformable convex function, or
equivalently
g(ημxμ +
(1 – ημ
)yμ
) ≤ ημg(xμ) + (1 – ημ)g(yμ) + ε, (x, y,η) ∈ V × [0, 1].
Then we define the function �μ : R̄× [0, 1] → R as follows:
�μ(y1, y2, y3, y4) = y1 – yμ4 y2 –(1 – yμ4
)y3 – ε, (2.5)
and T�μ ,w : R̄× [0, 1] → R as
T�μ ,w(y1, y2, y3) = y1 –(∫ 1
0ημw
(ημ
)dμη
)y2 –
(∫ 1
0
(1 – ημ
)w(η) dμη
)y3 – ε. (2.6)
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For
Lw = (1 – w)ε, (2.7)
it can be observed that � ∈F and
�μ
(g(ημxμ +
(1 – ημ
)yμ
), g
(xμ+
), g
(yμ
),ημ
)
= g(ημxμ +
(1 – ημ
)yμ
)– ημg
(xμ
)–
(1 – ημ
)g(yμ
)– ε ≤ 0,
or in another meaning g is an �-conformable convex function. In
particular, g is an �-conformable convex function according to �
for ε = 0 when g is a conformable convexfunction.
(ii) Let g : [v1, v2] ⊂ I → R be a μ-conformable convex function
μ ∈ (0, 1]; that is,
g(ημxμ +
(1 – ημ
)yμ
) ≤ ηg(xμ) + (1 – η)g(yμ), (x, y,η) ∈ V × [0, 1].
Then we define the functions �μ : R̄× [0, 1] → R as follows:
�μ(y1, y2, y3, y4) = y1 – y4y2 – (1 – y4)y3, (2.8)
and T�μ ,w : R̄× [0, 1] → R as
T�μ ,w(y1, y2, y3) = y1 –(∫ 1
0tw
(ημ
)dμη
)y2 –
(∫ 1
0(1 – η)w
(ημ
)dμη
)y3. (2.9)
For Lw = 0, we can observe that �μ ∈F and
�μ
(g(ημxμ +
(1 – ημ
)yμ
), g
(xμ
), g
(yμ
),ημ
)
= g(ημxμ +
(1 – ημ
)yμ
)– ηg
(xμ
)– (1 – η)g
(yμ
)– ε ≤ 0,
or equivalently g is an �μ-conformable convex function.(iii) Let
h : I → R be a function, which is not identically 0, where (0, 1) ⊆
I . Let g :
[v1, v2] ⊂ I → [0,∞) be an h-conformable convex function, or
let
g(ημxμ +
(1 – ημ
)yμ
) ≤ h(ημ)g(xμ) + h(1 – ημ)g(yμ), (x, y,η) ∈ V × [0, 1].
Then we define the functions �μ : R̄× [0, 1] → R as follows:
�μ(y1, y2, y3, y4) = y1 – h(y4)y3 – h(1 – yμ4
)y2, (2.10)
and T�μ ,w : R̄× [0, 1] → R as
T�μ ,w(y1, y2, y3) = y1 –(∫ 1
0h(ημ
)w
(ημ
)dμη
)y2
–(∫ 1
0h(1 – ημ
)w
(ημ
)dμη
)y3. (2.11)
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For Lw = (1 – w)ε, we can observe that �μ ∈F and
�μ
(g(ημxμ +
(1 – ημ
)yμ
), g
(xμ
), g
(yμ
),ημ
)
= g(ημxμ +
(1 – ημ
)yμ
)– h
(ημ
)g(xμ
)– h
(1 – ημ
)g(yμ
)– ε ≤ 0,
or equivalently we can say g is an �μ-conformable convex
function.
For the conformable operators, we recall some early findings in
the earlier literaturewhich may help us in finding our main
results. For example in [55], Sarikaya et al. investi-gated new
results for the conformable fractional operator, and their results
are as follows.
Theorem 2.6 ([55, Theorem 11]) Let μ ∈ (0, 1] and g : [v1, v2] ⊂
R→R be a μ-fractionaldifferentiable function on (v1, v2) with 0 ≤
v1 < v2. Then we have
g(
vμ1 + vμ2
2
)≤ μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx ≤ g(v
μ1 ) + g(v
μ2 )
2. (2.12)
Lemma 2.1 ([55, Lemma 3]) Let μ ∈ (0, 1] and g : [v1, v2] ⊂ R→R
be a μ-fractional differ-entiable function on (v1, v2) with 0 ≤ v1
< v2. If Dμ(g) is a μ-fractional integrable functionon [v1, v2],
then we have
g(vμ1 ) + g(vμ2 )
2–
∫ v2
v1g(xμ
)dμx
=vμ2 – v
μ1
2
∫ 1
0
(1 – 2ημ
)Dμ(g)
(ημvμ1 +
(1 – ημ
)vμ2
)dμη. (2.13)
Lemma 2.2 ([55, Lemma 4]) Let μ ∈ (0, 1] and g : [v1, v2] ⊂ R→R
be a μ-fractional differ-entiable function on (v1, v2) with 0 ≤ v1
< v2. If Dμ(g) is a μ-fractional integrable functionon [v1, v2],
then we have
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx – g
(vμ1 + v
μ2
2
)
=(vμ2 – v
μ1)∫ 1
0p(η)Dμ(g)
(ημvμ1 +
(1 – ημ
)vμ2
)dμη, (2.14)
where
P(η) =
⎧⎨
⎩ημ, 0 ≤ t ≤ 121/μ ,ημ – 1, 121/μ ≤ t ≤ 1.
In view of these indices, we investigate some new inequalities
of HH type for the � and�μ-convex functions involving conformable
fractional operators in this attempt. Specifi-cally, we investigate
some inequalities of trapezoidal and midpoint type.
3 Hermite–Hadamard inequalities for �-convex functionsThis
section deals with the investigation of HH-type inequalities for
�-convex functions.
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Theorem 3.1 Let g : [v1, v2] ⊂ R → R be a μ-fractional
differentiable function on (v1, v2)with 0 ≤ v1 < v2. If g is an
�-convex function on [v1, v2] for some � ∈F , then
�
(g(
vμ1 + vμ2
2
),
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx,
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx,
12
)+
∫ 1
0Lw(η) dη
≤ 0, (3.1)
T�,w(
2μvμ2 – v
μ1
∫ v2
v1g(xμ
)dμx, g
(vμ1
)+ g
(vμ2
), g
(vμ1
)+ g
(vμ2
))
+∫ 1
0Lw(η) dη ≤ 0. (3.2)
Proof The �-convexity of g leads to
�
(g(
x + y2
), g(x), g(y),
12
), x, y ∈ [v1, v2].
For the values x = ημvμ1 + (1 – ημ)vμ2 and y = (1 – ημ)v
μ1 + ημv
μ2 , where η ∈ [0, 1], we obtain
�
(g(
vμ1 + vμ2
2
), g
(ημvμ1 +
(1 – ημ
)vμ2
), g
((1 – ημ
)vμ1 + η
μvμ2),
12
)≤ 0.
Multiplying this inequality w(η) = 1 and making use of axiom
(Λ3), we get
�
(g(
vμ1 + vμ2
2
), g
(ημvμ1 +
(1 – ημ
)vμ2
), g
((1 – ημ
)vμ1 + η
μvμ2),
12
)+ Lw(η) ≤ 0.
Integrating over [0, 1] according to η and making use of axiom
(Λ1), we get
�
(∫ 1
0g(
vμ1 + vμ2
2
)dμη,
∫ 1
0g(ημvμ1 +
(1 – ημ
)vμ2
)dμη,
∫ 1
0g((
1 – ημ)vμ1 + η
μvμ2)
dμη,12
)+
∫ 1
0Lw(η) dμη ≤ 0,
that is,
�
(g(
vμ1 + vμ2
2
),
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx,
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx,
12
)+
∫ 1
0Lw(η) dη
≤ 0,
where we have used
∫ 1
0g(ημvμ1 +
(1 – ημ
)vμ2
)dμη =
∫ 1
0g((
1 – ημ)vμ1 + η
μvμ2)
dμη
=μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx.
This completely gives the proof of (3.1). On the other hand,
since g is �-convex, we have
�(g(ημvμ1 +
(1 – ημ
)vμ2
), g
(vμ1
), g
(vμ2
),η
) ≤ 0,�
(g((
1 – ημ)vμ1 + η
μvμ2), g
(vμ2
), g
(vμ1
),η
) ≤ 0.
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We make use of the linearity of � to get
�(g(ημvμ1 +
(1 – ημ
)vμ2
)+ g
((1 – ημ
)vμ1 + η
μvμ2), g
(vμ1
)+ g
(vμ2
), g
(vμ1
)+ g
(vμ2
),η
) ≤ 0.
Applying the axiom (Λ3) for w(η) = 1, we get
�(g(ημvμ1 +
(1 – ημ
)vμ2
)+ g
((1 – ημ
)vμ1 + η
μvμ2), g
(vμ1
)+ g
(vμ2
), g
(vμ1
)+ g
(vμ2
),η
)
+ Lw(η) ≤ 0.
Integrating over [0, 1] according to η and making use of axiom
(Λ2) we get
T�,w(
2μvμ2 – v
μ1
∫ v2
v1g(xμ
)dμx, g
(vμ1
)+ g
(vμ2
), g
(vμ1
)+ g
(vμ2
))+
∫ 1
0Lw(η) dη ≤ 0.
This completes the proof of (3.2). Thus, the proof of Theorem
3.1 is completed. �
Corollary 3.1 Theorem 3.1 with g to be ε-convex leads to
g(
vμ1 + vμ2
2
)– ε ≤ μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx ≤ g(v
μ1 ) + g(v
μ2 )
2+
ε
2. (3.3)
Proof By making use of w(η) = 1 in (1.5), we get
∫ 1
0Lw(η) dη = 0. (3.4)
Making use of (1.3), (3.1), and (3.4), we get
0 ≥ F(
g(
vμ1 + vμ2
2
),
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx,
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx,
12
)+
∫ 1
0Lw(η) dη
= g(
vμ1 + vμ2
2
)–
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx – ε,
or equivalently,
g(
vμ1 + vμ2
2
)– ε ≤ μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx.
Making use of w(η) = 1 in (1.4), we have
T�,w(y1, y2, y3) = y1 –(∫ 1
0t dη
)y2 –
(∫ 1
0(1 – η) dη
)y3 – ε
= y1 –y2 + y3
2– ε, y1, y2, y3 ∈ R. (3.5)
Now, from (3.2) and (3.5), we can deduce
0 ≥ T�,w(
2μvμ2 – v
μ1
∫ v2
v1g(xμ
)dμx, g
(vμ1
)+ g
(vμ2
), g
(vμ1
)+ g
(vμ2
))+
∫ 1
0Lw(η) dη
=2μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx –
(g(vμ1
)+ g
(vμ2
))– ε.
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This gives
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx ≤ g(v
μ1 ) + g(v
μ2 )
2+
ε
2.
This ends the proof of (3.3). �
Remark 3.1 Inequality (3.3) with ε = 0 becomes inequality
(2.12).
Corollary 3.2 Theorem 3.1 with g to be h-convex leads to
12h( 12 )
g(
vμ1 + vμ2
2
)≤ μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx ≤ g(v
μ1 ) + g(v
μ2 )
2
∫ 1
0h(η) dη. (3.6)
Proof Making use (1.5) and (3.1) with Lw(η) = 0, we have
0 ≥ F(
g(
vμ1 + vμ2
2
),
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx,
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx,
12
)+
∫ 1
0Lw(η) dη
= g(
vμ1 + vμ2
2
)– h
(12
)2μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx,
or equivalently,
12h( 12 )
g(
vμ1 + vμ2
2
)≤ μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx.
Now, by making use of w(η) = 1 in (1.4) and (3.2), we get
0 ≥ T�,w(
2μvμ2 – v
μ1
∫ v2
v1g(xμ
)dμx, g
(vμ1
)+ g
(vμ2
), g
(vμ1
)+ g
(vμ2
))+
∫ 1
0Lw(η) dη
=2μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx –
(g(vμ1
)+ g
(vμ2
))(∫ 1
0h(η) dη
).
This gives
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx ≤
(g(vμ1 ) + g(v
μ2 )
2
)(∫ 1
0h(η) dη
).
Thus, the proof of (3.6) is completed. �
4 Hermite–Hadamard inequalities for �μ-convex functionsHere, we
deal with the investigation of HH-type inequalities for�μ-convex
functions. Thissection is separated into two subsections: a section
for the trapezoidal formula inequalityand the other one for the
midpoint formula inequality of HH type, respectively.
4.1 Trapezoidal inequalities for �μ-convex functionsTheorem 4.1
Let g : [v1, v2] ⊂ R → R be a μ-fractional differentiable function
on (v1, v2)and Dμ(g) be a μ-fractional integrable function on [v1,
v2] with 0 ≤ v1 < v2. If |Dμ(g)| is
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an �μ-convex function on [v1, v2] for some �μ ∈ F and the
function η ∈ [0, 1] → Lw(ημ)belongs to L1[0, 1], where w(ημ) = |1 –
2ημ|, then we have the inequality
T�μ ,w(
2vμ2 – v
μ1
∣∣∣∣g(vμ1 ) + g(v
μ2 )
2–
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx
∣∣∣∣,
∣∣Dμ(g)(vμ1
)∣∣,∣∣Dμ(g)
(vμ2
)∣∣,η)
+∫ 1
0Lw(η) dμη ≤ 0. (4.1)
Proof The �μ-convexity of |Dμ(g)| leads to
�μ
(∣∣Dμ(g)(ημvμ1 +
(1 – ημ
)vμ2
)∣∣,∣∣Dμ(g)
(vμ1
)∣∣,∣∣Dμ(g)
(vμ2
)∣∣,η) ≤ 0.
By applying axiom (Λ̄3) for w(ημ) = |1 – 2ημ|, η ∈ [0, 1], we
can deduce
�μ
(w
(ημ
)∣∣Dμ(g)(ημvμ1 +
(1 – ημ
)vμ2
)∣∣, w(ημ
)∣∣Dμ(g)(vμ1
)∣∣, w(ημ
)∣∣Dμ(g)(vμ2
)∣∣,η) ≤ 0.
Integrating over [0, 1] according to η and by making use of
axiom (Λ̄2), we obtain
T�μ ,w(∫ 1
0w
(ημ
)∣∣Dμ(g)(ημvμ1 +
(1 – ημ
)vμ2
)∣∣dμη,∣∣Dμ(g)
(vμ1
)∣∣,∣∣Dμ(g)
(vμ2
)∣∣,η)
+∫ 1
0Lw(ημ) dμt ≤ 0.
From Lemma 2.1, we have
∣∣∣∣g(vμ1 ) + g(v
μ2 )
2–
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx
∣∣∣∣
≤ vμ2 – v
μ1
2
∫ 1
0w
(ημ
)∣∣Dμ(g)(ημvμ1 +
(1 – ημ
)vμ2
)∣∣dμη.
Since T�μ ,w is nondecreasing according to the first variable,
then we can deduce
T�μ ,w(
2vμ2 – v
μ1
∣∣∣∣g(vμ1 ) + g(v
μ2 )
2–
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx
∣∣∣∣,
∣∣Dμ(g)
(vμ1
)∣∣,∣∣Dμ(g)
(vμ2
)∣∣,η)
+∫ 1
0Lw(ημ) dμη ≤ 0,
which ends the proof of (4.1). �
Corollary 4.1 Theorem 4.1 with |Dμ(g)| to be ε conformable
convex leads to∣∣∣∣g(vμ1 ) + g(v
μ2 )
2–
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx
∣∣∣∣
≤ vμ2 – v
μ1
2
(23μ2 + 6 × 2μ2 – 8
6μ × 23μ2)(∣∣Dμ(g)
(vμ1
)∣∣ +∣∣Dμ(g)
(vμ2
)∣∣ +2μ – 1
2με
). (4.2)
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Proof We know that any ε-convex is �μ-convex. So, by making use
of w(ημ) = |1 – 2ημ|in (2.7) and by using Definition 2.2, we
get
∫ 1
0Lw(η) dμη = ε
∫ 1
0
(1 – w(η)
)dμη =
12μ
ε.
By making use of w(ημ) = |1 – 2ημ| in (2.6), we get
T�μ ,w(y1, y2, y3) = y1 –(∫ 1
0ημ
∣∣1 – 2ημ∣∣dμη
)y2 –
(∫ 1
0
(1 – ημ
)∣∣1 – 2ημ∣∣dμη
)y3 – ε
= y1 –(
23μ2 + 6 × 2μ2 – 86μ × 23μ2
)(y2 + y3) – ε
for y1, y2, y3 ∈ R. By making use of Theorem 4.1, we get
0 ≥ T�μ ,w(
2vμ2 – v
μ1
∣∣∣∣g(vμ1 ) + g(v
μ2 )
2–
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx
∣∣∣∣,
∣∣Dμ(g)
(vμ1
)∣∣,∣∣Dμ(g)
(vμ2
)∣∣,η)
+∫ 1
0Lw(ημ) dμη
=2
vμ2 – vμ1
∣∣∣∣g(vμ1 ) + g(v
μ2 )
2–
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx
∣∣∣∣
–(
23μ2 + 6 × 2μ2 – 86μ × 23μ2
)(∣∣Dμ(g)(vμ1
)∣∣ +∣∣Dμ(g)
(vμ2
)∣∣) – ε +1
2με.
This rearranges to the required inequality (4.2). �
Remark 4.1 Corollary 4.1 with ε = 0 becomes Theorem 13 in
[55].
Corollary 4.2 Theorem 4.1 with |Dμ(g)| to be μ-conformable
convex leads to∣∣∣∣g(vμ1 ) + g(v
μ2 )
2–
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx
∣∣∣∣
≤ vμ2 – v
μ1
2(μ + 1)(2μ + 1)
(1 +
μ
21/μ
)(∣∣Dμ(g)
(vμ1
)∣∣ +∣∣Dμ(g)
(vμ2
)∣∣). (4.3)
Proof We know that any μ-convex is �μ-convex. So, by making use
of w(ημ) = |1 – 2ημ|in (2.9), we get
T�μ ,w(y1, y2, y3) = y1 –(∫ 1
0t∣∣1 – 2ημ
∣∣dμη)
y2 –(∫ 1
0(1 – η)
∣∣1 – 2ημ∣∣dμη
)y3
= y1 –1
(μ + 1)(2μ + 1)
(1 +
μ
21/μ
)(y2 + y3)
for y1, y2, y3 ∈ R. Then, by applying Theorem 4.1, we have
0 ≥ T�μ ,w(
2vμ2 – v
μ1
∣∣∣∣g(vμ1 ) + g(v
μ2 )
2–
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx
∣∣∣∣,
∣∣Dμ(g)
(vμ1
)∣∣,∣∣Dμ(g)
(vμ2
)∣∣,η)
+∫ 1
0Lw(ημ) dμη
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=2
vμ2 – vμ1
∣∣∣∣g(vμ1 ) + g(v
μ2 )
2–
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx
∣∣∣∣
–1
(μ + 1)(2μ + 1)
(1 +
μ
21/μ
)(∣∣Dμ(g)(vμ1
)∣∣ +∣∣Dμ(g)
(vμ2
)∣∣).
This rearranges to the required inequality (4.3). �
Corollary 4.3 Theorem 4.1 with |Dμ(g)| to be h-conformable
convex leads to
∣∣∣∣g(vμ1 ) + g(v
μ2 )
2–
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx
∣∣∣∣
≤ vμ2 – v
μ1
2
(∫ 1
0h(ημ
)∣∣1 – 2ημ∣∣dμη
)(∣∣Dμ(g)(vμ1
)∣∣ +∣∣Dμ(g)
(vμ2
)∣∣). (4.4)
Proof It is known that every μ-convex is�μ-convex. So, by making
use of w(ημ) = |1–2ημ|in (2.11), we get
T�μ ,w(y1, y2, y3) = y1 –(∫ 1
0h(ημ
)∣∣1 – 2ημ∣∣dμη
)y2
–(∫ 1
0h(1 – ημ
)∣∣1 – 2ημ∣∣dμη
)y3
= y1 –(∫ 1
0h(ημ
)∣∣1 – 2ημ∣∣dμη
)(y2 + y3)
for y1, y2, y3 ∈ R. Then, by using Theorem 4.1, we get
0 ≥ T�μ ,w(
2vμ2 – v
μ1
∣∣∣∣g(vμ1 ) + g(v
μ2 )
2–
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx
∣∣∣∣,
∣∣Dμ(g)
(vμ1
)∣∣,∣∣Dμ(g)
(vμ2
)∣∣,η)
+∫ 1
0Lw(ημ) dμη
=2
vμ2 – vμ1
∣∣∣∣g(vμ1 ) + g(v
μ2 )
2–
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx
∣∣∣∣
–(∫ 1
0h(ημ
)∣∣1 – 2ημ∣∣dμη
)(∣∣Dμ(g)
(vμ1
)∣∣ +∣∣Dμ(g)
(vμ2
)∣∣).
This completes the proof of (4.4). �
Theorem 4.2 Let g : [v1, v2] ⊂ R → R be a μ-fractional
differentiable function on (v1, v2)and Dμ(g) be a μ-fractional
integrable function on [v1, v2] with 0 ≤ v1 < v2. If |Dμ(g)|
pp–1 is
an �μ-convex function on [v1, v2] for some �μ ∈F , then we
have
T�μ ,1(v1(g, p),
∣∣Dμ(g)(vμ1
)∣∣p
p–1 ,∣∣Dμ(g)
(vμ2
)∣∣p
p–1) ≤ 0, (4.5)
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where
v1(g, p) =(
2vμ2 – v
μ1
) pp–1
(1
2μ(p + 1)
{2 –
(1 –
12μ2–1
)p+1–
(1
2μ2–1– 1
)p+1}) –1p–1
×∣∣∣∣g(vμ1 ) + g(v
μ2 )
2–
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx
∣∣∣∣
pp–1
.
Proof By using the �μ-convexity of |Dμ(g)|p
p–1 , we have
�μ
(∣∣Dμ(g)(ημvμ1 +
(1 – ημ
)vμ2
)∣∣p
p–1 ,∣∣Dμ(g)
(vμ1
)∣∣p
p–1 ,∣∣Dμ(g)
(vμ2
)∣∣p
p–1 ,η) ≤ 0.
By making use of w(ημ) = 1 in axiom (Λ̄3), we obtain
T�μ ,1(∫ 1
0
∣∣Dμ(g)(ημvμ1 +
(1 – ημ
)vμ2
)∣∣p
p–1 dμη,∣∣Dμ(g)
(vμ1
)∣∣p
p–1 ,∣∣Dμ(g)
(vμ2
)∣∣p
p–1
)≤ 0.
Then, by making use Lemma of 2.2, we have
∣∣∣∣g(vμ1 ) + g(v
μ2 )
2–
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx
∣∣∣∣
≤ vμ2 – v
μ1
2
(1
2μ(p + 1)
{2 –
(1 –
12μ2–1
)p+1–
(1
2μ2–1– 1
)p+1}) 1p
×(∫ 1
0
∣∣Dμ(g)(ημvμ1 +
(1 – ημ
)vμ2
)∣∣p
p–1 dμη) p–1
p.
Since T�μ ,w is nondecreasing according to the first variable,
then we can deduce
T�μ ,1(v1(g, p),
∣∣Dμ(g)(vμ1
)∣∣p
p–1 ,∣∣Dμ(g)
(vμ2
)∣∣p
p–1) ≤ 0.
This completes the proof of (4.5). �
Corollary 4.4 Theorem 4.2 with |Dμ(g)|p
p–1 to be ε-conformable convex leads to
∣∣∣∣g(vμ1 ) + g(v
μ2 )
2–
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx
∣∣∣∣
≤ vμ2 – v
μ1
2
(1
2μ(p + 1)
{2 –
(1 –
12μ2–1
)p+1–
(1
2μ2–1– 1
)p+1}) 1p
×( |Dμ(g)(vμ1 )|
pp–1 + |Dμ(g)(vμ2 )|
pp–1
2μ+ ε
) p–1p
. (4.6)
Proof By making use of w(ημ) = |1 – 2ημ| in (2.6) and by
Definition 2.3, we get
T�μ ,w(y1, y2, y3) = y1 –(∫ 1
0ημ
∣∣1 – 2ημ
∣∣dμη
)y2 –
(∫ 1
0
(1 – ημ
)∣∣1 – 2ημ∣∣dμη
)y3 – ε
= y1 –y2 + y3
2μ– ε
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for y1, y2, y3 ∈ R. Then, by using Theorem 4.2, we have
0 ≥ T�μ ,1(v1(g, p),
∣∣Dμ(g)(vμ1
)∣∣p
p–1 ,∣∣Dμ(g)
(vμ2
)∣∣p
p–1)
– �
= v1(g, p) –|Dμ(g)(vμ1 )|
pp–1 + |Dμ(g)(vμ2 )|
pp–1
2μ– ε
=(
2vμ2 – v
μ1
) pp–1
(1
2μ(p + 1)
{2 –
(1 –
12μ2–1
)p+1–
(1
2μ2–1– 1
)p+1}) –1p–1
×∣∣∣∣g(vμ1 ) + g(v
μ2 )
2–
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx
∣∣∣∣
pp–1
–|Dμ(g)(vμ1 )|
pp–1 + |Dμ(g)(vμ2 )|
pp–1
2μ– ε.
This completes our proof. �
Remark 4.2 Corollary 4.4 with ε = 0 becomes Theorem 13 in
[55].
Corollary 4.5 Theorem 4.2 with |Dμ(g)| to be μ-convex leads
to
∣∣∣∣g(vμ1 ) + g(v
μ2 )
2–
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx
∣∣∣∣
≤ vμ2 – v
μ1
2
(1
2μ(p + 1)
{2 –
(1 –
12μ2–1
)p+1–
(1
2μ2–1– 1
)p+1}) 1p
×(
μ|Dμ(g)(vμ1 )|p
p–1 + |Dμ(g)(vμ2 )|p
p–1
μ(μ + 1)
) p–1p
. (4.7)
Proof By making use of w(ημ) = 1 in (2.9), we get
T�μ ,1(y1, y2, y3) = y1 –(∫ 1
0η dμη
)y2 –
(∫ 1
0(1 – η) dμη
)y3
= y1 –μy2 + y3μ(μ + 1)
for y1, y2, y3 ∈ R. Then, by using Theorem 4.2, we have
0 ≥ T�μ ,1(v1(g, p),
∣∣Dμ(g)(vμ1
)∣∣p
p–1 ,∣∣Dμ(g)
(vμ2
)∣∣p
p–1)
= v1(g, p) –μ|Dμ(g)(vμ1 )|
pp–1 + |Dμ(g)(vμ2 )|
pp–1
μ(μ + 1).
This rearranges to the required inequality (4.7). �
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Corollary 4.6 Theorem 4.2 with |Dμ(g)| to be h-convex leads
to∣∣∣∣g(vμ1 ) + g(v
μ2 )
2–
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx
∣∣∣∣
≤ vμ2 – v
μ1
2
(1
2μ(p + 1)
{2 –
(1 –
12μ2–1
)p+1–
(1
2μ2–1– 1
)p+1}) 1p
×(∫ 1
0h(ημ
)dμη
) p–1p (∣∣Dμ(g)
(vμ1
)∣∣p
p–1 +∣∣Dμ(g)
(vμ2
)∣∣p
p–1) p–1
p . (4.8)
Proof By making use of (2.11) with w(ημ) = 1, we have
T�μ ,1(y1, y2, y3) = y1 –(∫ 1
0h(ημ
)dμη
)y2 –
(∫ 1
0h(1 – ημ
)dμη
)y3
= y1 –(∫ 1
0h(ημ
)dμη
)(y2 + y3)
for y1, y2, y3 ∈ R. Then, by making use of Theorem 4.2, we
get
0 ≥ T�μ ,1(v1(g, p),
∣∣Dμ(g)
(vμ1
)∣∣p
p–1 ,∣∣Dμ(g)
(vμ2
)∣∣p
p–1)
= v1(g, p) –(∫ 1
0h(ημ
)dμη
)(∣∣Dμ(g)(vμ1
)∣∣p
p–1 +∣∣Dμ(g)
(vμ2
)∣∣p
p–1).
This rearranges to the required inequality (4.8). �
4.2 Midpoint formula inequalities for �μ-convex functionsTheorem
4.3 Let g : [v1, v2] ⊂ R → R be a μ-fractional differentiable
function on (v1, v2)and Dμ(g) be a μ-fractional integrable function
on [v1, v2] with 0 ≤ v1 < v2. If |Dμ(g)| is an�μ-convex function
on [v1, v2] for some �μ ∈F and the function η ∈ [0, 1] → Lw(η)
belongsto L1[0, 1], where w(η) = |P(η)| (P(η) is given in Lemma
2.2, then we have the inequality
T�μ ,w(
1vμ2 – v
μ1
∣∣∣∣
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx – g
(vμ1 + v
μ2
2
)∣∣∣∣,
∣∣Dμ(g)
(vμ1
)∣∣,∣∣Dμ(g)
(vμ2
)∣∣,η)
+∫ 1
0Lw(η) dμη ≤ 0. (4.9)
Proof By using the �μ-convexity of |Dμ(g)|, we have
�μ
(∣∣Dμ(g)(ημvμ1 +
(1 – ημ
)vμ2
)∣∣,∣∣Dμ(g)
(vμ1
)∣∣,∣∣Dμ(g)
(vμ2
)∣∣,η) ≤ 0.
Making use of axiom (Λ̄3) for w(η) = |P(η)|, η ∈ [0, 1], we
obtain
�μ
(w(η)
∣∣Dμ(g)(ημvμ1 +
(1 – ημ
)vμ2
)∣∣, w(η)∣∣Dμ(g)
(vμ1
)∣∣, w(η)∣∣Dμ(g)
(vμ2
)∣∣,η) ≤ 0.
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Integrating over [0, 1] according to η and by making use of
axiom (Λ̄2), we can obtain
T�μ ,w(∫ 1
0w(η)
∣∣Dμ(g)
(ημvμ1 +
(1 – ημ
)vμ2
)∣∣dμη,∣∣Dμ(g)
(vμ1
)∣∣,∣∣Dμ(g)
(vμ2
)∣∣,η)
+∫ 1
0Lw(η) dμη ≤ 0.
From Lemma 2.2, we have
∣∣∣∣
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx – g
(vμ1 + v
μ2
2
)∣∣∣∣
≤ (vμ2 – vμ1)∫ 1
0w(η)
∣∣Dμ(g)(ημvμ1 +
(1 – ημ
)vμ2
)∣∣dμη.
Since T�μ ,w is nondecreasing according to the first variable,
then we can deduce
T�,w(
1vμ2 – v
μ1
∣∣∣∣
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx – g
(vμ1 + v
μ2
2
)∣∣∣∣,
∣∣Dμ(g)
(vμ1
)∣∣,∣∣Dμ(g)
(vμ2
)∣∣,η)
+∫ 1
0Lw(η) dμη ≤ 0.
This completes the proof of (4.9). �
Corollary 4.7 Theorem 4.3 with |Dμ(g)| to be ε-convex leads
to∣∣∣∣
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx – g
(vμ1 + v
μ2
2
)∣∣∣∣
≤ (vμ2 – vμ1)( |Dμ(g)(vμ1 )| + |Dμ(g)(vμ2 )|
8μ+
4μ – 3μ
ε
). (4.10)
Proof By making use of w(ημ) = |P(η)| in (2.7) as well as
Definition 2.3, we get∫ 1
0Lw(η) dμη = ε
∫ 1
0
(1 –
∣∣P(η)
∣∣)dμη =
34μ
ε.
Then, by making use (2.6) with w(ημ) = |P(η)|, we get
T�μ ,w(y1, y2, y3) = y1 –(∫ 1
0ημ
∣∣P(η)∣∣dμη
)y2 –
(∫ 1
0
(1 – ημ
)∣∣P(η)∣∣dμη
)y3 – ε
= y1 –y2 + y3
8μ– ε,
for y1, y2, y3 ∈ R. Thus, by using Theorem 4.3, we have
0 ≥ T�,w(
1vμ2 – v
μ1
∣∣∣∣
μ
vμ2 – vμ1
∫ v2
ag(xμ
)dμx – g
(vμ1 + v
μ2
2
)∣∣∣∣,
∣∣Dμ(g)
(vμ1
)∣∣,∣∣Dμ(g)
(vμ2
)∣∣,η)
+∫ 1
0Lw(η) dμη
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=1
vμ2 – vμ1
∣∣∣∣
μ
vμ2 – vμ1
∫ v2
ag(xμ
)dμx – g
(vμ1 + v
μ2
2
)∣∣∣∣
–|Dμ(g)(vμ1 )| + |Dμ(g)(vμ2 )|
8μ– ε +
34μ
ε.
This completes the proof of (4.10). �
Remark 4.3 Corollary 4.7 with ε = 0 becomes Theorem 14 in
[55].
Corollary 4.8 Theorem 4.3 with |Dμ(g)| to be μ-convex leads
to∣∣∣∣
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx – g
(vμ1 + v
μ2
2
)∣∣∣∣
≤ (vμ2 – vμ1){ μ
(μ + 1)(2μ + 1)
(1 –
121/μ + 1
)∣∣Dμ(g)
(vμ1
)∣∣
+[
14μ
–μ
(μ + 1)(2μ + 1)
(1 –
121/μ + 1
)]∣∣Dμ(g)
(vμ2
)∣∣}
. (4.11)
Proof By making use of w(ημ) = |P(η)| in (2.9), we get
T�μ ,w(y1, y2, y3) = y1 –(∫ 1
0t∣∣P(η)
∣∣dμη
)y2 –
(∫ 1
0(1 – η)
∣∣P(η)
∣∣dμη
)y3
= y1 –μ
(μ + 1)(2μ + 1)
(1 –
121/μ + 1
)y2
–[
14μ
–μ
(μ + 1)(2μ + 1)
(1 –
121/μ + 1
)]y3
for y1, y2, y3 ∈ R. It follows from Theorem 4.3 that
0 ≥ T�μ ,w(
1vμ2 – v
μ1
∣∣∣∣
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx – g
(vμ1 + v
μ2
2
)∣∣∣∣,
∣∣Dμ(g)(vμ1
)∣∣,∣∣Dμ(g)
(vμ2
)∣∣,η)
+∫ 1
0Lw(ημ) dμη
=1
vμ2 – vμ1
∣∣∣∣
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx – g
(vμ1 + v
μ2
2
)∣∣∣∣
–μ
(μ + 1)(2μ + 1)
(1 –
121/μ + 1
)∣∣Dμ(g)(vμ1
)∣∣
–[
14μ
–μ
(μ + 1)(2μ + 1)
(1 –
121/μ + 1
)]∣∣Dμ(g)
(vμ2
)∣∣.
This rearranges to the required inequality (4.11). �
Corollary 4.9 Theorem 4.3 with |Dμ(g)| to be h-convex leads
to∣∣∣∣
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx – g
(vμ1 + v
μ2
2
)∣∣∣∣
≤ (vμ2 – vμ1)(∫ 1
0h(ημ
)∣∣1 – 2ημ∣∣dμη
)(∣∣Dμ(g)
(vμ1
)∣∣ +∣∣Dμ(g)
(vμ2
)∣∣). (4.12)
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Proof By making use of w(ημ) = |P(η)| in (2.11), we get
T�μ ,w(y1, y2, y3) = y1 –(∫ 1
0h(ημ
)∣∣P(η)∣∣dμη
)y2 –
(∫ 1
0h(1 – ημ
)∣∣P(η)∣∣dμη
)y3
= y1 –(∫ 1
0h(ημ
)∣∣P(η)∣∣dμη
)(y2 + y3)
for y1, y2, y3 ∈ R. Then, by using Theorem 4.3, we get
0 ≥ T�μ ,w(
1vμ2 – v
μ1
∣∣∣∣
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx – g
(vμ1 + v
μ2
2
)∣∣∣∣,
∣∣Dμ(g)
(vμ1
)∣∣,∣∣Dμ(g)
(vμ2
)∣∣,η)
+∫ 1
0Lw(ημ) dμη
=1
vμ2 – vμ1
∣∣∣∣
μ
vμ2 – vμ1
∫ v2
v1g(xμ
)dμx – g
(vμ1 + v
μ2
2
)∣∣∣∣
–(∫ 1
0h(ημ
)∣∣P(η)∣∣dμη
)(∣∣Dμ(g)
(vμ1
)∣∣ +∣∣Dμ(g)
(vμ2
)∣∣),
which rearranges to the required inequality (4.12). �
5 Application testIn this section we give some applications of
our theorems to the special means for thepositive numbers v1 > 0
and v2 > 0:
• Arithmetic mean:
A(v1, v2) =v1 + v2
2.
• Harmonic mean:
H = H(v1, v2) =2v1v2
v1 + v2, v1, v2 > 0.
• Logarithmic mean:
L(v1, v2) =v2 – v1
ln |v2| – ln |v1| , |v1| = |v2|, v1, v2 = 0, v1, v2 ∈ R.
• Generalized log-mean:
Lp(v1, v2) =[
v2p+1 – v1p+1
(p + 1)(v2 – v1)
] 1p
, p ∈ Z \ {–1, 0}, v1, v2 ∈ R, v1 = v2.
Proposition 5.1 Let μ ∈ (0, 1], v1, v2 ∈ R with 0 < v1 <
v2. Then we have
�
(A–1
(vμ1 , v
μ2),L–1
(vμ1 , v
μ2),L–1
(vμ1 , v
μ2),
12
)≤ 0, (5.1)
T�,w(
2L–1(vμ1 , v
μ2),
12H–1
(vμ1 , v
μ2),
12H–1
(vμ1 , v
μ2))
≤ 0. (5.2)
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Proof The assertion follows from Theorem 3.1 and a simple
computation applied to g(x) =1x , x ∈ [v1, v2], where g is convex
and therefore �-convex function on [v1, v2] according to� defined
in (1.3) with ε = 0. �
Proposition 5.2 Let μ ∈ (0, 1], v1, v2 ∈ R with 0 < v1 <
v2. Then we have
A–1(vμ1 , v
μ2) ≤L–1(vμ1 , vμ2
) ≤H–1(vμ1 , vμ2). (5.3)
Proof The assertion follows from Corollary 3.1 and a simple
computation applied to g(x) =1x , x ∈ [v1, v2], where it is easy to
check that g is convex and therefore ε-convex with ε = 0.�
Proposition 5.3 Let μ ∈ (0, 1], v1, v2 ∈ R with 0 < v1 <
v2. Then we have
�
(An
(vμ1 , v
μ2),Lnn
(vμ1 , v
μ2),Lnn
(vμ1 , v
μ2),
12
)≤ 0, (5.4)
T�,w(2Lnn
(vμ1 , v
μ2), vnμ1 + v
nμ2 , v
nμ1 + v
nμ2
). (5.5)
Proof The assertion follows from Theorem 3.1 and a simple
computation applied to g(x) =xn, x ∈ [v1, v2] with n ≥ 2, where g
is convex and therefore �-convex function on [v1, v2]according to �
defined in (1.3) with ε = 0. �
Proposition 5.4 Let μ ∈ (0, 1], v1, v2 ∈ R with 0 < v1 <
v2. Then we have
∣∣H–1
(vμ1 , v
μ2)
–L–1(vμ1 , v
μ2)∣∣ ≤ v
μ2 – v
μ1
2
(23μ2 + 6 × 2μ2 – 8
6μ × 23μ2)(
v–μ(1+μ)1 + v–μ(1+μ)2
). (5.6)
Proof The assertion follows from Corollary 4.1 and a simple
computation applied to g(x) =– 1x , x ∈ [v1, v2], where it is easy
to check that |Dμ(g)| is convex and therefore ε-convex withε = 0.
�
Proposition 5.5 Let μ ∈ (0, 1], v1, v2 ∈ R with 0 < v1 <
v2. Then we have
∣∣L–1
(vμ1 , v
μ2)
– A–1(vμ1 , v
μ2)∣∣ ≤ (vμ2 – vμ1
)(v–μ(1+μ)1 + v–μ(1+μ)2
8μ
). (5.7)
Proof The assertion follows from Corollary 4.7 and a simple
computation applied to g(x) =– 1x , x ∈ [v1, v2], where it is easy
to check that |Dμ(g)| is convex and therefore ε-convex withε = 0.
�
6 Three illustrative plotsIn this section, we give three plots
of three dimensions to the above propositions in theprevious
section.
• Fig. 1 represents Proposition 5.2 with μ = 12 , v1 = x, v2 =
y.• Fig. 2 represents Proposition 5.4 μ = 12 , v1 = x, v2 = y.•
Fig. 3 represents Proposition 5.5 μ = 12 , v1 = x, v2 = y.
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Page 22 of 25
Figure 1 Figure representation for Proposition 5.2
Figure 2 Figure representation for Proposition 5.4
Figure 3 Figure representation for Proposition 5.5
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Page 23 of 25
7 ConclusionIntroducing new definitions in the calculus will
always open new doors in the field of sci-ence and technology. The
use of these new definitions in mathematical analysis
alwaysrequires the presentation of integral inequalities related to
them in order to find the ex-istence and uniqueness of such
problems. One of the new definitions presented for localfractional
calculus is conformable fractional operator. In this study, we have
consideredthe Hermite–Hadamard integral inequalities in the context
of conformable fractional cal-culus. Also, we have introduced the
notion of �μ-convexity. For this, we have establishedsome
Hermite–Hadamard inequalities and related results in the contexts
of fractional cal-culus and conformable fractional calculus.
AcknowledgementsWe express our special thanks to the editor and
referees of this manuscript.
DeclarationsNot applicable.
FundingNot applicable.
Availability of data and materialsNot applicable.
Competing interestsThe authors declare no conflict of
interests.
Authors’ contributionsThe four authors have contributed equally
to the attempt. All four authors have read carefully and approved
the finalversion of the study.
Author details1Institute of Space Sciences, P.O. Box, MG-23,
Magurele-Bucharest, R 76900, Romania. 2Department of
Mathematics,Çankaya University, Ankara, Turkey. 3Department of
Medical Research, China Medical University Hospital, China
MedicalUniversity, Taichung, Taiwan. 4Department of Mathematics,
College of Education, University of Sulaimani, Sulaimani,Kurdistan
Region, Iraq. 5Facultad de Ciencias Exactas y Naturales, Escuela de
Ciencias Fisicas y Matematica, PontificiaUniversidad Católica del
Ecuador, Quito, Ecuador.
Publisher’s NoteSpringer Nature remains neutral with regard to
jurisdictional claims in published maps and institutional
affiliations.
Received: 23 April 2020 Accepted: 13 July 2020
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Some modifications in conformable fractional integral
inequalitiesAbstractMSCKeywords
IntroductionConformable fractional operators and
Fµ-convexityHermite-Hadamard inequalities for F-convex
functionsHermite-Hadamard inequalities for Fµ-convex
functionsTrapezoidal inequalities for Fµ-convex functionsMidpoint
formula inequalities for Fµ-convex functions
Application testThree illustrative
plotsConclusionAcknowledgementsDeclarationsFundingAvailability of
data and materialsCompeting interestsAuthors' contributionsAuthor
detailsPublisher's NoteReferences