Abstract
The prevalence of the use of integral inequalities has dramatically influenced the evolution of mathematical analysis. The use of these useful tools leads to faster advances in the presentation of fractional calculus. This article investigates the Hermite–Hadamard integral inequalities via the notion of Ϝ-convexity. After that, we introduce the notion of \(\digamma _{\mu}\)-convexity in the context of conformable operators. In view of this, we establish some Hermite–Hadamard integral inequalities (both trapezoidal 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.
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1 Introduction
For any \(v_{1}, v_{2}\in [a,b]\) and \(\ell \in [0,1]\), the real-valued function g on an interval \([a,b]\) is called a convex function if the following holds:
The theory and application of convexity has a close relationship with theory and application of inequalities or integral inequalities. The convex function (1.1) has been extended and generalized in several directions, such as pseudo-convex [1], quasi-convex [2], strongly convex [3], ϵ-convex [4], s-convex [5], h-convex [6, 7], \((\alpha ,m)\)-convex [8, 9], invex and preinvex [10–12], and other kinds of convex functions by a number of mathematicians; see [13–21] for more details.
Integral inequalities form an essential field of study among the field of mathematical analysis. They have been vital in providing bounds to solve some boundary value problems in fractional calculus, and in establishing the existence and uniqueness of solutions for certain fractional differential equations. Convexity plays an important role in the field of integral inequality due to the behavior of its definition. Also, there is a strong connection between convexity and integral inequality. For this reason, many known integral inequalities have been established in the literature. The Hermite–Hadamard (HH) inequality is the most well known one: for an \(L^{1}\) convex function \(g:\mathcal{I}\subseteq \mathtt{R}\to \mathtt{R}\) with \(v_{1}, v_{2}\in \mathcal{I}\), \(v_{1}< v_{2}\), the HH inequality is defined as follows:
A huge number of researchers in the field of applied mathematics have dedicated their interest to generalize, improve, refine, counterpart, and extend HH inequality (1.2) for various types of convex functions; see e.g. [22–30].
Recently, Samet [31] introduced a new notion of convexity for certain functions that depends on some axioms. This often generalizes various types of convexity e.g. ϵ-convex functions, α-convex functions, h-convex functions, and so on. Also, for further details, visit [16, 32].
Throughout our study, we suppose that \(\mathcal{I}\subseteq \mathtt{R}\) (R the set of real numbers), \(\mathcal{V}= \{(\eta _{1},\eta _{2}); \eta _{\ell }\in [v_{1},v_{2}], \ell =1,2 \}\) and \(\bar{\mathfrak{R}}= \{(\eta _{1},\eta _{2},\eta _{3}); \eta _{i} \in \mathtt{R}, \ell =1,2,3 \}\). Then the family of \(\mathcal{F}\) of functions \(\digamma :\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) satisfies the major axioms [31]:
- (\(\varLambda _{1}\)):
-
If \(\mathbf{y}_{\ell }\in L^{1}(0,1)\), \(\ell =1,2,3\), then for every \(\gamma \in [0,1]\) we have
$$\begin{aligned}& \int _{0}^{1} \digamma \bigl( \mathbf{y}_{1}(\eta ),\mathbf{y}_{2}(\eta ), \mathbf{y}_{3}(\eta ),\gamma \bigr)\,d\eta \\& \quad =\digamma \biggl( \int _{0}^{1} \mathbf{y}_{1}(\eta )\,d\eta , \int _{0}^{1} \mathbf{y}_{2}(\eta )\,d\eta , \int _{0}^{1} \mathbf{y}_{3}(\eta )\,d\eta ,\gamma \biggr). \end{aligned}$$ - (\(\varLambda _{2}\)):
-
For every \(u\in L^{1}(0,1)\), \(w\in L^{\infty }(0,1)\), and \((z_{1},z_{2})\in \mathtt{R}^{2}\), we have
$$ \int _{0}^{1}\digamma \bigl(w(\eta ) u(\eta ),w( \eta )z_{1},w(\eta )z_{2}, \eta \bigr)\,d\eta = \mathcal{T}_{\digamma ,w} \biggl( \int _{0}^{1} w(\eta ) u(\eta )\,d\eta ,z_{1},z_{2} \biggr), $$where \(\mathcal{T}_{\digamma ,w}:\bar{\mathfrak{R}}\to \mathtt{R}\) is a function depending on \((F,w)\). Moreover, it is a nondecreasing function according to the first variable.
- (\(\varLambda _{3}\)):
-
For any \((w,\mathbf{y}_{1},\mathbf{y}_{2},\mathbf{y}_{3})\in \mathtt{R}^{4}\), \(\mathbf{y}_{4}\in [0,1]\), we have
$$ w\digamma (\mathbf{y}_{1},\mathbf{y}_{2}, \mathbf{y}_{3},\mathbf{y}_{4}) =\digamma (w \mathbf{y}_{1},w\mathbf{y}_{2},w\mathbf{y}_{3}, \mathbf{y}_{4})+L_{w}, $$where \(L_{w}\in \mathtt{R}\) is a constant (depending on w).
Definition 1.1
Let \(g: [v_{1},v_{2}]\subseteq \mathtt{R}\to \mathtt{R}\) with \(v_{1}< v_{2}\) be a function, then we say that g is a convex function according to \(\digamma \in \mathcal{F}\) (or briefly Ϝ-convex function) iff
Remark 1.1
Suppose that \((v_{1},v_{2})\in \mathtt{R}^{2}\) with \(v_{1}< v_{2}\),
(i) if \(g:[v_{1},v_{2}]\subset \mathtt{R}\to \mathtt{R}\) is an ε-convex function, or equivalently [26]
then we define the functions \(\digamma :\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as follows:
and \(\mathcal{T}_{\digamma ,w}:\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as
For
we can observe that \(\digamma \in \mathcal{F}\) and
and this tells us g is an Ϝ-convex function. In a particular case, we take \(\varepsilon =0\) to show that g is an Ϝ-convex function according to Ϝ when g is assumed to be a convex function.
(ii) If \(g:[v_{1},v_{2}]\subset \mathtt{R}\to \mathtt{R}\) is a μ-convex function with \(\mu \in (0,1]\), or equivalently
Then we define the function \(\digamma :\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as follows:
and \(\mathcal{T}_{\digamma ,w}:\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as
For \(L_{w}=0\), we can observe that \(\digamma \in \mathcal{F}\) and
or g is an Ϝ-convex function.
(iii) If \(h:\mathcal{I}\to \mathtt{R}\) is a function and it is not identically 0, where \((0,1)\subseteq \mathcal{I}\). Also, suppose that \(g:[v_{1},v_{2}]\subset \mathcal{I}\to [0,\infty )\) is an h-convex function, that is,
Then we define the functions \(\digamma :\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as follows:
and \(\mathcal{T}_{\digamma ,w}:\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as
For \(L_{w}=(1-w)\varepsilon \), we can observe that \(\digamma \in \mathcal{F}\) and
or we can say that g is an Ϝ-convex function.
In recent years, many possible inequalities have been proposed in the context of fractional calculus including the midpoint and trapezoidal formula inequalities and inequalities for ε-convexity, α-convexity, \((\alpha ,m)\)-convexity, and h-convexity; see [26, 31, 33] for more details.
2 Conformable fractional operators and \(\digamma _{\mu }\)-convexity
In the last fifteen years, the definition of fractional calculus has been more appropriate to describe historical dependence processes than the local limit definitions of integer ordinary differential equations (ODEs) or partial differential equations (PDEs), and has received more and more attention in many mathematical and physical fields, see for details [34–44]. Differential equations of fractional order are more accurate than differential equations 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 have dedicated themselves to the study of fractional calculus. The most commonly used fractional calculus definitions are Riemann–Liouville definition, Caputo definition, and conformable fractional definition in basic mathematical and engineering application research. In the present paper, we deal with the conformable fractional definition [45–47] in order to obtain our results.
In this section, we recall some preliminaries and properties on conformable fractional calculus. For further details and applications, see the previously published articles [33, 45–54].
Definition 2.1
([47])
Let \(g: [0,\infty )\to \mathtt{R}\), then the μth order conformable derivative of g at η is defined by
For μ-differentiable function g in some \((0,\mu ), \mu >0\), \(\lim_{t\to 0^{+}}g^{(\mu )}(\eta )\) exist, define
Furthermore, if g is differentiable, then we have
Observe that we can write \(g^{(\mu )}(\eta )\) for \(\frac{d_{\mu }}{d_{\mu }\eta }(g(\eta ))\) or simply \(D_{\mu }(g)(\eta )\) to denote a μth order conformable derivative of g at η. Furthermore, if the μth order conformable derivative of g exists, then we can simply say g is μ-differentiable.
Theorem 2.1
([48])
Assume that\(\mu \in (0,1]\)andf, gare twoμ-differentiable functions at a point\(\eta >0\). Then we have:
-
1.
\(D_{\mu}(v_{1}f+v_{2}g)=v_{1}D_{\mu}(f)+v_{2}D_{\mu}(g)\)for all\(v_{1},v_{2}\in \mathtt{R}\),
-
2.
\(D_{\mu}(fg)=fD_{\mu}(g)+gD_{\mu}(f)\),
-
3.
\(D_{\mu}(\frac{f}{g} )=\frac{gD_{\mu}(f)-fD_{\mu}(g)}{g^{2}}\),
-
4.
\(D_{\mu }(c)=0\)for each constant function, namely\(g(\eta )=c\),
-
5.
\(D_{\mu } (1 )=0\),
-
6.
\(D_{\mu } (\frac{1}{\mu }\eta ^{\mu } )=1\).
Some basic properties of conformable operator are now stated, which are useful in what follows.
Definition 2.2
([47])
Assume that \(\mu \in (0,1]\), \(0\leq v_{1}\leq v_{2}\), and \(g: [v_{1},v_{2}]\subset \mathtt{R}\to \mathtt{R}\), then we say that a function g is μ-fractional integrable on the interval \([v_{1},v_{2}]\) if the following integral
exists and is finite.
Remark 2.1
-
(a)
We indicate by \(L_{\mu }^{1}([v_{1},v_{2}])\) all μ-fractional integrable functions on an interval \([v_{1},v_{2}]\).
-
(b)
The usual Riemann improper integral has the form
$$ I_{\mu }^{v_{1}} (g) (\eta )=I_{1}^{v_{1}} \bigl(\eta ^{\mu -1}g \bigr)= \int _{v_{1}}^{t} x^{\mu -1}g(x)\,dx,\quad \mu \in (0,1]. $$(2.4)
Theorem 2.2
Let\(g:(v_{1},v_{2})\to \mathtt{R}\)be differentiable and\(\mu \in (0,1]\). Then, for all\(\eta >v_{1}\), we have
Theorem 2.3
([51])
Suppose that\(g:[v_{1},\infty )\to \mathtt{R}\)such that\(g^{(n)}\)is continuous. Then, for each\(\eta >v_{1}\), we have
which is called the inverse property.
Theorem 2.4
Let\(g:[v_{1},v_{2}]\subset \mathtt{R}\to \mathtt{R}\)be two functions withfgis differentiable. Then
Theorem 2.5
Let\(f,g:[v_{1},v_{2}]\subset \mathtt{R}\to \mathtt{R}\)be a continuous function on\([v_{1},v_{2}]\)and with\(0\leq v_{1}\leq v_{2}\). Then
It is time to define the concept of \(\digamma _{\mu }\)-convexity on conformable integrals, namely the family of \(\digamma _{\mu }\).
The family of \(\digamma _{\mu }\) of functions \(\digamma _{\mu }:\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) satisfies the major axioms:
- (\(\bar{\varLambda }_{1}\)):
-
If \(\mathbf{y}_{\ell }\in L^{1}(0,1)\), \(\ell =1,2,3\), then for every \(\gamma \in [0,1]\) we have
$$\begin{aligned} &\int _{0}^{1} \digamma _{\mu }\bigl( \mathbf{y}_{1}(\eta ),\mathbf{y}_{2}( \eta ), \mathbf{y}_{3}(\eta ),\gamma \bigr)\,d\eta \\ &\quad =\digamma _{\mu } \biggl( \int _{0}^{1} \mathbf{y}_{1}(\eta )\,d\eta , \int _{0}^{1} \mathbf{y}_{2}( \eta )\,d\eta , \int _{0}^{1} \mathbf{y}_{3}(\eta )\,d\eta ,\gamma \biggr). \end{aligned}$$ - (\(\bar{\varLambda }_{2}\)):
-
For every \(u\in L^{1}(0,1)\), \(w\in L^{\infty }(0,1)\), and \((z_{1},z_{2})\in \mathtt{R}^{2}\), we have
$$ \int _{0}^{1} \digamma _{\mu }\bigl(w( \eta ) u(\eta ),w(\eta )z_{1},w( \eta )z_{2},\eta \bigr)\,d\eta =\mathcal{T}_{\digamma _{\mu },w} \biggl( \int _{0}^{1} w(\eta ) u(\eta )\,d\eta ,z_{1},z_{2} \biggr), $$where \(\mathcal{T}_{\digamma _{\mu },w}:\bar{\mathfrak{R}}\to \mathtt{R}\) is a nondecreasing function according to the first variable which depends on \((\digamma _{\mu },w)\).
- (\(\bar{\varLambda }_{3}\)):
-
For any \((w,\mathbf{y}_{1},\mathbf{y}_{2},\mathbf{y}_{3})\in \mathtt{R}^{4}\), \(\mathbf{y}_{4}\in [0,1]\), we have
$$ w\digamma _{\mu }(\mathbf{y}_{1},\mathbf{y}_{2}, \mathbf{y}_{3}, \mathbf{y}_{4})=\digamma _{\mu }(w\mathbf{y}_{1},w\mathbf{y}_{2},w \mathbf{y}_{3},\mathbf{y}_{4})+L_{w}, $$where \(L_{w}\in \mathtt{R}\) is a constant (depending on w).
Definition 2.3
Let \(\mu \in (0,1]\) and \(g: [v_{1},v_{2}]\subset \mathtt{R}\to \mathbb{R}\) with \(v_{1}< v_{2}\) be a function, then we say g is a conformable convex function according to \(\digamma _{\mu }\in \mathcal{F}\) (or briefly \(\digamma _{\mu }\)-conformable convex function) if
Remark 2.2
Suppose that \((v_{1},v_{2})\in \mathtt{R}^{2}\) with \(v_{1}< v_{2}\).
(i) Let \(g:[v_{1},v_{2}]\subset \mathtt{R}\to \mathtt{R}\) be an ε-conformable convex function, or equivalently
Then we define the function \(\digamma _{\mu }:\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as follows:
and \(\mathcal{T}_{\digamma _{\mu },w}:\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as
For
it can be observed that \(\digamma \in \mathcal{F}\) and
or in another meaning g is an Ϝ-conformable convex function. In particular, g is an Ϝ-conformable convex function according to Ϝ for \(\varepsilon =0\) when g is a conformable convex function.
(ii) Let \(g:[v_{1},v_{2}]\subset \mathcal{I}\to \mathtt{R}\) be a μ-conformable convex function \(\mu \in (0,1]\); that is,
Then we define the functions \(\digamma _{\mu }:\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as follows:
and \(\mathcal{T}_{\digamma _{\mu },w}:\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as
For \(L_{w}=0\), we can observe that \(\digamma _{\mu }\in \mathcal{F}\) and
or equivalently g is an \(\digamma _{\mu }\)-conformable convex function.
(iii) Let \(h:\mathcal{I}\to \mathtt{R}\) be a function, which is not identically 0, where \((0,1)\subseteq \mathcal{I}\). Let \(g:[v_{1},v_{2}]\subset \mathcal{I}\to [0,\infty )\) be an h-conformable convex function, or let
Then we define the functions \(\digamma _{\mu }:\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as follows:
and \(\mathcal{T}_{\digamma _{\mu },w}:\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as
For \(L_{w}=(1-w)\varepsilon \), we can observe that \(\digamma _{\mu }\in \mathcal{F}\) and
or equivalently we can say g is an \(\digamma _{\mu }\)-conformable convex function.
For the conformable operators, we recall some early findings in the earlier literature which may help us in finding our main results. For example in [55], Sarikaya et al. investigated new results for the conformable fractional operator, and their results are as follows.
Theorem 2.6
([55, Theorem 11])
Let\(\mu \in (0,1]\)and\(g: [v_{1},v_{2}]\subset \mathtt{R}\to \mathbb{R}\)be aμ-fractional differentiable function on\((v_{1},v_{2})\)with\(0\leq v_{1}< v_{2}\). Then we have
Lemma 2.1
([55, Lemma 3])
Let\(\mu \in (0,1]\)and\(g: [v_{1},v_{2}]\subset \mathtt{R}\to \mathbb{R}\)be aμ-fractional differentiable function on\((v_{1},v_{2})\)with\(0\leq v_{1}< v_{2}\). If\(D_{\mu }(g)\)is aμ-fractional integrable function on\([v_{1},v_{2}]\), then we have
Lemma 2.2
([55, Lemma 4])
Let\(\mu \in (0,1]\)and\(g: [v_{1},v_{2}]\subset \mathtt{R}\to \mathbb{R}\)be aμ-fractional differentiable function on\((v_{1},v_{2})\)with\(0\leq v_{1}< v_{2}\). If\(D_{\mu }(g)\)is aμ-fractional integrable function on\([v_{1},v_{2}]\), then we have
where
In view of these indices, we investigate some new inequalities of HH type for the Ϝ and \(\digamma _{\mu }\)-convex functions involving conformable fractional operators in this attempt. Specifically, we investigate some inequalities of trapezoidal and midpoint type.
3 Hermite–Hadamard inequalities for Ϝ-convex functions
This section deals with the investigation of HH-type inequalities for Ϝ-convex functions.
Theorem 3.1
Let\(g:[v_{1},v_{2}]\subset \mathtt{R}\to \mathtt{R}\)be aμ-fractional differentiable function on\((v_{1},v_{2})\)with\(0\leq v_{1}< v_{2}\). Ifgis anϜ-convex function on\([v_{1},v_{2}]\)for some\(\digamma \in \mathcal{F}\), then
Proof
The Ϝ-convexity of g leads to
For the values \(x=\eta ^{\mu }v_{1}^{\mu }+ (1-\eta ^{\mu } )v_{2}^{\mu }\) and \(y= (1-\eta ^{\mu } )v_{1}^{\mu }+\eta ^{\mu }v_{2}^{\mu }\), where \(\eta \in [0,1]\), we obtain
Multiplying this inequality \(w(\eta )=1\) and making use of axiom (\(\varLambda _{3}\)), we get
Integrating over \([0,1]\) according to η and making use of axiom (\(\varLambda _{1}\)), we get
that is,
where we have used
This completely gives the proof of (3.1). On the other hand, since g is Ϝ-convex, we have
We make use of the linearity of Ϝ to get
Applying the axiom (\(\varLambda _{3}\)) for \(w(\eta )=1\), we get
Integrating over \([0,1]\) according to η and making use of axiom (\(\varLambda _{2}\)) we get
This completes the proof of (3.2). Thus, the proof of Theorem 3.1 is completed. □
Corollary 3.1
Theorem 3.1withgto beε-convex leads to
Proof
By making use of \(w(\eta )=1\) in (1.5), we get
Making use of (1.3), (3.1), and (3.4), we get
or equivalently,
Making use of \(w(\eta )=1\) in (1.4), we have
Now, from (3.2) and (3.5), we can deduce
This gives
This ends the proof of (3.3). □
Remark 3.1
Inequality (3.3) with \(\varepsilon =0\) becomes inequality (2.12).
Corollary 3.2
Theorem 3.1withgto beh-convex leads to
Proof
Making use (1.5) and (3.1) with \(L_{w(\eta )}=0\), we have
or equivalently,
Now, by making use of \(w(\eta )=1\) in (1.4) and (3.2), we get
This gives
Thus, the proof of (3.6) is completed. □
4 Hermite–Hadamard inequalities for \(\digamma _{\mu }\)-convex functions
Here, we deal with the investigation of HH-type inequalities for \(\digamma _{\mu }\)-convex functions. This section is separated into two subsections: a section for the trapezoidal formula inequality and the other one for the midpoint formula inequality of HH type, respectively.
4.1 Trapezoidal inequalities for \(\digamma _{\mu }\)-convex functions
Theorem 4.1
Let\(g:[v_{1},v_{2}]\subset \mathtt{R}\to \mathtt{R}\)be aμ-fractional differentiable function on\((v_{1},v_{2})\)and\(D_{\mu }(g)\)be aμ-fractional integrable function on\([v_{1},v_{2}]\)with\(0\leq v_{1}< v_{2}\). If\(|D_{\mu }(g)|\)is an\(\digamma _{\mu }\)-convex function on\([v_{1},v_{2}]\)for some\(\digamma _{\mu }\in \mathcal{F}\)and the function\(\eta \in [0,1]\to L_{w(\eta ^{\mu })} \)belongs to\(L_{1}[0,1]\), where\(w(\eta ^{\mu })= \vert 1-2\eta ^{\mu } \vert \), then we have the inequality
Proof
The \(\digamma _{\mu }\)-convexity of \(|D_{\mu }(g)|\) leads to
By applying axiom (\(\bar{\varLambda }_{3}\)) for \(w(\eta ^{\mu })= \vert 1-2\eta ^{\mu } \vert \), \(\eta \in [0,1]\), we can deduce
Integrating over \([0,1]\) according to η and by making use of axiom (\(\bar{\varLambda }_{2}\)), we obtain
From Lemma 2.1, we have
Since \(\mathcal{T}_{\digamma _{\mu },w}\) is nondecreasing according to the first variable, then we can deduce
which ends the proof of (4.1). □
Corollary 4.1
Theorem 4.1 with \(|D_{\mu }(g)|\) to be ε conformable convex leads to
Proof
We know that any ε-convex is \(\digamma _{\mu }\)-convex. So, by making use of \(w(\eta ^{\mu })= \vert 1-2\eta ^{\mu } \vert \) in (2.7) and by using Definition 2.2, we get
By making use of \(w(\eta ^{\mu })= \vert 1-2\eta ^{\mu } \vert \) in (2.6), we get
for \(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\in \mathtt{R}\). By making use of Theorem 4.1, we get
This rearranges to the required inequality (4.2). □
Remark 4.1
Corollary 4.1 with \(\varepsilon =0\) becomes Theorem 13 in [55].
Corollary 4.2
Theorem 4.1with\(|D_{\mu }(g)|\)to beμ-conformable convex leads to
Proof
We know that any μ-convex is \(\digamma _{\mu }\)-convex. So, by making use of \(w(\eta ^{\mu })= \vert 1-2\eta ^{\mu } \vert \) in (2.9), we get
for \(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\in \mathtt{R}\). Then, by applying Theorem 4.1, we have
This rearranges to the required inequality (4.3). □
Corollary 4.3
Theorem 4.1with\(|D_{\mu }(g)|\)to beh-conformable convex leads to
Proof
It is known that every μ-convex is \(\digamma _{\mu }\)-convex. So, by making use of \(w(\eta ^{\mu })= \vert 1-2\eta ^{\mu } \vert \) in (2.11), we get
for \(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\in \mathtt{R}\). Then, by using Theorem 4.1, we get
This completes the proof of (4.4). □
Theorem 4.2
Let\(g:[v_{1},v_{2}]\subset \mathtt{R}\to \mathtt{R}\)be aμ-fractional differentiable function on\((v_{1},v_{2})\)and\(D_{\mu }(g)\)be aμ-fractional integrable function on\([v_{1},v_{2}]\)with\(0\leq v_{1}< v_{2}\). If\(|D_{\mu }(g)|^{\frac{p}{p-1}}\)is an\(\digamma _{\mu }\)-convex function on\([v_{1},v_{2}]\)for some\(\digamma _{\mu }\in \mathcal{F}\), then we have
where
Proof
By using the \(\digamma _{\mu }\)-convexity of \(|D_{\mu }(g)|^{\frac{p}{p-1}}\), we have
By making use of \(w(\eta ^{\mu })=1\) in axiom (\(\bar{\varLambda }_{3}\)), we obtain
Then, by making use Lemma of 2.2, we have
Since \(\mathcal{T}_{\digamma _{\mu },w}\) is nondecreasing according to the first variable, then we can deduce
This completes the proof of (4.5). □
Corollary 4.4
Theorem 4.2with\(|D_{\mu }(g)|^{\frac{p}{p-1}}\)to beε-conformable convex leads to
Proof
By making use of \(w(\eta ^{\mu })= \vert 1-2\eta ^{\mu } \vert \) in (2.6) and by Definition 2.3, we get
for \(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\in \mathtt{R}\). Then, by using Theorem 4.2, we have
This completes our proof. □
Remark 4.2
Corollary 4.4 with \(\varepsilon =0\) becomes Theorem 13 in [55].
Corollary 4.5
Theorem 4.2with\(|D_{\mu }(g)|\)to beμ-convex leads to
Proof
By making use of \(w(\eta ^{\mu })=1\) in (2.9), we get
for \(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\in \mathtt{R}\). Then, by using Theorem 4.2, we have
This rearranges to the required inequality (4.7). □
Corollary 4.6
Theorem 4.2with\(|D_{\mu }(g)|\)to beh-convex leads to
Proof
By making use of (2.11) with \(w(\eta ^{\mu })=1\), we have
for \(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\in \mathtt{R}\). Then, by making use of Theorem 4.2, we get
This rearranges to the required inequality (4.8). □
4.2 Midpoint formula inequalities for \(\digamma _{\mu }\)-convex functions
Theorem 4.3
Let\(g:[v_{1},v_{2}]\subset \mathtt{R}\to \mathtt{R}\)be aμ-fractional differentiable function on\((v_{1},v_{2})\)and\(D_{\mu }(g)\)be aμ-fractional integrable function on\([v_{1},v_{2}]\)with\(0\leq v_{1}< v_{2}\). If\(|D_{\mu }(g)|\)is an\(\digamma _{\mu }\)-convex function on\([v_{1},v_{2}]\)for some\(\digamma _{\mu }\in \mathcal{F}\)and the function\(\eta \in [0,1]\to L_{w(\eta )}\)belongs to\(L_{1}[0,1]\), where\(w(\eta )=|P(\eta )|\) (\(P(\eta )\)is given in Lemma 2.2, then we have the inequality
Proof
By using the \(\digamma _{\mu }\)-convexity of \(|D_{\mu }(g)|\), we have
Making use of axiom (\(\bar{\varLambda }_{3}\)) for \(w(\eta )= \vert P(\eta ) \vert \), \(\eta \in [0,1]\), we obtain
Integrating over \([0,1]\) according to η and by making use of axiom (\(\bar{\varLambda }_{2}\)), we can obtain
From Lemma 2.2, we have
Since \(\mathcal{T}_{\digamma _{\mu },w}\) is nondecreasing according to the first variable, then we can deduce
This completes the proof of (4.9). □
Corollary 4.7
Theorem 4.3with\(|D_{\mu }(g)|\)to beε-convex leads to
Proof
By making use of \(w(\eta ^{\mu })=|P(\eta )|\) in (2.7) as well as Definition 2.3, we get
Then, by making use (2.6) with \(w(\eta ^{\mu })=|P(\eta )|\), we get
for \(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\in \mathtt{R}\). Thus, by using Theorem 4.3, we have
This completes the proof of (4.10). □
Remark 4.3
Corollary 4.7 with \(\varepsilon =0\) becomes Theorem 14 in [55].
Corollary 4.8
Theorem 4.3with\(|D_{\mu }(g)|\)to beμ-convex leads to
Proof
By making use of \(w(\eta ^{\mu })=|P(\eta )|\) in (2.9), we get
for \(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\in \mathtt{R}\). It follows from Theorem 4.3 that
This rearranges to the required inequality (4.11). □
Corollary 4.9
Theorem 4.3with\(|D_{\mu }(g)|\)to beh-convex leads to
Proof
By making use of \(w(\eta ^{\mu })=|P(\eta )|\) in (2.11), we get
for \(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\in \mathtt{R}\). Then, by using Theorem 4.3, we get
which rearranges to the required inequality (4.12). □
5 Application test
In this section we give some applications of our theorems to the special means for the positive numbers \(v_{1}>0\) and \(v_{2}>0\):
-
Arithmetic mean:
$$ \mathcal{A}(v_{1},v_{2})=\frac{v_{1}+v_{2}}{2}. $$ -
Harmonic mean:
$$ \mathcal{H}=\mathcal{H}(v_{1},v_{2})= \frac{2v_{1}v_{2}}{v_{1}+v_{2}},\quad v_{1},v_{2}>0. $$ -
Logarithmic mean:
$$ \mathcal{L}(v_{1},v_{2})=\frac{v_{2}-v_{1}}{\ln \vert v_{2} \vert -\ln \vert v_{1} \vert },\quad \vert v_{1} \vert \neq \vert v_{2} \vert , v_{1},v_{2}\neq 0, v_{1},v_{2}\in \mathtt{R}. $$ -
Generalized log-mean:
$$ \mathcal{L}_{p}(v_{1},v_{2})= \biggl[ \frac{{v_{2}}^{p+1}-{v_{1}}^{p+1}}{(p+1)(v_{2}-v_{1})} \biggr]^{ \frac{1}{p}}, \quad p\in \mathbb{Z}\setminus \{-1,0\}, v_{1},v_{2} \in \mathtt{R}, v_{1}\neq v_{2}. $$
Proposition 5.1
Let\(\mu \in (0,1]\), \(v_{1}, v_{2} \in \mathtt{R}\)with\(0< v_{1}< v_{2}\). Then we have
Proof
The assertion follows from Theorem 3.1 and a simple computation applied to \(g(x)=\frac{1}{x}\), \(x\in [v_{1},v_{2}]\), where g is convex and therefore Ϝ-convex function on \([v_{1},v_{2}]\) according to Ϝ defined in (1.3) with \(\varepsilon =0\). □
Proposition 5.2
Let\(\mu \in (0,1]\), \(v_{1}, v_{2} \in \mathtt{R}\)with\(0< v_{1}< v_{2}\). Then we have
Proof
The assertion follows from Corollary 3.1 and a simple computation applied to \(g(x)=\frac{1}{x}\), \(x\in [v_{1},v_{2}]\), where it is easy to check that g is convex and therefore ε-convex with \(\varepsilon =0\). □
Proposition 5.3
Let\(\mu \in (0,1]\), \(v_{1}, v_{2} \in \mathtt{R}\)with\(0< v_{1}< v_{2}\). Then we have
Proof
The assertion follows from Theorem 3.1 and a simple computation applied to \(g(x)=x^{n}\), \(x\in [v_{1},v_{2}]\) with \(n\ge 2\), where g is convex and therefore Ϝ-convex function on \([v_{1},v_{2}]\) according to Ϝ defined in (1.3) with \(\varepsilon =0\). □
Proposition 5.4
Let\(\mu \in (0,1]\), \(v_{1}, v_{2} \in \mathtt{R}\)with\(0< v_{1}< v_{2}\). Then we have
Proof
The assertion follows from Corollary 4.1 and a simple computation applied to \(g(x)=-\frac{1}{x}\), \(x\in [v_{1},v_{2}]\), where it is easy to check that \(|D_{\mu }(g)|\) is convex and therefore ε-convex with \(\varepsilon =0\). □
Proposition 5.5
Let\(\mu \in (0,1]\), \(v_{1}, v_{2} \in \mathtt{R}\)with\(0< v_{1}< v_{2}\). Then we have
Proof
The assertion follows from Corollary 4.7 and a simple computation applied to \(g(x)=-\frac{1}{x}\), \(x\in [v_{1},v_{2}]\), where it is easy to check that \(|D_{\mu }(g)|\) is convex and therefore ε-convex with \(\varepsilon =0\). □
6 Three illustrative plots
In this section, we give three plots of three dimensions to the above propositions in the previous section.
7 Conclusion
Introducing new definitions in the calculus will always open new doors in the field of science and technology. The use of these new definitions in mathematical analysis always requires the presentation of integral inequalities related to them in order to find the existence and uniqueness of such problems. One of the new definitions presented for local fractional calculus is conformable fractional operator. In this study, we have considered the Hermite–Hadamard integral inequalities in the context of conformable fractional calculus. Also, we have introduced the notion of \(\digamma _{\mu}\)-convexity. For this, we have established some Hermite–Hadamard inequalities and related results in the contexts of fractional calculus and conformable fractional calculus.
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Baleanu, D., Mohammed, P.O., Vivas-Cortez, M. et al. Some modifications in conformable fractional integral inequalities. Adv Differ Equ 2020, 374 (2020). https://doi.org/10.1186/s13662-020-02837-0
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DOI: https://doi.org/10.1186/s13662-020-02837-0