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:

$$ g\bigl(\ell v_{1}+(1-\ell )v_{2}\bigr) \leq \ell g(v_{1})+(1-\ell )g(v_{2}). $$
(1.1)

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:

$$ g \biggl(\frac{v_{1}+v_{2}}{2} \biggr)\leq \frac{1}{v_{2}-v_{1}} \int _{v_{1}}^{v_{2}} g(x)\,dx \leq \frac{g(v_{1})+g(v_{2})}{2}. $$
(1.2)

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

$$ \bar{\digamma }\bigl(g\bigl(\eta x+(1-\eta )y\bigr),g(x),g(y),\eta \bigr)\leq 0,\quad (x,y,\eta )\in \mathcal{V}\times [0,1]. $$

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]

$$ g\bigl(\eta x+(1-\eta )y\bigr)\leq \eta g(x)+(1-\eta )g(y)+\varepsilon ,\quad (x,y, \eta )\in \mathcal{V}\times [0,1], $$

then we define the functions \(\digamma :\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as follows:

$$ \digamma (\mathbf{y}_{1},\mathbf{y}_{2}, \mathbf{y}_{3},\mathbf{y}_{4})=\mathbf{y}_{1} -\mathbf{y}_{4}\mathbf{y}_{2}-(1-\mathbf{y}_{4}) \mathbf{y}_{3}-\varepsilon , $$
(1.3)

and \(\mathcal{T}_{\digamma ,w}:\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as

$$ \mathcal{T}_{\digamma ,w}(\mathbf{y}_{1}, \mathbf{y}_{2},\mathbf{y}_{3})=\mathbf{y}_{1}- \biggl( \int _{0}^{1} t w(\eta )\,d\eta \biggr) \mathbf{y}_{2} - \biggl( \int _{0}^{1} (1-\eta )w(\eta )\,d\eta \biggr) \mathbf{y}_{3}-\varepsilon . $$
(1.4)

For

$$ L_{w}=(1-w)\varepsilon , $$
(1.5)

we can observe that \(\digamma \in \mathcal{F}\) and

$$ \digamma \bigl(g\bigl(\eta x+(1-\eta )y\bigr),g(x),g(y),\eta \bigr)=g\bigl(\eta x+(1-\eta )y\bigr)-\eta g(x)-(1-\eta )g(y)-\varepsilon \leq 0, $$

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

$$ g\bigl(\eta x+(1-\eta )y\bigr) \leq \eta ^{\mu}g(x)+ \bigl(1-\eta ^{\mu}\bigr)g(y), \quad (x,y,\eta )\in \mathcal{V}\times [0,1]. $$

Then we define the function \(\digamma :\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as follows:

$$ \digamma (\mathbf{y}_{1},\mathbf{y}_{2}, \mathbf{y}_{3},\mathbf{y}_{4})=\mathbf{y}_{1} -\mathbf{y}_{4}^{\mu}\mathbf{y}_{2}- \bigl(1- \mathbf{y}_{4}^{\mu}\bigr)\mathbf{y}_{3}, $$
(1.6)

and \(\mathcal{T}_{\digamma ,w}:\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as

$$ \mathcal{T}_{\digamma ,w}(\mathbf{y}_{1}, \mathbf{y}_{2},\mathbf{y}_{3}) =\mathbf{y}_{1}- \biggl( \int _{0}^{1} \eta ^{\mu}w(\eta )\,d\eta \biggr)\mathbf{y}_{2} - \biggl( \int _{0}^{1} \bigl(1-\eta ^{\mu}\bigr)w(\eta )\,d\eta \biggr)\mathbf{y}_{3}. $$
(1.7)

For \(L_{w}=0\), we can observe that \(\digamma \in \mathcal{F}\) and

$$ \digamma \bigl(g\bigl(\eta x+(1-\eta )y\bigr),g(x),g(y),\eta \bigr) =g\bigl( \eta x+(1-\eta )y\bigr)-\eta ^{\mu}g(x)- \bigl(1-\eta ^{\mu}\bigr)g(y)-\varepsilon \leq 0, $$

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,

$$ g\bigl(\eta x+(1-\eta )y\bigr)\leq h(\eta )g(x)+h(1-\eta )g(y),\quad (x,y,\eta )\in \mathcal{V}\times [0,1]. $$

Then we define the functions \(\digamma :\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as follows:

$$ \digamma (\mathbf{y}_{1},\mathbf{y}_{2}, \mathbf{y}_{3},\mathbf{y}_{4})=\mathbf{y}_{1} -h(\mathbf{y}_{4})\mathbf{y}_{3}-h(1- \mathbf{y}_{4})\mathbf{y}_{2}, $$
(1.8)

and \(\mathcal{T}_{\digamma ,w}:\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as

$$ \mathcal{T}_{\digamma ,w}(\mathbf{y}_{1}, \mathbf{y}_{2},\mathbf{y}_{3}) =\mathbf{y}_{1}- \biggl( \int _{0}^{1} h(\eta )w(\eta )\,d\eta \biggr) \mathbf{y}_{2} - \biggl( \int _{0}^{1} h(1-\eta )w(\eta )\,d\eta \biggr) \mathbf{y}_{3}. $$
(1.9)

For \(L_{w}=(1-w)\varepsilon \), we can observe that \(\digamma \in \mathcal{F}\) and

$$ \digamma \bigl(g\bigl(\eta x+(1-\eta )y\bigr),g(x),g(y),\eta \bigr)=g\bigl(\eta x+(1-\eta )y\bigr) -h(\eta )g(x)-h(1-\eta )g(y)-\varepsilon \leq 0, $$

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

$$ D_{\mu }(g) (\eta )=\lim_{\epsilon \to 0} \frac{g (\eta +\epsilon \eta ^{1-\mu } )-g(\eta )}{\epsilon },\quad \mu \in (0,1), \eta >0. $$
(2.1)

For μ-differentiable function g in some \((0,\mu ), \mu >0\), \(\lim_{t\to 0^{+}}g^{(\mu )}(\eta )\) exist, define

$$ g^{(\mu )}(0)=\lim_{t\to 0^{+}}g^{(\mu )}(\eta ). $$

Furthermore, if g is differentiable, then we have

$$ D_{\mu }(g) (\eta )=\eta ^{1-\mu }g^{\prime }( \eta ), \quad \text{where } g^{ \prime }(\eta )=\lim_{\epsilon \to 0} \frac{g (\eta +\epsilon )-g(\eta )}{\epsilon }. $$
(2.2)

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. 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. 2.

    \(D_{\mu}(fg)=fD_{\mu}(g)+gD_{\mu}(f)\),

  3. 3.

    \(D_{\mu}(\frac{f}{g} )=\frac{gD_{\mu}(f)-fD_{\mu}(g)}{g^{2}}\),

  4. 4.

    \(D_{\mu }(c)=0\)for each constant function, namely\(g(\eta )=c\),

  5. 5.

    \(D_{\mu } (1 )=0\),

  6. 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

$$ \int _{v_{1}}^{v_{2}} g(\eta )\, d_{\mu }\eta = \int _{v_{1}}^{v_{2}} g( \eta )\eta ^{\mu -1}\,d\eta $$
(2.3)

exists and is finite.

Remark 2.1

  1. (a)

    We indicate by \(L_{\mu }^{1}([v_{1},v_{2}])\) all μ-fractional integrable functions on an interval \([v_{1},v_{2}]\).

  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

([47, 48])

Let\(g:(v_{1},v_{2})\to \mathtt{R}\)be differentiable and\(\mu \in (0,1]\). Then, for all\(\eta >v_{1}\), we have

$$ I_{\mu }^{v_{1}} D_{\mu }^{v_{1}} (g) ( \eta )=g(\eta )-g({v_{1}}). $$

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

$$ D_{\mu }^{v_{1}} I_{\mu }^{v_{1}} (g) ( \eta )=g(\eta ),\quad \mu \in (n,n+1], $$

which is called the inverse property.

Theorem 2.4

([47, 48])

Let\(g:[v_{1},v_{2}]\subset \mathtt{R}\to \mathtt{R}\)be two functions withfgis differentiable. Then

$$ \int _{v_{1}}^{v_{2}} g(x) D_{\mu }^{v_{1}} (h) (x)\,d_{\mu }x=g h|_{v_{1}}^{v_{2}} - \int _{v_{1}}^{v_{2}} h(x) D_{\mu }^{v_{1}} (g) (x)\,d_{\mu }x. $$

Theorem 2.5

([47, 48])

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

$$\begin{aligned} \bigl\vert I_{\mu }^{v_{1}} (g) (\eta ) \bigr\vert \leq I_{\mu }^{v_{1}} \vert f \vert (\eta ),\quad \mu \in (0,1]. \end{aligned}$$

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

$$\begin{aligned} \digamma _{\mu } \bigl(g \bigl(\eta ^{\mu }x^{\mu }+ \bigl(1-\eta ^{\mu } \bigr)y^{\mu } \bigr), g \bigl(x^{\mu } \bigr),g \bigl(y^{\mu } \bigr), \eta ^{\mu } \bigr)\leq 0,\quad (x,y,\eta )\in \mathcal{V}\times [0,1]. \end{aligned}$$

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

$$ g \bigl(\eta ^{\mu }x^{\mu }+ \bigl(1-\eta ^{\mu } \bigr)y^{\mu } \bigr) \leq \eta ^{\mu }g \bigl(x^{\mu } \bigr) + \bigl(1-\eta ^{\mu } \bigr)g \bigl(y^{\mu } \bigr)+\varepsilon , \quad (x,y,\eta )\in \mathcal{V} \times [0,1]. $$

Then we define the function \(\digamma _{\mu }:\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as follows:

$$ \digamma _{\mu }(\mathbf{y}_{1}, \mathbf{y}_{2},\mathbf{y}_{3}, \mathbf{y}_{4})= \mathbf{y}_{1} -\mathbf{y}_{4}^{\mu } \mathbf{y}_{2}- \bigl(1-\mathbf{y}_{4}^{\mu } \bigr)\mathbf{y}_{3}-\varepsilon , $$
(2.5)

and \(\mathcal{T}_{\digamma _{\mu },w}:\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as

$$ \mathcal{T}_{\digamma _{\mu },w}(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}) =\mathbf{y}_{1}- \biggl( \int _{0}^{1} \eta ^{\mu } w\bigl( \eta ^{\mu }\bigr)\,d_{\mu }\eta \biggr)\mathbf{y}_{2} - \biggl( \int _{0}^{1} \bigl(1-\eta ^{\mu } \bigr)w(\eta )\,d_{\mu }\eta \biggr)\mathbf{y}_{3}- \varepsilon . $$
(2.6)

For

$$ L_{w}=(1-w)\varepsilon , $$
(2.7)

it can be observed that \(\digamma \in \mathcal{F}\) and

$$\begin{aligned}& \digamma _{\mu } \bigl(g \bigl(\eta ^{\mu }x^{\mu }+ \bigl(1-\eta ^{\mu } \bigr)y^{\mu } \bigr), g \bigl(x^{\mu }+ \bigr),g \bigl(y^{\mu } \bigr), \eta ^{\mu } \bigr) \\& \quad =g \bigl(\eta ^{\mu }x^{\mu }+ \bigl(1-\eta ^{\mu } \bigr)y^{\mu } \bigr)- \eta ^{\mu }g \bigl(x^{\mu }\bigr) - \bigl(1-\eta ^{\mu } \bigr)g \bigl(y^{\mu }\bigr)- \varepsilon \leq 0, \end{aligned}$$

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,

$$ g \bigl(\eta ^{\mu }x^{\mu }+ \bigl(1-\eta ^{\mu } \bigr)y^{\mu } \bigr) \leq \eta g\bigl(x^{\mu }\bigr) +(1- \eta )g\bigl(y^{\mu }\bigr),\quad (x,y,\eta )\in \mathcal{V}\times [0,1]. $$

Then we define the functions \(\digamma _{\mu }:\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as follows:

$$ \digamma _{\mu }(\mathbf{y}_{1}, \mathbf{y}_{2},\mathbf{y}_{3}, \mathbf{y}_{4})= \mathbf{y}_{1} -\mathbf{y}_{4} \mathbf{y}_{2}- (1- \mathbf{y}_{4} )\mathbf{y}_{3}, $$
(2.8)

and \(\mathcal{T}_{\digamma _{\mu },w}:\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as

$$ \mathcal{T}_{\digamma _{\mu },w}(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}) =\mathbf{y}_{1}- \biggl( \int _{0}^{1} t w\bigl(\eta ^{\mu } \bigr)\,d_{\mu }\eta \biggr)\mathbf{y}_{2} - \biggl( \int _{0}^{1} (1-\eta )w\bigl(\eta ^{\mu }\bigr)\,d_{\mu }\eta \biggr)\mathbf{y}_{3}. $$
(2.9)

For \(L_{w}=0\), we can observe that \(\digamma _{\mu }\in \mathcal{F}\) and

$$\begin{aligned}& \digamma _{\mu } \bigl(g \bigl(\eta ^{\mu }x^{\mu }+ \bigl(1-\eta ^{\mu } \bigr)y^{\mu } \bigr), g \bigl(x^{\mu }\bigr),g\bigl(y^{\mu }\bigr),\eta ^{\mu } \bigr) \\& \quad =g \bigl(\eta ^{\mu }x^{\mu }+ \bigl(1-\eta ^{\mu } \bigr)y^{\mu } \bigr)- \eta g\bigl(x^{\mu } \bigr) -(1-\eta )g\bigl(y^{\mu }\bigr)-\varepsilon \leq 0, \end{aligned}$$

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

$$ g \bigl(\eta ^{\mu }x^{\mu }+ \bigl(1-\eta ^{\mu } \bigr)y^{\mu } \bigr) \leq h\bigl(\eta ^{\mu }\bigr)g \bigl(x^{\mu }\bigr) +h \bigl(1-\eta ^{\mu } \bigr)g \bigl(y^{\mu }\bigr), \quad (x,y,\eta )\in \mathcal{V}\times [0,1]. $$

Then we define the functions \(\digamma _{\mu }:\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as follows:

$$ \digamma _{\mu }(\mathbf{y}_{1}, \mathbf{y}_{2},\mathbf{y}_{3}, \mathbf{y}_{4})= \mathbf{y}_{1} -h(\mathbf{y}_{4})\mathbf{y}_{3}-h \bigl(1-\mathbf{y}_{4}^{\mu } \bigr)\mathbf{y}_{2}, $$
(2.10)

and \(\mathcal{T}_{\digamma _{\mu },w}:\bar{\mathfrak{R}}\times [0,1]\to \mathtt{R}\) as

$$\begin{aligned} \mathcal{T}_{\digamma _{\mu },w}(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}) =&\mathbf{y}_{1}- \biggl( \int _{0}^{1} h\bigl(\eta ^{\mu } \bigr)w\bigl( \eta ^{\mu }\bigr)\,d_{\mu }\eta \biggr) \mathbf{y}_{2} \\ &{} - \biggl( \int _{0}^{1} h \bigl(1-\eta ^{\mu } \bigr)w\bigl(\eta ^{\mu }\bigr)\,d_{\mu }\eta \biggr) \mathbf{y}_{3}. \end{aligned}$$
(2.11)

For \(L_{w}=(1-w)\varepsilon \), we can observe that \(\digamma _{\mu }\in \mathcal{F}\) and

$$\begin{aligned}& \digamma _{\mu }\bigl(g \bigl(\eta ^{\mu }x^{\mu }+ \bigl(1-\eta ^{\mu } \bigr)y^{\mu } \bigr),g \bigl(x^{\mu }\bigr),g\bigl(y^{\mu }\bigr),\eta ^{\mu }\bigr) \\& \quad =g \bigl(\eta ^{\mu }x^{\mu }+ \bigl(1-\eta ^{\mu } \bigr)y^{\mu } \bigr) -h\bigl( \eta ^{\mu } \bigr)g\bigl(x^{\mu }\bigr)-h \bigl(1-\eta ^{\mu } \bigr)g \bigl(y^{\mu }\bigr)- \varepsilon \leq 0, \end{aligned}$$

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

$$ g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr)\leq \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x\leq \frac{g (v_{1}^{\mu } )+g (v_{2}^{\mu } )}{2}. $$
(2.12)

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

$$ \begin{aligned}[b] &\frac{g (v_{1}^{\mu } )+g (v_{2}^{\mu } )}{2}- \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x \\ &\quad =\frac{v_{2}^{\mu }-v_{1}^{\mu }}{2} \int _{0}^{1} \bigl(1-2\eta ^{\mu } \bigr) D_{\mu }(g) \bigl(\eta ^{\mu }v_{1}^{\mu }+ \bigl(1-\eta ^{\mu } \bigr)v_{2}^{\mu } \bigr)\,d_{\mu }\eta . \end{aligned} $$
(2.13)

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

$$ \begin{aligned}[b] &\frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x -g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr) \\ &\quad = \bigl(v_{2}^{\mu }-v_{1}^{\mu } \bigr) \int _{0}^{1} p(\eta ) D_{\mu }(g) \bigl(\eta ^{\mu }v_{1}^{\mu }+ \bigl(1-\eta ^{\mu } \bigr)v_{2}^{\mu } \bigr)\,d_{\mu } \eta , \end{aligned} $$
(2.14)

where

$$ P(\eta )= \textstyle\begin{cases} \eta ^{\mu}, & 0\leq t\leq \frac{1}{2^{1/\mu}}, \\ \eta ^{\mu}-1, & \frac{1}{2^{1/\mu}}\leq t\leq 1. \end{cases} $$

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

$$\begin{aligned}& \digamma \biggl(g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr), \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x,\frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x,\frac{1}{2} \biggr)+ \int _{0}^{1} L_{w(\eta )}\,d\eta \\& \quad \leq 0, \end{aligned}$$
(3.1)
$$\begin{aligned}& \mathcal{T}_{\digamma ,w} \biggl(\frac{2\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x, g \bigl(v_{1}^{\mu } \bigr)+g \bigl(v_{2}^{\mu } \bigr),g \bigl(v_{1}^{\mu } \bigr)+g \bigl(v_{2}^{\mu } \bigr) \biggr)+ \int _{0}^{1} L_{w(\eta )}\,d\eta \leq 0. \end{aligned}$$
(3.2)

Proof

The Ϝ-convexity of g leads to

$$ \digamma \biggl(g \biggl(\frac{x+y}{2} \biggr),g(x),g(y), \frac{1}{2} \biggr),\quad x,y\in [v_{1},v_{2}]. $$

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

$$ \digamma \biggl(g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr),g \bigl(\eta ^{\mu }v_{1}^{\mu }+ \bigl(1-\eta ^{\mu } \bigr)v_{2}^{\mu } \bigr),g \bigl( \bigl(1-\eta ^{\mu } \bigr)v_{1}^{\mu }+\eta ^{\mu }v_{2}^{\mu } \bigr), \frac{1}{2} \biggr) \leq 0. $$

Multiplying this inequality \(w(\eta )=1\) and making use of axiom (\(\varLambda _{3}\)), we get

$$ \digamma \biggl(g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr),g \bigl(\eta ^{\mu }v_{1}^{\mu }+ \bigl(1-\eta ^{\mu } \bigr)v_{2}^{\mu } \bigr),g \bigl( \bigl(1-\eta ^{\mu } \bigr)v_{1}^{\mu }+\eta ^{\mu }v_{2}^{\mu } \bigr), \frac{1}{2} \biggr)+L_{w(\eta )} \leq 0. $$

Integrating over \([0,1]\) according to η and making use of axiom (\(\varLambda _{1}\)), we get

$$\begin{aligned} &\digamma \biggl( \int _{0}^{1} g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr)\,d_{\mu }\eta , \int _{0}^{1} g \bigl(\eta ^{\mu }v_{1}^{\mu }+ \bigl(1- \eta ^{\mu } \bigr)v_{2}^{\mu } \bigr)\,d_{\mu }\eta , \\ &\quad \int _{0}^{1} g \bigl( \bigl(1-\eta ^{\mu } \bigr)v_{1}^{\mu }+\eta ^{\mu }v_{2}^{\mu } \bigr)\,d_{\mu }\eta , \frac{1}{2} \biggr)+ \int _{0}^{1} L_{w(\eta )}\,d_{ \mu } \eta \leq 0, \end{aligned}$$

that is,

$$\begin{aligned} &\digamma \biggl(g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr), \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x,\frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x,\frac{1}{2} \biggr)+ \int _{0}^{1} L_{w(\eta )}\,d\eta \\ &\quad \leq 0, \end{aligned}$$

where we have used

$$\begin{aligned} \int _{0}^{1} g \bigl(\eta ^{\mu }v_{1}^{\mu }+ \bigl(1-\eta ^{\mu } \bigr)v_{2}^{\mu } \bigr)\,d_{\mu }\eta &= \int _{0}^{1} g \bigl( \bigl(1-\eta ^{\mu } \bigr)v_{1}^{\mu }+\eta ^{\mu }v_{2}^{\mu } \bigr)\,d_{\mu }\eta \\ &= \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x. \end{aligned}$$

This completely gives the proof of (3.1). On the other hand, since g is Ϝ-convex, we have

$$\begin{aligned}& \digamma \bigl(g \bigl(\eta ^{\mu }v_{1}^{\mu }+ \bigl(1-\eta ^{\mu } \bigr)v_{2}^{\mu } \bigr),g \bigl(v_{1}^{\mu } \bigr) ,g \bigl(v_{2}^{\mu } \bigr),\eta \bigr) \leq 0, \\& \digamma \bigl(g \bigl( \bigl(1-\eta ^{\mu } \bigr)v_{1}^{\mu }+ \eta ^{\mu }v_{2}^{\mu } \bigr),g \bigl(v_{2}^{\mu } \bigr) ,g \bigl(v_{1}^{\mu } \bigr),\eta \bigr) \leq 0. \end{aligned}$$

We make use of the linearity of Ϝ to get

$$ \digamma \bigl(g \bigl(\eta ^{\mu }v_{1}^{\mu }+ \bigl(1-\eta ^{\mu } \bigr)v_{2}^{\mu } \bigr)+g \bigl( \bigl(1-\eta ^{\mu } \bigr)v_{1}^{\mu }+ \eta ^{\mu }v_{2}^{\mu } \bigr),g \bigl(v_{1}^{\mu } \bigr)+ g \bigl(v_{2}^{\mu } \bigr),g \bigl(v_{1}^{\mu } \bigr)+g \bigl(v_{2}^{\mu } \bigr), \eta \bigr) \leq 0. $$

Applying the axiom (\(\varLambda _{3}\)) for \(w(\eta )=1\), we get

$$\begin{aligned}& \digamma \bigl(g \bigl(\eta ^{\mu }v_{1}^{\mu }+ \bigl(1-\eta ^{\mu } \bigr)v_{2}^{\mu } \bigr)+g \bigl( \bigl(1-\eta ^{\mu } \bigr)v_{1}^{\mu }+ \eta ^{\mu }v_{2}^{\mu } \bigr),g \bigl(v_{1}^{\mu } \bigr)+ g \bigl(v_{2}^{\mu } \bigr),g \bigl(v_{1}^{\mu } \bigr)+g \bigl(v_{2}^{\mu } \bigr), \eta \bigr) \\& \quad {}+L_{w(\eta )} \leq 0. \end{aligned}$$

Integrating over \([0,1]\) according to η and making use of axiom (\(\varLambda _{2}\)) we get

$$ \mathcal{T}_{\digamma ,w} \biggl(\frac{2\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x, g \bigl(v_{1}^{\mu } \bigr)+g \bigl(v_{2}^{\mu } \bigr),g \bigl(v_{1}^{\mu } \bigr)+g \bigl(v_{2}^{\mu } \bigr) \biggr)+ \int _{0}^{1} L_{w(\eta )}\,d\eta \leq 0. $$

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

$$ g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr)-\varepsilon \leq \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x \leq \frac{g (v_{1}^{\mu } )+g (v_{2}^{\mu } )}{2}+ \frac{\varepsilon }{2}. $$
(3.3)

Proof

By making use of \(w(\eta )=1\) in (1.5), we get

$$ \int _{0}^{1} L_{w(\eta )}\,d\eta =0. $$
(3.4)

Making use of (1.3), (3.1), and (3.4), we get

$$\begin{aligned} 0&\geq F \biggl(g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr), \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x,\frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x,\frac{1}{2} \biggr)+ \int _{0}^{1} L_{w(\eta )}\,d\eta \\ &=g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr)- \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }}\int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x-\varepsilon , \end{aligned}$$

or equivalently,

$$ g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr)-\varepsilon \leq \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x. $$

Making use of \(w(\eta )=1\) in (1.4), we have

$$ \begin{aligned}[b] \mathcal{T}_{\digamma ,w} ( \mathbf{y}_{1},\mathbf{y}_{2}, \mathbf{y}_{3} ) &=\mathbf{y}_{1}- \biggl( \int _{0}^{1} t \,d\eta \biggr) \mathbf{y}_{2} - \biggl( \int _{0}^{1} (1-\eta )\,d\eta \biggr) \mathbf{y}_{3}-\varepsilon \\ &=\mathbf{y}_{1}-\frac{\mathbf{y}_{2}+\mathbf{y}_{3}}{2}-\varepsilon ,\quad \mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3} \in \mathtt{R}. \end{aligned} $$
(3.5)

Now, from (3.2) and (3.5), we can deduce

$$\begin{aligned} 0&\geq \mathcal{T}_{\digamma ,w} \biggl( \frac{2\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr) \,d_{\mu }x,g \bigl(v_{1}^{\mu } \bigr)+g \bigl(v_{2}^{\mu } \bigr),g \bigl(v_{1}^{\mu } \bigr)+g \bigl(v_{2}^{\mu } \bigr) \biggr)+ \int _{0}^{1} L_{w(\eta )}\,d\eta \\ &=\frac{2\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x - \bigl(g \bigl(v_{1}^{\mu } \bigr)+g \bigl(v_{2}^{\mu } \bigr) \bigr)-\varepsilon . \end{aligned}$$

This gives

$$ \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x \leq \frac{g (v_{1}^{\mu } )+g (v_{2}^{\mu } )}{2}+ \frac{\varepsilon }{2}. $$

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

$$ \frac{1}{2h (\frac{1}{2} )} g \biggl( \frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr) \leq \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x \leq \frac{g (v_{1}^{\mu } )+g (v_{2}^{\mu } )}{2} \int _{0}^{1} h(\eta )\,d\eta . $$
(3.6)

Proof

Making use (1.5) and (3.1) with \(L_{w(\eta )}=0\), we have

$$\begin{aligned} 0&\geq F \biggl(g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr), \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x,\frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x,\frac{1}{2} \biggr)+ \int _{0}^{1} L_{w(\eta )}\,d\eta \\ &=g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr)-h \biggl(\frac{1}{2} \biggr) \frac{2\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x, \end{aligned}$$

or equivalently,

$$ \frac{1}{2h (\frac{1}{2} )}g \biggl( \frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr)\leq \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x. $$

Now, by making use of \(w(\eta )=1\) in (1.4) and (3.2), we get

$$\begin{aligned} 0&\geq \mathcal{T}_{\digamma ,w} \biggl( \frac{2\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr) \,d_{\mu }x,g \bigl(v_{1}^{\mu } \bigr)+g \bigl(v_{2}^{\mu } \bigr),g \bigl(v_{1}^{\mu } \bigr)+g \bigl(v_{2}^{\mu } \bigr) \biggr)+ \int _{0}^{1} L_{w(\eta )}\,d\eta \\ &=\frac{2\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x - \bigl(g \bigl(v_{1}^{\mu } \bigr)+g \bigl(v_{2}^{\mu } \bigr) \bigr) \biggl( \int _{0}^{1} h(\eta )\,d\eta \biggr). \end{aligned}$$

This gives

$$ \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x \leq \biggl( \frac{g (v_{1}^{\mu } )+g (v_{2}^{\mu } )}{2} \biggr) \biggl( \int _{0}^{1} h(\eta )\,d\eta \biggr). $$

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

$$\begin{aligned}& \mathcal{T}_{\digamma _{\mu },w} \biggl(\frac{2}{v_{2}^{\mu }-v_{1}^{\mu }} \biggl\vert \frac{g (v_{1}^{\mu } ) +g (v_{2}^{\mu } )}{2}- \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr) \,d_{\mu }x \biggr\vert , \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert , \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert ,\eta \biggr) \\& \quad {}+ \int _{0}^{1} L_{w(\eta )}\,d_{\mu } \eta \leq 0. \end{aligned}$$
(4.1)

Proof

The \(\digamma _{\mu }\)-convexity of \(|D_{\mu }(g)|\) leads to

$$ \digamma _{\mu } \bigl( \bigl\vert D_{\mu }(g) \bigl(\eta ^{\mu }v_{1}^{\mu }+\bigl(1- \eta ^{\mu }\bigr)v_{2}^{\mu } \bigr) \bigr\vert , \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert , \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert , \eta \bigr)\leq 0. $$

By applying axiom (\(\bar{\varLambda }_{3}\)) for \(w(\eta ^{\mu })= \vert 1-2\eta ^{\mu } \vert \), \(\eta \in [0,1]\), we can deduce

$$ \digamma _{\mu } \bigl(w\bigl(\eta ^{\mu }\bigr) \bigl\vert D_{\mu }(g) \bigl(\eta ^{\mu }v_{1}^{\mu }+ \bigl(1-\eta ^{\mu }\bigr)v_{2}^{\mu } \bigr) \bigr\vert ,w\bigl(\eta ^{\mu }\bigr) \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert ,w\bigl(\eta ^{\mu }\bigr) \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert ,\eta \bigr)\leq 0. $$

Integrating over \([0,1]\) according to η and by making use of axiom (\(\bar{\varLambda }_{2}\)), we obtain

$$\begin{aligned} &\mathcal{T}_{\digamma _{\mu },w} \biggl( \int _{0}^{1} w\bigl(\eta ^{\mu } \bigr) \bigl\vert D_{\mu }(g) \bigl(\eta ^{\mu }v_{1}^{\mu }+ \bigl(1-\eta ^{\mu }\bigr)v_{2}^{\mu } \bigr) \bigr\vert \,d_{\mu }\eta , \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert , \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert ,\eta \biggr) \\ &\quad {}+ \int _{0}^{1} L_{w(\eta ^{\mu })}\,d_{\mu }t \leq 0. \end{aligned}$$

From Lemma 2.1, we have

$$\begin{aligned} &\biggl\vert \frac{g (v_{1}^{\mu } )+g (v_{2}^{\mu } )}{2}- \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x \biggr\vert \\ &\quad \leq \frac{v_{2}^{\mu }-v_{1}^{\mu }}{2} \int _{0}^{1} w\bigl( \eta ^{\mu } \bigr) \bigl\vert D_{\mu }(g) \bigl(\eta ^{\mu }v_{1}^{\mu }+ \bigl(1-\eta ^{\mu }\bigr)v_{2}^{\mu } \bigr) \bigr\vert \,d_{\mu }\eta . \end{aligned}$$

Since \(\mathcal{T}_{\digamma _{\mu },w}\) is nondecreasing according to the first variable, then we can deduce

$$\begin{aligned}& \mathcal{T}_{\digamma _{\mu },w} \biggl(\frac{2}{v_{2}^{\mu }-v_{1}^{\mu }} \biggl\vert \frac{g (v_{1}^{\mu } ) +g (v_{2}^{\mu } )}{2}- \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr) \,d_{\mu }x \biggr\vert , \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert , \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert ,\eta \biggr) \\& \quad {}+ \int _{0}^{1} L_{w(\eta ^{\mu })}\,d_{\mu } \eta \leq 0, \end{aligned}$$

which ends the proof of (4.1). □

Corollary 4.1

Theorem  4.1 with \(|D_{\mu }(g)|\) to be ε conformable convex leads to

$$\begin{aligned}& \biggl\vert \frac{g (v_{1}^{\mu } )+g (v_{2}^{\mu } )}{2} - \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x \biggr\vert \\& \quad \leq \frac{v_{2}^{\mu }-v_{1}^{\mu }}{2} \biggl( \frac{2^{3\mu ^{2}}+6\times 2^{\mu ^{2}}-8}{6\mu \times 2^{3\mu ^{2}}} \biggr) \biggl( \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert + \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert +\frac{2\mu -1}{2\mu }\varepsilon \biggr). \end{aligned}$$
(4.2)

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

$$ \int _{0}^{1} L_{w(\eta )}\,d_{\mu } \eta =\varepsilon \int _{0}^{1} \bigl(1-w( \eta ) \bigr)\,d_{\mu }\eta =\frac{1}{2\mu }\varepsilon . $$

By making use of \(w(\eta ^{\mu })= \vert 1-2\eta ^{\mu } \vert \) in (2.6), we get

$$\begin{aligned} \mathcal{T}_{\digamma _{\mu },w}(\mathbf{y}_{1},\mathbf{y}_{2}, \mathbf{y}_{3})&=\mathbf{y}_{1} - \biggl( \int _{0}^{1} \eta ^{\mu } \bigl\vert 1-2\eta ^{\mu } \bigr\vert \,d_{\mu }\eta \biggr) \mathbf{y}_{2} - \biggl( \int _{0}^{1} \bigl(1-\eta ^{\mu } \bigr) \bigl\vert 1-2\eta ^{\mu } \bigr\vert \,d_{\mu }\eta \biggr)\mathbf{y}_{3} -\varepsilon \\ &=\mathbf{y}_{1}- \biggl( \frac{2^{3\mu ^{2}}+6\times 2^{\mu ^{2}}-8}{6\mu \times 2^{3\mu ^{2}}} \biggr)( \mathbf{y}_{2}+\mathbf{y}_{3})-\varepsilon \end{aligned}$$

for \(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\in \mathtt{R}\). By making use of Theorem 4.1, we get

$$\begin{aligned} 0 \geq& \mathcal{T}_{\digamma _{\mu },w} \biggl( \frac{2}{v_{2}^{\mu }-v_{1}^{\mu }} \biggl\vert \frac{g (v_{1}^{\mu } )+g (v_{2}^{\mu } )}{2}- \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr) \,d_{\mu }x \biggr\vert , \\ & \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert , \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert ,\eta \biggr) + \int _{0}^{1} L_{w(\eta ^{\mu })}\,d_{\mu } \eta \\ =&\frac{2}{v_{2}^{\mu }-v_{1}^{\mu }} \biggl\vert \frac{g (v_{1}^{\mu } )+g (v_{2}^{\mu } )}{2} - \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}}g \bigl(x^{\mu } \bigr)\,d_{\mu }x \biggr\vert \\ &{}- \biggl( \frac{2^{3\mu ^{2}}+6\times 2^{\mu ^{2}}-8}{6\mu \times 2^{3\mu ^{2}}} \biggr) \bigl( \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert + \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert \bigr) - \varepsilon +\frac{1}{2\mu }\varepsilon . \end{aligned}$$

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

$$\begin{aligned}& \biggl\vert \frac{g (v_{1}^{\mu } )+g (v_{2}^{\mu } )}{2} - \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x \biggr\vert \\& \quad \leq \frac{v_{2}^{\mu }-v_{1}^{\mu }}{2(\mu +1)(2\mu +1)} \biggl(1+ \frac{\mu }{2^{1/\mu }} \biggr) \bigl( \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert + \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert \bigr). \end{aligned}$$
(4.3)

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

$$\begin{aligned} \mathcal{T}_{\digamma _{\mu },w}(\mathbf{y}_{1},\mathbf{y}_{2}, \mathbf{y}_{3})&=\mathbf{y}_{1}- \biggl( \int _{0}^{1} t \bigl\vert 1-2\eta ^{\mu } \bigr\vert \,d_{\mu }\eta \biggr) \mathbf{y}_{2} - \biggl( \int _{0}^{1} (1-\eta ) \bigl\vert 1-2\eta ^{\mu } \bigr\vert \,d_{\mu }\eta \biggr)\mathbf{y}_{3} \\ &=\mathbf{y}_{1}-\frac{1}{(\mu +1)(2\mu +1)} \biggl(1+ \frac{\mu }{2^{1/\mu }} \biggr) (\mathbf{y}_{2}+\mathbf{y}_{3}) \end{aligned}$$

for \(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\in \mathtt{R}\). Then, by applying Theorem 4.1, we have

$$\begin{aligned} 0 \geq& \mathcal{T}_{\digamma _{\mu },w} \biggl( \frac{2}{v_{2}^{\mu }-v_{1}^{\mu }} \biggl\vert \frac{g (v_{1}^{\mu } ) +g (v_{2}^{\mu } )}{2}- \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr) \,d_{\mu }x \biggr\vert , \\ &\bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert , \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert ,\eta \biggr) + \int _{0}^{1} L_{w(\eta ^{\mu })}\,d_{\mu } \eta \\ =&\frac{2}{v_{2}^{\mu }-v_{1}^{\mu }} \biggl\vert \frac{g (v_{1}^{\mu } )+g (v_{2}^{\mu } )}{2} - \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x \biggr\vert \\ &{}-\frac{1}{(\mu +1)(2\mu +1)} \biggl(1+\frac{\mu }{2^{1/\mu }} \biggr) \bigl( \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert + \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert \bigr). \end{aligned}$$

This rearranges to the required inequality (4.3). □

Corollary 4.3

Theorem 4.1with\(|D_{\mu }(g)|\)to beh-conformable convex leads to

$$\begin{aligned} & \biggl\vert \frac{g (v_{1}^{\mu } )+g (v_{2}^{\mu } )}{2} - \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x \biggr\vert \\ &\quad \leq \frac{v_{2}^{\mu }-v_{1}^{\mu }}{2} \biggl( \int _{0}^{1} h\bigl(\eta ^{\mu } \bigr) \bigl\vert 1-2\eta ^{\mu } \bigr\vert \,d_{\mu }\eta \biggr) \bigl( \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert + \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert \bigr). \end{aligned}$$
(4.4)

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

$$\begin{aligned} \mathcal{T}_{\digamma _{\mu },w}(\mathbf{y}_{1},\mathbf{y}_{2}, \mathbf{y}_{3}) &=\mathbf{y}_{1}- \biggl( \int _{0}^{1} h\bigl(\eta ^{\mu } \bigr) \bigl\vert 1-2\eta ^{\mu } \bigr\vert \,d_{\mu }\eta \biggr)\mathbf{y}_{2} \\ &\quad {}- \biggl( \int _{0}^{1} h \bigl(1-\eta ^{\mu } \bigr) \bigl\vert 1-2\eta ^{\mu } \bigr\vert \,d_{\mu }\eta \biggr)\mathbf{y}_{3} \\ &=\mathbf{y}_{1}- \biggl( \int _{0}^{1} h\bigl(\eta ^{\mu } \bigr) \bigl\vert 1-2\eta ^{\mu } \bigr\vert \,d_{\mu }\eta \biggr) (\mathbf{y}_{2}+\mathbf{y}_{3}) \end{aligned}$$

for \(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\in \mathtt{R}\). Then, by using Theorem 4.1, we get

$$\begin{aligned} 0 \geq& \mathcal{T}_{\digamma _{\mu },w} \biggl( \frac{2}{v_{2}^{\mu }-v_{1}^{\mu }} \biggl\vert \frac{g (v_{1}^{\mu } ) +g (v_{2}^{\mu } )}{2}- \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr) \,d_{\mu }x \biggr\vert , \\ &\bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert , \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert ,\eta \biggr)+ \int _{0}^{1} L_{w(\eta ^{\mu })}\,d_{\mu } \eta \\ =&\frac{2}{v_{2}^{\mu }-v_{1}^{\mu }} \biggl\vert \frac{g (v_{1}^{\mu } )+g (v_{2}^{\mu } )}{2} - \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}}g \bigl(x^{\mu } \bigr)\,d_{\mu }x \biggr\vert \\ &{}- \biggl( \int _{0}^{1}h\bigl(\eta ^{\mu }\bigr) \bigl\vert 1-2\eta ^{\mu } \bigr\vert \,d_{\mu }\eta \biggr) \bigl( \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert + \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert \bigr). \end{aligned}$$

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

$$ \mathcal{T}_{\digamma _{\mu },1} \bigl(v_{1}(g,p), \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert ^{ \frac{p}{p-1}} \bigr) \leq 0, $$
(4.5)

where

$$\begin{aligned} v_{1}(g,p)&= \biggl(\frac{2}{v_{2}^{\mu }-v_{1}^{\mu }} \biggr)^{ \frac{p}{p-1}} \biggl(\frac{1}{2\mu (p+1)} \biggl\{ 2- \biggl(1-\frac{1}{2^{\mu ^{2}-1}} \biggr)^{p+1} - \biggl(\frac{1}{2^{\mu ^{2}-1}}-1 \biggr)^{p+1} \biggr\} \biggr)^{ \frac{-1}{p-1}} \\ &\quad {}\times \biggl\vert \frac{g (v_{1}^{\mu } )+g (v_{2}^{\mu } )}{2} - \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr) \,d_{\mu }x \biggr\vert ^{\frac{p}{p-1}}. \end{aligned}$$

Proof

By using the \(\digamma _{\mu }\)-convexity of \(|D_{\mu }(g)|^{\frac{p}{p-1}}\), we have

$$ \digamma _{\mu } \bigl( \bigl\vert D_{\mu }(g) \bigl(\eta ^{\mu }v_{1}^{\mu }+\bigl(1- \eta ^{\mu }\bigr)v_{2}^{\mu } \bigr) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert ^{\frac{p}{p-1}}, \eta \bigr)\leq 0. $$

By making use of \(w(\eta ^{\mu })=1\) in axiom (\(\bar{\varLambda }_{3}\)), we obtain

$$ \mathcal{T}_{\digamma _{\mu },1} \biggl( \int _{0}^{1} \bigl\vert D_{\mu }(g) \bigl(\eta ^{\mu }v_{1}^{\mu }+\bigl(1-\eta ^{\mu }\bigr)v_{2}^{\mu } \bigr) \bigr\vert ^{ \frac{p}{p-1}}\,d_{\mu }\eta , \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert ^{\frac{p}{p-1}} \biggr) \leq 0. $$

Then, by making use Lemma of 2.2, we have

$$\begin{aligned}& \biggl\vert \frac{g (v_{1}^{\mu } )+g (v_{2}^{\mu } )}{2}- \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x \biggr\vert \\& \quad \leq \frac{v_{2}^{\mu }-v_{1}^{\mu }}{2} \biggl(\frac{1}{2\mu (p+1)} \biggl\{ 2- \biggl(1-\frac{1}{2^{\mu ^{2}-1}} \biggr)^{p+1} - \biggl( \frac{1}{2^{\mu ^{2}-1}}-1 \biggr)^{p+1} \biggr\} \biggr)^{\frac{1}{p}} \\& \qquad {}\times \biggl( \int _{0}^{1} \bigl\vert D_{\mu }(g) \bigl(\eta ^{\mu }v_{1}^{\mu }+\bigl(1-\eta ^{\mu }\bigr)v_{2}^{\mu } \bigr) \bigr\vert ^{\frac{p}{p-1}}\,d_{\mu }\eta \biggr)^{\frac{p-1}{p}}. \end{aligned}$$

Since \(\mathcal{T}_{\digamma _{\mu },w}\) is nondecreasing according to the first variable, then we can deduce

$$ \mathcal{T}_{\digamma _{\mu },1} \bigl(v_{1}(g,p), \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert ^{ \frac{p}{p-1}} \bigr) \leq 0. $$

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

$$\begin{aligned}& \biggl\vert \frac{g (v_{1}^{\mu } )+g (v_{2}^{\mu } )}{2} - \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x \biggr\vert \\& \quad \leq \frac{v_{2}^{\mu }-v_{1}^{\mu }}{2} \biggl(\frac{1}{2\mu (p+1)} \biggl\{ 2- \biggl(1-\frac{1}{2^{\mu ^{2}-1}} \biggr)^{p+1} - \biggl( \frac{1}{2^{\mu ^{2}-1}}-1 \biggr)^{p+1} \biggr\} \biggr)^{\frac{1}{p}} \\& \qquad {}\times \biggl( \frac{ \vert D_{\mu }(g) (v_{1}^{\mu } ) \vert ^{\frac{p}{p-1}} + \vert D_{\mu }(g) (v_{2}^{\mu } ) \vert ^{\frac{p}{p-1}}}{2\mu }+\varepsilon \biggr) ^{\frac{p-1}{p}}. \end{aligned}$$
(4.6)

Proof

By making use of \(w(\eta ^{\mu })= \vert 1-2\eta ^{\mu } \vert \) in (2.6) and by Definition 2.3, we get

$$\begin{aligned} \mathcal{T}_{\digamma _{\mu },w}(\mathbf{y}_{1},\mathbf{y}_{2}, \mathbf{y}_{3})&=\mathbf{y}_{1} - \biggl( \int _{0}^{1} \eta ^{\mu } \bigl\vert 1-2\eta ^{\mu } \bigr\vert \,d_{\mu }\eta \biggr) \mathbf{y}_{2} - \biggl( \int _{0}^{1} \bigl(1-\eta ^{\mu } \bigr) \bigl\vert 1-2\eta ^{\mu } \bigr\vert \,d_{\mu }\eta \biggr)\mathbf{y}_{3} -\varepsilon \\ &=\mathbf{y}_{1}-\frac{\mathbf{y}_{2}+\mathbf{y}_{3}}{2\mu }- \varepsilon \end{aligned}$$

for \(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\in \mathtt{R}\). Then, by using Theorem 4.2, we have

$$\begin{aligned} 0 &\geq \mathcal{T}_{\digamma _{\mu },1} \bigl(v_{1}(g,p), \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert ^{ \frac{p}{p-1}} \bigr)-\epsilon \\ &=v_{1}(g,p)- \frac{ \vert D_{\mu }(g) (v_{1}^{\mu } ) \vert ^{\frac{p}{p-1}}+ \vert D_{\mu }(g) (v_{2}^{\mu } ) \vert ^{\frac{p}{p-1}}}{2\mu }- \varepsilon \\ &= \biggl(\frac{2}{v_{2}^{\mu }-v_{1}^{\mu }} \biggr)^{\frac{p}{p-1}} \biggl( \frac{1}{2\mu (p+1)} \biggl\{ 2- \biggl(1-\frac{1}{2^{\mu ^{2}-1}} \biggr)^{p+1} - \biggl(\frac{1}{2^{\mu ^{2}-1}}-1 \biggr)^{p+1} \biggr\} \biggr)^{ \frac{-1}{p-1}} \\ &\quad {}\times \biggl\vert \frac{g (v_{1}^{\mu } )+g (v_{2}^{\mu } )}{2} - \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr) \,d_{\mu }x \biggr\vert ^{\frac{p}{p-1}} \\ &\quad {}- \frac{ \vert D_{\mu }(g) (v_{1}^{\mu } ) \vert ^{\frac{p}{p-1}}+ \vert D_{\mu }(g) (v_{2}^{\mu } ) \vert ^{\frac{p}{p-1}}}{2\mu }- \varepsilon . \end{aligned}$$

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

$$\begin{aligned}& \biggl\vert \frac{g (v_{1}^{\mu } )+g (v_{2}^{\mu } )}{2} - \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x \biggr\vert \\& \quad \leq \frac{v_{2}^{\mu }-v_{1}^{\mu }}{2} \biggl(\frac{1}{2\mu (p+1)} \biggl\{ 2- \biggl(1-\frac{1}{2^{\mu ^{2}-1}} \biggr)^{p+1} - \biggl( \frac{1}{2^{\mu ^{2}-1}}-1 \biggr)^{p+1} \biggr\} \biggr)^{\frac{1}{p}} \\& \qquad {}\times \biggl( \frac{\mu \vert D_{\mu }(g) (v_{1}^{\mu } ) \vert ^{\frac{p}{p-1}} + \vert D_{\mu }(g) (v_{2}^{\mu } ) \vert ^{\frac{p}{p-1}}}{\mu (\mu +1)} \biggr)^{\frac{p-1}{p}}. \end{aligned}$$
(4.7)

Proof

By making use of \(w(\eta ^{\mu })=1\) in (2.9), we get

$$\begin{aligned} \mathcal{T}_{\digamma _{\mu },1}(\mathbf{y}_{1},\mathbf{y}_{2}, \mathbf{y}_{3})&=\mathbf{y}_{1}- \biggl( \int _{0}^{1} \eta \,d_{\mu }\eta \biggr)\mathbf{y}_{2} - \biggl( \int _{0}^{1} (1-\eta )\,d_{\mu }\eta \biggr) \mathbf{y}_{3} \\ &=\mathbf{y}_{1}- \frac{\mu \mathbf{y}_{2}+\mathbf{y}_{3}}{\mu (\mu +1)} \end{aligned}$$

for \(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\in \mathtt{R}\). Then, by using Theorem 4.2, we have

$$\begin{aligned} 0 &\geq \mathcal{T}_{\digamma _{\mu },1} \bigl(v_{1}(g,p), \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert ^{ \frac{p}{p-1}} \bigr) \\ &=v_{1}(g,p)- \frac{\mu \vert D_{\mu }(g) (v_{1}^{\mu } ) \vert ^{\frac{p}{p-1}}+ \vert D_{\mu }(g) (v_{2}^{\mu } ) \vert ^{\frac{p}{p-1}}}{\mu (\mu +1)}. \end{aligned}$$

This rearranges to the required inequality (4.7). □

Corollary 4.6

Theorem 4.2with\(|D_{\mu }(g)|\)to beh-convex leads to

$$\begin{aligned}& \biggl\vert \frac{g (v_{1}^{\mu } )+g (v_{2}^{\mu } )}{2} - \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x \biggr\vert \\& \quad \leq \frac{v_{2}^{\mu }-v_{1}^{\mu }}{2} \biggl(\frac{1}{2\mu (p+1)} \biggl\{ 2- \biggl(1-\frac{1}{2^{\mu ^{2}-1}} \biggr)^{p+1} - \biggl( \frac{1}{2^{\mu ^{2}-1}}-1 \biggr)^{p+1} \biggr\} \biggr)^{\frac{1}{p}} \\& \qquad {}\times \biggl( \int _{0}^{1} h\bigl(\eta ^{\mu } \bigr)\,d_{\mu }\eta \biggr)^{ \frac{p-1}{p}} \bigl( \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert ^{\frac{p}{p-1}} + \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert ^{\frac{p}{p-1}} \bigr) ^{\frac{p-1}{p}}. \end{aligned}$$
(4.8)

Proof

By making use of (2.11) with \(w(\eta ^{\mu })=1\), we have

$$\begin{aligned} \mathcal{T}_{\digamma _{\mu },1}(\mathbf{y}_{1},\mathbf{y}_{2}, \mathbf{y}_{3})&=\mathbf{y}_{1} - \biggl( \int _{0}^{1} h\bigl(\eta ^{\mu } \bigr)\,d_{\mu }\eta \biggr)\mathbf{y}_{2} - \biggl( \int _{0}^{1} h \bigl(1-\eta ^{\mu } \bigr)\,d_{\mu }\eta \biggr)\mathbf{y}_{3} \\ &=\mathbf{y}_{1}- \biggl( \int _{0}^{1} h\bigl(\eta ^{\mu } \bigr)\,d_{\mu }\eta \biggr) ( \mathbf{y}_{2}+ \mathbf{y}_{3}) \end{aligned}$$

for \(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\in \mathtt{R}\). Then, by making use of Theorem 4.2, we get

$$\begin{aligned} 0 &\geq \mathcal{T}_{\digamma _{\mu },1} \bigl(v_{1}(g,p), \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert ^{\frac{p}{p-1}}, \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert ^{\frac{p}{p-1}} \bigr) \\ &=v_{1}(g,p)- \biggl( \int _{0}^{1} h\bigl(\eta ^{\mu } \bigr)\,d_{\mu }\eta \biggr) \bigl( \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert ^{ \frac{p}{p-1}} + \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert ^{ \frac{p}{p-1}} \bigr). \end{aligned}$$

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

$$\begin{aligned}& \mathcal{T}_{\digamma _{\mu },w} \biggl(\frac{1}{v_{2}^{\mu }-v_{1}^{\mu }} \biggl\vert \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x-g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr) \biggr\vert , \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert , \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert ,\eta \biggr) \\& \quad {}+ \int _{0}^{1} L_{w(\eta )}\,d_{\mu } \eta \leq 0. \end{aligned}$$
(4.9)

Proof

By using the \(\digamma _{\mu }\)-convexity of \(|D_{\mu }(g)|\), we have

$$ \digamma _{\mu } \bigl( \bigl\vert D_{\mu }(g) \bigl(\eta ^{\mu }v_{1}^{\mu }+\bigl(1- \eta ^{\mu }\bigr)v_{2}^{\mu } \bigr) \bigr\vert , \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert , \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert , \eta \bigr)\leq 0. $$

Making use of axiom (\(\bar{\varLambda }_{3}\)) for \(w(\eta )= \vert P(\eta ) \vert \), \(\eta \in [0,1]\), we obtain

$$ \digamma _{\mu } \bigl(w(\eta ) \bigl\vert D_{\mu }(g) \bigl(\eta ^{\mu }v_{1}^{\mu }+\bigl(1-\eta ^{\mu }\bigr)v_{2}^{\mu } \bigr) \bigr\vert ,w( \eta ) \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert ,w(\eta ) \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert ,\eta \bigr)\leq 0. $$

Integrating over \([0,1]\) according to η and by making use of axiom (\(\bar{\varLambda }_{2}\)), we can obtain

$$\begin{aligned} &\mathcal{T}_{\digamma _{\mu },w} \biggl( \int _{0}^{1} w(\eta ) \bigl\vert D_{\mu }(g) \bigl(\eta ^{\mu } v_{1}^{\mu }+ \bigl(1-\eta ^{\mu }\bigr)v_{2}^{\mu } \bigr) \bigr\vert \,d_{\mu }\eta , \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert , \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert , \eta \biggr) \\ &\quad {}+ \int _{0}^{1} L_{w(\eta )}\,d_{\mu } \eta \leq 0. \end{aligned}$$

From Lemma 2.2, we have

$$\begin{aligned} &\biggl\vert \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x-g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr) \biggr\vert \\ &\quad \leq \bigl(v_{2}^{\mu }-v_{1}^{\mu } \bigr) \int _{0}^{1} w(\eta ) \bigl\vert D_{\mu }(g) \bigl(\eta ^{\mu }v_{1}^{\mu }+ \bigl(1-\eta ^{\mu }\bigr)v_{2}^{\mu } \bigr) \bigr\vert \,d_{\mu }\eta . \end{aligned}$$

Since \(\mathcal{T}_{\digamma _{\mu },w}\) is nondecreasing according to the first variable, then we can deduce

$$\begin{aligned}& \mathcal{T}_{\digamma ,w} \biggl(\frac{1}{v_{2}^{\mu }-v_{1}^{\mu }} \biggl\vert \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x-g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr) \biggr\vert , \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert , \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert ,\eta \biggr) \\& \quad {}+ \int _{0}^{1} L_{w(\eta )}\,d_{\mu } \eta \leq 0. \end{aligned}$$

This completes the proof of (4.9). □

Corollary 4.7

Theorem 4.3with\(|D_{\mu }(g)|\)to beε-convex leads to

$$\begin{aligned} &\biggl\vert \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x-g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr) \biggr\vert \\ &\quad \leq \bigl(v_{2}^{\mu }-v_{1}^{\mu } \bigr) \biggl( \frac{ \vert D_{\mu }(g) (v_{1}^{\mu } ) \vert + \vert D_{\mu }(g) (v_{2}^{\mu } ) \vert }{8\mu } +\frac{4\mu -3}{\mu }\varepsilon \biggr). \end{aligned}$$
(4.10)

Proof

By making use of \(w(\eta ^{\mu })=|P(\eta )|\) in (2.7) as well as Definition 2.3, we get

$$ \int _{0}^{1} L_{w(\eta )}\,d_{\mu } \eta =\varepsilon \int _{0}^{1} \bigl(1- \bigl\vert P( \eta ) \bigr\vert \bigr)\,d_{\mu }\eta =\frac{3}{4\mu }\varepsilon . $$

Then, by making use (2.6) with \(w(\eta ^{\mu })=|P(\eta )|\), we get

$$\begin{aligned} \mathcal{T}_{\digamma _{\mu },w}(\mathbf{y}_{1},\mathbf{y}_{2}, \mathbf{y}_{3})&=\mathbf{y}_{1}- \biggl( \int _{0}^{1} \eta ^{\mu } \bigl\vert P(\eta ) \bigr\vert \,d_{\mu }\eta \biggr) \mathbf{y}_{2}- \biggl( \int _{0}^{1} \bigl(1-\eta ^{\mu } \bigr) \bigl\vert P( \eta ) \bigr\vert \,d_{\mu }\eta \biggr) \mathbf{y}_{3}-\varepsilon \\ &=\mathbf{y}_{1}-\frac{\mathbf{y}_{2}+\mathbf{y}_{3}}{8\mu }- \varepsilon , \end{aligned}$$

for \(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\in \mathtt{R}\). Thus, by using Theorem 4.3, we have

$$\begin{aligned} 0 \geq& \mathcal{T}_{\digamma ,w} \biggl( \frac{1}{v_{2}^{\mu }-v_{1}^{\mu }} \biggl\vert \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{a}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x-g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr) \biggr\vert , \\ & \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert , \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert , \eta \biggr) + \int _{0}^{1} L_{w(\eta )}\,d_{\mu } \eta \\ =&\frac{1}{v_{2}^{\mu }-v_{1}^{\mu }} \biggl\vert \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{a}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x-g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr) \biggr\vert \\ &{}- \frac{ \vert D_{\mu }(g) (v_{1}^{\mu } ) \vert + \vert D_{\mu }(g) (v_{2}^{\mu } ) \vert }{8\mu } -\varepsilon +\frac{3}{4\mu } \varepsilon . \end{aligned}$$

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

$$\begin{aligned} &\biggl\vert \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x-g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr) \biggr\vert \\ &\quad \leq \bigl(v_{2}^{\mu }-v_{1}^{\mu } \bigr) \biggl\{ \frac{\mu }{(\mu +1)(2\mu +1)} \biggl(1-\frac{1}{2^{1/\mu }+1} \biggr) \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert \\ &\qquad {}+ \biggl[\frac{1}{4\mu }-\frac{\mu }{(\mu +1)(2\mu +1)} \biggl(1- \frac{1}{2^{1/\mu }+1} \biggr) \biggr] \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert \biggr\} . \end{aligned}$$
(4.11)

Proof

By making use of \(w(\eta ^{\mu })=|P(\eta )|\) in (2.9), we get

$$\begin{aligned} \mathcal{T}_{\digamma _{\mu },w}(\mathbf{y}_{1},\mathbf{y}_{2}, \mathbf{y}_{3})&=\mathbf{y}_{1}- \biggl( \int _{0}^{1} t \bigl\vert P(\eta ) \bigr\vert \,d_{\mu }\eta \biggr)\mathbf{y}_{2} - \biggl( \int _{0}^{1} (1-\eta ) \bigl\vert P(\eta ) \bigr\vert \,d_{\mu }\eta \biggr)\mathbf{y}_{3} \\ &=\mathbf{y}_{1}-\frac{\mu }{(\mu +1)(2\mu +1)} \biggl(1- \frac{1}{2^{1/\mu }+1} \biggr)\mathbf{y}_{2} \\ &\quad {}- \biggl[ \frac{1}{4\mu }- \frac{\mu }{(\mu +1)(2\mu +1)} \biggl(1-\frac{1}{2^{1/\mu }+1} \biggr) \biggr] \mathbf{y}_{3} \end{aligned}$$

for \(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\in \mathtt{R}\). It follows from Theorem 4.3 that

$$\begin{aligned} 0 \geq &\mathcal{T}_{\digamma _{\mu },w} \biggl( \frac{1}{v_{2}^{\mu }-v_{1}^{\mu }} \biggl\vert \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x-g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr) \biggr\vert , \\ & \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert , \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert , \eta \biggr) + \int _{0}^{1} L_{w(\eta ^{\mu })}\,d_{\mu } \eta \\ =&\frac{1}{v_{2}^{\mu }-v_{1}^{\mu }} \biggl\vert \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x-g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr) \biggr\vert \\ &{}-\frac{\mu }{(\mu +1)(2\mu +1)} \biggl(1- \frac{1}{2^{1/\mu }+1} \biggr) \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert \\ &{}- \biggl[\frac{1}{4\mu }-\frac{\mu }{(\mu +1)(2\mu +1)} \biggl(1- \frac{1}{2^{1/\mu }+1} \biggr) \biggr] \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert . \end{aligned}$$

This rearranges to the required inequality (4.11). □

Corollary 4.9

Theorem 4.3with\(|D_{\mu }(g)|\)to beh-convex leads to

$$\begin{aligned}& \biggl\vert \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}}g \bigl(x^{\mu } \bigr)\,d_{\mu }x -g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr) \biggr\vert \\& \quad \leq \bigl(v_{2}^{\mu }-v_{1}^{\mu } \bigr) \biggl( \int _{0}^{1}h\bigl(\eta ^{\mu }\bigr) \bigl\vert 1-2\eta ^{\mu } \bigr\vert \,d_{\mu }\eta \biggr) \bigl( \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert + \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert \bigr). \end{aligned}$$
(4.12)

Proof

By making use of \(w(\eta ^{\mu })=|P(\eta )|\) in (2.11), we get

$$\begin{aligned} \mathcal{T}_{\digamma _{\mu },w}(\mathbf{y}_{1},\mathbf{y}_{2}, \mathbf{y}_{3})&=\mathbf{y}_{1} - \biggl( \int _{0}^{1} h\bigl(\eta ^{\mu } \bigr) \bigl\vert P( \eta ) \bigr\vert \,d_{\mu }\eta \biggr) \mathbf{y}_{2} - \biggl( \int _{0}^{1} h \bigl(1-\eta ^{\mu } \bigr) \bigl\vert P(\eta ) \bigr\vert \,d_{\mu }\eta \biggr) \mathbf{y}_{3} \\ &=\mathbf{y}_{1}- \biggl( \int _{0}^{1} h\bigl(\eta ^{\mu } \bigr) \bigl\vert P(\eta ) \bigr\vert \,d_{\mu }\eta \biggr) ( \mathbf{y}_{2}+\mathbf{y}_{3}) \end{aligned}$$

for \(\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\in \mathtt{R}\). Then, by using Theorem 4.3, we get

$$\begin{aligned} 0 \geq& \mathcal{T}_{\digamma _{\mu },w} \biggl( \frac{1}{v_{2}^{\mu }-v_{1}^{\mu }} \biggl\vert \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x-g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr) \biggr\vert , \\ & \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert , \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert ,\eta \biggr) + \int _{0}^{1} L_{w(\eta ^{\mu })}\,d_{\mu } \eta \\ =&\frac{1}{v_{2}^{\mu }-v_{1}^{\mu }} \biggl\vert \frac{\mu }{v_{2}^{\mu }-v_{1}^{\mu }} \int _{v_{1}}^{v_{2}} g \bigl(x^{\mu } \bigr)\,d_{\mu }x-g \biggl(\frac{v_{1}^{\mu }+v_{2}^{\mu }}{2} \biggr) \biggr\vert \\ &{}- \biggl( \int _{0}^{1} h\bigl(\eta ^{\mu } \bigr) \bigl\vert P(\eta ) \bigr\vert \,d_{\mu }\eta \biggr) \bigl( \bigl\vert D_{\mu }(g) \bigl(v_{1}^{\mu } \bigr) \bigr\vert + \bigl\vert D_{\mu }(g) \bigl(v_{2}^{\mu } \bigr) \bigr\vert \bigr), \end{aligned}$$

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

$$\begin{aligned}& \digamma \biggl(\mathcal{A}^{-1} \bigl(v_{1}^{\mu },v_{2}^{\mu }\bigr), \mathcal{L}^{-1}\bigl(v_{1}^{\mu },v_{2}^{\mu } \bigr), \mathcal{L}^{-1}\bigl(v_{1}^{\mu },v_{2}^{\mu } \bigr),\frac{1}{2} \biggr)\leq 0, \end{aligned}$$
(5.1)
$$\begin{aligned}& \mathcal{T}_{\digamma ,w} \biggl(2\mathcal{L}^{-1} \bigl(v_{1}^{\mu },v_{2}^{\mu }\bigr), \frac{1}{2}\mathcal{H}^{-1}\bigl(v_{1}^{\mu },v_{2}^{\mu } \bigr), \frac{1}{2} \mathcal{H}^{-1}\bigl(v_{1}^{\mu },v_{2}^{\mu } \bigr) \biggr) \leq 0. \end{aligned}$$
(5.2)

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

$$ \mathcal{A}^{-1}\bigl(v_{1}^{\mu }, v_{2}^{\mu }\bigr)\le \mathcal{L}^{-1} \bigl(v_{1}^{ \mu }, v_{2}^{\mu }\bigr) \leq \mathcal{H}^{-1}\bigl(v_{1}^{\mu }, v_{2}^{\mu }\bigr). $$
(5.3)

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

$$\begin{aligned}& \digamma \biggl(\mathcal{A}^{n} \bigl(v_{1}^{\mu },v_{2}^{\mu }\bigr), \mathcal{L}_{n}^{n}\bigl(v_{1}^{\mu },v_{2}^{\mu } \bigr), \mathcal{L}_{n}^{n}\bigl(v_{1}^{\mu },v_{2}^{\mu } \bigr), \frac{1}{2} \biggr)\leq 0, \end{aligned}$$
(5.4)
$$\begin{aligned}& \mathcal{T}_{\digamma ,w} \bigl(2\mathcal{L}_{n}^{n} \bigl(v_{1}^{\mu },v_{2}^{\mu }\bigr), v_{1}^{n\mu }+v_{2}^{n\mu }, v_{1}^{n\mu }+v_{2}^{n\mu } \bigr). \end{aligned}$$
(5.5)

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

$$ \bigl\vert \mathcal{H}^{-1}\bigl(v_{1}^{\mu }, v_{2}^{\mu }\bigr)-\mathcal{L}^{-1} \bigl(v_{1}^{ \mu }, v_{2}^{\mu }\bigr) \bigr\vert \leq \frac{v_{2}^{\mu }-v_{1}^{\mu }}{2} \biggl( \frac{2^{3\mu ^{2}}+6\times 2^{\mu ^{2}}-8}{6\mu \times 2^{3\mu ^{2}}} \biggr) \bigl(v_{1}^{-\mu (1+\mu )}+v_{2}^{-\mu (1+\mu )} \bigr). $$
(5.6)

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

$$ \bigl\vert \mathcal{L}^{-1}\bigl(v_{1}^{\mu }, v_{2}^{\mu }\bigr)-\mathcal{A}^{-1} \bigl(v_{1}^{ \mu }, v_{2}^{\mu }\bigr) \bigr\vert \leq \bigl(v_{2}^{\mu }-v_{1}^{\mu } \bigr) \biggl(\frac{v_{1}^{-\mu (1+\mu )}+v_{2}^{-\mu (1+\mu )}}{8\mu } \biggr). $$
(5.7)

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.

  • Fig. 1 represents Proposition 5.2 with \(\mu =\frac{1}{2}\), \(v_{1}=x\), \(v_{2}=y\).

    Figure 1
    figure 1

    Figure representation for Proposition 5.2

    Full size image

  • Fig. 2 represents Proposition 5.4\(\mu =\frac{1}{2}\), \(v_{1}=x\), \(v_{2}=y\).

    Figure 2
    figure 2

    Figure representation for Proposition 5.4

    Full size image

  • Fig. 3 represents Proposition 5.5\(\mu =\frac{1}{2}\), \(v_{1}=x\), \(v_{2}=y\).

    Figure 3
    figure 3

    Figure representation for Proposition 5.5

    Full size image

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.