1 Introduction and preliminaries

A function \(\varUpsilon :\mathcal{W}\rightarrow \mathbb{R}\) on an interval of real line, for all \(w_{1},w_{2}\in \mathcal{W}\) and \(\kappa \in [0,1]\), is called convex if the following inequality holds:

$$ \varUpsilon \bigl(\kappa w_{1}+(1-\kappa )w_{2}\bigr)\leq \kappa \varUpsilon (w_{1})+(1- \kappa ) \varUpsilon (w_{2}). $$
(1)

Due to the importance of convex functions, many authors have given results not only for convex functions but also for their generalizations. The Hermite–Hadamard inequality [9] on a real interval was defined by

$$ \varUpsilon \biggl(\frac{w_{1}+w_{2}}{2} \biggr)\leq \frac{1}{w_{2}-w_{1}} \int ^{w_{2}}_{w_{1}}\varUpsilon (u)\,du\leq \frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} $$
(2)

for all \(w_{1},w_{2}\in \mathcal{W}\) with \(w_{1}< w_{2}\). Then Fejér [8] proved the following inequality:

$$\begin{aligned} \varUpsilon \biggl(\frac{w_{1}+w_{2}}{2} \biggr) \int _{w_{1}}^{w_{2}} \curlyvee (u)\,du \leq& \frac{1}{w_{2}-w_{1}} \int ^{w_{2}}_{w_{1}} \varUpsilon (u)\curlyvee (u)\,du \\ \leq& \frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \int _{w_{1}}^{w_{2}} \curlyvee (u)\,du, \end{aligned}$$
(3)

where \(\curlyvee :[w_{1},w_{2}]\rightarrow \mathbb{R}\) is nonnegative, integrable, and symmetric to \((w_{1}+w_{2})/2\), called Hermite–Hadamard–Fejér inequality. Inequalities (2) and (3) have been further generalized in different ways not only for classical integral but also for other generalized integrals such as Riemann–Liouville fractional integral, Katugampola, ψ-Riemann–Liouville, and conformable fractional integrals etc. For more results and details see [1, 4–7, 17–23, 26–30].

Definition 1.1

([11, 12])

Suppose an interval \(\mathcal{W}\subset (0, \infty )=\mathbb{R}_{+}\) and \(p\in \mathbb{R}\setminus \{0\}\). Then a function \(\varUpsilon :\mathcal{W}\rightarrow \mathbb{R}\) is called p-convex if

$$ \varUpsilon \bigl(\bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p}\bigr]^{\frac{1}{p}} \bigr)\leq \kappa \varUpsilon (w_{1})+(1-\kappa )\varUpsilon (w_{2}) $$
(4)

holds for all \(w_{1},w_{2}\in \mathcal{W}\) and \(\kappa \in [0,1]\). If inequality (4) is in opposite order, then ϒ is called p-concave function.

Definition 1.2

([14])

Let \(\varUpsilon \in L[w_{1},w_{2}]\). The left- and right-sided Riemann–Liouville fractional integrals \(J^{\alpha }_{w_{1}+}\varUpsilon \) and \(J^{\alpha }_{w_{2}-}\varUpsilon \) of order \(\alpha \in \mathbb{C}\) with \(\mathbb{R}(\alpha )>0\) and \(w_{2} > w_{1}\geq 0\) are given by

$$ J^{\alpha }_{w_{1}+}\varUpsilon (u)=\frac{1}{\varGamma (\alpha )} \int _{w_{1}}^{u}(u-v)^{ \alpha -1}\varUpsilon (v)\,dv,\quad u>w_{1}, $$

and

$$ J^{\alpha }_{w_{2}-}\varUpsilon (u)=\frac{1}{\varGamma (\alpha )} \int _{u}^{w_{2}}(v-u)^{ \alpha -1}\varUpsilon (v)\,dv,\quad u< w_{2}, $$

respectively, where \(\varGamma (\cdot )\) is the gamma function.

Abdeljawad [2] defined the conformable fractional integral as follows.

Definition 1.3

([2])

Let \(\alpha \in (n,n+1]\) and \(\gamma =\alpha -n\). Then the left- and right-sided conformable fractional integrals of order \(\alpha >0\) are given by

$$ J_{\alpha }^{w_{1}}\varUpsilon (u)=\frac{1}{n!} \int _{w_{1}}^{u}(u-v)^{n}(v-w_{1})^{ \gamma -1} \varUpsilon (v)\,dv, $$

and

$$ ^{w_{2}}J_{\alpha }\varUpsilon (u)=\frac{1}{n!} \int _{u}^{w_{2}}(v-u)^{n}(w_{2}-v)^{ \gamma -1} \varUpsilon (v)\,dv, $$

respectively.

Note that for \(\alpha =n+1\) then \(\gamma =1\), where \(n=0,1,2,\ldots \) , and in this case conformable fractional integrals become Riemann–Liouville fractional integrals.

The classical beta function and hypergeometric function are defined, respectively, by

$$ \beta (w_{1},w_{2})= \int _{0}^{1}u^{w_{1}-1}(1-u)^{w_{2}-1} \,du $$

and

$$ {}_{2}F_{1}(w_{1},w_{2};u;v)= \frac{1}{\beta (w_{2},u-w_{2})} \int _{0}^{1}u^{w_{2}-1}(1-u)^{u-w_{2}-1}(1-vu)^{-w_{1}} \,du, $$

with \(u>w_{2}>0\), \(|v|<1\).

The incomplete beta function is defined as follows:

$$ \beta _{u}(w_{1},w_{2})= \int _{0}^{u}v^{w_{1}-1}(1-v)^{w_{2}-1}\,dv,\quad u \in [0,1]. $$

The relationship between the classical beta function and the incomplete beta function is given as follows:

$$ \beta (w_{1},w_{2})=\beta _{u}(w_{1},w_{2})+ \beta _{1-u}(w_{1},w_{2}). $$

2 Hermite–Hadamard type inequalities

In this section we prove some Hermite–Hadamard type inequalities for p-convex functions via conformable fractional integral.

Theorem 2.1

Let\(\varUpsilon :[w_{1},w_{2}]\subset (0,\infty )\rightarrow \mathbb{R}\)be ap-convex function such that\(\varUpsilon \in L[w_{1},w_{2}]\)and\(\alpha >0\). Then

  1. (i)

    for\(p>0\), we have

    $$\begin{aligned} \begin{aligned}[b] &\varUpsilon \biggl( \biggl[ \frac{w_{1}^{p}+w_{2}^{p}}{2} \biggr]^{1/p} \biggr) \\ &\quad \leq \frac{\varGamma (\alpha +1)}{2\varGamma (\alpha -n)(w_{2}^{p}-w_{1}^{p})^{\alpha }} \bigl[J^{w_{1}^{p}}_{\alpha }(\varUpsilon \circ \phi ) \bigl(w_{2}^{p}\bigr)+{}^{w_{2}^{p}}J_{ \alpha }( \varUpsilon \circ \phi ) \bigl(w_{1}^{p}\bigr) \bigr] \\ &\quad \leq \frac{\varUpsilon (w_{1}^{p})+\varUpsilon (w_{2}^{p})}{2}, \end{aligned} \end{aligned}$$
    (5)

    here\(\phi (u)=u^{\frac{1}{p}}\)for all\(u\in [w_{1}^{p},w_{2}^{p}]\);

  2. (ii)

    for\(p<0\), we have

    $$\begin{aligned} \begin{aligned}[b] &\varUpsilon \biggl( \biggl[ \frac{w_{1}^{p}+w_{2}^{p}}{2} \biggr]^{1/p} \biggr) \\ &\quad \leq \frac{\varGamma (\alpha +1)}{2\varGamma (\alpha -n)(w_{1}^{p}-w_{2}^{p})^{\alpha }} \bigl[{}^{w_{1}^{p}}J_{\alpha }(\varUpsilon \circ \phi ) \bigl(w_{2}^{p}\bigr)+ J^{w_{2}^{p}}_{ \alpha }( \varUpsilon \circ \phi ) \bigl(w_{1}^{p}\bigr) \bigr] \\ &\quad \leq \frac{\varUpsilon (w_{1}^{p})+\varUpsilon (w_{2}^{p})}{2}, \end{aligned} \end{aligned}$$
    (6)

    here\(\phi (u)=u^{\frac{1}{p}}\)for all\(u\in [w_{2}^{p},w_{1}^{p}]\).

Proof

(i) Since ϒ is a p-convex function on \([w_{1},w_{2}]\), we have

$$ \varUpsilon \biggl( \biggl[\frac{x^{p}+y^{p}}{2} \biggr]^{\frac{1}{p}} \biggr) \leq \frac{\varUpsilon (x)+\varUpsilon (y)}{2}. $$

Taking \(x^{p}=\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p}\) and \(y^{p}=(1-\kappa )w_{1}^{p}+\kappa w_{2}^{p}\) with \(\kappa \in [0,1]\), we get

$$ \varUpsilon \biggl( \biggl[\frac{w_{1}^{p}+w_{2}^{p}}{2} \biggr]^{ \frac{1}{p}} \biggr)\leq \frac{\varUpsilon ( [\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} ]^{\frac{1}{p}} ) +\varUpsilon ( [(1-\kappa )w_{1}^{p}+\kappa w_{2}^{p} ]^{\frac{1}{p}} )}{2}. $$
(7)

Multiplying (7) by \(\frac{1}{n!}\kappa ^{n}(1-\kappa )^{\alpha -n-1}\), with \(\kappa \in (0,1)\), \(\alpha >0\), on both sides and then integrating about κ over \([0,1]\), we find

$$\begin{aligned} \begin{aligned}[b] &\frac{2}{n!}\varUpsilon \biggl( \biggl[\frac{w_{1}^{p}+w_{2}^{p}}{2} \biggr]^{\frac{1}{p}} \biggr) \int _{0}^{1}\kappa ^{n}(1-\kappa )^{ \alpha -n-1}\,d\kappa \\ &\quad \leq \frac{1}{n!} \int _{0}^{1}\kappa ^{n}(1-\kappa )^{\alpha -n-1} \varUpsilon \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr)\,d\kappa \\ &\qquad {} +\frac{1}{n!} \int _{0}^{1}\kappa ^{n}(1-\kappa )^{\alpha -n-1} \varUpsilon \bigl( \bigl[(1-\kappa )w_{1}^{p}+ \kappa w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr)\,d\kappa \\ &\quad =I_{1}+I_{2}. \end{aligned} \end{aligned}$$
(8)

By setting \(u=\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p}\), we have

$$\begin{aligned} \begin{aligned}[b] I_{1}&=\frac{1}{n!} \int _{0}^{1}\kappa ^{n}(1-\kappa )^{\alpha -n-1} \varUpsilon \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr)\,d\kappa \\ &=\frac{1}{n!} \int _{w_{2}^{p}}^{w_{1}^{p}} \biggl( \frac{u-w_{2}^{p}}{w_{1}^{p}-w_{2}^{p}} \biggr)^{n} \biggl(1- \frac{u-w_{2}^{p}}{w_{1}^{p}-w_{2}^{p}} \biggr)^{\alpha -n-1}( \varUpsilon \circ \phi ) (u)\frac{du}{w_{1}^{p}-w_{2}^{p}} \\ &=\frac{1}{n! (w_{2}^{p}-w_{1}^{p} )^{\alpha }} \int _{w_{1}^{p}}^{w_{2}^{p}} \bigl(w_{2}^{p}-u \bigr)^{n} \bigl(u-w_{1}^{p} \bigr)^{\alpha -n-1}( \varUpsilon \circ \phi ) (u)\,du \\ &=\frac{1}{(w_{2}^{p}-w_{1}^{p})^{\alpha }}\, J^{w_{1}^{p}}_{\alpha }( \varUpsilon \circ \phi ) \bigl(w_{2}^{p}\bigr). \end{aligned} \end{aligned}$$
(9)

Similarly, by setting \(u=\kappa w_{2}^{p}+(1-\kappa )w_{1}^{p}\), we have

$$\begin{aligned} \begin{aligned}[b] I_{2}&=\frac{1}{n!} \int _{0}^{1}\kappa ^{n}(1-\kappa )^{\alpha -n-1} \varUpsilon \bigl( \bigl[\kappa w_{2}^{p}+(1- \kappa )w_{1}^{p} \bigr]^{ \frac{1}{p}} \bigr)\,d\kappa \\ &=\frac{1}{n!} \int _{w_{1}^{p}}^{w_{2}^{p}} \biggl( \frac{u-w_{1}^{p}}{w_{2}^{p}-w_{1}^{p}} \biggr)^{n} \biggl(1- \frac{u-w_{1}^{p}}{w_{2}^{p}-w_{1}^{p}} \biggr)^{\alpha -n-1}( \varUpsilon \circ \phi ) (u)\frac{du}{w_{2}^{p}-w_{1}^{p}} \\ &=\frac{1}{n! (w_{2}^{p}-w_{1}^{p} )^{\alpha }} \int _{w_{1}^{p}}^{w_{2}^{p}} \bigl(u-w_{1}^{p} \bigr)^{n} \bigl(w_{2}^{p}-u \bigr)^{\alpha -n-1}( \varUpsilon \circ \phi ) (u)\,du \\ &=\frac{1}{(w_{2}^{p}-w_{1}^{p})^{\alpha }}{}^{w_{2}^{p}}J_{\alpha }( \varUpsilon \circ \phi ) \bigl(w_{1}^{p}\bigr). \end{aligned} \end{aligned}$$
(10)

Thus, by putting values of \(I_{1}\) and \(I_{2}\) in (8), the first inequality of (5) is achieved. For another inequality, we note that

$$ \varUpsilon \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr)+\varUpsilon \bigl( \bigl[\kappa w_{2}^{p}+(1- \kappa )w_{1}^{p} \bigr]^{\frac{1}{p}} \bigr) \leq \bigl[ \varUpsilon (w_{1})+ \varUpsilon (w_{2}) \bigr]. $$
(11)

Multiplying (11) by \(\frac{1}{n!}\kappa ^{n}(1-\kappa )^{\alpha -n-1}\), with \(\kappa \in (0,1)\), \(\alpha >0\), on both sides and then integrating about κ over \([0,1]\), we achieve the second inequality of (5). This completes the proof.

(ii) Proof is identical to that of (i). □

Remark 2.1

In Theorem 2.1:

  1. 1.

    If we let \(p=1\) in (i), we get Theorem 2.1 in [25].

  2. 2.

    If we let \(p=-1\) in (ii), we get Theorem 2.1 in [3].

  3. 3.

    If we let \(p=1\) and \(\alpha =n+1\) in (i), we get Theorem 2 in [24].

  4. 4.

    If we let \(p=-1\) and \(\alpha =n+1\) in (ii), we get Theorem 4 in [13].

Lemma 2.1

Let\(\varUpsilon :[w_{1},w_{2}]\subset (0,\infty )\rightarrow \mathbb{R}\)be a differentiable function on\((w_{1},w_{2})\)with\(w_{1}< w_{2}\)such that\(\varUpsilon '\in L[w_{1},w_{2}]\)and\(\alpha >0\). Then

  1. (i)

    for\(p>0\), we have

    $$\begin{aligned} \begin{aligned}[b] &{}_{1}\Delta _{\varUpsilon }(w_{1},w_{2};\alpha ;\beta ;J) \\ &\quad =\frac{w_{2}^{p}-w_{1}^{p}}{2p} \int _{0}^{1} \bigl(\beta _{1-\kappa }(n+1, \alpha -n)-\beta _{\kappa }(n+1,\alpha -n) \bigr) \\ &\qquad {}\times A_{\kappa }^{ \frac{1}{p}-1} \varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa , \end{aligned} \end{aligned}$$
    (12)

    here\(A_{\kappa }= [\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} ]\)and

    $$\begin{aligned} \begin{aligned} &{}_{1}\Delta _{\varUpsilon }(w_{1},w_{2}; \alpha ;\beta ;J) \\ &\quad =\beta (n+1,\alpha -n) \biggl( \frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \biggr) \\ &\qquad {}- \frac{n!}{2(w_{2}^{p}-w_{1}^{p})^{\alpha }} \bigl[J^{w_{1}^{p}}_{ \alpha }(\varUpsilon \circ \phi ) \bigl(w_{2}^{p}\bigr)+{}^{w_{2}^{p}}J_{\alpha }( \varUpsilon \circ \phi ) \tbinom{1}{p} \bigr]; \end{aligned} \end{aligned}$$
  2. (ii)

    for\(p<0\), we have

    $$\begin{aligned} \begin{aligned}[b] &{}_{2}\Delta _{\varUpsilon }(w_{1},w_{2};\alpha ;\beta ;J) \\ &\quad =\frac{w_{1}^{p}-w_{2}^{p}}{2p} \int _{0}^{1} \bigl(\beta _{\kappa }(n+1, \alpha -n)-\beta _{1-\kappa }(n+1,\alpha -n) \bigr) \\ &\qquad {}\times B_{\kappa }^{ \frac{1}{p}-1} \varUpsilon ' \bigl( \bigl[\kappa w_{2}^{p}+(1- \kappa )w_{1}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa , \end{aligned} \end{aligned}$$
    (13)

    here\(B_{\kappa }= [\kappa w_{2}^{p}+(1-\kappa )w_{1}^{p} ]\)and

    $$\begin{aligned} \begin{aligned} &{}_{2}\Delta _{\varUpsilon }(w_{1},w_{2}; \alpha ;\beta ;J) \\ &\quad =\beta (n+1,\alpha -n) \biggl( \frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \biggr) \\ &\qquad {} - \frac{n!}{2(w_{1}^{p}-w_{2}^{p})^{\alpha }} \bigl[{}^{w_{1}^{p}}J_{ \alpha }(\varUpsilon \circ \phi ) \bigl(w_{2}^{p}\bigr)+ J^{w_{2}^{p}}_{\alpha }( \varUpsilon \circ \phi ) \bigl(w_{1}^{p}\bigr) \bigr]. \end{aligned} \end{aligned}$$

Proof

(i) Consider

$$\begin{aligned} \begin{aligned}[b] &\int _{0}^{1} \bigl(\beta _{1-\kappa }(n+1, \alpha -n)-\beta _{ \kappa }(n+1,\alpha -n) \bigr) A_{\kappa }^{\frac{1}{p}-1} \varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr)\,d\kappa \\ &\quad = \int _{0}^{1}\beta _{1-\kappa }(n+1,\alpha -n)A_{\kappa }^{ \frac{1}{p}-1} \varUpsilon ' \bigl( \bigl[ \kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &\qquad {} - \int _{0}^{1}\beta _{\kappa }(n+1,\alpha -n) A_{\kappa }^{ \frac{1}{p}-1} \varUpsilon ' \bigl( \bigl[ \kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &\quad =I_{1}-I_{2}. \end{aligned} \end{aligned}$$
(14)

Then, by integration by parts, we have

$$\begin{aligned} \begin{aligned}[b] I_{1}&= \int _{0}^{1}\beta _{1-\kappa }(n+1,\alpha -n)A_{\kappa }^{ \frac{1}{p}-1} \varUpsilon ' \bigl( \bigl[ \kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &= \int _{0}^{1} \biggl( \int _{0}^{1-\kappa }u^{n}(1-u)^{\alpha -n-1} \,du \biggr)A_{\kappa }^{\frac{1}{p}-1} \varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &=\frac{p}{w_{2}^{p}-w_{1}^{p}}\beta (n+1,\alpha -n)\varUpsilon (w_{2}) \\ &\quad {} -\frac{p}{w_{2}^{p}-w_{1}^{p}} \int _{0}^{1}(1-\kappa )^{n}\kappa ^{ \alpha -n-1} \varUpsilon \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &=\frac{p}{w_{2}^{p}-w_{1}^{p}}\beta (n+1,\alpha -n)\varUpsilon (w_{2}) \\ &\quad {} -\frac{p}{w_{2}^{p}-w_{1}^{p}} \int _{w_{2}^{p}}^{w_{1}^{p}} \biggl(1- \frac{x-w_{2}^{p}}{w_{1}^{p}-w_{2}^{p}} \biggr)^{n} \biggl( \frac{x-w_{2}^{p}}{w_{1}^{p}-w_{2}^{p}} \biggr)^{\alpha -n-1} \frac{(\varUpsilon \circ \phi )(x)}{w_{1}^{p}-w_{2}^{p}}dx \\ &=\frac{p}{w_{2}^{p}-w_{1}^{p}}\beta (n+1,\alpha -n)\varUpsilon (w_{2})- \frac{n!}{(w_{2}^{p}-w_{1}^{p})^{\alpha +1}}{}^{w_{2}^{p}}J_{\alpha }( \varUpsilon \circ \phi ) \bigl(w_{1}^{p}\bigr). \end{aligned} \end{aligned}$$
(15)

Similarly, we have

$$\begin{aligned} \begin{aligned}[b] I_{2}&= \int _{0}^{1}\beta _{\kappa }(n+1,\alpha -n)\varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &= \int _{0}^{1} \biggl( \int _{0}^{\kappa }u^{n}(1-u)^{\alpha -n-1} \,du \biggr)\varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &=-\frac{p}{w_{2}^{p}-w_{1}^{p}}\beta (n+1,\alpha -n)\varUpsilon (w_{1}) \\ &\quad {} +\frac{p}{w_{2}^{p}-w_{1}^{p}} \int _{0}^{1}\kappa ^{n}(1-\kappa )^{ \alpha -n-1} \varUpsilon \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &=-\frac{p}{w_{2}^{p}-w_{1}^{p}}\beta (n+1,\alpha -n)\varUpsilon (w_{1}) \\ &\quad {} +\frac{p}{w_{2}^{p}-w_{1}^{p}} \int _{w_{2}^{p}}^{w_{1}^{p}} \biggl( \frac{x-w_{2}^{p}}{w_{1}^{p}-w_{2}^{p}} \biggr)^{n} \biggl(1- \frac{x-w_{2}^{p}}{w_{1}^{p}-w_{2}^{p}} \biggr)^{\alpha -n-1} \frac{(\varUpsilon \circ \phi )(x)}{w_{1}^{p}-w_{2}^{p}}dx \\ &=-\frac{p}{w_{2}^{p}-w_{1}^{p}}\beta (n+1,\alpha -n)\varUpsilon (w_{1})+ \frac{n!}{(w_{2}^{p}-w_{1}^{p})^{\alpha +1}}\, J^{w_{1}^{p}}_{\alpha }( \varUpsilon \circ \phi ) \bigl(w_{2}^{p}\bigr). \end{aligned} \end{aligned}$$
(16)

By substituting values of \(I_{1}\) and \(I_{2}\) in (14) and then multiplying by \(\frac{w_{2}^{p}-w_{1}^{p}}{2}\), we get (12).

(ii) Proof is similar to that of (i). □

Remark 2.2

By taking \(p=-1\) in Lemma 2.1, we obtain Lemma 2.1 in [3].

Theorem 2.2

Let\(\varUpsilon :[w_{1},w_{2}]\subset (0,\infty )\rightarrow \mathbb{R}\)be a differentiable function on\((w_{1},w_{2})\)with\(w_{1}< w_{2}\)such that\(\varUpsilon '\in L[w_{1},w_{2}]\)and\(\alpha >0\). If\(|\varUpsilon '|^{q}\), where\(q\geq 1\), is ap-convex function, then

  1. (i)

    for\(p>0\), we have

    $$ \bigl\vert {}_{1}\Delta _{\varUpsilon }(w_{1},w_{2}; \alpha ;\beta ;J) \bigr\vert \leq \frac{w_{2}^{p}-w_{1}^{p}}{2p}\lambda ^{1-1/q} \bigl(\lambda _{1} \bigl\vert \varUpsilon '(w_{1}) \bigr\vert ^{q}+\lambda _{2} \bigl\vert \varUpsilon '(w_{2}) \bigr\vert ^{q} \bigr)^{1/q}, $$
    (17)

    here

    $$\begin{aligned}& \lambda =\beta (n+1,\alpha -n+1)-\beta (n+1,\alpha -n)+\beta (n+2, \alpha -n), \\& \lambda _{1}=\frac{w_{2}^{1-p}}{2}\,{}_{2}F_{1} \biggl(1-\frac{1}{p},2;3;1-\frac{w_{1}^{p}}{w_{2}^{p}} \biggr) \quad \textit{and}\quad \lambda _{2}=\frac{w_{2}^{1-p}}{2}\,{}_{2}F_{1} \biggl(1-\frac{1}{p},1;3;1-\frac{w_{1}^{p}}{w_{2}^{p}} \biggr); \end{aligned}$$
  2. (ii)

    for\(p<0\), we have

    $$ \bigl\vert {}_{2}\Delta _{\varUpsilon }(w_{1},w_{2}; \alpha ;\beta ;J) \bigr\vert \leq \frac{w_{1}^{p}-w_{2}^{p}}{2p}\lambda _{3}^{1-1/q} \bigl(\lambda _{4} \bigl\vert \varUpsilon '(w_{1}) \bigr\vert ^{q}+ \lambda _{5} \bigl\vert \varUpsilon '(w_{2}) \bigr\vert ^{q} \bigr)^{1/q}, $$
    (18)

    here

    $$\begin{aligned}& \lambda _{3}=\beta (n+1,\alpha -n+1)-\beta (n+2,\alpha -n), \\& \lambda _{4}=\frac{w_{2}^{p-1}}{2}\,{}_{2}F_{1} \biggl(1-\frac{1}{p},1;3;1-\frac{w_{2}^{p}}{w_{1}^{p}} \biggr)\quad \textit{and}\quad \lambda _{5}=\frac{w_{2}^{p-1}}{2}\,{}_{2}F_{1} \biggl(1-\frac{1}{p},2;3;1-\frac{w_{2}^{p}}{w_{1}^{p}} \biggr). \end{aligned}$$

Proof

(i) Let \(A_{\kappa }= [\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} ]\). Applying Lemma 2.1, power mean inequality, and p-convexity of \(|\varUpsilon '|^{q}\), we find

$$\begin{aligned} \begin{aligned}[b] & \bigl\vert {}_{1}\Delta _{\varUpsilon }(w_{1},w_{2};\alpha ;\beta ;J) \bigr\vert \\ &\quad = \biggl\vert \frac{w_{2}^{p}-w_{1}^{p}}{2p} \int _{0}^{1} \bigl\{ \beta _{1- \kappa }(n+1,\alpha -n)-\beta _{\kappa }(n+1,\alpha -n) \bigr\} \\ &\qquad {}\times A_{ \kappa }^{\frac{1}{p}-1}\varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \biggr\vert \\ &\quad \leq \frac{w_{2}^{p}-w_{1}^{p}}{2p} \biggl( \int _{0}^{1} \bigl\{ \beta _{1-\kappa }(n+1,\alpha -n)-\beta _{\kappa }(n+1,\alpha -n) \bigr\} \,d\kappa \biggr)^{1-1/q} \\ &\qquad {} \times \biggl( \int _{0}^{1}A_{\kappa }^{\frac{1}{p}-1} \bigl\vert \varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr) \bigr\vert ^{q}\,d\kappa \biggr)^{1/q} \\ &\quad \leq \frac{w_{2}^{p}-w_{1}^{p}}{2p}\lambda ^{1-1/q} \biggl( \int _{0}^{1}A_{ \kappa }^{\frac{1}{p}-1} \bigl[\kappa \bigl\vert \varUpsilon '(w_{1}) \bigr\vert ^{q}+(1- \kappa ) \bigl\vert \varUpsilon '(w_{2}) \bigr\vert ^{q} \bigr]\,d\kappa \biggr)^{1/q} \\ &\quad =\frac{w_{2}^{p}-w_{1}^{p}}{2p}\lambda ^{1-1/q} \bigl(\lambda _{1} \bigl\vert \varUpsilon '(w_{1}) \bigr\vert ^{q}+\lambda _{2} \bigl\vert \varUpsilon '(w_{2}) \bigr\vert ^{q} \bigr)^{1/q}, \end{aligned} \end{aligned}$$
(19)

where

$$\begin{aligned}& \begin{aligned} \lambda &= \int _{0}^{1} \bigl(\beta _{1-\kappa }(n+1, \alpha -n)- \beta _{\kappa }(n+1,\alpha -n) \bigr)\,d\kappa \\ &= \int _{0}^{1} \biggl( \int _{0}^{1-\kappa }u^{n}(1-u)^{\alpha -n-1} \,du \biggr)\,d\kappa + \int _{0}^{1} \biggl( \int _{0}^{\kappa }u^{n}(1-u)^{ \alpha -n-1} \,du \biggr)\,d\kappa \\ &=\kappa \biggl( \int _{0}^{1-\kappa }u^{n}(1-u)^{\alpha -n-1} \,du \biggr)\bigg|_{0}^{1}+ \int _{0}^{1}\kappa (1-\kappa )^{n} \kappa ^{ \alpha -n-1}\,d\kappa \\ &\quad {} +\kappa \biggl( \int _{0}^{\kappa }u^{n}(1-u)^{\alpha -n-1} \,du \biggr) \bigg|_{0}^{1}+ \int _{0}^{1}\kappa ^{n+1}(1-\kappa )^{\alpha -n-1}\,d\kappa \\ &=\beta (n+1,\alpha -n+1)-\beta (n+1,\alpha -n)+\beta (n+2,\alpha -n), \end{aligned} \\& \lambda _{1}= \int _{0}^{1}\kappa A_{\kappa }^{\frac{1}{p}-1} \,d\kappa = \frac{w_{2}^{1-p}}{2}\,{}_{2}F_{1} \biggl( 1- \frac{1}{p},2;3;1-\frac{w_{1}^{p}}{w_{2}^{p}} \biggr) , \end{aligned}$$

and

$$ \lambda _{2}= \int _{0}^{1}(1-\kappa )A_{\kappa }^{\frac{1}{p}-1} \,d\kappa =\frac{w_{2}^{1-p}}{2}\,{}_{2}F_{1} \biggl( 1- \frac{1}{p},1;3;1-\frac{w_{1}^{p}}{w_{2}^{p}} \biggr). $$

Hence the proof is completed.

(ii) Proof is similar to that of (i). □

Remark 2.3

By letting \(p=-1\) in Theorem 2.2, we obtain Theorem 2.2 in [3].

Now, for the next two results, we consider the case when \(p>0\) and leave the case when \(p<0\) for the reader.

Theorem 2.3

Let\(\varUpsilon :[w_{1},w_{2}]\subset (0,\infty )\rightarrow \mathbb{R}\)be a differentiable function on\((w_{1},w_{2})\)with\(w_{1}< w_{2}\)such that\(\varUpsilon '\in L[w_{1},w_{2}]\)and\(\alpha >0\). If\(|\varUpsilon '|^{q}\), where\(q\geq 1\), is ap-convex function, then for\(p>0\), we have

$$\begin{aligned} \begin{aligned}[b] & \bigl\vert {}_{1}\Delta _{\varUpsilon }(w_{1},w_{2};\alpha ;\beta ;J) \bigr\vert \\ &\quad \leq \frac{w_{2}^{p}-w_{1}^{p}}{2p}\mu ^{1-1/q} \bigl((\mu _{1}- \mu _{2}) \bigl\vert \varUpsilon '(w_{1}) \bigr\vert ^{q}+(\mu _{3}-\mu _{4}) \bigl\vert \varUpsilon '(w_{2}) \bigr\vert ^{q} \bigr)^{1/q}, \end{aligned} \end{aligned}$$
(20)

here

$$\begin{aligned}& \mu =\frac{w_{2}^{1-p}}{2}\,{}_{2}F_{1}\biggl(1- \frac{1}{p},1;2;1-\frac{w_{1}^{p}}{w_{2}^{p}}\biggr), \\& \mu _{1}=\frac{1}{2}\beta (n+1,\alpha -n+2), \\& \mu _{2}=\frac{1}{2} \bigl(\beta (n+1,\alpha -n)-\beta (n+3,\alpha -n) \bigr), \\& \mu _{3}=\beta (n+2,\alpha -n+1)-\frac{1}{2}\beta (n+1, \alpha -n+2), \end{aligned}$$

and

$$ \mu _{4}=\frac{1}{2}\beta (n+1,\alpha -n)+ \frac{1}{2}\beta (n+3, \alpha -n)-\beta (n+2,\alpha -n). $$

Proof

Applying Lemma 2.1, power mean inequality, and p-convexity of \(|\varUpsilon '|^{q}\), we have

$$\begin{aligned} \begin{aligned}[b] & \bigl\vert {}_{1}\Delta _{\varUpsilon }(w_{1},w_{2};\alpha ;\beta ;J) \bigr\vert \\ &\quad = \biggl\vert \frac{w_{2}^{p}-w_{1}^{p}}{2p} \int _{0}^{1} \bigl\{ \beta _{1- \kappa }(n+1,\alpha -n)-\beta _{\kappa }(n+1,\alpha -n) \bigr\} \\ &\qquad {}\times A_{ \kappa }^{\frac{1}{p}-1}\varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \biggr\vert \\ &\quad \leq \frac{w_{2}^{p}-w_{1}^{p}}{2p} \biggl( \int _{0}^{1}A_{\kappa }^{ \frac{1}{p}-1} \,d\kappa \biggr)^{1-1/q} \\ &\qquad {} \times \biggl( \int _{0}^{1} \bigl\{ \beta _{1-\kappa }(n+1,\alpha -n)- \beta _{\kappa }(n+1,\alpha -n) \bigr\} \bigl\vert \varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr) \bigr\vert ^{q}\,d\kappa \biggr)^{1/q} \\ &\quad \leq \frac{w_{2}^{p}-w_{1}^{p}}{2p}\mu ^{1-1/q} \biggl( \int _{0}^{1} \bigl\{ \beta _{1-\kappa }(n+1,\alpha -n)-\beta _{\kappa }(n+1, \alpha -n) \bigr\} \\ &\qquad {} \times \bigl[\kappa \bigl\vert \varUpsilon '(w_{1}) \bigr\vert ^{q}+(1-\kappa ) \bigl\vert \varUpsilon '(w_{2}) \bigr\vert ^{q} \bigr]\,d \kappa \biggr)^{1/q} \\ &\quad =\frac{w_{2}^{p}-w_{1}^{p}}{2p}\mu ^{1-1/q} \bigl((\mu _{1}-\mu _{2}) \bigl\vert \varUpsilon '(w_{1}) \bigr\vert ^{q}+(\mu _{3}-\mu _{4}) \bigl\vert \varUpsilon '(w_{2}) \bigr\vert ^{q} \bigr)^{1/q}, \end{aligned} \end{aligned}$$
(21)

where

$$\begin{aligned}& \mu = \int _{0}^{1}A_{\kappa }^{\frac{1}{p}-1} \,d\kappa = \frac{w_{2}^{1-p}}{2}\,{}_{2}F_{1}\biggl(1- \frac{1}{p},1;2;1-\frac{w_{1}^{p}}{w_{2}^{p}}\biggr), \\& \mu _{1}= \int _{0}^{1}\kappa \beta _{1-\kappa }(n+1, \alpha -n)\,d\kappa =\frac{1}{2}\beta (n+1,\alpha -n+2), \\& \mu _{2}= \int _{0}^{1}\kappa \beta _{\kappa }(n+1, \alpha -n)= \frac{1}{2} \bigl(\beta (n+1,\alpha -n)-\beta (n+3,\alpha -n) \bigr), \\& \mu _{3}= \int _{0}^{1}(1-\kappa )\beta _{1-\kappa }(n+1,\alpha -n)\,d\kappa =\beta (n+2,\alpha -n+1)- \frac{1}{2}\beta (n+1,\alpha -n+2), \end{aligned}$$

and

$$\begin{aligned} \mu _{4} =& \int _{0}^{1}(1-\kappa )\beta _{\kappa }(n+1,\alpha -n)\,d\kappa \\ =&\frac{1}{2}\beta (n+1,\alpha -n)+\frac{1}{2}\beta (n+3, \alpha -n)-\beta (n+2,\alpha -n). \end{aligned}$$

Hence the proof is completed. □

Theorem 2.4

Let\(\varUpsilon :[w_{1},w_{2}]\subset (0,\infty )\rightarrow \mathbb{R}\)be a differentiable function on\((w_{1},w_{2})\)with\(w_{1}< w_{2}\)such that\(\varUpsilon '\in L[w_{1},w_{2}]\)and\(\alpha >0\). If\(|\varUpsilon '|^{q}\), where\(q,l> 1\)with\(\frac{1}{q}+\frac{1}{l}=1\), is ap-convex function, then

$$\begin{aligned} \begin{aligned}[b] & \bigl\vert {}_{1}\Delta _{\varUpsilon }(w_{1},w_{2};\alpha ;\beta ;J) \bigr\vert \\ &\quad \leq \frac{w_{2}^{p}-w_{1}^{p}}{2p}\nu ^{\frac{1}{l}} \bigl(\nu _{1} \bigl\vert \varUpsilon '(w_{1}) \bigr\vert ^{q}+\nu _{2} \bigl\vert \varUpsilon '(w_{2}) \bigr\vert ^{q} \bigr)^{1/q}, \end{aligned} \end{aligned}$$
(22)

here

$$\begin{aligned}& \nu =2 \int _{0}^{\frac{1}{2}} \biggl( \int _{\kappa }^{1-\kappa }u^{n}(1-u)^{ \alpha -n-1} \,du \biggr)\,d\kappa , \\& \nu _{1}=\frac{w_{2}^{q(1-p)}}{2}\,{}_{2}F_{1} \biggl( q\biggl(1-\frac{1}{p}\biggr),2;3;1-\frac{w_{1}^{p}}{w_{2}^{p}} \biggr) , \\& \nu _{2}=\frac{w_{2}^{q(1-p)}}{2}\,{}_{2}F_{1} \biggl( q\biggl(1-\frac{1}{p}\biggr),1;3;1-\frac{w_{1}^{p}}{w_{2}^{p}} \biggr). \end{aligned}$$

Proof

Let \(A_{\kappa }= [\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} ]\). Applying Lemma 2.1, Hölder’s inequality, and p-convexity of \(|\varUpsilon '|^{q}\), we have

$$\begin{aligned} \begin{aligned}[b] & \bigl\vert {}_{1}\Delta _{\varUpsilon }(w_{1},w_{2};\alpha ;\beta ;J) \bigr\vert \\ &\quad = \biggl\vert \frac{w_{2}^{p}-w_{1}^{p}}{2p} \int _{0}^{1} \bigl\{ \beta _{1- \kappa }(n+1,\alpha -n)-\beta _{\kappa }(n+1,\alpha -n) \bigr\} \\ &\qquad \times{}A_{ \kappa }^{\frac{1}{p}-1}\varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \biggr\vert \\ &\quad \leq \frac{w_{2}^{p}-w_{1}^{p}}{2p} \biggl( \int _{0}^{1} \bigl\vert \beta _{1-\kappa }(n+1,\alpha -n)-\beta _{\kappa }(n+1,\alpha -n) \bigr\vert ^{l}\,d\kappa \biggr)^{\frac{1}{l}} \\ &\qquad {} \times \biggl( \int _{0}^{1}A_{\kappa }^{q(\frac{1}{p}-1)} \bigl\vert \varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr) \bigr\vert ^{q}\,d\kappa \biggr)^{1/q} \\ &\quad \leq \frac{w_{2}^{p}-w_{1}^{p}}{2p}\nu ^{\frac{1}{l}} \biggl( \int _{0}^{1}A_{ \kappa }^{q(\frac{1}{p}-1)} \bigl[\kappa \bigl\vert \varUpsilon '(w_{1}) \bigr\vert ^{q}+(1- \kappa ) \bigl\vert \varUpsilon '(w_{2}) \bigr\vert ^{q} \bigr]\,d\kappa \biggr)^{1/q} \\ &\quad =\frac{w_{2}^{p}-w_{1}^{p}}{2p}\nu ^{\frac{1}{p}} \bigl(\nu _{1} \bigl\vert \varUpsilon '(w_{1}) \bigr\vert ^{q}+\nu \bigl\vert \varUpsilon '(w_{2}) \bigr\vert ^{q} \bigr)^{1/q}, \end{aligned} \end{aligned}$$
(23)

where

$$\begin{aligned}& \begin{aligned} \nu &= \int _{0}^{1} \bigl\vert \beta _{1-\kappa }(n+1,\alpha -n)-\beta _{ \kappa }(n+1,\alpha -n) \bigr\vert ^{l}\,d\kappa \\ &= \int _{0}^{\frac{1}{2}} \bigl(\beta _{1-\kappa }(n+1, \alpha -n)- \beta _{\kappa }(n+1,\alpha -n) \bigr)^{l}\,d \kappa \\ &\quad {} + \int _{\frac{1}{2}}^{1} \bigl(\beta _{\kappa }(n+1, \alpha -n)-\beta _{1- \kappa }(n+1,\alpha -n) \bigr)^{l}\,d\kappa \\ &= \int _{0}^{\frac{1}{2}} \biggl( \int _{\kappa }^{1-\kappa }u^{n}(1-u)^{ \alpha -n-1} \,du \biggr)^{l}\,d\kappa + \int _{\frac{1}{2}}^{1} \biggl( \int _{1-\kappa }^{\kappa }u^{n}(1-u)^{\alpha -n-1} \,du \biggr)^{l}\,d\kappa \\ &=2 \int _{0}^{\frac{1}{2}} \biggl( \int _{\kappa }^{1-\kappa }u^{n}(1-u)^{ \alpha -n-1} \,du \biggr)^{l}\,d\kappa , \end{aligned} \\& \nu _{1}= \int _{0}^{1}\kappa A_{\kappa }^{q(\frac{1}{p}-1)} \,d\kappa = \frac{w_{2}^{q(1-p)}}{2}\,{}_{2}F_{1} \biggl( q \biggl(1-\frac{1}{p}\biggr),2;3;1-\frac{w_{1}^{p}}{w_{2}^{p}} \biggr) , \end{aligned}$$

and

$$ \nu _{2}= \int _{0}^{1}(1-\kappa )A_{\kappa }^{q(\frac{1}{p}-1)} \,d\kappa =\frac{w_{2}^{q(1-p)}}{2}\,{}_{2}F_{1} \biggl( q \biggl(1-\frac{1}{p}\biggr),1;3;1-\frac{w_{1}^{p}}{w_{2}^{p}} \biggr) . $$

Hence the proof is completed. □

3 Hermite–Hadamard–Fejér type inequalities

In this section we prove some Hermite–Hadamard–Fejér type inequalities for p-convex functions via conformable fractional integral. First we give the following useful definition.

Definition 3.1

([15])

Let \(p\in \mathbb{R}\setminus \{0\}\). A function \(\curlyvee :[w_{1},w_{2}]\subseteq (0,\infty )\rightarrow \mathbb{R} \) is called p-symmetric around \([\frac{w_{1}^{p}+w_{2}^{p}}{2} ]^{1/p}\) if

$$ \curlyvee (x)=\curlyvee \bigl( \bigl[w_{1}^{p}+w_{2}^{p}-x^{p} \bigr]^{\frac{1}{p}} \bigr) $$

holds for all \(x\in [w_{1},w_{2}]\).

Now we prove the following identity.

Lemma 3.1

Let\(p\in \mathbb{R}\setminus \{0\}\). If\(\curlyvee :[w_{1},w_{2}]\subseteq (0,\infty )\rightarrow \mathbb{R} \)is integrable andp-symmetric around\([\frac{w_{1}^{p}+w_{2}^{p}}{2} ]^{1/p}\), then

  1. (i)

    for\(p>0\), we have

    $$ \begin{aligned}[b] J_{\alpha }^{w_{1}^{p}}(\curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr)&={}^{w_{2}^{p}}J_{ \alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr) \\ &= \frac{1}{2} \bigl[J_{ \alpha }^{w_{1}^{p}}(\curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr)+{}^{w_{2}^{p}}J_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr) \bigr], \end{aligned} $$
    (24)

    with\(\alpha >0\)and\(\phi (u)=u^{\frac{1}{p}}\), for all\(u\in [w_{1}^{p},w_{2}^{p}]\);

  2. (ii)

    for\(p<0\), we have

    $$ \begin{aligned}[b] J_{\alpha }^{w_{2}^{p}}(\curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr)&={}^{w_{1}^{p}}J_{ \alpha }( \curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr) \\ &= \frac{1}{2} \bigl[J_{ \alpha }^{w_{2}^{p}}(\curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr)+{}^{w_{1}^{p}}J_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr) \bigr], \end{aligned} $$
    (25)

    with\(\alpha >0\)and\(\phi (u)=u^{\frac{1}{p}}\), for all\(u\in [w_{2}^{p},w_{1}^{p}]\).

Proof

(i) Since ⋎ is p-symmetric around \([\frac{w_{1}^{p}+w_{2}^{p}}{2} ]^{1/p}\), then by definition we have \(\curlyvee (x^{\frac{1}{p}})=\curlyvee ( [w_{1}^{p}+w_{2}^{p}-x ]^{\frac{1}{p}} )\) for all \(x\in [w_{1}^{p},w_{2}^{p}]\). In the following integral, setting \(u=w_{1}^{p}+w_{2}^{p}-x\) gives

$$\begin{aligned} \begin{aligned}[b] J^{w_{1}^{p}}_{\alpha }(\curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr)&= \frac{1}{n!} \int _{w_{1}^{p}}^{w_{2}^{p}} \bigl(w_{2}^{p}-u \bigr)^{n} \bigl(u-w_{1}^{p} \bigr)^{\alpha -n-1}\curlyvee \bigl(u^{\frac{1}{p}}\bigr)\,du \\ &=\frac{1}{n!} \int _{w_{1}^{p}}^{w_{2}^{p}} \bigl(x-w_{1}^{p} \bigr)^{n} \bigl(w_{2}^{p}-x \bigr)^{\alpha -n-1}\curlyvee \bigl( \bigl[w_{1}^{p}+w_{2}^{p}-x \bigr]^{\frac{1}{p}} \bigr)dx \\ &=\frac{1}{n!} \int _{w_{1}^{p}}^{w_{2}^{p}} \bigl(x-w_{1}^{p} \bigr)^{n} \bigl(w_{2}^{p}-x \bigr)^{\alpha -n-1}\curlyvee \bigl(x^{ \frac{1}{p}} \bigr)dx \\ &={}^{w_{2}^{p}}J_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr). \end{aligned} \end{aligned}$$
(26)

This completes the proof.

(ii) Proof is similar to that of (i). □

Remark 3.1

In Lemma 3.1:

  1. 1.

    By taking \(\alpha =n+1\), we obtain Lemma 1 in [16].

  2. 2.

    By taking \(\alpha =n+1\) and \(p=1\), we find Lemma 3 of [10].

Corollary 3.1

Under the assumptions of Lemma 3.1:

  1. 1.

    If\(p=1\)in (i), then we get

    $$ J_{\alpha }^{w_{1}}\curlyvee (w_{2})={}^{w_{2}}J_{\alpha } \curlyvee (w_{1}) =\frac{1}{2} \bigl[J_{\alpha }^{w_{1}} \curlyvee (w_{2})+{}^{w_{2}}J_{ \alpha }\curlyvee (w_{1}) \bigr]. $$
    (27)
  2. 2.

    If\(p=-1\)in (ii), then we get

    $$\begin{aligned} \begin{aligned}[b] {}^{1/w_{1}}J_{\alpha }(\curlyvee \circ \phi ) (1/w_{2})&=J_{\alpha }^{1/w_{2}}( \curlyvee \circ \phi ) (1/w_{1}) \\ &=\frac{1}{2} \bigl[{}^{1/w_{1}}J_{\alpha }(\curlyvee \circ \phi ) (1/w_{2})+ J^{1/w_{2}}_{\alpha }(\curlyvee \circ \phi ) (1/w_{1}) \bigr]. \end{aligned} \end{aligned}$$
    (28)

Theorem 3.2

Let\(p\in \mathbb{R}\setminus \{0\}\). Let\(\varUpsilon :[w_{1},w_{2}]\subset (0,\infty )\rightarrow \mathbb{R}\)be ap-convex function with\(w_{1}< w_{2}\)and\(\varUpsilon \in L[w_{1},w_{2}]\). If\(\curlyvee :[w_{1},w_{2}]\subseteq \mathbb{R}\setminus \{0\} \rightarrow \mathbb{R}\)is nonnegative, integrable, andp-symmetric around\([\frac{w_{1}^{p}+w_{2}^{p}}{2} ]^{1/p}\). Then

  1. (i)

    for\(p>0\), the following inequalities hold:

    $$\begin{aligned} \begin{aligned}[b] &\varUpsilon \biggl( \biggl[ \frac{w_{1}^{p}+w_{2}^{p}}{2} \biggr]^{1/p} \biggr) \bigl[J^{w_{1}^{p}}_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr)+ {}^{w_{2}^{p}}J_{\alpha }(\curlyvee \circ \phi ) \bigl(w_{1}^{p} \bigr) \bigr] \\ &\quad \leq \bigl[J^{w_{1}^{p}}_{\alpha }\bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{2}^{p}\bigr)+ {}^{w_{2}^{p}}J_{\alpha } \bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{1}^{p} \bigr) \bigr] \\ &\quad \leq \frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \bigl[J^{w_{1}^{p}}_{ \alpha }(\curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr)+{}^{w_{2}^{p}}J_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr) \bigr], \end{aligned} \end{aligned}$$
    (29)

    with\(\alpha >0\)and\(\phi (x)=x^{\frac{1}{p}}\), for all\(x\in [w_{1}^{p},w_{2}^{p}]\);

  2. (ii)

    for\(p<0\), the following inequalities hold:

    $$\begin{aligned} \begin{aligned}[b] &\varUpsilon \biggl( \biggl[ \frac{w_{1}^{p}+w_{2}^{p}}{2} \biggr]^{1/p} \biggr) \bigl[J^{w_{2}^{p}}_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr)+ {}^{w_{1}^{p}}J_{\alpha }(\curlyvee \circ \phi ) \bigl(w_{2}^{p} \bigr) \bigr] \\ &\quad \leq \bigl[J^{w_{2}^{p}}_{\alpha }\bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{1}^{p}\bigr)+ {}^{w_{1}^{p}}J_{\alpha } \bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{2}^{p} \bigr) \bigr] \\ &\quad \leq \frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \bigl[J^{w_{2}^{p}}_{ \alpha }(\curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr)+{}^{w_{1}^{p}}J_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr) \bigr], \end{aligned} \end{aligned}$$
    (30)

    with\(\alpha >0\)and\(\phi (x)=x^{\frac{1}{p}}\), for all\(x\in [w_{2}^{p},w_{1}^{p}]\).

Proof

(i) Since ϒ is a p-convex function on \([w_{1},w_{2}]\), we have

$$ \varUpsilon \biggl( \biggl[\frac{x^{p}+y^{p}}{2} \biggr]^{\frac{1}{p}} \biggr) \leq \frac{\varUpsilon (x)+\varUpsilon (y)}{2}. $$

Taking \(x^{p}=\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p}\) and \(y^{p}=(1-\kappa )w_{1}^{p}+\kappa w_{2}^{p}\) with \(\kappa \in [0,1]\), we get

$$ \varUpsilon \biggl( \biggl[\frac{w_{1}^{p}+w_{2}^{p}}{2} \biggr]^{ \frac{1}{p}} \biggr)\leq \frac{\varUpsilon ( [\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} ]^{\frac{1}{p}} ) +\varUpsilon ( [(1-\kappa )w_{1}^{p}+\kappa w_{2}^{p} ]^{\frac{1}{p}} )}{2}. $$
(31)

Multiplying (31) by \(\frac{1}{n!}\kappa ^{n}(1-\kappa )^{\alpha -n-1}\curlyvee ( [\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} ]^{\frac{1}{p}} )\) on both sides, \(\alpha >0\) and then integrating about κ over \([0,1]\), we obtain

$$\begin{aligned} \begin{aligned}[b] &\frac{2}{n!}\varUpsilon \biggl( \biggl[ \frac{w_{1}^{p}+w_{2}^{p}}{2} \biggr]^{\frac{1}{p}} \biggr) \int _{0}^{1}\kappa ^{n}(1-\kappa )^{ \alpha -n-1} \curlyvee \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &\quad \leq \frac{1}{n!} \int _{0}^{1}\kappa ^{n}(1-\kappa )^{\alpha -n-1} \varUpsilon \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr) \curlyvee \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &\qquad {} +\frac{1}{n!} \int _{0}^{1}\kappa ^{n}(1-\kappa )^{\alpha -n-1} \varUpsilon \bigl( \bigl[(1-\kappa )w_{1}^{p}+ \kappa w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr) \curlyvee \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa . \end{aligned} \end{aligned}$$
(32)

Since ⋎ is nonnegative, integrable, and p-symmetric with respect to \([\frac{w_{1}^{p}+w_{2}^{p}}{2} ]^{1/p}\), then

$$ \curlyvee \bigl( \bigl[\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr)= \curlyvee \bigl( \bigl[\kappa w_{2}^{p}+(1- \kappa )w_{1}^{p} \bigr]^{\frac{1}{p}} \bigr). $$

Also choosing \(u=\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p}\) leads to

$$\begin{aligned} &\frac{2}{n! (w_{2}^{p}-w_{1}^{p} )^{\alpha }}\varUpsilon \biggl( \biggl[ \frac{w_{1}^{p}+w_{2}^{p}}{2} \biggr]^{\frac{1}{p}} \biggr) \int _{w_{1}^{p}}^{w_{2}^{p}} \bigl(w_{2}^{p}-u \bigr)^{n} \bigl(u-w_{1}^{p} \bigr)^{\alpha -n-1} \curlyvee \bigl(u^{\frac{1}{p}}\bigr)\,du \\ &\quad \leq \frac{1}{n! (w_{2}^{p}-w_{1}^{p} )^{\alpha }} \biggl[ \int _{w_{1}^{p}}^{w_{2}^{p}} \bigl(w_{2}^{p}-u \bigr)^{n} \bigl(u-w_{1}^{p} \bigr)^{\alpha -n-1} \varUpsilon \bigl(u^{\frac{1}{p}}\bigr)\curlyvee \bigl(u^{ \frac{1}{p}}\bigr)\,du \\ &\qquad {} + \int _{w_{1}^{p}}^{w_{2}^{p}} \bigl(w_{2}^{p}-u \bigr)^{n} \bigl(u-w_{1}^{p} \bigr)^{\alpha -n-1} \varUpsilon \bigl( \bigl[w_{1}^{p}+w_{2}^{p}-u \bigr]^{\frac{1}{p}} \bigr)\curlyvee \bigl(u^{\frac{1}{p}}\bigr)\,du \biggr] \\ &\quad =\frac{1}{n! (w_{2}^{p}-w_{1}^{p} )^{\alpha }} \biggl[ \int _{w_{1}^{p}}^{w_{2}^{p}} \bigl(w_{2}^{p}-u \bigr)^{n} \bigl(u-w_{1}^{p} \bigr)^{\alpha -n-1} \varUpsilon \bigl(u^{\frac{1}{p}}\bigr)\curlyvee \bigl(u^{\frac{1}{p}}\bigr)\,du \\ &\qquad {} + \int _{w_{1}^{p}}^{w_{2}^{p}} \bigl(u-w_{1}^{p} \bigr)^{n} \bigl(w_{2}^{p}-u \bigr)^{\alpha -n-1} \varUpsilon \bigl(u^{\frac{1}{p}}\bigr)\curlyvee \bigl( \bigl[w_{1}^{p}+w_{2}^{p}-u \bigr]^{\frac{1}{p}} \bigr)\,du \biggr]. \end{aligned}$$
(33)

Therefore, by Lemma 3.1 we have

$$\begin{aligned} \begin{aligned}[b] &\frac{1}{ (w_{2}^{p}-w_{1}^{p} )^{\alpha }}\varUpsilon \biggl( \biggl[\frac{w_{1}^{p}+w_{2}^{p}}{2} \biggr]^{\frac{1}{p}} \biggr) \bigl[J^{w_{1}^{p}}_{\alpha }(\curlyvee \circ \phi ) \bigl(b^{p}\bigr)+ {}^{w_{2}^{p}}J_{\alpha }(\curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr) \bigr] \\ &\quad \leq \frac{1}{ (w_{2}^{p}-w_{1}^{p} )^{\alpha }} \bigl[J^{w_{1}^{p}}_{ \alpha }\bigl( \varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{2}^{p} \bigr)+{}^{w_{2}^{p}}J_{ \alpha }\bigl(\varUpsilon (\curlyvee \circ \phi ) \bigr) \bigl(w_{1}^{p}\bigr) \bigr]. \end{aligned} \end{aligned}$$
(34)

This completes the first inequality of (29). For the second inequality, we first note that if ϒ is a p-convex function, then we have

$$ \varUpsilon \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr)+\varUpsilon \bigl( \bigl[\kappa w_{2}^{p}+(1- \kappa )w_{1}^{p} \bigr]^{\frac{1}{p}} \bigr) \leq \bigl[ \varUpsilon (w_{1})+ \varUpsilon (w_{2}) \bigr]. $$
(35)

Multiplying (35) by \(\frac{1}{n!}\kappa ^{n}(1-\kappa )^{\alpha -n-1}\curlyvee ( [\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} ]^{\frac{1}{p}} )\) on both sides, \(\alpha >0\) and then integrating about κ over \([0,1]\), we obtain

$$\begin{aligned} \begin{aligned}[b] &\frac{1}{n!} \int _{0}^{1}\kappa ^{n}(1-\kappa )^{\alpha -n-1} \varUpsilon \bigl( \bigl[\kappa _{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr) \curlyvee \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &\qquad {} +\frac{1}{n!} \int _{0}^{1}\kappa ^{n}(1-\kappa )^{\alpha -n-1} \varUpsilon \bigl( \bigl[\kappa w_{2}^{p}+(1- \kappa )w_{1}^{p} \bigr]^{ \frac{1}{p}} \bigr) \curlyvee \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &\quad \leq \bigl[\varUpsilon (w_{1})+\varUpsilon (w_{2}) \bigr] \frac{1}{n!} \int _{0}^{1}\kappa ^{n}(1-\kappa )^{\alpha -n-1}\curlyvee \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa. \end{aligned} \end{aligned}$$
(36)

That is,

$$\begin{aligned} \begin{aligned}[b] &\frac{1}{ (w_{2}^{p}-w_{1}^{p} )^{\alpha }} \bigl[J^{w_{1}^{p}}_{ \alpha } \bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{2}^{p} \bigr)+{}^{w_{2}^{p}}J_{ \alpha }\bigl(\varUpsilon (\curlyvee \circ \phi ) \bigr) \bigl(w_{1}^{p}\bigr) \bigr] \\ &\quad \leq \frac{1}{ (w_{2}^{p}-w_{1}^{p} )^{\alpha }} \biggl[ \frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \biggr] \bigl[J^{w_{1}^{p}}_{ \alpha }(\curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr)+{}^{w_{2}^{p}}J_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr) \bigr]. \end{aligned} \end{aligned}$$
(37)

This completes the proof.

(ii) Proof is similar to that of (i). □

Remark 3.2

In Theorem 3.2:

  1. 1.

    If \(\alpha =n+1\), we obtain Theorem 9 in [16].

  2. 2.

    If \(\alpha =n+1\) and \(p=1\), we find Theorem 4 in [10].

Corollary 3.3

Under the assumptions of Theorem 3.2:

  1. 1.

    If\(p=1\), then

    $$\begin{aligned} \begin{aligned}[b] &\varUpsilon \biggl(\frac{w_{1}+w_{2}}{2} \biggr) \bigl[J^{w_{1}}_{ \alpha }\curlyvee (w_{2})+{}^{w_{2}}J_{\alpha } \curlyvee (w_{1}) \bigr] \\ &\quad \leq \bigl[J^{w_{1}}_{\alpha }(\varUpsilon \curlyvee ) (w_{2})+{}^{w_{2}}J_{ \alpha }(\varUpsilon \curlyvee ) (w_{1}) \bigr] \\ &\quad \leq \frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \bigl[J^{w_{1}}_{ \alpha }\curlyvee (w_{2})+{}^{w_{2}}J_{\alpha }\curlyvee (w_{1}) \bigr]. \end{aligned} \end{aligned}$$
    (38)
  2. 2.

    If\(p=-1\), then

    $$\begin{aligned} \begin{aligned}[b] &\varUpsilon \biggl(\frac{2w_{1}w_{2}}{w_{1}+w_{2}} \biggr) \bigl[{}^{1/w_{1}}J_{ \alpha }(\curlyvee \circ \phi ) (1/w_{2})+J^{1/w_{2}}_{\alpha }( \curlyvee \circ \phi ) (1/w_{1}) \bigr] \\ &\quad \leq \bigl[{}^{1/w_{1}}J_{\alpha }\bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) (1/w_{2})+J^{1/w_{2}}_{ \alpha }\bigl( \varUpsilon (\curlyvee \circ \phi )\bigr) (1/w_{1}) \bigr] \\ &\quad \leq \frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \bigl[{}^{1/w_{1}}J_{ \alpha }( \curlyvee \circ \phi ) (1/w_{2})+J^{1/w_{2}}_{\alpha }( \curlyvee \circ \phi ) (1/w_{1}) \bigr]. \end{aligned} \end{aligned}$$
    (39)

Remark 3.3

In Corollary 3.3(1), if we take \(\alpha =n+1\), we get inequality (3).

Lemma 3.2

Let\(p\in \mathbb{R}\setminus \{0\}\)and\(\alpha >0\). Let\(\varUpsilon :[w_{1},w_{2}]\subset (0,\infty )\rightarrow \mathbb{R}\)be a differentiable mapping and\(\varUpsilon \in L[w_{1},w_{2}]\). If\(\curlyvee :[w_{1},w_{2}]\subseteq \mathbb{R}\setminus \{0\} \rightarrow \mathbb{R}\)is nonnegative, integrable, andp-symmetric around\([\frac{w_{1}^{p}+w_{2}^{p}}{2} ]^{1/p}\), then

  1. (i)

    for\(p>0\), the following inequality holds:

    $$\begin{aligned} \begin{aligned}[b] &\frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \bigl[J^{w_{1}^{p}}_{ \alpha }(\curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr)+{}^{w_{2}^{p}}J_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr) \bigr] \\ &\qquad {} - \bigl[J^{w_{1}^{p}}_{\alpha }\bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{2}^{p}\bigr)+ {}^{w_{2}^{p}}J_{\alpha }\bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{1}^{p}\bigr) \bigr] \\ &\quad \leq \frac{1}{n!} \int _{w_{1}^{p}}^{w_{2}^{p}} \biggl[ \int _{w_{1}^{p}}^{t}\bigl(w_{2}^{p}-s \bigr)^{n}\bigl(s-w_{1}^{p} \bigr)^{ \alpha -n-1}(\curlyvee \circ \phi ) (s)\,ds \\ &\qquad {} - \int _{t}^{w_{2}^{p}}\bigl(s-w_{1}^{p} \bigr)^{n}\bigl(w_{2}^{p}-s \bigr)^{\alpha -n-1}( \curlyvee \circ \phi ) (s)\,ds \biggr](\varUpsilon \circ \phi )'(t) \,dt, \end{aligned} \end{aligned}$$
    (40)

    where\(\phi (x)=x^{1/p}\)for all\(x\in [w^{p}_{1},w^{p}_{2}]\);

  2. (ii)

    for\(p<0\), the following inequality holds:

    $$\begin{aligned} \begin{aligned}[b] &\frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \bigl[J^{w_{2}^{p}}_{ \alpha }(\curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr)+{}^{w_{1}^{p}}J_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr) \bigr] \\ &\qquad {} - \bigl[J^{w_{2}^{p}}_{\alpha }\bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{1}^{p}\bigr)+ {}^{w_{1}^{p}}J_{\alpha }\bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{2}^{p}\bigr) \bigr] \\ &\quad \leq \frac{1}{n!} \int _{w_{2}^{p}}^{w_{1}^{p}} \biggl[ \int _{w_{2}^{p}}^{t}\bigl(w_{1}^{p}-s \bigr)^{n}\bigl(s-w_{2}^{p} \bigr)^{ \alpha -n-1}(\curlyvee \circ \phi ) (s)\,ds \\ &\qquad {} - \int _{t}^{w_{1}^{p}}\bigl(s-w_{2}^{p} \bigr)^{n}\bigl(w_{1}^{p}-s \bigr)^{\alpha -n-1}( \curlyvee \circ \phi ) (s)\,ds \biggr](\varUpsilon \circ \phi )'(t) \,dt, \end{aligned} \end{aligned}$$
    (41)

    where\(\phi (x)=x^{1/p}\)for all\(x\in [w^{p}_{2},w^{p}_{1}]\).

Proof

(i) Note that

$$\begin{aligned} \begin{aligned}[b] I&= \int _{w_{1}^{p}}^{w_{2}^{p}} \biggl( \int _{w_{1}^{p}}^{t}\bigl(w_{2}^{p}-s \bigr)^{n}\bigl(s-w_{1}^{p} \bigr)^{ \alpha -n-1}(\curlyvee \circ \phi ) (s)\,ds \biggr) (\varUpsilon \circ \phi )'(t)\,dt \\ &\quad {} - \int _{w_{1}^{p}}^{w_{2}^{p}} \biggl( \int _{t}^{w_{2}^{p}}\bigl(s-w_{1}^{p} \bigr)^{n}\bigl(w_{2}^{p}-s \bigr)^{ \alpha -n-1}(\curlyvee \circ \phi ) (s)\,ds \biggr) (\varUpsilon \circ \phi )'(t) \,dt \\ &=I_{1}-I_{2}. \end{aligned} \end{aligned}$$
(42)

Integrating by parts and using Lemma 3.1, we get

$$\begin{aligned} \begin{aligned}[b] I_{1}&= \biggl( \int _{w_{1}^{p}}^{t}\bigl(w_{2}^{p}-s \bigr)^{n}\bigl(s-w_{1}^{p} \bigr)^{ \alpha -n-1}(\curlyvee \circ \phi ) (s)\,ds \biggr) (\varUpsilon \circ \phi ) (t) \bigg|_{w_{1}^{p}}^{w_{2}^{p}} \\ &\quad {} - \int _{w_{1}^{p}}^{w_{2}^{p}}\bigl(w_{2}^{p}-t \bigr)^{n}\bigl(t-w_{1}^{p} \bigr)^{\alpha -n-1}( \curlyvee \circ \phi ) (t) (\varUpsilon \circ \phi ) (t) \,dt \\ &=n! \bigl[(\varUpsilon \circ \phi ) \bigl(w_{2}^{p} \bigr)J^{w_{1}^{p}}_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr)-J^{w_{1}^{p}}_{\alpha } \bigl(\varUpsilon ( \curlyvee \circ \phi )\bigr) \bigl(w_{2}^{p} \bigr) \bigr] \\ &=n! \biggl[\frac{(\varUpsilon \circ \phi )(w_{2}^{p})}{2} \bigl\{ {}^{w_{2}^{p}}J_{ \alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr)+ J^{w_{1}^{p}}_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{2}^{p} \bigr) \bigr\} -J^{w_{1}^{p}}_{\alpha }\bigl( \varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{2}^{p}\bigr) \biggr]. \end{aligned} \end{aligned}$$
(43)

Similarly,

$$\begin{aligned} \begin{aligned}[b] I_{2}&= \biggl( \int _{t}^{w_{2}^{p}}\bigl(s-w_{1}^{p} \bigr)^{n}\bigl(w_{2}^{p}-s \bigr)^{ \alpha -n-1}(\curlyvee \circ \phi ) (s)\,ds \biggr) (\varUpsilon \circ \phi ) (t) \bigg|_{w_{1}^{p}}^{w_{2}^{p}} \\ &\quad {} + \int _{w_{1}^{p}}^{w_{2}^{p}}\bigl(t-w_{1}^{p} \bigr)^{n}\bigl(w_{2}^{p}-t \bigr)^{\alpha -n-1}( \curlyvee \circ \phi ) (t) (\varUpsilon \circ \phi ) (t) \,dt \\ &=n! \bigl[-(\varUpsilon \circ \phi ) \bigl(w_{1}^{p} \bigr){}^{w_{2}^{p}}J_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr)-{}^{w_{2}^{p}}J_{\alpha } \bigl(\varUpsilon ( \curlyvee \circ \phi )\bigr) \bigl(w_{1}^{p} \bigr) \bigr] \\ &=n! \biggl[\frac{-(\varUpsilon \circ \phi )(w_{1}^{p})}{2} \bigl\{ {}^{w_{2}^{p}}J_{ \alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p} \bigr)+J^{w_{1}^{p}}_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr) \bigr\} \\ &\quad {}+{}^{w_{2}^{p}}J_{\alpha } \bigl( \varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{1}^{p} \bigr) \biggr]. \end{aligned} \end{aligned}$$
(44)

Thus from (43) and (44) we get

$$\begin{aligned} \begin{aligned}[b] I&=I_{1}-I_{2} \\ &=n! \biggl[\frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \bigl[J^{w_{1}^{p}}_{ \alpha }( \curlyvee \circ \phi ) \bigl(w_{2}^{p} \bigr)+{}^{w_{2}^{p}}J_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr) \bigr] \\ &\quad {} - \bigl[J^{w_{1}^{p}}_{\alpha }\bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{2}^{p}\bigr)+ {}^{w_{2}^{p}}J_{\alpha }\bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{1}^{p}\bigr) \bigr] \biggr]. \end{aligned} \end{aligned}$$
(45)

Multiplying (45) by \(\frac{1}{n!}\), we obtain (40).

(ii) Proof is similar to that of (i). □

Remark 3.4

In Lemma 3.2:

  1. 1.

    If we take \(\alpha =n+1\), we get Lemma 2 in [16].

  2. 2.

    If we take \(\alpha =n+1\) and \(p=1\), we get Lemma 4 in [10].

Lemma 3.2 also holds for convex functions and harmonically convex functions just by taking \(p=1\) and \(p=-1\), respectively. Also, from Lemma 3.2 we can establish more useful results.