In this paper, by using a new identity we establish some trapezoidal type inequalities for functions whose modulus of the first derivatives are \( \left( s,m\right)\)-preinvex via Caputo fractional derivatives.
One of the famous inequalities for the class of convex functions is the so-called Hermite-Hadamard inequality (see [1, 2]), which can be stated as follows: For every convex function \(f\) on the interval \(\left[ a,b\right]\) with \(a<b\), we have \[\label{1.1} f\left( \frac{a+b}{2}\right) \leq \frac{1}{b-a}\int_{a}^{b}f\left( x\right) dx\leq \frac{f\left( a\right) +f\left( b\right) }{2}. \tag{1}\] Several generalizations and variants of inequality (1), have been obtained see [3– 14] and references therein.
In [15], Dragomir and Agarwal gave some results which are in relationship with inequality (1) as follows: \[\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a} \overset{b}{\underset{a}{\int }}f\left( u\right) du\right\vert \leq \frac{ b-a}{8}\left( \left\vert f^{\prime }\left( a\right) \right\vert +\left\vert f^{\prime }\left( b\right) \right\vert \right) , \tag{2}\] and \[\label{1.2} \left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a} \overset{b}{\underset{a}{\int }}f\left( u\right) du\right\vert \leq \frac{ b-a}{2\left( p+1\right) ^{\frac{1}{p}}}\left( \frac{\left\vert f^{\prime }\left( a\right) \right\vert ^{q}+\left\vert f^{\prime }\left( b\right) \right\vert ^{q}}{2}\right) ^{\frac{1}{q}}. \tag{3}\] Pearce and Pečarić [16], gave a variant of the result given in the inequality (3) as follows: \[\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a} \overset{b}{\underset{a}{\int }}f\left( u\right) du\right\vert \leq \frac{ b-a}{4}\left( \left\vert f^{\prime }\left( a\right) \right\vert ^{q}+\left\vert f^{\prime }\left( b\right) \right\vert ^{q}\right) . \tag{4}\] Noor [17], established the anologue preinvex of inequality (1) as follows: \[f\left( \frac{2a+\eta \left( b,a\right) }{2}\right) \leq \frac{1}{\eta \left( b,a\right) }\overset{a+\eta \left( b,a\right) }{\underset{a}{\int }} f\left( u\right) du\leq \frac{f\left( a\right) +f\left( b\right) }{2}. \tag{5}\] In [18], Barani et al., investigated some trapezium type inequalities for differentiable preinvex functions \[\left\vert \frac{f\left( a\right) +f\left( a+\eta \left( b,a\right) \right) }{2}-\frac{1}{\eta \left( b,a\right) }\int_{a}^{a+\eta \left( b,a\right) }f\left( x\right) dx\right\vert \leq \frac{\left\vert \eta \left( b,a\right) \right\vert }{8}\left( \left\vert f^{\prime }\left( a\right) \right\vert +\left\vert f^{\prime }\left( b\right) \right\vert \right) , \tag{6}\] and \[\left\vert \frac{f\left( a\right) +f\left( a+\eta \left( b,a\right) \right) }{2}-\frac{1}{\eta \left( b,a\right) }\int_{a}^{a+\eta \left( b,a\right) }f\left( x\right) dx\right\vert \leq \frac{\left\vert \eta \left( b,a\right) \right\vert }{2\left( p+1\right) ^{\frac{1}{p}}}\left( \left\vert f^{\prime }\left( a\right) \right\vert ^{q}+\left\vert f^{\prime }\left( b\right) \right\vert ^{q}\right) ^{\frac{1}{q}}. \tag{7}\]
Recently, Farid et al. [19], gave the followig Hermite-Hadamard inequality for Caputo fractional derivatives \[\begin{aligned} &f^{\left( n\right) }\left( \frac{a+b}{2}\right) \leq \frac{\Gamma \left( n-\alpha +1\right) }{2\left( b-a\right) ^{n-\alpha }}\left( \left( ^{c}D_{a^{+}}^{\alpha }f^{\left( n\right) }\right) \left( b\right) +\left( -1\right) ^{n}\left( ^{c}D_{b^{-}}^{\alpha }f^{\left( n\right) }\right) \left( a\right) \right) \leq \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( b\right) }{2}. \end{aligned} \tag{8}\] Also, in the same paper, they established the following result \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( b\right) }{2}-\frac{\Gamma \left( n-\alpha +1\right) }{2\left( b-a\right) ^{n-\alpha }}\left( \left( ^{c}D_{a^{+}}^{\alpha }f^{\left( n\right) }\right) \left( b\right) +\left( -1\right) ^{n}\left( ^{c}D_{b^{-}}^{\alpha }f^{\left( n\right) }\right) \left( a\right) \right) \right\vert \notag\\ &\leq \frac{b-a}{2\left( n-\alpha +1\right) }\left( 1-\frac{1}{ 2^{n-\alpha }}\right) \left( \left\vert f^{\left( n+1\right) }\left( a\right) \right\vert +\left\vert f^{\left( n+1\right) }\left( b\right) \right\vert \right) . \end{aligned} \tag{9}\]
Motivated by the results of the paper [19], in this paper, we use a new identity in order to establish some trapezoidal type inequalities for functions whose modulus of the first derivatives are \((s,m)\)-preinvex in the second sens via fractional derivatives at Caputo meaning.
This section recalls some known definitions
Definition 1. [20] A function \(f:I\rightarrow \mathbb{R}\) is said to be convex, if \[f\left( tx+\left( 1-t\right) y\right) \leq tf\left( x\right) +\left( 1-t\right) f(y), \tag{10}\] holds for all \(x,y\in I\) and all \(t\in \lbrack 0,1]\).
Definition 2. [21] A function \(f:I=[a,b]\subset \left[ 0,+\infty \right) \rightarrow \mathbb{R}\) is called \(s\)-convex, if \[f\left( tx+\left( 1-t\right) y\right) \leq t^{s}f\left( x\right) +\left( 1-t\right) ^{s}f\left( y\right), \tag{11}\] holds for all \(x,y\in I\) and all \(t\in \lbrack 0,1]\).
Definition 3. [22] A function \(f:\left[ 0,b\right] \rightarrow \mathbb{R}\) is said to be \(m\) -convex, where \(m\in \left( 0,1\right]\),if \[f\left( tx+m\left( 1-t\right) y\right) \leq tf\left( x\right) +m\left( 1-t\right) f(y), \tag{12}\] holds for all \(x,y\in I\), and \(t\in \lbrack 0,1]\).
Definition 4. [23] A function \(f:\left[ 0,b\right] \rightarrow \mathbb{R}\) is said to be \(% \left( s,m\right)\)-convex, where \(s,m\in \left( 0,1\right]\),if \[f\left( tx+m\left( 1-t\right) y\right) \leq t^{s}f\left( x\right) +m\left( 1-t\right) ^{s}f(y), \tag{13}\] holds for all \(x,y\in I\), and \(t\in \lbrack 0,1]\).
Definition 5. [24] A set \(K\subseteq \mathbb{R}^{n}\) is said an invex with respect to the bi-function \(\eta :K\times K\rightarrow \mathbb{R}^{n}\), if for all \(x,y\in K\), we have \[x+t\eta \left( y,x\right) \in K. \tag{14}\]
Definition 6. [24] A function \(f:K\rightarrow \mathbb{R}\) is said to be preinvex, if \[f\left( x+t\eta \left( y,x\right) \right) \leq \left( 1-t\right) f\left( x\right) +tf\left( y\right), \tag{15}\] holds for all \(x,y\in K\) and all \(t\in \lbrack 0,1]\).
Definition 7. [25] A function \(f:K=[a,a+\eta \left( b,a\right) ]\subset \left[ 0,+\infty \right) \rightarrow \mathbb{R}\) is called \(s\)-preinvex, if \[f\left( x+t\eta \left( y,x\right) \right) \leq \left( 1-t\right) ^{s}f\left( x\right) +t^{s}f\left( y\right), \tag{16}\] holds for all \(x,y\in K\) and all \(t\in \lbrack 0,1]\).
Definition 8. [26] A function \(f:K\subset \left[ 0,b^{\ast }\right] \ \rightarrow \mathbb{R}\) is said to be \(m\)-preinvex with respect to \(\eta\), where \(b^{\ast }>0\) and \(% m\in \left( 0,1\right]\), if \[f\left( x+t\eta \left( y,x\right) \right) \ \leq \left( 1-t\right) f\left( x\right) +mtf\left( \frac{y}{m}\right), \tag{17}\] holds for all \(x,y\in K\), and \(t\in \lbrack 0,1]\).
Definition 9. [27] A function \(f:K\subset \left[ 0,b^{\ast }\right] \ \rightarrow \mathbb{R}\) is said to be \(\left( s,m\right)\)-preinvex with respect to \(\eta\) for some fixed \(s,m\in \left( 0,1\right]\), where \(b^{\ast }>0\), if \[f\left( x+t\eta \left( y,x\right) \right) \ \leq \left( 1-t\right) ^{s}f\left( x\right) +mt^{s}f\left( \frac{y}{m}\right), \tag{18}\] holds for all \(x,y\in K\), and \(t\in \lbrack 0,1]\).
Condition C. ( [28]) Let \(K\) be an invex set with respect to the bi-function \(\eta \left( .,.\right)\) then for any \(a,b\in K\) and \(t\in \left[ 0.1\right]\), we have \[\eta \left( a,a+t\eta (b,a)\right) =-t\eta (b,a)\text{ and }\eta \left( b,a+t\eta (b,a)\right) =\left( 1-t\right) \eta \left( b,a\right) . \tag{19}\]
From Condition C, it follows that \[\eta \left( a+t_{2}\eta (b,a),a+t_{1}\eta (b,a)\right) =\left( t_{2}-t_{1}\right) \eta \left( b,a\right), \tag{20}\] holds for any \(a,b\in K\) and \(t_{1},t_{2}\in \left[ 0.1\right]\).
Definition 10. [29] Let \(\alpha >0\) and \(\alpha \notin \left\{ 1,2,3,…\right\} ,n=\left[ \alpha \right] +1,f\in AC^{n}\left[ a,b\right]\) which is the space of functions having \(n^{th}\) derivatives absolutely continuous. The right-side and left-side Caputo fractional derivatives of order \(\alpha\) are defined as follows: \[\left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( x\right) =\frac{1}{\Gamma \left( n-\alpha \right) }\overset{x}{\underset{a}{\int }}\left( x-t\right) ^{n-\alpha -1}f^{\left( n\right) }\left( t\right) dt,x>a, \tag{21}\] and \[\left( ^{c}D_{b^{-}}^{\alpha }f\right) \left( x\right) =\frac{\left( -1\right) ^{n}}{\Gamma \left( n-\alpha \right) }\overset{b}{\underset{x}{ \int }}\left( t-x\right) ^{n-\alpha -1}f^{\left( n\right) }\left( t\right) dt,b>x. \tag{22}\]
If \(\alpha =n\in \left\{ 1,2,3,…\right\}\) and usual derivative \(f^{\left( n\right) }\left( x\right)\) of order \(n\) exists, then Caputo fractional derivative \(\left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( x\right)\) coincides with \(f^{\left( n\right) }\left( x\right)\) whereas \(\left( ^{c}D_{b^{-}}^{\alpha }f\right) \left( x\right)\) coincides with \(f^{\left( n\right) }\left( x\right)\) with exactness to a constant multiplier \(\left( -1\right) ^{n}\).
In particular we have for \(n=1\) and \(\alpha =0\) \[\left( ^{c}D_{a^{+}}^{0}f\right) \left( x\right) =\left( ^{c}D_{b^{-}}^{0}f\right) \left( x\right) =f\left( x\right) . \tag{23}\]
Definition 11. [30] The incomplete beta function is defined for all complex numbers \(x\) and \(y\) with \({Re}(x)>0\) and \({Re}(y)>0\) and \(0\leq a<1\) as follows \[B_{a}\left( x,y\right) =\overset{a}{\underset{0}{\int }}t^{x-1}\left( 1-t\right) ^{y-1}dt. \tag{24}\]
Lemma 1. [31] For any \(0\leq a<b\) in \(\mathbb{R}\) and \(0<\alpha \leq 1\), we have \[b^{\alpha }-a^{\alpha }\leq \left( b-a\right) ^{\alpha }. \tag{25}\]
Lemma 2. Let \(f:\left[ a,a+\eta \left( b,a\right) \right] \subset \left[ 0,b^{\ast }% \right] \rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion\) be the function such that \(f\in C^{n+1}\left( \left[ a,a+\eta \left( b,a\right) \right] \right)\) with \(\eta \left( b,a\right) >0\) and \(b^{\ast }>0\). Then the following equality for Caputo fractional derivatives holds \[\begin{aligned} &\frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( a+\eta \left( b,a\right) \right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) \left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( a+\eta \left( b,a\right) \right) +\left( -1\right) ^{n}\left( ^{c}D_{\left( a+\eta \left( b,a\right) \right) ^{-}}^{\alpha }f\right) \left( a\right) \right] }{2\left( \eta \left( b,a\right) \right) ^{n-\alpha }} \notag\\ &=\frac{\eta \left( b,a\right) }{2}\overset{1}{\underset{0}{\int }}\left( t^{n-\alpha }-\left( 1-t\right) ^{n-\alpha }\right) f^{\left( n+1\right) }\left( a+t\eta \left( b,a\right) \right) dt. \end{aligned} \tag{26}\]
Proof. Let \[I_{1}=\overset{1}{\underset{0}{\int }}t^{n-\alpha }f^{\left( n+1\right) }\left( a+t\eta \left( b,a\right) \right) dt, \tag{27}\] and \[I_{2}=\overset{1}{\underset{0}{\int }}\left( 1-t\right) ^{n-\alpha }f^{\left( n+1\right) }\left( a+t\eta \left( b,a\right) \right) dt. \tag{28}\] Integrating by parts \(I_{1}\) we get \[\begin{aligned} I_{1} =&\frac{1}{\eta \left( b,a\right) }f^{\left( n\right) }\left( a+\eta \left( b,a\right) \right) -\frac{n-\alpha }{\eta \left( b,a\right) }\overset {1}{\underset{0}{\int }}t^{n-\alpha -1}f^{\left( n\right) }\left( a+t\eta \left( b,a\right) \right) dt , \notag\\ =&\frac{1}{\eta \left( b,a\right) }f^{\left( n\right) }\left( a+\eta \left( b,a\right) \right) -\frac{n-\alpha }{\left( \eta \left( b,a\right) \right) ^{2}}\overset{a+\eta \left( b,a\right) }{\underset{a}{\int }}\left( \frac{u-a}{\eta \left( b,a\right) }\right) ^{n-\alpha -1}f^{\left( n\right) }\left( u\right) du , \notag\\ =&\frac{1}{\eta \left( b,a\right) }f^{\left( n\right) }\left( a+\eta \left( b,a\right) \right) -\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) }{\left( \eta \left( b,a\right) \right) ^{n-\alpha +1}} \left( -1\right) ^{n}\left( ^{c}D_{\left( a+\eta \left( b,a\right) \right) ^{-}}^{\alpha }f\right) \left( a\right) . \label{3.1} \end{aligned} \tag{29}\] Similarly, we have \[\begin{aligned} I_{2} =&-\frac{1}{\eta \left( b,a\right) }f^{\left( n\right) }\left( a\right) +\frac{n-\alpha }{\eta \left( b,a\right) }\overset{1}{\underset{0}{ \int }}\left( 1-t\right) ^{n-\alpha -1}f^{\left( n\right) }\left( a+t\eta \left( b,a\right) \right) dt , \notag\\ =&-\frac{1}{\eta \left( b,a\right) }f^{\left( n\right) }\left( a\right) + \frac{n-\alpha }{\left( \eta \left( b,a\right) \right) ^{n-\alpha +1}} \overset{a+\eta \left( b,a\right) }{\underset{a}{\int }}\left( a+\eta \left( b,a\right) -u\right) ^{n-\alpha -1}f^{\left( n\right) }\left( u\right) du , \notag\\ =&-\frac{1}{\eta \left( b,a\right) }f^{\left( n\right) }\left( a\right) + \frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) }{\left( \eta \left( b,a\right) \right) ^{n-\alpha +1}}\left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( a+\eta \left( b,a\right) \right) . \label{3.2} \end{aligned} \tag{30}\] Subtracting (30) from (29), and then multiplying the resulting equality by \(\frac{\eta \left( b,a\right) }{2}\) we get the desired result. ◻
Theorem 1. Let \(f:\left[ a,a+\eta \left( b,a\right) \right] \subset \left[ 0,b^{\ast }% \right] \rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion\) be the function such that \(f\in C^{n+1}\left( \left[ a,a+\eta \left( b,a\right) \right] \right)\) with \(b^{\ast }>0\) and \(\eta \left( b,a\right) >0\). If \(\left\vert f^{\left( n+1\right) }\right\vert\) is \(\left( s,m\right)\)-preinvex for some fixed \(s,m\in \left( 0,1\right]\), then the following inequality for Caputo fractional derivatives holds \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( a+\eta \left( b,a\right) \right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) \left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( a+\eta \left( b,a\right) \right) +\left( -1\right) ^{n}\left( ^{c}D_{\left( a+\eta \left( b,a\right) \right) ^{-}}^{\alpha }f\right) \left( a\right) \right] }{2\left( \eta \left( b,a\right) \right) ^{n-\alpha } }\right\vert \notag\\ &\leq \frac{\eta \left( b,a\right) }{2}\left( \left\vert f^{\left( n+1\right) }\left( a\right) \right\vert +m\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert \right) \left( \frac{1}{n-\alpha +s+1}\left( 1-\frac{1}{2^{n-\alpha +s}}\right) \right. \notag\\ &\quad+\left. B_{\frac{1}{2}}\left( s+1,n-\alpha +1\right) -B_{\frac{1}{2} }\left( n-\alpha +1,s+1\right) \right) , \end{aligned} \tag{31}\] where \(B_{\frac{1}{2}}\left( .,.\right)\) is the incomplete beta function.
Proof. From Lemma 2, properties of modulus, and \(\left( s,m\right)\)-preinvexity of \(\left\vert f^{\left( n+1\right) }\right\vert\) we have \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( a+\eta \left( b,a\right) \right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) \left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( a+\eta \left( b,a\right) \right) +\left( -1\right) ^{n}\left( ^{c}D_{\left( a+\eta \left( b,a\right) \right) ^{-}}^{\alpha }f\right) \left( a\right) \right] }{2\left( \eta \left( b,a\right) \right) ^{n-\alpha } }\right\vert \notag\\ &\leq \frac{\eta \left( b,a\right) }{2}\left( \overset{\frac{1}{2}}{ \underset{0}{\int }}\left( \left( 1-t\right) ^{n-\alpha }-t^{n-\alpha }\right) \left\vert f^{\left( n+1\right) }\left( a+t\eta \left( b,a\right) \right) \right\vert dt\right.\notag \\ &\quad+\left. \overset{1}{\underset{\frac{1}{2}}{\int }}\left( t^{n-\alpha }-\left( 1-t\right) ^{n-\alpha }\right) \left\vert f^{\left( n+1\right) }\left( a+t\eta \left( b,a\right) \right) \right\vert dt\right) \notag\\ &\leq \frac{\eta \left( b,a\right) }{2}\left( \overset{\frac{1}{2}}{ \underset{0}{\int }}\left( \left( 1-t\right) ^{n-\alpha }-t^{n-\alpha }\right) \left( \left( 1-t\right) ^{s}\left\vert f^{\left( n+1\right) }\left( a\right) \right\vert +mt^{s}\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert \right) dt\right. \notag\\ &\quad+\left. \overset{1}{\underset{\frac{1}{2}}{\int }}\left( t^{n-\alpha }-\left( 1-t\right) ^{n-\alpha }\right) \left( \left( 1-t\right) ^{s}\left\vert f^{\left( n+1\right) }\left( a\right) \right\vert +mt^{s}\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert \right) dt\right) \notag\\ &=\frac{\eta \left( b,a\right) }{2}\left( \left\vert f^{\left( n+1\right) }\left( a\right) \right\vert \overset{\frac{1}{2}}{\underset{0}{\int }} \left( \left( 1-t\right) ^{n-\alpha }-t^{n-\alpha }\right) \left( 1-t\right) ^{s}dt\right. +m\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert \overset{\frac{1}{2}}{\underset{0}{\int }}\left( \left( 1-t\right) ^{n-\alpha }-t^{n-\alpha }\right) t^{s}dt \notag\\ &\quad+\left\vert f^{\left( n+1\right) }\left( a\right) \right\vert \overset{1}{ \underset{\frac{1}{2}}{\int }}\left( t^{n-\alpha }-\left( 1-t\right) ^{n-\alpha }\right) \left( 1-t\right) ^{s}dt +\left. m\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert \overset{1}{\underset{\frac{1}{2}}{\int }}\left( t^{n-\alpha }-\left( 1-t\right) ^{n-\alpha }\right) t^{s}dt\right),\notag\\ &=\frac{\eta \left( b,a\right) }{2}\left( \overset{\frac{1}{2}}{\underset{0 }{\int }}\left( \left( 1-t\right) ^{n-\alpha }-t^{n-\alpha }\right) \left( 1-t\right) ^{s}dt\right. +\left. \overset{\frac{1}{2}}{\underset{0}{\int }}\left( \left( 1-t\right) ^{n-\alpha }-t^{n-\alpha }\right) t^{s}dt\right) \notag\\ & \quad\times\left( \left\vert f^{\left( n+1\right) }\left( a\right) \right\vert +m\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert \right) \notag\\ &=\frac{\eta \left( b,a\right) }{2}\left( \left\vert f^{\left( n+1\right) }\left( a\right) \right\vert +m\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert \right) \left( \frac{1}{n-\alpha +s+1} \left( 1-\frac{1}{2^{n-\alpha +s}}\right) \right. \notag\\ &\quad+\bigg. B_{\frac{1}{2}}\left( s+1,n-\alpha +1\right) -B_{\frac{1}{2} }\left( n-\alpha +1,s+1\right) \bigg) . \end{aligned} \tag{32}\] The proof is completed. ◻
Corollary 1. In Theorem 1, if we choose \(\eta \left( b,a\right) =b-a\), we obtain \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( b\right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) }{2\left( b-a\right) ^{n-\alpha }}\left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( b\right) +\left( -1\right) ^{n}\left( ^{c}D_{b^{-}}^{\alpha }f\right) \left( a\right) \right] \right\vert \notag\\ &\leq \frac{b-a}{2}\left( \left\vert f^{\left( n+1\right) }\left( a\right) \right\vert +m\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert \right) \left( \frac{1}{n-\alpha +s+1}\left( 1-\frac{1}{ 2^{n-\alpha +s}}\right) \right.\notag \\ &\quad+\big. B_{\frac{1}{2}}\left( s+1,n-\alpha +1\right) -B_{\frac{1}{2} }\left( n-\alpha +1,s+1\right) \bigg) . \end{aligned} \tag{33}\]
Corollary 2. In Theorem 1, if we take \(m=1\), we obtain \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( a+\eta \left( b,a\right) \right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) \left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( a+\eta \left( b,a\right) \right) +\left( -1\right) ^{n}\left( ^{c}D_{\left( a+\eta \left( b,a\right) \right) ^{-}}^{\alpha }f\right) \left( a\right) \right] }{2\left( \eta \left( b,a\right) \right) ^{n-\alpha } }\right\vert \notag\\ &\leq \frac{\eta \left( b,a\right) }{2}\left( \left\vert f^{\left( n+1\right) }\left( a\right) \right\vert +\left\vert f^{\left( n+1\right) }\left( b\right) \right\vert \right) \left( \frac{1}{n-\alpha +s+1}\left( 1- \frac{1}{2^{n-\alpha +s}}\right) \right.\notag \\ &\quad+\bigg. B_{\frac{1}{2}}\left( s+1,n-\alpha +1\right) -B_{\frac{1}{2} }\left( n-\alpha +1,s+1\right) \bigg) . \end{aligned} \tag{34}\] Moreover, if we take \(\eta \left( b,a\right) =b-a\), we get \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( b\right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) }{2\left( b-a\right) ^{n-\alpha }}\left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( b\right) +\left( -1\right) ^{n}\left( ^{c}D_{b^{-}}^{\alpha }f\right) \left( a\right) \right] \right\vert \notag\\ &\leq \frac{b-a}{2}\left( \left\vert f^{\left( n+1\right) }\left( a\right) \right\vert +\left\vert f^{\left( n+1\right) }\left( b\right) \right\vert \right) \left( \frac{1}{n-\alpha +s+1}\left( 1-\frac{1}{2^{n-\alpha +s}} \right) \right. \notag\\ &\quad+\bigg. B_{\frac{1}{2}}\left( s+1,n-\alpha +1\right) -B_{\frac{1}{2} }\left( n-\alpha +1,s+1\right) \bigg) . \end{aligned} \tag{35}\]
Corollary 3. In Theorem 1, if we take \(s=1\), we obtain \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( a+\eta \left( b,a\right) \right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) \left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( a+\eta \left( b,a\right) \right) +\left( -1\right) ^{n}\left( ^{c}D_{\left( a+\eta \left( b,a\right) \right) ^{-}}^{\alpha }f\right) \left( a\right) \right] }{2\left( \eta \left( b,a\right) \right) ^{n-\alpha } }\right\vert \notag\\ &\leq \frac{\eta \left( b,a\right) }{2^{n-\alpha +1}}\left( \frac{ 2^{n-\alpha }-1}{n-\alpha +1}\right) \left( \left\vert f^{\left( n+1\right) }\left( a\right) \right\vert +m\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert \right) . \end{aligned} \tag{36}\] Moreover, if we take \(\eta \left( b,a\right) =b-a\), we get \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( b\right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) }{2\left( b-a\right) ^{n-\alpha }}\left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( b\right) +\left( -1\right) ^{n}\left( ^{c}D_{b^{-}}^{\alpha }f\right) \left( a\right) \right] \right\vert \notag\\ &\leq \frac{b-a}{2^{n-\alpha +1}}\left( \frac{2^{n-\alpha }-1}{n-\alpha +1 }\right) \left( \left\vert f^{\left( n+1\right) }\left( a\right) \right\vert +m\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert \right) . \end{aligned} \tag{37}\]
Corollary 4. In Theorem 1, if we take \(s=m=1\), we obtain \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( a+\eta \left( b,a\right) \right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) \left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( a+\eta \left( b,a\right) \right) +\left( -1\right) ^{n}\left( ^{c}D_{\left( a+\eta \left( b,a\right) \right) ^{-}}^{\alpha }f\right) \left( a\right) \right] }{2\left( \eta \left( b,a\right) \right) ^{n-\alpha } }\right\vert \notag\\ &\leq \frac{\eta \left( b,a\right) }{2^{n-\alpha +1}}\left( \frac{ 2^{n-\alpha }-1}{n-\alpha +1}\right) \left( \left\vert f^{\left( n+1\right) }\left( a\right) \right\vert +\left\vert f^{\left( n+1\right) }\left( b\right) \right\vert \right) . \end{aligned} \tag{38}\] Moreover, if we take \(\eta \left( b,a\right) =b-a\), we get Theorem 2.4 from [19].
Theorem 2. Let \(f:\left[ a,a+\eta \left( b,a\right) \right] \subset \left[ 0,b^{\ast }% \right] \rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion\) be the function such that \(f\in C^{n+1}\left( \left[ a,a+\eta \left( b,a\right) \right] \right)\) with \(b^{\ast }>0\) and \(\eta \left( b,a\right) >0\). If \(\left\vert f^{\left( n+1\right) }\right\vert ^{q}\) is \(\left( s,m\right)\)-preinvex for some fixed \(s,m\in \left( 0,1\right]\) where \(q>1\) with \(p^{-1}+q^{-1}=1\), then the following inequality for Caputo fractional derivatives holds \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( a+\eta \left( b,a\right) \right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) \left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( a+\eta \left( b,a\right) \right) +\left( -1\right) ^{n}\left( ^{c}D_{\left( a+\eta \left( b,a\right) \right) ^{-}}^{\alpha }f\right) \left( a\right) \right] }{2\left( \eta \left( b,a\right) \right) ^{n-\alpha } }\right\vert \notag\\ &\leq \frac{\eta \left( b,a\right) }{2}\left( \frac{1}{pn-p\alpha +1} \right) ^{\frac{1}{p}}\left( \frac{\left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+m\left\vert f^{\left( n+1\right) }\left( \frac{b}{ m}\right) \right\vert ^{q}}{s+1}\right) ^{\frac{1}{q}}. \end{aligned} \tag{39}\]
Proof. From Lemma 2, properties of modulus, Hölder’s inequality, Lemma 1, and \(% \left( s,m\right)\)-preinvexity of \(\left\vert f^{\left( n+1\right) }\right\vert ^{q}\), we have \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( a+\eta \left( b,a\right) \right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) \left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( a+\eta \left( b,a\right) \right) +\left( -1\right) ^{n}\left( ^{c}D_{\left( a+\eta \left( b,a\right) \right) ^{-}}^{\alpha }f\right) \left( a\right) \right] }{2\left( \eta \left( b,a\right) \right) ^{n-\alpha } }\right\vert \notag\\ &\leq \frac{\eta \left( b,a\right) }{2}\left( \overset{1}{\underset{0}{ \int }}\left\vert t^{n-\alpha }-\left( 1-t\right) ^{n-\alpha }\right\vert ^{p}dt\right) ^{\frac{1}{p}}\left( \overset{1}{\underset{0}{\int }} \left\vert f^{\left( n+1\right) }\left( a+t\eta \left( b,a\right) \right) \right\vert ^{q}dt\right) ^{\frac{1}{q}} \notag\\ &=\frac{\eta \left( b,a\right) }{2}\left( \overset{\frac{1}{2}}{\underset{0 }{\int }}\left( \left( 1-t\right) ^{n-\alpha }-t^{n-\alpha }\right) ^{p}dt+ \overset{1}{\underset{\frac{1}{2}}{\int }}\left( t^{n-\alpha }-\left( 1-t\right) ^{n-\alpha }\right) ^{p}dt\right) ^{\frac{1}{p}} \times \left( \overset{1}{\underset{0}{\int }}\left\vert f^{\left( n+1\right) }\left( a+t\eta \left( b,a\right) \right) \right\vert ^{q}dt\right) ^{\frac{1}{q}} \notag\\ &\leq \frac{\eta \left( b,a\right) }{2}\left( \overset{\frac{1}{2}}{ \underset{0}{\int }}\left( 1-2t\right) ^{pn-p\alpha }dt+\overset{1}{ \underset{\frac{1}{2}}{\int }}\left( 2t-1\right) ^{pn-p\alpha }dt\right) ^{ \frac{1}{p}} \notag\\ &\quad \times \left( \overset{1}{\underset{0}{\int }}\left( \left( 1-t\right) ^{s}\left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+mt^{s}\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert ^{q}\right) dt\right) ^{\frac{1}{q}} \notag\\ &=\frac{\eta \left( b,a\right) }{2}\left( \frac{1}{pn-p\alpha +1}\right) ^{\frac{1}{p}}\left( \frac{\left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+m\left\vert f^{\left( n+1\right) }\left( \frac{b}{m} \right) \right\vert ^{q}}{s+1}\right) ^{\frac{1}{q}}. \end{aligned} \tag{40}\] The proof is completed. ◻
Corollary 5. In Theorem 2, if we choose \(\eta \left( b,a\right) =b-a\), we obtain \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( b\right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) }{2\left( b-a\right) ^{n-\alpha }}\left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( b\right) +\left( -1\right) ^{n}\left( ^{c}D_{b^{-}}^{\alpha }f\right) \left( a\right) \right] \right\vert \notag \\ &\leq \frac{b-a}{2}\left( \frac{1}{pn-p\alpha +1}\right) ^{\frac{1}{p} }\left( \frac{\left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+m\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert ^{q}}{s+1}\right) ^{\frac{1}{q}}. \end{aligned} \tag{41}\]
Corollary 6. In Theorem 2, if we take \(m=1\), we obtain \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( a+\eta \left( b,a\right) \right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) \left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( a+\eta \left( b,a\right) \right) +\left( -1\right) ^{n}\left( ^{c}D_{\left( a+\eta \left( b,a\right) \right) ^{-}}^{\alpha }f\right) \left( a\right) \right] }{2\left( \eta \left( b,a\right) \right) ^{n-\alpha } }\right\vert \notag\\ &\leq \frac{\eta \left( b,a\right) }{2}\left( \frac{1}{pn-p\alpha +1} \right) ^{\frac{1}{p}}\left( \frac{\left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+\left\vert f^{\left( n+1\right) }\left( b\right) \right\vert ^{q}}{s+1}\right) ^{\frac{1}{q}}. \end{aligned} \tag{42}\] Moreover, if we take \(\eta \left( b,a\right) =b-a\), we get \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( b\right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) }{2\left( b-a\right) ^{n-\alpha }}\left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( b\right) +\left( -1\right) ^{n}\left( ^{c}D_{b^{-}}^{\alpha }f\right) \left( a\right) \right] \right\vert \notag\\ &\leq \frac{b-a}{2}\left( \frac{1}{pn-p\alpha +1}\right) ^{\frac{1}{p} }\left( \frac{\left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+m\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert ^{q}}{s+1}\right) ^{\frac{1}{q}}. \end{aligned} \tag{43}\]
Corollary 7. In Theorem 2, if we take \(s=1\), we obtain \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( a+\eta \left( b,a\right) \right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) \left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( a+\eta \left( b,a\right) \right) +\left( -1\right) ^{n}\left( ^{c}D_{\left( a+\eta \left( b,a\right) \right) ^{-}}^{\alpha }f\right) \left( a\right) \right] }{2\left( \eta \left( b,a\right) \right) ^{n-\alpha } }\right\vert \notag\\ &\leq \frac{\eta \left( b,a\right) }{2}\left( \frac{1}{pn-p\alpha +1} \right) ^{\frac{1}{p}}\left( \frac{\left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+m\left\vert f^{\left( n+1\right) }\left( \frac{b}{ m}\right) \right\vert ^{q}}{2}\right) ^{\frac{1}{q}}. \end{aligned} \tag{44}\] Moreover, if we take \(\eta \left( b,a\right) =b-a\), we get \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( b\right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) }{2\left( b-a\right) ^{n-\alpha }}\left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( b\right) +\left( -1\right) ^{n}\left( ^{c}D_{b^{-}}^{\alpha }f\right) \left( a\right) \right] \right\vert \notag\\ &\leq \frac{b-a}{2}\left( \frac{1}{pn-p\alpha +1}\right) ^{\frac{1}{p} }\left( \frac{\left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+m\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert ^{q}}{2}\right) ^{\frac{1}{q}}. \end{aligned} \tag{45}\]
Corollary 8. In Theorem 2, if we take \(s=m=1\), we obtain \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( a+\eta \left( b,a\right) \right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) \left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( a+\eta \left( b,a\right) \right) +\left( -1\right) ^{n}\left( ^{c}D_{\left( a+\eta \left( b,a\right) \right) ^{-}}^{\alpha }f\right) \left( a\right) \right] }{2\left( \eta \left( b,a\right) \right) ^{n-\alpha } }\right\vert \notag\\ &\leq \frac{\eta \left( b,a\right) }{2}\left( \frac{1}{pn-p\alpha +1} \right) ^{\frac{1}{p}}\left( \frac{\left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+\left\vert f^{\left( n+1\right) }\left( b\right) \right\vert ^{q}}{2}\right) ^{\frac{1}{q}}. \end{aligned} \tag{46}\] Moreover, if we take \(\eta \left( b,a\right) =b-a\), we get \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( b\right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) }{2\left( b-a\right) ^{n-\alpha }}\left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( b\right) +\left( -1\right) ^{n}\left( ^{c}D_{b^{-}}^{\alpha }f\right) \left( a\right) \right] \right\vert \notag\\ &\leq \frac{b-a}{2}\left( \frac{1}{pn-p\alpha +1}\right) ^{\frac{1}{p} }\left( \frac{\left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+\left\vert f^{\left( n+1\right) }\left( b\right) \right\vert ^{q}}{2} \right) ^{\frac{1}{q}}. \end{aligned} \tag{47}\]
Theorem 3. Let \(f:\left[ a,a+\eta \left( b,a\right) \right] \subset \left[ 0,b^{\ast }% \right] \rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion\) be the function such that \(f\in C^{n+1}\left( \left[ a,a+\eta \left( b,a\right) \right] \right)\) with \(b^{\ast }>0\) and \(\eta \left( b,a\right) >0\). If \(\left\vert f^{\left( n+1\right) }\right\vert ^{q}\) is \(\left( s,m\right)\)-preinvex for some fixed \(s,m\in \left( 0,1\right]\), where \(% q\geq 1\), then the following inequality for Caputo fractional derivatives holds \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( a+\eta \left( b,a\right) \right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) \left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( a+\eta \left( b,a\right) \right) +\left( -1\right) ^{n}\left( ^{c}D_{\left( a+\eta \left( b,a\right) \right) ^{-}}^{\alpha }f\right) \left( a\right) \right] }{2\left( \eta \left( b,a\right) \right) ^{n-\alpha } }\right\vert \notag\\ &\leq \frac{\eta \left( b,a\right) }{2}\left( \frac{2^{n-\alpha }-1}{ \left( n-\alpha +1\right) 2^{n-\alpha -1}}\right) ^{1-\frac{1}{q}}\left( \left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+m\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert ^{q}\right) ^{\frac{1}{q}} \\\notag &\quad\times \left( \frac{2^{n-\alpha +s}-1}{\left( n-\alpha +s+1\right) 2^{n-\alpha +s}}+B_{\frac{1}{2}}\left( s+1,n-\alpha +1\right) -B_{\frac{1}{2} }\left( n-\alpha +1,s+1\right) \right) ^{\frac{1}{q}}, \end{aligned} \tag{48}\] where \(B_{\frac{1}{2}}\left( .,.\right)\) is the incomplete beta function.
Proof. From Lemma 2, properties of modulus, power mean inequality, and \(\left( s,m\right)\)-preinvexity of \(\left\vert f^{\left( n+1\right) }\right\vert ^{q}\), we have \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( a+\eta \left( b,a\right) \right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) \left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( a+\eta \left( b,a\right) \right) +\left( -1\right) ^{n}\left( ^{c}D_{\left( a+\eta \left( b,a\right) \right) ^{-}}^{\alpha }f\right) \left( a\right) \right] }{2\left( \eta \left( b,a\right) \right) ^{n-\alpha } }\right\vert \notag\\ &\leq \frac{\eta \left( b,a\right) }{2}\left( \overset{1}{\underset{0}{ \int }}\left\vert t^{n-\alpha }-\left( 1-t\right) ^{n-\alpha }\right\vert dt\right) ^{1-\frac{1}{q}} \times \left( \overset{1}{\underset{0}{\int }}\left\vert t^{n-\alpha }-\left( 1-t\right) ^{n-\alpha }\right\vert \left\vert f^{\left( n+1\right) }\left( a+t\eta \left( b,a\right) \right) \right\vert ^{q}dt\right) ^{\frac{1 }{q}} \notag\\ &\leq \frac{\eta \left( b,a\right) }{2}\left( \overset{1}{\underset{0}{ \int }}\left\vert t^{n-\alpha }-\left( 1-t\right) ^{n-\alpha }\right\vert dt\right) ^{1-\frac{1}{q}} \notag\\ &\quad\times \left( \overset{1}{\underset{0}{\int }}\left\vert t^{n-\alpha }-\left( 1-t\right) ^{n-\alpha }\right\vert \left( \left( 1-t\right) ^{s}\left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+mt^{s}\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert ^{q}\right) dt\right) ^{\frac{1}{q}} \notag\\ &=\frac{\eta \left( b,a\right) }{2}\left( \overset{\frac{1}{2}}{\underset{0 }{\int }}\left( \left( 1-t\right) ^{n-\alpha }-t^{n-\alpha }\right) dt+ \overset{1}{\underset{\frac{1}{2}}{\int }}\left( t^{n-\alpha }-\left( 1-t\right) ^{n-\alpha }\right) dt\right) ^{1-\frac{1}{q}} \notag\\ &\quad\times \left( \overset{\frac{1}{2}}{\underset{0}{\int }}\left( \left( 1-t\right) ^{n-\alpha }-t^{n-\alpha }\right) \left( \left( 1-t\right) ^{s}\left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+mt^{s}\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert ^{q}\right) dt\right. \notag\\ &\quad+\left. \overset{1}{\underset{\frac{1}{2}}{\int }}\left( t^{n-\alpha }-\left( 1-t\right) ^{n-\alpha }\right) \left( \left( 1-t\right) ^{s}\left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+mt^{s}\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert ^{q}\right) dt\right) ^{\frac{1}{q}} \notag\\ &=\frac{\eta \left( b,a\right) }{2}\left( \frac{2^{n-\alpha }-1}{\left( n-\alpha +1\right) 2^{n-\alpha -1}}\right) ^{1-\frac{1}{q}}\left( \left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+m\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert ^{q}\right) ^{\frac{1}{q}} \notag\\ &\quad\times \left( \overset{\frac{1}{2}}{\underset{0}{\int }}\left( \left( 1-t\right) ^{n-\alpha }-t^{n-\alpha }\right) \left( 1-t\right) ^{s}dt+ \overset{\frac{1}{2}}{\underset{0}{\int }}\left( \left( 1-t\right) ^{n-\alpha }-t^{n-\alpha }\right) t^{s}dt\right) ^{\frac{1}{q}} \notag\\ &=\frac{\eta \left( b,a\right) }{2}\left( \frac{2^{n-\alpha }-1}{\left( n-\alpha +1\right) 2^{n-\alpha -1}}\right) ^{1-\frac{1}{q}}\left( \left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+m\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert ^{q}\right) ^{\frac{1}{q}} \notag\\ &\quad\times \left( \frac{2^{n-\alpha +s}-1}{\left( n-\alpha +s+1\right) 2^{n-\alpha +s}}+B_{\frac{1}{2}}\left( s+1,n-\alpha +1\right) -B_{\frac{1}{2} }\left( n-\alpha +1,s+1\right) \right) ^{\frac{1}{q}}. \end{aligned} \tag{49}\] The proof is completed. ◻
Corollary 9. In Theorem 3, if we choose \(\eta \left( b,a\right) =b-a\), we obtain \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( b\right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) }{2\left( b-a\right) ^{n-\alpha }}\left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( b\right) +\left( -1\right) ^{n}\left( ^{c}D_{b^{-}}^{\alpha }f\right) \left( a\right) \right] \right\vert \notag\\ &\leq \frac{b-a}{2}\left( \frac{2^{n-\alpha }-1}{\left( n-\alpha +1\right) 2^{n-\alpha -1}}\right) ^{1-\frac{1}{q}}\left( \left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+m\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert ^{q}\right) ^{\frac{1}{q}} \notag\\ &\quad\times \left( \frac{2^{n-\alpha +s}-1}{\left( n-\alpha +s+1\right) 2^{n-\alpha +s}}+B_{\frac{1}{2}}\left( s+1,n-\alpha +1\right) -B_{\frac{1}{2} }\left( n-\alpha +1,s+1\right) \right) ^{\frac{1}{q}}. \end{aligned} \tag{50}\]
Corollary 10. In Theorem 3, if we take \(m=1\), we obtain \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( a+\eta \left( b,a\right) \right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) \left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( a+\eta \left( b,a\right) \right) +\left( -1\right) ^{n}\left( ^{c}D_{\left( a+\eta \left( b,a\right) \right) ^{-}}^{\alpha }f\right) \left( a\right) \right] }{2\left( \eta \left( b,a\right) \right) ^{n-\alpha } }\right\vert \notag\\ &\leq \frac{\eta \left( b,a\right) }{2}\left( \frac{2^{n-\alpha }-1}{ \left( n-\alpha +1\right) 2^{n-\alpha -1}}\right) ^{1-\frac{1}{q}}\left( \left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+\left\vert f^{\left( n+1\right) }\left( b\right) \right\vert ^{q}\right) ^{\frac{1}{q}} \notag\\ &\quad\times \left( \frac{2^{n-\alpha +s}-1}{\left( n-\alpha +s+1\right) 2^{n-\alpha +s}}+B_{\frac{1}{2}}\left( s+1,n-\alpha +1\right) -B_{\frac{1}{2} }\left( n-\alpha +1,s+1\right) \right) ^{\frac{1}{q}}. \end{aligned} \tag{51}\] Moreover, if we take \(\eta \left( b,a\right) =b-a\), we get \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( b\right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) }{2\left( b-a\right) ^{n-\alpha }}\left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( b\right) +\left( -1\right) ^{n}\left( ^{c}D_{b^{-}}^{\alpha }f\right) \left( a\right) \right] \right\vert \notag\\ &\leq \frac{b-a}{2}\left( \frac{2^{n-\alpha }-1}{\left( n-\alpha +1\right) 2^{n-\alpha -1}}\right) ^{1-\frac{1}{q}}\left( \left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+\left\vert f^{\left( n+1\right) }\left( b\right) \right\vert ^{q}\right) ^{\frac{1}{q}} \notag\\ &\quad\times \left( \frac{2^{n-\alpha +s}-1}{\left( n-\alpha +s+1\right) 2^{n-\alpha +s}}+B_{\frac{1}{2}}\left( s+1,n-\alpha +1\right) -B_{\frac{1}{2} }\left( n-\alpha +1,s+1\right) \right) ^{\frac{1}{q}}. \end{aligned} \tag{52}\]
Corollary 11. In Theorem 3, if we take \(s=1\), we obtain \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( a+\eta \left( b,a\right) \right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) \left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( a+\eta \left( b,a\right) \right) +\left( -1\right) ^{n}\left( ^{c}D_{\left( a+\eta \left( b,a\right) \right) ^{-}}^{\alpha }f\right) \left( a\right) \right] }{2\left( \eta \left( b,a\right) \right) ^{n-\alpha } }\right\vert \notag\\ &\leq \frac{\eta \left( b,a\right) }{2}\left( \frac{2^{n-\alpha }-1}{ \left( n-\alpha +1\right) 2^{n-\alpha -1}}\right) \left( \frac{\left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+m\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert ^{q}}{2}\right) ^{\frac{1 }{q}}. \end{aligned} \tag{53}\] Moreover, if we take \(\eta \left( b,a\right) =b-a\), we get \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( b\right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) }{2\left( b-a\right) ^{n-\alpha }}\left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( b\right) +\left( -1\right) ^{n}\left( ^{c}D_{b^{-}}^{\alpha }f\right) \left( a\right) \right] \right\vert \notag\\ &\leq \frac{b-a}{2}\left( \frac{2^{n-\alpha }-1}{\left( n-\alpha +1\right) 2^{n-\alpha -1}}\right) \left( \frac{\left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+m\left\vert f^{\left( n+1\right) }\left( \frac{b}{m}\right) \right\vert ^{q}}{2}\right) ^{\frac{1 }{q}}. \end{aligned} \tag{54}\]
Corollary 12. In Theorem 3, if we take \(s=m=1\), we obtain \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( a+\eta \left( b,a\right) \right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) \left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( a+\eta \left( b,a\right) \right) +\left( -1\right) ^{n}\left( ^{c}D_{\left( a+\eta \left( b,a\right) \right) ^{-}}^{\alpha }f\right) \left( a\right) \right] }{2\left( \eta \left( b,a\right) \right) ^{n-\alpha } }\right\vert \notag\\ &\leq \frac{\eta \left( b,a\right) }{2}\left( \frac{2^{n-\alpha }-1}{ \left( n-\alpha +1\right) 2^{n-\alpha -1}}\right) \left( \frac{\left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+\left\vert f^{\left( n+1\right) }\left( b\right) \right\vert ^{q}}{2}\right) ^{\frac{1}{q}}. \end{aligned} \tag{55}\] Moreover, if we take \(\eta \left( b,a\right) =b-a\), we get \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( b\right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) }{2\left( b-a\right) ^{n-\alpha }}\left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( b\right) +\left( -1\right) ^{n}\left( ^{c}D_{b^{-}}^{\alpha }f\right) \left( a\right) \right] \right\vert \notag\\ &\leq \frac{b-a}{2}\left( \frac{2^{n-\alpha }-1}{\left( n-\alpha +1\right) 2^{n-\alpha -1}}\right) \left( \frac{\left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+\left\vert f^{\left( n+1\right) }\left( b\right) \right\vert ^{q}}{2}\right) ^{\frac{1}{q}}. \end{aligned} \tag{56}\]
Hadamard, J. (1893). Study on the properties of entire functions and in particular of a function considered by Riemann. Journal of Pure and Applied Mathematics , 9 , 171-215.
Hermite, C. (1883). Sur deux limites d’une intégrale définie. Mathesis, 3(82), 6.
Abdeljawad, T., Mohammed, P. O., & Kashuri, A. (2020). New modified conformable fractional integral inequalities of Hermite–Hadamard type with applications. Journal of Function Spaces, 2020(1), 4352357.
Dahmani, Z. (2014). New classes of integral inequalities of fractional order. Le Matematiche, 69(1), 237-247.
Farid, G., Javed, A., Rehman, A. U., & Qureshi, M. I. (2017). On Hadamard-type inequalities for differentiable functions via Caputo k-fractional derivatives. Cogent Mathematics, 4(1), 1355429.
Farid, G. (2020). On Caputo fractional derivatives via convexity. Kragujevac Journal of Mathematics, 44(3), 393-399.
Houas, M., Dahmani, Z., & Sarikaya, M. Z. (2018). Some integral inequalities for (k, s)–Riemann-Liouville fractional operators. Journal of Interdisciplinary Mathematics, 21(7-8), 1575-1585.
Kashuri, A., Meftah, B., & Mohammed, P. O. (2020). Some weighted Simpson type inequalities for differentiable s–convex functions and their applications: Some weighted Simpson type inequalities. Journal of Fractional Calculus and Nonlinear Systems, 1(1), 75-94.
Meftah, B., & Mekalfa, K. (2021). Some weighted trapezoidal inequalities for differentiable log-convex functions. Journal of Interdisciplinary Mathematics, 24(3), 505-517.
Meftah, B., & Mekhalfa, K. (2020). Some weighted trapezoidal inequalities for prequasiinvex functions. Communications in Optimization Theory, Article ID 20.
Meftah, B., Benssaad, M., Kaidouchi, W., & Ghomrani, S. (2021). Conformable fractional Hermite-Hadamard type inequalities for product of two harmonic s-convex functions. Proceedings of the American Mathematical Society, 149(4), 1495-1506.
Sarikaya, M. Z., Set, E., Yaldiz, H., & Başak, N. (2013). Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities. Mathematical and Computer Modelling, 57(9-10), 2403-2407.
Sarıkaya, M. Z., Dahmanı, Z., Kırıs, M. E., & Ahmad, F. (2016). (k, s)-Riemann-Liouville fractional integral and applications. Hacettepe Journal of Mathematics and Statistics, 45(1), 77-89.
Zhao, S., Butt, S. I., Nazeer, W., Nasir, J., Umar, M., & Liu, Y. (2020). Some Hermite–Jensen–Mercer type inequalities for k-Caputo-fractional derivatives and related results. Advances in Difference Equations, 2020(1), 262.
Dragomir, S. S., & Agarwal, R. (1998). Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula. Applied Mathematics Letters, 11(5), 91-95.
Pearce, C. E., & Pečarić, J. (2000). Inequalities for differentiable mappings with application to special means and quadrature formulae. Applied Mathematics Letters, 13(2), 51-55.
Noor, M. A. (2007). Hermite-Hadamard integral inequalities for log-preinvex functions. Journal of Mathematical Analysis and Approximation Theory, 2(2), 126-131.
Barani, A., Ghazanfari, A. G., & Dragomir, S. S. (2012). Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex. Journal of Inequalities and Applications, 2012, 1-9.
Farid, G., Naqvi, S., & Javed, A. (2017). Hadamard and Fejer Hadamard inequalities and related results via Caputo fractional derivatives. Bulletin of Mathematical Analysis and Applications, 9(3), 16-30.
Peajcariaac, J. E., & Tong, Y. L. (1992). Convex Functions, Partial Orderings, and Statistical Applications. Academic Press.
Breckner, W. W. (1978). Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Räumen. Publications de l’Institut Mathématique (Beograd)(NS), 23(37), 13-20.
Toader, G. (1984, October). Some generalizations of the convexity. In Proceedings of the Colloquium on Approximation and Optimization (Vol. 329, p. 338). Cluj-Napoca, Romania: University of Cluj-Napoca.
Eftekhari, N. (2014). Some remarks on (s, m)-convexity in the second sense. Journal of Mathematical Inequalities, 8(3), 489-495.
Weir, T., & Mond, B. (1988). Pre-invex functions in multiple objective optimization. Journal of Mathematical Analysis and Applications, 136(1), 29-38.
Li, J. Y. (2010). On Hadamard-type inequalities for s-preinvex functions. Journal of Chongqing Normal University (Natural Science), 27(4), 003.
Latif, M. A., & Shoaib, M. (2015). Hermite–Hadamard type integral inequalities for differentiable m-preinvex and (a, m)-preinvex functions. Journal of the Egyptian Mathematical Society, 23(2), 236-241.
Meftah, B. (2016). Hermite-Hadamard’s inequalities for functions whose first derivatives are (s, m)-preinvex in the second sense. Journal of New Theory, (10), 54-65.
Mohan, S. R., & Neogy, S. K. (1995). On invex sets and preinvex functions. Journal of Mathematical Analysis and Applications, 189(3), 901-908.
Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations (Vol. 204). elsevier.
Rainville, E. D. (1971). Special Functions. Reprint of 1960 first edition. Chelsea Publishing Co., Bronx, N.Y.
Wang, J., Zhu, C., & Zhou, Y. (2013). New generalized Hermite-Hadamard type inequalities and applications to special means. Journal of Inequalities and Applications, 2013, 1-15.
