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 \[\begin{aligned} &\left\vert \frac{f^{\left( n\right) }\left( a\right) +f^{\left( n\right) }\left( b\right) }{2}-\frac{\left( n-\alpha \right) \Gamma \left( n-\alpha \right) }{2\left( b-a\right) ^{n-\alpha }}\left[ \left( ^{c}D_{a^{+}}^{\alpha }f\right) \left( b\right) +\left( -1\right) ^{n}\left( ^{c}D_{b^{-}}^{\alpha }f\right) \left( a\right) \right] \right\vert \notag\\ &\leq \frac{b-a}{2}\left( \frac{2^{n-\alpha }-1}{\left( n-\alpha +1\right) 2^{n-\alpha -1}}\right) \left( \frac{\left\vert f^{\left( n+1\right) }\left( a\right) \right\vert ^{q}+\left\vert f^{\left( n+1\right) }\left( b\right) \right\vert ^{q}}{2}\right) ^{\frac{1}{q}}. \end{aligned} \tag{56}\]
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