Abstract
The authors discover a general k-fractional integral identity with multi-parameters for twice differentiable functions. By using this integral equation, the authors derive some new bounds on Hermite–Hadamard’s and Simpson’s inequalities for generalized \((m,h)\)-preinvex functions through k-fractional integrals. By taking the special parameter values for various suitable choices of function h, some interesting results are also obtained.
Similar content being viewed by others
1 Introduction
The subsequent inequalities are notable in the literature as Hermite–Hadamard’s inequality and Simpson’s inequality, respectively.
Theorem 1.1
Suppose that \(f:I\subseteq \mathbb{R}\rightarrow \mathbb{R}\) is a convex function defined on the interval I of real numbers and \(a,b\in I\) along with \(a< b\). The following double inequality holds:
Theorem 1.2
Assume that \(f:[a,b]\rightarrow \mathbb{R}\) is a four times continuously differentiable mapping on \((a,b)\) and \(\Vert f^{(4)} \Vert _{\infty }=\mathrm{sup}_{x\in (a,b)} \vert f^{(4)}(x) \vert <\infty \). Then the following inequality holds:
Hermite–Hadamard’s inequalities and Simpson’s inequalities have remained an area of great interest owing to their extensive applications in mathematics and other sciences. Many researchers generalized these inequalities. For recent results, for example, see [1–8] and the references mentioned in these papers.
In 2013, Sarikaya et al. established the subsequent interesting Hermite–Hadamard’s inequalities by utilizing Riemann–Liouville fractional integrals.
Theorem 1.3
([9])
Let \(f:[a,b]\rightarrow \mathbb{R}\) be a positive function along with \(0\leq a< b\), and let \(f\in L^{1}[a,b]\). Suppose that f is a convex function on \([a,b]\), then the following inequalities for fractional integrals hold:
where the symbols \(J^{\mu }_{a^{+}} f\) and \(J^{\mu }_{b^{-}} f\) denote respectively the left-sided and right-sided Riemann–Liouville fractional integrals of order \(\mu >0\) defined by
and
Here, \(\Gamma (\mu )\) is the gamma function and its definition is \(\Gamma (\mu )=\int _{0}^{\infty }e^{-t}t^{\mu -1}\,\mathrm{d}t\). It is to be noted that \(J^{0}_{a^{+}}f(x)=J^{0}_{b^{-}}f(x)=f(x)\).
In the case of \(\mu =1\), the fractional integral recaptures the classical integral.
Because of the extensive application of Riemann–Liouville fractional integrals, some authors extended their studies to fractional Hermite–Hadamard’s inequalities via mappings of different classes. For example, refer to [10–12] for convex mappings, to [13] for s-convex mappings, to [14] for \((s,m)\)-convex mappings, to [15] for s-Godunova–Levin mappings, to [16] for harmonically convex mappings, to [17] for preinvex mappings, to [18] for MT m -preinvex mappings, to [19] for h-convex mappings, to [20] for r-convex mappings, and see the references cited therein.
In 2012, Mubeen and Habibullah introduced the following class of fractional integrals.
Definition 1.1
([21])
Let \(f\in L^{1}[a,b]\), then the Riemann–Liouville k-fractional integrals \({}_{k}J^{\mu }_{a^{+}}f(x)\) and \({}_{k}J^{\mu }_{b^{-}}f(x)\) of order \(\mu >0\) are given as
and
respectively, where \(k>0\) and \(\Gamma _{k}(\mu )\) is the k-gamma function defined by \(\Gamma _{k}(\mu )=\int ^{\infty }_{0} t^{\mu -1} e^{-\frac{t^{k}}{k}}\,\mathrm{d}t\). Furthermore, \(\Gamma _{k}(\mu +k)=\mu \Gamma _{k}(\mu )\) and \({}_{k}J^{0}_{a^{+}}f(x)={}_{k}J^{0}_{b^{-}}f(x)=f(x)\).
The concept of Riemann–Liouville k-fractional integral is an important generalization of Riemann–Liouville fractional integrals. We would like to stress here that for \(k\neq 1\) the properties of Riemann–Liouville k-fractional integrals are very dissimilar to those of classical Riemann–Liouville fractional integrals. Due to this, the Riemann–Liouville k-fractional integrals have aroused many researchers’ interest. Properties and estimations for the integral inequality related to this operator can be sought out in [22–27] and the references cited therein.
The main purpose of the current paper is to establish some new bounds on Hermite–Hadamard’s and Simpson’s inequalities for mappings whose absolute values of second derivatives are generalized \((m,h)\)-preinvex. To do this, the authors derive a general k-fractional integral identity along with multi parameters for twice differentiable mappings. By using this integral identity, the authors derive some new inequalities of Simpson and Hermite–Hadamard type for these mappings.
To end this section, we restate some special functions and definitions as follows.
Let us consider the following special functions:
-
(1)
The beta function:
$$ \begin{aligned} \beta (x,y)=\frac{\Gamma (x)\Gamma (y)}{\Gamma (x+y)}= \int ^{1}_{0}t^{x-1}(1-t)^{y-1}\,\mathrm{d}t, \quad x,y>0; \end{aligned} $$ -
(2)
The incomplete beta function:
$$ \begin{aligned} \beta (a;x,y)= \int ^{a}_{0}t^{x-1}(1-t)^{y-1}\,\mathrm{d}t, \quad 0< a< 1,x,y>0; \end{aligned} $$ -
(3)
The hypergeometric function:
$$ {}_{2}F_{1}(a,b;c;z)= \frac{1}{\beta (b,c-b)} \int ^{1}_{0} t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}\,\mathrm{d}t,\quad c>b>0,\vert z \vert < 1. $$
Definition 1.2
([28])
A function \(f : [0,\infty )\rightarrow \mathbb{R}\) is named s-convex in the second sense along with \(s\in (0,1]\) if
holds for all \(x,y\in [0,\infty )\) and \(\alpha ,\beta \geq 0\) along with \(\alpha +\beta =1\).
Definition 1.3
([29])
A function \(f :A \subseteq \mathbb{R}\rightarrow \mathbb{R}\) is called s-Godunova–Levin function of the second kind along with \(s\in [0,1]\) if
holds for all \(x,y\in A\) and \(t\in (0,1)\).
Definition 1.4
([30])
A function \(f:A\subseteq \mathbb{R}\rightarrow \mathbb{R}\) is named \(tgs\)-convex on A if f is non-negative and
holds for all \(x, y \in A \) and \(t\in (0,1)\).
Definition 1.5
([31])
A function \(f: A\subseteq \mathbb{R}\rightarrow \mathbb{R}\) is called MT-convex if f is non-negative and
holds for all \(x,y\in A\) and \(t\in (0, 1)\).
Definition 1.6
([32])
A set \(A\subseteq \mathbb{R}^{n}\) is called m-invex with respect to the mapping \(\eta :A \times A\times (0, 1]\rightarrow \mathbb{R}^{n}\) for some fixed \(m\in (0, 1]\) if \(mx+\lambda \eta (y, x, m) \in A\) holds for all \(x, y \in A\) and \(\lambda \in [0, 1]\).
Definition 1.7
([33])
Let \(A \subseteq \mathbb{R}\) be an open m-invex subset with respect to \(\eta : A \times A \times (0,1]\rightarrow \mathbb{R}\), and let \(h_{1}, h_{2} :[0,1]\rightarrow \mathbb{R}_{0}\). A function \(f:A\rightarrow \mathbb{R}\) is said to be generalized \((m, h_{1}, h_{2})\)-preinvex if
is valid for all \(x, y \in A\) and \(t \in [0, 1]\). If inequality (1.2) reverses, then f is said to be generalized \((m, h_{1}, h_{2})\)-preincave on A.
Clearly, if we take \(h_{1}(t)=h(1-t)\), \(h_{2}(t)=h(t)\) in Definition 1.7, then f becomes generalized \((m,h)\)-preinvex functions as follows.
Definition 1.8
Let \(A\subseteq \mathbb{R}\) be an open m-invex subset with respect to \(\eta : A\times A\times (0,1] \rightarrow \mathbb{R}\), and let \(h :[0,1]\rightarrow \mathbb{R}_{0}\). A function \(f:A\rightarrow \mathbb{R}\) is called generalized \((m,h)\)-preinvex if
is valid for all \(x, y \in A\) and \(t \in [0,1]\).
Remark 1.1
Let us discuss some special cases of Definition 1.8 as follows:
-
(i)
choosing \(h(t)=1\), we obtain the definition of generalized \((m,P)\)-preinvex functions;
-
(ii)
choosing \(h(t)=t^{s}\) for \(s\in (0,1]\), we obtain the definition of generalized \((m,s)\)-Breckner-preinvex functions;
-
(iii)
choosing \(h(t)=t^{-s}\) for \(s\in (0,1)\), we obtain the definition of generalized \((m,s)\)-Godunova–Levin–Dragomir-preinvex functions;
-
(iv)
choosing \(h(t)=t(1-t)\), we obtain the definition of generalized \((m, tgs)\)-preinvex functions;
-
(v)
choosing \(h(t)=\frac{\sqrt{t}}{2\sqrt{1-t}}\), we obtain the definition of generalized m-MT-preinvex functions.
It is worth mentioning here that, as far as we know, all the special cases considered above are new in the literature.
2 Main results
In order to derive our main results, we need the subsequent identity.
Lemma 2.1
Let \(A\subseteq \mathbb{R}\) be an open m-invex subset with respect to \(\eta : A\times A\times (0,1] \rightarrow \mathbb{R}\setminus \{0\}\) for some fixed \(m \in (0,1]\), and let \(a, b \in A\), \(a< b\) with \(\eta (b,a,m)>0\). Assume that \(f:A\rightarrow \mathbb{R}\) is a twice differentiable function on A such that \(f''\) is integrable on \([ma,ma+\eta (b,a,m) ]\). Then the following identity for Riemann–Liouville k-fractional integrals along with \(x \in [a, b]\), \(\lambda \in [0, 1]\), \(\mu > 0\), and \(k>0\) exists:
where
and \(\Gamma _{k}\) is the k-gamma function.
Proof
By integration by parts and replacing the variable, we can state
Similarly, we get
Multiplying both sides of (2.2) and (2.3) by \(\frac{\eta ^{\frac{\mu }{k}+2}(x,a,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)}\) and \(\frac{(-1)^{\frac{\mu }{k}+2}\eta ^{\frac{\mu }{k}+2}(x,b,m)}{(\frac{\mu }{k}+1)\eta (b,a,m)}\), respectively, and adding the resulting identities together, we obtain the desired result. □
Remark 2.1
In Lemma 2.1, if we put \(k=1\) and \(\eta (b,a,m)=b-ma\) along with \(m=1\), then we get the following identity:
which is proved by İşcan in [34]. Further, if we put \(\mu =1, \lambda =\frac{1}{2}\), and \(x=a\) or \(x=b\), then the above identity recaptures Lemma 1 in [35].
Using Lemma 2.1, we now state the following theorem.
Theorem 2.1
Let \(A\subseteq \mathbb{R}\) be an open m-invex subset with respect to \(\eta : A\times A\times (0,1] \rightarrow \mathbb{R}\setminus \{0\}\) for some fixed \(m \in (0,1]\), and let \(a, b \in A\), \(a< b\) with \(\eta (b,a,m)>0\). Assume that \(f:A\rightarrow \mathbb{R}\) is a twice differentiable function on A such that \(f''\) is integrable on \([ma,ma+\eta (b,a,m) ]\). If \(\vert f'' \vert ^{q}\) for \(q\geq 1\) is a generalized \((m,h)\)-preinvex function with respect to η and \(h:[0,1]\rightarrow \mathbb{R}_{0}\), then the following inequality for k-fractional integrals with \(x \in [a,b]\), \(\lambda \in [0, 1]\), \(\mu > 0\), \(k>0\) exists:
where
and
Proof
Applying Lemma 2.1 and the power mean inequality, we have
Since \(\vert f'' \vert ^{q}\) is generalized \((m,h)\)-preinvex on \([ma,ma+\eta (b,a,m) ]\), we get
and
where \(C_{0}(k,\mu ,\lambda )\), \(C_{1}(k,\mu ,\lambda ;h)\), and \(C_{2}(k,\mu ,\lambda ;h)\) are defined by (2.5)–(2.7), respectively. Hence, if we use (2.9) and (2.10) in (2.8), we can get the desired result. This completes the proof. □
Let us point out some special cases of Theorem 2.1.
I. If \(h(t)=t^{s}\) in Theorem 2.1, then we have the following results.
Corollary 2.1
In Theorem 2.1, if \(\vert f'' \vert ^{q}\) for \(q\geq 1\) is generalized \((m,s)\)-Breckner-preinvex functions, then, for \(s\in (0,1]\) and \(m\in (0,1]\), we have
where we use the fact that
and \(C_{0}(k,\mu ,\lambda )\) is defined by (2.5).
Corollary 2.2
In Theorem 2.1, if the mapping \(\eta (b,a,m)=b-ma\) along with \(m=1\), taking \(x=\frac{a+b}{2}\), then for \(s\in (0,1]\), we have the following inequality for s-convex functions:
Remark 2.2
In Corollary 2.2,
-
(i)
taking \(\lambda =\frac{1}{2}\), we obtain
$$\begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},\frac{1}{2},1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}} \biggl(k,\mu , \frac{1}{2} \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}T_{1} \biggl(k,\mu ,\frac{1}{2};s \biggr)+ \bigl\vert f''(a) \bigr\vert ^{q}T_{2} \biggl(k,\mu ,\frac{1}{2};s \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}T_{1} \biggl(k, \mu ,\frac{1}{2};s \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}T_{2} \biggl(k,\mu ,\frac{1}{2};s \biggr) \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$ -
(ii)
taking \(\lambda =\frac{1}{3}\), we obtain
$$\begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},\frac{1}{3},1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr] \\ &\quad \quad {}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}} \biggl(k,\mu ,\frac{1}{3} \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}T_{1} \biggl(k,\mu ,\frac{1}{3};s \biggr)+ \bigl\vert f''(a) \bigr\vert ^{q}T_{2} \biggl(k,\mu ,\frac{1}{3};s \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}T_{1} \biggl(k, \mu ,\frac{1}{3};s \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}T_{2} \biggl(k,\mu ,\frac{1}{3};s \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$Specially, if we put \(k=\mu =s=1\), then we obtain
$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{162} \biggl\{ \biggl[\frac{59}{96} \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+\frac{37}{96} \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[\frac{59}{96} \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \frac{37}{96} \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} \\ &\quad \leq \frac{(b-a)^{2}}{162} \biggl\{ \biggl(\frac{59\vert f''(a) \vert ^{q}+133\vert f''(b) \vert ^{q}}{192} \biggr)^{\frac{1}{q}} \\ &\quad \quad {} + \biggl(\frac{59\vert f''(b) \vert ^{q}+133\vert f''(a) \vert ^{q}}{192} \biggr)^{\frac{1}{q}} \biggr\} . \end{aligned}$$(2.11)It is noted that the result of the first inequality in (2.11) is proved by İşcan in [34], which is better than the result presented by Sarikaya et al. in [36, Theorem 6].
Remark 2.3
In Corollary 2.2, if we take \(k=1\) and \(\lambda =0,1\), we have the results (f) and (h) in [34, Corollary 2.3], respectively. Further, if we take \(\mu =1\), we have the results (g) and (i) in [34, Corollary 2.3], respectively.
II. If \(h(t)=t^{-s}\) in Theorem 2.1, then we have the following results.
Corollary 2.3
In Theorem 2.1, if \(\vert f'' \vert ^{q}\) for \(q\geq 1\) is generalized \((m,s)\)-Godunova–Levin–Dragomir-preinvex functions, then, for \(s\in (0,1)\) and \(m\in (0,1]\), we have
where we use the fact that
and \(C_{0}(k,\mu ,\lambda )\) is defined by (2.5).
Corollary 2.4
In Theorem 2.1, if the mapping \(\eta (b,a,m)=b-ma\) together with \(m=1\), taking \(x=\frac{a+b}{2}\), for \(s\in (0,1)\), we have the following inequality for s-Godunova–Levin functions:
Remark 2.4
In Corollary 2.4,
-
(a)
if \(\lambda =\frac{1}{3}\), then we obtain
$$\begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},\frac{1}{3},1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}} \biggl(k,\mu , \frac{1}{3} \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1} \biggl(k,\mu ,\frac{1}{3};s \biggr)+ \bigl\vert f''(a) \bigr\vert ^{q}U_{2} \biggl(k,\mu ,\frac{1}{3};s \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1} \biggl(k, \mu ,\frac{1}{3};s \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}U_{2} \biggl(k,\mu ,\frac{1}{3};s \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$Specially, if we put \(k=1=\mu \), then we obtain a Simpson-type inequality:
$$\begin{aligned} & \biggl\vert I_{f} \biggl(1,1; \frac{a+b}{2},\frac{1}{3},1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{8}{81} \biggr)^{1-\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2}\biggr) \biggr\vert ^{q}U_{1} \biggl(1,1, \frac{1}{3};s \biggr) \\ &\quad \quad {}+ \bigl\vert f''(a) \bigr\vert ^{q}U_{2} \biggl(1,1,\frac{1}{3};s \biggr)\biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1} \biggl(1,1,\frac{1}{3};s \biggr) + \bigl\vert f''(b)\bigr\vert ^{q}U_{2} \biggl(1,1,\frac{1}{3};s \biggr) \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$ -
(b)
if \(\lambda =\frac{1}{2}\), then we obtain
$$\begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},\frac{1}{2},1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}} \biggl(k,\mu , \frac{1}{2} \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1} \biggl(k,\mu ,\frac{1}{2};s \biggr)+ \bigl\vert f''(a) \bigr\vert ^{q}U_{2} \biggl(k,\mu ,\frac{1}{2};s \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1} \biggl(k, \mu ,\frac{1}{2};s \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}U_{2} \biggl(k,\mu ,\frac{1}{2};s \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$Specially, if we put \(k=1=\mu \), then we obtain an averaged midpoint-trapezoid-type inequality:
$$\begin{aligned} & \biggl\vert I_{f} \biggl(1,1; \frac{a+b}{2},\frac{1}{2},1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{1}{6} \biggr)^{1-\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1} \biggl(1,1, \frac{1}{2};s \biggr)+ \bigl\vert f''(a) \bigr\vert ^{q}U_{2} \biggl(1,1,\frac{1}{2};s \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}U_{1} \biggl(1,1,\frac{1}{2};s \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}U_{2} \biggl(1,1,\frac{1}{2};s \biggr) \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$ -
(c)
if \(\lambda =0\), then we obtain
$$\begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},0,1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert f \biggl(\frac{a+b}{2} \biggr)-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} \biggl(\frac{1}{\frac{\mu }{k}+2} \biggr)^{1-\frac{1}{q}} \biggl\{ \biggl[\frac{\vert f''(\frac{a+b}{2}) \vert ^{q}}{\frac{\mu }{k}-s+2}+ \bigl\vert f''(a) \bigr\vert ^{q}\beta \biggl(\frac{\mu }{k}+2,1-s \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[\frac{\vert f''(\frac{a+b}{2}) \vert ^{q}}{\frac{\mu }{k}-s+2}+ \bigl\vert f''(b) \bigr\vert ^{q}\beta \biggl(\frac{\mu }{k}+2,1-s \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$Specially, if we put \(k=1=\mu \), then we obtain a midpoint-type inequality:
$$\begin{aligned} & \biggl\vert I_{f} \biggl(1,1; \frac{a+b}{2},0,1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert f \biggl(\frac{a+b}{2} \biggr)-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{1}{3} \biggr)^{1-\frac{1}{q}} \biggl\{ \biggl[\frac{\vert f''(\frac{a+b}{2}) \vert ^{q}}{3-s}+ \bigl\vert f''(a) \bigr\vert ^{q}\beta (3,1-s ) \biggr]^{\frac{1}{q}} \\ &\quad \quad {}+ \biggl[\frac{\vert f''(\frac{a+b}{2}) \vert ^{q}}{3-s}+ \bigl\vert f''(b) \bigr\vert ^{q}\beta (3,1-s ) \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$ -
(d)
if \(\lambda =1\), then we obtain
$$\begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},1,1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{f(a)+f(b)}{2}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} \biggl[\frac{\frac{\mu }{k} (\frac{\mu }{k}+3 )}{2 (\frac{\mu }{k}+2 )} \biggr]^{1-\frac{1}{q}} \\ &\quad \quad {} \times \biggl\{ \biggl[\frac{\frac{\mu }{k} (\frac{\mu }{k}-s+3 )\vert f''(\frac{a+b}{2}) \vert ^{q}}{(2-s) (\frac{\mu }{k}-s+2 )} \\ &\quad \quad {} + \bigl\vert f''(a) \bigr\vert ^{q} \biggl( \biggl(\frac{\mu }{k}+1 \biggr)\beta (2,1-s)-\beta \biggl(\frac{\mu }{k}+2,1-s\biggr) \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[\frac{\frac{\mu }{k} (\frac{\mu }{k}-s+3 )\vert f''(\frac{a+b}{2}) \vert ^{q}}{(2-s) (\frac{\mu }{k}-s+2 )} \\ &\quad \quad {} + \bigl\vert f''(b) \bigr\vert ^{q} \biggl( \biggl(\frac{\mu }{k}+1 \biggr)\beta (2,1-s)-\beta \biggl(\frac{\mu }{k}+2,1-s \biggr) \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$Specially, if we put \(k=1=\mu \), then we obtain a trapezoid-type inequality:
$$\begin{aligned} & \biggl\vert I_{f} \biggl(1,1; \frac{a+b}{2},1,1,a,b \biggr) \biggr\vert \\ &\quad = \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{2}{3} \biggr)^{1-\frac{1}{q}} \biggl\{ \biggl[\frac{(4-s)\vert f''(\frac{a+b}{2}) \vert ^{q}}{(2-s)(3-s)} + \bigl\vert f''(a) \bigr\vert ^{q} \bigl(2\beta (2,1-s)-\beta (3,1-s) \bigr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[\frac{(4-s)\vert f''(\frac{a+b}{2}) \vert ^{q}}{(2-s)(3-s)} + \bigl\vert f''(b) \bigr\vert ^{q} \bigl(2\beta (2,1-s)-\beta (3,1-s) \bigr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
III. If \(h(t)=t(1-t)\) in Theorem 2.1, then we have the following results.
Corollary 2.5
In Theorem 2.1, if \(\vert f'' \vert ^{q}\) for \(q\geq 1\) is generalized \((m, tgs)\)-preinvex functions, then, for \(m\in (0,1]\), we have
where we use the fact that
and \(C_{0}(k,\mu ,\lambda )\) is defined by (2.5).
Corollary 2.6
In Theorem 2.1, if the mapping \(\eta (b,a,m)=b-ma\) along with \(m=1\), taking \(x=\frac{a+b}{2}\), we get the following inequality for \(tgs\)-convex functions:
Remark 2.5
In Corollary 2.6,
-
(a)
if \(\lambda =\frac{1}{3}\), then we obtain
$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )}C_{0}^{1-\frac{1}{q}} \biggl(k,\mu , \frac{1}{3} \biggr)\theta ^{\frac{1}{q}} \biggl(k,\mu ,\frac{1}{3} \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$Specially, if we put \(k=1=\mu \), then we obtain a Simpson-type inequality:
$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{162}{ \biggl(\frac{23}{160} \biggr)}^{\frac{1}{q}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$ -
(b)
if \(\lambda =\frac{1}{2}\), then we obtain
$$ \begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )}C_{0}^{1-\frac{1}{q}} \biggl(k,\mu , \frac{1}{2} \biggr)\theta ^{\frac{1}{q}} \biggl(k,\mu ,\frac{1}{2} \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$Specially, if we put \(k=1=\mu \), then we derive an averaged midpoint-trapezoid-type inequality:
$$\begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{96}{ \biggl(\frac{1}{5} \biggr)}^{\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\quad \quad {}+ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$ -
(c)
if \(\lambda =0\), then we obtain
$$ \begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} \biggl(\frac{1}{\frac{\mu }{k}+2} \biggr)^{1-\frac{1}{q}} \biggl[\frac{1}{(\frac{\mu }{k}+3)(\frac{\mu }{k}+4)} \biggr]^{\frac{1}{q}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$Specially, if we put \(k=1=\mu \), then we derive a midpoint-type inequality:
$$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{48} \biggl(\frac{3}{20} \biggr)^{\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\quad \quad {}+ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$ -
(d)
if \(\lambda =1\), then we obtain
$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} \biggl[\frac{\frac{\mu }{k} (\frac{\mu }{k}+3 )}{2 (\frac{\mu }{k}+2 )} \biggr]^{1-\frac{1}{q}} \biggl[\frac{\frac{\mu }{k} (\frac{\mu ^{2}}{k^{2}}+\frac{8\mu }{k}+19 )}{12 (\frac{\mu }{k}+3 ) (\frac{\mu }{k}+4 )} \biggr]^{\frac{1}{q}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$Specially, if we put \(k=1=\mu \), then we obtain a trapezoid-type inequality:
$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{24} \biggl(\frac{7}{40} \biggr)^{\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\quad \quad {}+ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
IV. If \(h(t)=\frac{\sqrt{t}}{2\sqrt{1-t}}\) in Theorem 2.1, then we have the following results.
Corollary 2.7
In Theorem 2.1, if \(\vert f'' \vert ^{q}\) for \(q\geq 1\) is generalized m-MT-preinvex functions, then, for \(m\in (0,1]\), we have
where we use the fact that
and \(C_{0}(k,\mu ,\lambda )\) is defined by (2.5).
Corollary 2.8
In Theorem 2.1, if the mapping \(\eta (b,a,m)=b-ma\) together with \(m=1\), taking \(x=\frac{a+b}{2}\), we get the following inequality for MT-convex functions:
Remark 2.6
In Corollary 2.8,
-
(a)
if \(\lambda =\frac{1}{3}\), then we obtain
$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )}C_{0}^{1-\frac{1}{q}} \biggl(k,\mu ,\frac{1}{3} \biggr) \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1} \biggl(k,\mu ,\frac{1}{3} \biggr) \\ &\quad \quad {}+ \bigl\vert f''(a) \bigr\vert ^{q}\Phi _{2} \biggl(k,\mu ,\frac{1}{3} \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1} \biggl(k,\mu ,\frac{1}{3} \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}\Phi _{2} \biggl(k,\mu ,\frac{1}{3} \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$Specially, if we put \(k=1=\mu \), then we derive a Simpson-type inequality:
$$ \begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} { \biggl(\frac{8}{81} \biggr)}^{1-\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1} \biggl(1,1,\frac{1}{3} \biggr)+ \bigl\vert f''(a) \bigr\vert ^{q}\Phi _{2} \biggl(1,1,\frac{1}{3} \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1} \biggl(1,1,\frac{1}{3} \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}\Phi _{2} \biggl(1,1,\frac{1}{3} \biggr) \biggr]^{\frac{1}{q}} \biggr\} , \end{aligned} $$where
$$ \Phi _{1} \biggl(1,1,\frac{1}{3} \biggr)=\frac{71\sqrt{2}}{648}+\frac{1}{8}\arcsin {\frac{\sqrt{3}}{3}}- \frac{\pi }{32} $$and
$$ \Phi _{2} \biggl(1,1,\frac{1}{3} \biggr)=\frac{19\sqrt{2}}{648}+\frac{1}{12}\arcsin {\frac{1}{3}}+ \frac{1}{16}\arctan {2\sqrt{2}}-\frac{\pi }{32}; $$ -
(b)
if \(\lambda =\frac{1}{2}\), then we obtain
$$\begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} C_{0}^{1-\frac{1}{q}} \biggl(k,\mu , \frac{1}{2} \biggr) \biggl\{ \biggl[ \biggl\vert f''\biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1} \biggl(k,\mu ,\frac{1}{2} \biggr) \\ &\quad \quad {}+ \bigl\vert f''(a) \bigr\vert ^{q}\Phi _{2} \biggl(k,\mu ,\frac{1}{2} \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}\Phi _{1} \biggl(k,\mu ,\frac{1}{2} \biggr)+ \bigl\vert f''(b)\bigr\vert ^{q}\Phi _{2} \biggl(k,\mu ,\frac{1}{2} \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$Specially, if we put \(k=1=\mu \), then we derive an averaged midpoint-trapezoid-type inequality:
$$\begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{96} \biggl(\frac{3\pi }{16} \biggr)^{\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\quad \quad {}+ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$ -
(c)
if \(\lambda =0\), then we obtain
$$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} \biggl(\frac{1}{\frac{\mu }{k}+2} \biggr)^{1-\frac{1}{q}} \biggl(\frac{1}{2} \biggr)^{\frac{1}{q}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}\beta \biggl( \frac{\mu }{k}+\frac{5}{2},\frac{1}{2} \biggr)+ \bigl\vert f''(a) \bigr\vert ^{q}\beta \biggl( \frac{\mu }{k}+\frac{3}{2},\frac{3}{2} \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}\beta \biggl( \frac{\mu }{k}+\frac{5}{2},\frac{1}{2} \biggr)+ \bigl\vert f''(b) \bigr\vert ^{q}\beta \biggl( \frac{\mu }{k}+\frac{3}{2},\frac{3}{2} \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$Specially, if we put \(k=1=\mu \), then we obtain a midpoint-type inequality:
$$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{48} \biggl(\frac{3}{2} \biggr)^{\frac{1}{q}} \\ &\quad\quad {} \times \biggl\{ \biggl[\frac{5\pi }{16} \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+\frac{\pi }{16} \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\quad \quad {}+ \biggl[\frac{5\pi }{16} \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+\frac{\pi }{16} \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$ -
(d)
if \(\lambda =1\), then we obtain
$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )} \biggl(\frac{\frac{\mu }{k}(\frac{\mu }{k}+3)}{2(\frac{\mu }{k}+2)} \biggr)^{1-\frac{1}{q}} \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q} \biggl(\frac{3(\frac{\mu }{k}+1)\pi }{16}-\frac{1}{2} \beta \biggl(\frac{\mu }{k}+\frac{5}{2},\frac{1}{2} \biggr) \biggr) \\ &\quad \quad {} + \bigl\vert f''(a) \bigr\vert ^{q} \biggl(\frac{(\frac{\mu }{k}+1)\pi }{16}-\frac{1}{2}\beta \biggl( \frac{\mu }{k}+\frac{3}{2},\frac{3}{2} \biggr) \biggr) \biggr]^{\frac{1}{q}} \\ &\quad \quad {} + \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q} \biggl( \frac{3(\frac{\mu }{k}+1)\pi }{16}-\frac{1}{2}\beta \biggl(\frac{\mu }{k}+ \frac{5}{2},\frac{1}{2} \biggr) \biggr) \\ &\quad \quad {} + \bigl\vert f''(b) \bigr\vert ^{q} \biggl(\frac{(\frac{\mu }{k}+1)\pi }{16}-\frac{1}{2}\beta \biggl( \frac{\mu }{k}+\frac{3}{2},\frac{3}{2} \biggr) \biggr) \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$Specially, if we put \(k=1=\mu \), then we obtain a trapezoid-type inequality:
$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{2}{3} \biggr)^{1-\frac{1}{q}} \\ &\quad \quad {} \times \biggl\{ \biggl[\frac{7\pi }{32} \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+\frac{3\pi }{32} \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\qquad {} + \biggl[\frac{7\pi }{32} \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+\frac{3\pi }{32} \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
Now, we get ready to state the second theorem as follows.
Theorem 2.2
Let \(A\subseteq \mathbb{R}\) be an open m-invex subset with respect to \(\eta : A\times A\times (0,1] \rightarrow \mathbb{R}\setminus \{0\}\) for some fixed \(m \in (0,1]\), and let \(a, b \in A\), \(a< b\) with \(\eta (b,a,m)>0\). Assume that \(f:A\rightarrow \mathbb{R}\) is a twice differentiable function on A such that \(f''\) is integrable on \([ma,ma+\eta (b,a,m) ]\). If \(\vert f'' \vert ^{q}\) for \(q>1\) is a generalized \((m,h)\)-preinvex function with respect to η and \(h:[0,1]\rightarrow \mathbb{R}_{0}\), then the following inequality for k-fractional integrals with \(x \in [a,b]\), \(\lambda \in [0, 1]\), \(\mu > 0\), \(k>0\) exists:
where \(p=\frac{q}{q-1}\) and
Proof
Using Lemma 2.1 and Hölder’s inequality, we have
Since \(\vert f'' \vert ^{q}\) is generalized \((m,h)\)-preinvex on \([ma,ma+\eta (b,a,m)]\), we get
and
Hence, if we use (2.14)–(2.16) in (2.13), we can get the desired result. This completes the proof. □
Let us point out some special cases of Theorem 2.2.
I. If \(h(t)=t^{s}\) in Theorem 2.2, then we have the following results.
Corollary 2.9
In Theorem 2.2, if we use the generalized \((m,s)\)-Breckner-preinvexity of \(\vert f'' \vert ^{q}\) along with \(q>1\) and \(p=\frac{q}{q-1}\), then, for \(s\in (0,1]\) and \(m\in (0,1]\), we have the following inequality:
Corollary 2.10
In Theorem 2.2, if the mapping \(\eta (b,a,m)=b-ma\) together with \(m=1\), choosing \(x=\frac{a+b}{2}\), for \(s\in (0,1]\), we have the following inequality for s-convex functions:
Remark 2.7
In Corollary 2.10,
-
(i)
if \(\lambda =\frac{1}{2}\), then we obtain
$$ \begin{aligned} & \biggl\vert \frac{2^{\frac{\mu }{k}-1}}{(b-a)^{\frac{\mu }{k}-1}}I_{f} \biggl(\mu ,k;\frac{a+b}{2},\frac{1}{2},1,a,b \biggr) \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8 (\frac{\mu }{k}+1 )}C_{3}^{\frac{1}{p}} \biggl(k,\mu , \frac{1}{2},p \biggr) \biggl\{ \biggl[\frac{\vert f''(x) \vert ^{q}+\vert f''(a) \vert ^{q}}{s+1} \biggr]^{\frac{1}{q}} + \biggl[\frac{\vert f''(x) \vert ^{q}+\vert f''(b) \vert ^{q}}{s+1} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned} $$ -
(ii)
if \(k=1\) and \(\lambda =\frac{1}{3},1\), then we have the results (c) and (f) of Corollary 2.3 in [34], respectively. Further, if we choose \(\mu =1\), then we have the results (d) and (g) of Corollary 2.3 in [34], respectively.
II. If \(h(t)=t^{-s}\) in Theorem 2.2, then we have the following results.
Corollary 2.11
In Theorem 2.2, if we use the generalized \((m,s)\)-Godunova–Levin–Dragomir-preinvexity of \(\vert f'' \vert ^{q}\) along with \(q>1\) and \(p=\frac{q}{q-1}\), then, for \(s\in (0,1)\) and \(m\in (0,1]\), we have the following inequality:
Corollary 2.12
In Theorem 2.2, if the mapping \(\eta (b,a,m)=b-ma\) along with \(m=1\), choosing \(x=\frac{a+b}{2}\), for \(s\in (0,1)\), we have the following inequality for s-Godunova–Levin functions:
Remark 2.8
In Corollary 2.12,
-
(a)
if \(\lambda =\frac{1}{3}\), then we obtain
$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)(1-s)^{\frac{1}{q}}}C_{3}^{\frac{1}{p}} \biggl(k,\mu , \frac{1}{3},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$Specially, if we put \(k=1=\mu \), then we derive a Simpson-type inequality:
$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16(1-s)^{\frac{1}{q}}}C_{3}^{\frac{1}{p}} \biggl(1,1, \frac{1}{3},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$ -
(b)
if \(\lambda =\frac{1}{2}\), then we obtain
$$\begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)(1-s)^{\frac{1}{q}}}C_{3}^{\frac{1}{p}} \biggl(k,\mu , \frac{1}{2},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$Specially, if we put \(k=1=\mu \), then we derive an averaged midpoint-trapezoid-type inequality:
$$\begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16(1-s)^{\frac{1}{q}}}{\beta ^{\frac{1}{p}}(1+p,1+p)} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$ -
(c)
if \(\lambda =0\), then we obtain
$$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)(1-s)^{\frac{1}{q}}}{ \biggl[\frac{1}{p (\frac{\mu }{k}+1 )+1} \biggr]}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$Specially, if we put \(k=1=\mu \), then we derive a midpoint-type inequality:
$$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16(1-s)^{\frac{1}{q}}}{ \biggl(\frac{1}{2p+1} \biggr)}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$ -
(d)
if \(\lambda =1\), then we obtain
$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)(1-s)^{\frac{1}{q}}}C_{3}^{\frac{1}{p}}(k,\mu ,1,p) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$Specially, if we put \(k=1=\mu \), then we derive a trapezoid-type inequality:
$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16(1-s)^{\frac{1}{q}}}{ \biggl[2^{1+2p}\beta \biggl(\frac{1}{2};1+p,1+p \biggr) \biggr]}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
III. If \(h(t)=t(1-t)\) in Theorem 2.2, then we have the following results.
Corollary 2.13
In Theorem 2.2, if we use the generalized \((m,tgs)\)-preinvexity of \(\vert f'' \vert ^{q}\) along with \(q>1\) and \(p=\frac{q}{q-1}\), then, for \(m\in (0,1]\), we have the following inequality:
where \(C_{3}(k,\mu ,\lambda ,p)\) is defined by (2.16).
Corollary 2.14
In Theorem 2.2, if the mapping \(\eta (b,a,m)=b-ma\) together with \(m=1\), choosing \(x=\frac{a+b}{2}\), we get the following inequality for \(tgs\)-convex functions:
Remark 2.9
In Corollary 2.14,
-
(a)
if \(\lambda =\frac{1}{3}\), then we obtain
$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}} f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}} \biggl(k,\mu ,\frac{1}{3},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$Specially, if we put \(k=1=\mu \), then we derive a Simpson-type inequality:
$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}} \biggl(1,1, \frac{1}{3},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$ -
(b)
if \(\lambda =\frac{1}{2}\), then we obtain
$$ \begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}} \biggl(k,\mu ,\frac{1}{2},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$Specially, if we put \(k=1=\mu \), then we derive an averaged midpoint-trapezoid-type inequality:
$$ \begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}\beta ^{\frac{1}{p}}(1+p,1+p) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned} $$ -
(c)
if \(\lambda =0\), then we obtain
$$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}{ \biggl[\frac{1}{p (\frac{\mu }{k}+1 )+1} \biggr]}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$Specially, if we put \(k=1=\mu \), then we derive a midpoint-type inequality:
$$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}{ \biggl(\frac{1}{2p+1} \biggr)}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$ -
(d)
if \(\lambda =1\), then we obtain
$$ \begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}}(k, \mu ,1,p) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$Specially, if we put \(k=1=\mu \), then we derive a trapezoid-type inequality:
$$ \begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{1}{6} \biggr)^{\frac{1}{q}}{ \biggl[2^{1+2p}\beta \biggl(\frac{1}{2};1+p,1+p \biggr) \biggr]}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$
IV. If \(h(t)=\frac{\sqrt{t}}{2\sqrt{1-t}}\) in Theorem 2.2, then we have the following results.
Corollary 2.15
In Theorem 2.2, if we use the generalized m-MT-preinvexity of \(\vert f'' \vert ^{q}\) along with \(q>1\) and \(p=\frac{q}{q-1}\), then, for \(m\in (0,1]\), we have the following inequality:
where we use the fact that
Corollary 2.16
In Theorem 2.2, if the mapping \(\eta (b,a,m)=b-ma\) together with \(m=1\), choosing \(x=\frac{a+b}{2}\), we get the following inequality for MT-convex functions:
Remark 2.10
In Corollary 2.16,
-
(a)
if \(\lambda =\frac{1}{3}\), then we obtain
$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr] \\ &\quad \quad {}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}} \biggl(k,\mu ,\frac{1}{3},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$Specially, if we put \(k=1=\mu \), then we derive a Simpson-type inequality:
$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[f(a)+4f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}} \biggl(1,1,\frac{1}{3},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl( \frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}}\biggr\} ; \end{aligned}$$ -
(b)
if \(\lambda =\frac{1}{2}\), then we obtain
$$\begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr] \\ &\quad \quad {}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}} \biggl(k,\mu ,\frac{1}{2},p \biggr) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$Specially, if we put \(k=1=\mu \), then we derive an averaged midpoint-trapezoid-type inequality:
$$\begin{aligned} & \biggl\vert \frac{1}{4} \biggl[f(a)+2f \biggl(\frac{a+b}{2} \biggr)+f(b) \biggr]-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}\beta ^{\frac{1}{p}}(1+p,1+p) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned}$$ -
(c)
if \(\lambda =0\), then we obtain
$$\begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}{ \biggl[\frac{1}{p (\frac{\mu }{k}+1 )+1} \biggr]}^{\frac{1}{p}} \\ &\quad\quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$Specially, if we put \(k=1=\mu \), then we derive a midpoint-type inequality:
$$ \begin{aligned} & \biggl\vert f \biggl(\frac{a+b}{2} \biggr)- \frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}{ \biggl(\frac{1}{2p+1} \biggr)}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} ; \end{aligned} $$ -
(d)
if \(\lambda =1\), then we obtain
$$ \begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{2^{\frac{\mu }{k}-1}\Gamma _{k}(\mu +k)}{(b-a)^{\frac{\mu }{k}}} \bigl[{}_{k}J^{\mu }_{(\frac{a+b}{2})^{-}}f(a)+{}_{k}J^{\mu }_{(\frac{a+b}{2})^{+}}f(b) \bigr] \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{8(\frac{\mu }{k}+1)} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}C_{3}^{\frac{1}{p}}(k, \mu ,1,p) \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned} $$Specially, if we put \(k=1=\mu \), then we derive a trapezoid-type inequality:
$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a} \int ^{b}_{a} f(x)\,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{(b-a)^{2}}{16} \biggl(\frac{\pi }{4} \biggr)^{\frac{1}{q}}{ \biggl[2^{1+2p}\beta \biggl(\frac{1}{2};1+p,1+p \biggr) \biggr]}^{\frac{1}{p}} \\ &\quad \quad {} \times \biggl\{ \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(a) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} + \biggl[ \biggl\vert f'' \biggl(\frac{a+b}{2} \biggr) \biggr\vert ^{q}+ \bigl\vert f''(b) \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \biggr\} . \end{aligned}$$
References
Du, T.S., Li, Y.J., Yang, Z.Q.: A generalization of Simpson’s inequality via differentiable mapping using extended \((s, m)\)-convex functions. Appl. Math. Comput. 293, 358–369 (2017)
Hussain, S., Qaisar, S.: More results on Simpson’s type inequality through convexity for twice differentiable continuous mappings. SpringerPlus 5, Article ID 77 (2016)
Hwang, D.Y., Dragomir, S.S.: Extensions of the Hermite–Hadamard inequality for r-preinvex functions on an invex set. Bull. Aust. Math. Soc. 95(3), 412–423 (2017)
Khan, M.A., Ali, T., Dragomir, S.S., Sarikaya, M.Z.: Hermite–Hadamard type inequalities for conformable fractional integrals. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. (2017). https://doi.org/10.1007/s13398-017-0408-5
Latif, M.A.: On some new inequalities of Hermite–Hadamard type for functions whose derivatives are s-convex in the second sense in the absolute value. Ukr. Math. J. 67(10), 1552–1571 (2016)
Latif, M.A., Dragomir, S.S.: Generalization of Hermite–Hadamard type inequalities for n-times differentiable functions through preinvexity. Georgian Math. J. 23(1), 97–104 (2016)
Qi, F., Xi, B.Y.: Some integral inequalities of Simpson type for GA-ε-convex functions. Georgian Math. J. 20, 775–788 (2013)
Wu, S.H., Baloch, I.A., İşcan, İ.: On harmonically \((p,h,m)\)-preinvex functions. J. Funct. Spaces 2017, Article ID 2148529 (2017)
Sarikaya, M.Z., Set, E., Yaldiz, H., Başak, N.: Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities. Math. Comput. Model. 57, 2403–2407 (2013)
Dragomir, S.S., Bhatti, M.I., Iqbal, M., Muddassar, M.: Some new Hermite–Hadamard’s type fractional integral inequalities. J. Comput. Anal. Appl. 18(4), 655–661 (2015)
Hwang, S.R., Tseng, K.L., Hsu, K.C.: New inequalities for fractional integrals and their applications. Turk. J. Math. 40, 471–486 (2016)
Iqbal, M., Bhatti, M.I., Nazeer, K.: Generalization of inequalities analogous to Hermite–Hadamard inequality via fractional integrals. Bull. Korean Math. Soc. 52(3), 707–716 (2015)
Set, E., İşcan, İ., Kara, H.H.: Hermite–Hadamard–Fejer type inequalities for s-convex function in the second sense via fractional integrals. Filomat 30(12), 3131–3138 (2016)
Anastassiou, G.A.: Generalised fractional Hermite–Hadamard inequalities involving m-convexity and \((s,m)\)-convexity. Facta Univ., Ser. Math. Inform. 28(2), 107–126 (2013)
Awan, M.U., Noor, M.A., Mihai, M.V., Noor, K.I.: Fractional Hermite–Hadamard inequalities for differentiable s-Godunova–Levin functions. Filomat 30(12), 3235–3241 (2016)
İşcan, İ., Wu, S.H.: Hermite–Hadamard type inequalities for harmonically convex functions via fractional integrals. Appl. Math. Comput. 238, 237–244 (2014)
Qaisar, S., Iqbal, M., Muddassar, M.: New Hermite–Hadamard’s inequalities for preinvex functions via fractional integrals. J. Comput. Anal. Appl. 20(7), 1318–1328 (2016)
Kashuri, A., Liko, R.: Generalizations of Hermite–Hadamard and Ostrowski type inequalities for MT m -preinvex functions. Proyecciones 36(1), 45–80 (2017)
Matłoka, M.: Some inequalities of Hadamard type for mappings whose second derivatives are h-convex via fractional. J. Fract. Calc. Appl. 6(1), 110–119 (2015)
Wang, J., Deng, J., Fečkan, M.: Hermite–Hadamard-type inequalities for r-convex functions based on the use of Riemann–Liouville fractional integrals. Ukr. Math. J. 65(2), 193–211 (2013)
Mubeen, S., Habibullah, G.M.: k-fractional integrals and application. Int. J. Contemp. Math. Sci. 7(2), 89–94 (2012)
Agarwal, P.: Some inequalities involving Hadamard-type k-fractional integral operators. Math. Methods Appl. Sci. 40(11), 3882–3891 (2017)
Ali, A., Gulshan, G., Hussain, R., Latif, A., Muddassar, M.: Generalized inequalities of the type of Hermite–Hadamard–Fejer with quasi-convex functions by way of k-fractional derivatives. J. Comput. Anal. Appl. 22(7), 1208–1219 (2017)
Tomar, M., Mubeen, S., Choi, J.: Certain inequalities associated with Hadamard k-fractional integral operators. J. Inequal. Appl. 2016, Article ID 234 (2016)
Romero, L.G., Luque, L.L., Dorrego, G.A., Cerutti, R.A.: On the k-Riemann–Liouville fractional derivative. Int. J. Contemp. Math. Sci. 8(1), 41–51 (2013)
Set, E., Tomar, M., Sarikaya, M.Z.: On generalized Grüss type inequalities for k-fractional integrals. Appl. Math. Comput. 269, 29–34 (2015)
Tariboon, J., Ntouyas, S.K., Tomar, M.: Some new integral inequalities for k-fractional integrals. Malaya J. Mat. 4(1), 100–110 (2016)
Hudzik, H., Maligranda, L.: Some remarks on s-convex functions. Aequ. Math. 48, 100–111 (1994)
Noor, M.A., Noor, K.I., Awan, M.U., Khan, S.: Fractional Hermite–Hadamard inequalities for some new classes of Godunova–Levin functions. Appl. Math. Inf. Sci. 8(6), 2865–2872 (2014)
Tunç, M., Göv, E., Şanal, Ü: On \(tgs\)-convex function and their inequalities. Facta Univ., Ser. Math. Inform. 30(5), 679–691 (2015)
Tunç, M., Şubaş, Y., Karabayir, I.: On some Hadamard type inequalities for MT-convex functions. Int. J. Open Probl. Comput. Sci. Math. 6(2), 102–113 (2013)
Du, T.S., Liao, J.G., Li, Y.J.: Properties and integral inequalities of Hadamard–Simpson type for the generalized \((s,m)\)-preinvex functions. J. Nonlinear Sci. Appl. 9, 3112–3126 (2016)
Peng, C., Zhou, C., Du, T.S.: Riemann–Liouville fractional Simpson’s inequalities through generalized \((m,h_{1},h_{2})\)-preinvexity. Ital. J. Pure Appl. Math. 38, 345–367 (2017)
İşcan, İ.: Generalization of different type integral inequalities for s-convex functions via fractional integrals. Appl. Anal. 93(9), 1846–1862 (2014)
Özdemir, M.E., Avci, M., Kavurmaci, H.: Hermite–Hadamard-type inequalities via \((\alpha ,m)\)-convexity. Comput. Math. Appl. 61, 2614–2620 (2011)
Sarikaya, M.Z., Set, E., Özdemir, M.E.: On new inequalities of Simpson’s type for functions whose second derivatives absolute values are convex. J. Appl. Math. Stat. Inform. 9(1), 37–45 (2013)
Acknowledgements
This work was partially supported by the National Natural Science Foundation of China under Grant No. 61374028.
Author information
Authors and Affiliations
Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Zhang, Y., Du, TS., Wang, H. et al. Extensions of different type parameterized inequalities for generalized \((m,h)\)-preinvex mappings via k-fractional integrals. J Inequal Appl 2018, 49 (2018). https://doi.org/10.1186/s13660-018-1639-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-018-1639-5