Abstract
In this paper some new general fractional integral inequalities for convex and m-convex functions by involving an extended Mittag-Leffler function are presented. These results produce inequalities for several kinds of fractional integral operators. Some interesting special cases of our main results are also pointed out.
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1 Introduction, definitions, and preliminaries
Convex functions are very important in the field of integral inequalities. A lot of fractional integral inequalities and novel results have been established due to convex functions (for more details, see [1, 8, 13, 14]).
Definition 1
A function \(f: I\rightarrow\mathbb{R}\), where I is an interval in \(\mathbb{R}\), is said to be a convex function if
holds for \(t\in[0,1]\) and \(x,y\in I\).
A convex function \(f: I\rightarrow\mathbb{R}\) is also equivalently defined by the Hadamard inequality
where \(a,b\in I\), \(a< b\).
The concept of m-convexity was introduced in [17] and since then many properties, especially inequalities, have been obtained for this class of functions (see [3, 6, 7, 18]).
Definition 2
A function \(f:[0,b]\rightarrow\mathbb{R}\), \(b>0\) is called m-convex, where \(m\in[0,1]\), if for every \(x,y\in[0,b]\) and \(t\in[0,1]\), we have
For \(m=1\), we recapture the definition of convex functions, and for \(m=0\), the definition of star-shaped functions defined on \([0,b]\). We recall that a function \(f:[0,b]\to\mathbb{R}\) is called star-shaped if
If we denote by \(K_{m}(b)\) the set of m-convex functions defined on \([0,b]\) for which \(f(0)<0\), then
whenever \(m\in(0,1)\). Note that in the class \(K_{1}(b)\) there are only convex functions \(f:[0,b]\to\mathbb{R}\) for which \(f(0)\leq0\) (see [4]), while \(k_{0}(b)\) contains star-shaped functions.
Example 1.1
([6])
The function \(f:[0,\infty)\to\mathbb{R}\), given by
is a \(\frac{16}{17}\)-convex function but it is not m-convex for any \(m\in(\frac{16}{17},1]\).
For more results and inequalities related to m-convex functions, one can consult, for example, [3, 6, 7] along with the references therein.
Recently in [2] Andrić et al. defined an extended generalized Mittag-Leffler function \(E_{\mu,\alpha,l}^{\gamma,\delta,k,c}(\cdot;p)\) as follows.
Definition 3
([2])
Let \(\mu,\alpha,l,\gamma,c\in\mathbb{C}\), \(\Re(\mu),\Re(\alpha ),\Re(l)>0\), \(\Re(c)>\Re(\gamma)>0\) with \(p\geq0\), \(\delta>0\), and \(0< k\leq\delta+\Re(\mu)\). Then the extended generalized Mittag-Leffler function \(E_{\mu,\alpha,l}^{\gamma,\delta,k,c}(t;p)\) is defined by
where \(\beta_{p}\) is the generalized beta function defined by
and \((c)_{nk}\) is the Pochhammer symbol defined as \((c)_{nk}=\frac {\Gamma(c+nk)}{\Gamma(c)}\).
In [2] properties of the generalized Mittag-Leffler function are discussed, and it is given that \(E_{\mu,\alpha,l}^{\gamma,\delta,k,c}(t;p)\) is absolutely convergent for \(k<\delta+\Re(\mu)\). Let S be the sum of series of absolute terms of the Mittag-Leffler function \(E_{\mu,\alpha,l}^{\gamma,\delta,k,c}(t;p)\), then we have \(\vert E_{\mu,\alpha,l}^{\gamma,\delta,k,c}(t;p) \vert \leq S\). We use this property of Mittag-Leffler function in our results where we need.
The corresponding left and right sided extended generalized fractional integral operators are defined as follows.
Definition 4
([2])
Let \(\omega,\mu,\alpha,l,\gamma,c\in\mathbb{C}\), \(\Re(\mu),\Re(\alpha),\Re(l)>0\), \(\Re(c)>\Re(\gamma)>0\) with \(p\geq0\), \(\delta>0\) and \(0< k\leq \delta+\Re(\mu)\). Let \(f\in L_{1}[a,b]\) and \(x\in[a,b]\). Then the extended generalized fractional integral operators \(\epsilon_{\mu,\alpha,l,\omega,a^{+}}^{\gamma,\delta,k,c}f \) and \(\epsilon_{\mu,\alpha,l,\omega,b^{-}}^{\gamma,\delta,k,c}f\) are defined by
and
From extended generalized fractional integral operators, we have
Hence
and similarly
We use the following notations in our results:
and
For more information related to Mittag-Leffler functions and corresponding fractional integral operators, the readers are referred to [9–12, 15, 16, 19].
In this paper we give general fractional integral inequalities for convex and m-convex functions by involving an extended Mittag-Leffler function and deduce some results already published in [1, 5, 6, 8, 13]. Also we give a Hadamard type inequality for convex and m-convex functions by involving an extended Mittag-Leffler function.
2 Main results
Here we give some fractional integral inequalities for convex and m-convex functions via an extended generalized Mittag-Leffler function and corresponding fractional integral operators given in (3) and (4). The following lemma is useful to establish the results.
Lemma 2.1
Let \(f:[a,mb]\rightarrow\mathbb{R}\) be a differentiable function such that \(f'\in L_{1}[a,mb]\) with \(0\leq a< mb\). Also let \(g:[a,mb]\rightarrow\mathbb{R}\) be a continuous function on \([a,mb]\), then the following identity for extended generalized fractional integral operators holds:
Proof
On integrating by parts one can have
and
Subtracting (9) from (8), we get (7) which is the required identity. □
If we take \(m=1\) in (7), then we get the following identity for a convex function.
Corollary 2.2
Let \(f:[a,b]\subseteq[0,\infty)\rightarrow\mathbb{R}\) be a differentiable function such that \(f'\in L_{1}[a,b]\) with \(a< b\). Also let \(g:[a,b]\rightarrow\mathbb{R}\) be continuous on \([a,b]\), then the following identity for extended generalized fractional integral operators holds:
We use identity (7) to establish the following fractional integral inequality.
Theorem 2.3
Let \(f:[a,mb]\rightarrow\mathbb{R}\) be a differentiable function such that \(f'\in L_{1}[a,mb]\) with \(0\leq a< mb\). Also let \(g:[a,mb]\rightarrow\mathbb{R}\) be a continuous function on \([a,mb]\). If \(|f'|\) is an m-convex function on \([a,mb]\), then the following inequality for extended generalized fractional integral operators holds:
for \(k<\delta+\Re(\mu)\) and \(\Vert g \Vert_{\infty}= \sup_{t\in[a,mb]}|g(t)|\).
Proof
From Lemma 2.1, we have
Using absolute convergence of the Mittag-Leffler function and \(\Vert g \Vert_{\infty}= \sup_{t\in[a,b]}|g(t)|\), we have
Since \(|f'|\) is an m-convex function, we have
for \(t\in[a,mb]\).
After simple calculation of the above inequality, we get (11) which is required. □
If we take \(m=1\) in (11), then we get the following result for a convex function.
Corollary 2.4
Let \(f:[a,b]\subseteq[0,\infty)\rightarrow\mathbb{R}\) be a differentiable function such that \(f'\in L_{1}[a,b]\) with \(a< b\). Also let \(g:[a,b]\rightarrow\mathbb{R}\) be a continuous function on \([a,b]\). If \(|f'|\) is a convex function on \([a,b]\), then the following inequality for extended generalized fractional integral operators holds:
for \(k<\delta+\Re(\mu)\) and \(\Vert g \Vert_{\infty}= \sup_{t\in[a,b]}|g(t)|\).
Remark 2.5
In Theorem 2.3.
-
(i)
If we put \(p=0\), then we get [6, Theorem 3.2].
-
(ii)
If we put \(\omega=p=0\) and \(m=1\), then we get [13, Theorem 6].
-
(iii)
If we take \(\omega=p=0\), \(m=1\), \(\alpha=\frac{\mu }{k}\), and \(g(s)=1\), then we get [8, Corollary 2.3].
-
(iv)
For \(g(s)=1\) along with \(\omega=p=0\), \(m=1\), and \(\alpha =\mu\), we get [13, Corollary 2].
Remark 2.6
In Corollary 2.4.
-
(i)
If we put \(p=0\), then we get [1, Theorem 3.2].
-
(ii)
If we put \(\omega=p=0\), then we get [13, Theorem 6].
-
(iii)
For \(\omega=p=0\), \(\alpha=\frac{\mu}{k}\), and \(g(s)=1\), we get [8, Corollary 2.3].
-
(iv)
For \(g(s)=1\) along with \(\omega=p=0\), we get [13, Corollary 2].
Next we give the following fractional integral inequality.
Theorem 2.7
Let \(f:[a,mb]\rightarrow\mathbb{R}\) be a differentiable function such that \(f\in L_{1}[a,mb]\) with \(0\leq a< mb\). Also let \(g:[a,mb]\rightarrow\mathbb{R}\) be a continuous function on \([a,mb]\). If \(|f'|^{q}\) is a convex function on \([a,mb]\), then for \(q>0\) the following inequality for extended generalized fractional integral operators holds:
for \(k<\delta+\Re(\mu)\) and \(\Vert g \Vert_{\infty}= \sup_{t\in[a,mb]}|g(t)|\) and \(\frac{1}{p}+\frac{1}{q}=1\).
Proof
From Lemma 2.1 and by using Hölder’s inequality, we have
Using absolute convergence of the Mittag-Leffler function and \(\Vert g \Vert_{\infty}= \sup_{t\in[a,b]}|g(t)|\), we have
Since \(|f'(t)|^{q}\) is an m-convex function, we have
After simple calculation of the above inequality, we get (17) which is required. □
If we take \(m=1\) in (17), then we get the following result for a convex function.
Corollary 2.8
Let \(f:[a,b]\subseteq[0,\infty)\rightarrow\mathbb{R}\) be a differentiable function such that \(f'\in L_{1}[a,b]\) with \(a< b\). Also let \(g:[a,b]\rightarrow\mathbb{R}\) be a continuous function on \([a,b]\). If \(|f'|^{q}\) is a convex function on \([a,b]\), then for \(q>0\) the following inequality for extended generalized fractional integral operators holds:
for \(k<\delta+\Re(\mu)\) and \(\Vert g \Vert_{\infty}= \sup_{t\in[a,b]}|g(t)|\) and \(\frac{1}{p}+\frac{1}{q}=1\).
Remark 2.9
In Theorem 2.7.
-
(i)
If we put \(p=0\), then we get [6, Theorem 3.6].
-
(ii)
If we put \(\omega=p=0\) and \(m=1\), then we get [13, Theorem 7].
-
(iii)
If we take \(\omega=p=0\), \(m=1\) along with \(\alpha=\frac {\mu}{k}\), then we get [8, Theorem 2.5].
-
(iv)
If we take \(g(s)=1\), \(m=1\), and \(\omega=p=0\), then we get [5, Theorem 2.3].
-
(v)
If we put \(\omega=p=0\), \(m=1\), and \(\alpha=1\), then we get [5, Corollary 3].
Remark 2.10
In Corollary 2.8.
-
(i)
If we put \(p=0\), then we get [1, Theorem 3.5].
-
(ii)
If we put \(\omega=p=0\), then we get [13, Theorem 7].
-
(iii)
If we put \(\omega=p=0\), \(\alpha=1\), then we get [13, Corollary 3].
-
(iv)
If we take \(\omega=p=0\) along with \(\alpha=\frac{\mu }{k}\), then we get [8, Theorem 2.5].
-
(v)
If we take \(g(s)=1\) and \(\omega=p=0\), then we get [5, Theorem 2.3].
In the next result we give Hadamard type inequalities for m-convex functions via an extended Mittag-Leffler function.
Theorem 2.11
Let \(f:[a,mb]\rightarrow\mathbb{R}\) be a function such that \(f\in L_{1}[a,mb]\) with \(0\leq a< mb\). If f is m-convex on \([a,mb]\), then the following inequalities for extended generalized fractional integral operators hold:
where \(\omega'=\frac{2^{\mu}\omega}{(mb-a)^{\mu}}\).
Proof
Since f is an m-convex function, we have
Also from m-convexity of f, we have
Multiplying (24) by \(t^{\alpha-1}E_{\mu,\alpha,l}^{\gamma,\delta,k, c}(\omega t^{\mu}; p)\) on both sides and then integrating over \([0,1]\), we have
Putting \(u=\frac{t}{2}a+\frac{2-t}{2}mb\) and \(v=\frac {2-t}{2m}a+\frac{t}{2}b\) in (26), we have
By using (3), (4), and (5) we get the first inequality of (23).
Now multiplying (25) by \(t^{\alpha-1}E_{\mu,\alpha,l}^{\gamma,\delta,k, c}(\omega t^{\mu}; p)\) on both sides and then integrating over \([0,1]\), we have
Putting \(u=\frac{t}{2}a+m\frac{2-t}{2}b\) and \(v=\frac {2-t}{2m}a+\frac{t}{2}b\) in (27), we have
By using (3), (4), and (6), we get the second inequality of (23). □
If we take \(m=1\) in (23), then we get the following Hadamard type inequality for a convex function.
Corollary 2.12
Let \(f:[a,b]\subseteq[0,\infty)\rightarrow\mathbb{R}\) be a function such that \(f\in L_{1}[a,b]\) with \(a< b\). If f is convex on \([a,b]\), then the following inequalities for extended generalized fractional integral operators hold:
where \(\omega'=\frac{2^{\mu}\omega}{(b-a)^{\mu}}\).
Remark 2.13
In Theorem 2.11.
-
(i)
If we put \(p=0\), then we get [6, Theorem 3.10].
-
(ii)
If we put \(\omega=p=0\), \(m=1\), and \(\alpha=1\), then we get the classical Hadamard inequality.
Remark 2.14
In Corollary 2.12.
3 Concluding remarks
We have investigated more general fractional integral inequalities. By selecting specific values of parameters quite interesting results can be obtained. For example selecting \(p=0\), fractional integral inequalities for fractional integral operators defined by Salim and Faraj in [12], selecting \(l=\delta=1\), fractional integral inequalities for fractional integral operators defined by Rahman et al. in [11], selecting \(p=0\) and \(l=\delta=1\), fractional integral inequalities for fractional integral operators defined by Shukla and Prajapati in [15] (see also [16]), selecting \(p=0\) and \(l=\delta=k=1\), fractional integral inequalities for fractional integral operators defined by Prabhakar in [10], selecting \(p=\omega=0\), fractional integral inequalities for Riemann–Liouville fractional integral operators.
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We thank the editor and referees for their careful reading and valuable suggestions to make the article reader friendly. The research work of Ghulam Farid is supported by COMSATS University Islamabad.
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Farid, G., Khan, K.A., Latif, N. et al. General fractional integral inequalities for convex and m-convex functions via an extended generalized Mittag-Leffler function. J Inequal Appl 2018, 243 (2018). https://doi.org/10.1186/s13660-018-1830-8
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DOI: https://doi.org/10.1186/s13660-018-1830-8