Abstract
The focus of the present study is to prove some new Pólya-Szegö type integral inequalities involving the generalized Riemann-Liouville k-fractional integral operator. These inequalities are used then to establish some fractional integral inequalities of Chebyshev type.
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1 Introduction and motivation
The celebrated functionals were introduced by the Chebyshev in his famous paper [1] and were subsequently rediscovered in various inequalities (for the celebrated functionals) by numerous authors, including Anastassiou [2], Belarbi and Dahmani [3], Dahmani et al. [4], Dragomir [5], Kalla and Rao [6], Lakshmikantham and Vatsala [7], Ntouyas et al. [8], Öǧünmez and Özkan [9], Sudsutad et al. [10], Sulaiman [11]; and, for very recent work, see also Wang et al. [12]. This type of functionals is usually defined as
where f and g are two integrable functions which are synchronous on \([a, b]\), i.e.,
for any \(x, y \in[a, b]\).
The well-known Grüss inequality [13] is defined by
where f and g are two integrable functions which are synchronous on \([a, b]\) and satisfy the following inequalities:
for all \(x, y \in[a, b]\) and for some \(m, M, n, N \in\mathbb{R}\).
Pólya and Szegö [14] introduced the following inequality:
Dragomir and Diamond [15] by using the Pólya and Szegö inequality, proved that
where f and g are two positive integrable functions which are synchronous on \([a, b]\), and
for all \(x, y \in[a, b]\) and for some \(m, M, n, N \in\mathbb{R}\).
Recently, k-extensions of some familiar fractional integral operator like Riemann-Liouville have been investigated by many authors in interesting and useful manners (see [16–18], and [19]). Here, we begin with the following.
Definition 1.1
Let \(k>0\), then the generalized k-gamma and k-beta functions defined by [20]
where \((x)_{n,k}\), is the Pochhammer k-symbol defined by
Definition 1.2
The k-gamma function is defined by
It is well known that the Mellin transform of the exponential function \(e^{-\frac{t^{k}}{k}}\) is the k-gamma function. Clearly
Definition 1.3
If \(k>0\), let \(f \in L^{1}(a,b)\), \(a\geq0\), then the Riemann-Liouville k-fractional integral \(R^{\alpha}_{a,k}\) of order \(\alpha>0\) for a real-valued continuous function \(f(t)\) is defined by ([21]; see also [22])
For \(k=1\), (1.9) is reduced to the classical Riemann-Liouville fractional integral.
Definition 1.4
If \(k>0\), let \(f \in L^{1,r}[a,b]\), \(a\geq0\), \(r\in \mathbb{R}\setminus\{-1 \}\) then the generalized Riemann-Liouville k-fractional integral \(R^{\alpha,r}_{a,k}\) of order \(\alpha>0\) for a real-valued continuous function \(f(t)\) is defined by ([19])
where \(\Gamma_{k}\) is the Euler gamma k-function.
The generalized Riemann-Liouville k-fractional integral (1.10) has the properties
and
In this paper, we derive some new Pólya-Szegö type inequalities by making use of the generalized Riemann-Liouville k-fractional integral operators and then use them to establish some Chebyshev type integral inequalities.
We organize the paper as follows: in Section 2, we prove some generalized Pólya-Szegö type integral inequalities involving the generalized Riemann-Liouville k-fractional integral operators that we need to establish main theorems in the sequel and Section 3 contains some Chebyshev type integral inequalities via generalized Riemann-Liouville k-fractional integral operators.
2 Some Pólya-Szegö types inequalities
In this section, we prove some Pólya-Szegö type integral inequalities for positive integrable functions involving the generalized Riemann-Liouville k-fractional integral operator (1.10).
Lemma 2.1
Let f and g be two positive integrable functions on \([a,\infty)\). Assume that there exist four positive integrable functions \(\varphi_{1}\), \(\varphi_{2}\), \(\psi_{1}\), and \(\psi_{2}\) on \([a, \infty)\) such that:
- \((H_{1})\) :
-
\(0<\varphi_{1}(\tau) \leq f(\tau)\leq\varphi_{2}(\tau)\), \(0<\psi_{1}(\tau) \leq g(\tau)\leq\psi_{2}(\tau)\) (\(\tau\in[a,t]\), \(t>a \)).
Then, for \(t>a\), \(k>0\), \(a\geq0\), \(\alpha>0\), and \(r\in\mathbb {R}\setminus\{-1 \}\), the following inequality holds:
Proof
From \((H_{1})\), for \(\tau\in[a,t]\), \(t>a\), we have
which yields
Analogously, we have
from which one has
Multiplying (2.3) and (2.5), we obtain
or
The inequality (2.6) can be written as
Now, multiplying both sides of (2.7) by \(\frac{ (1+r )^{1-\frac{\alpha}{k}} (t^{r+1}-\tau^{r+1})^{\frac{\alpha }{k}-1}}{k\Gamma_{k}(\alpha)} \) and integrating with respect to τ from a to t, we get
Applying the AM-GM inequality, i.e., \(a+b\geq2\sqrt{ab}\), \(a,b\in\mathbb{R}^{+}\), we have
which leads to
Therefore, we obtain the inequality (2.1) as required. □
Lemma 2.2
Let f and g be two positive integrable functions on \([a,\infty)\). Assume that there exist four positive integrable functions \(\varphi_{1}\), \(\varphi_{2}\), \(\psi_{1}\), and \(\psi_{2}\) satisfying \((H_{1})\) on \([a, \infty)\). Then, for \(t>a\), \(k>0\), \(a\geq0\), \(\alpha>0\), \(\beta>0\), and \(r\in \mathbb{R}\setminus\{-1 \}\), the following inequality holds:
Proof
To prove (2.8), using the condition \((H_{1})\), we obtain
and
which imply that
Multiplying both sides of (2.11) by \(\psi_{1}(\rho)\psi_{2}(\rho )g^{2}(\rho)\), we have
Multiplying both sides of (2.12) by
and double integrating with respect to τ and ρ from a to t, we have
Applying the AM-GM inequality, we get
which leads to the desired inequality in (2.8). The proof is completed. □
Lemma 2.3
Let f and g be two positive integrable functions on \([a,\infty)\). Assume that there exist four positive integrable functions \(\varphi_{1}\), \(\varphi_{2}\), \(\psi_{1}\), and \(\psi_{2}\) satisfying \((H_{1})\) on \([a, \infty)\). Then, for \(t>a\), \(k>0\), \(a\geq0\), \(\alpha>0\), \(\beta>0\), and \(r\in \mathbb{R}\setminus\{-1 \}\), the following inequality holds:
Proof
From (2.2), we have
which implies
By (2.4), we get
from which one has
Multiplying (2.14) and (2.15), we get the desired inequality in (2.13). □
Corollary 2.1
Let f and g be two positive integrable functions on \([0,\infty)\) satisfying
- \((H_{2})\) :
-
\(0< m\leq f(\tau)\leq M<\infty\), \(0< n\leq g(\tau)\leq N<\infty\) (\(\tau\in[a,t]\), \(t>a\)).
Then, for \(t>a\), \(k>0\), \(a\geq0\), \(\alpha>0\), \(\beta>0\), and \(r\in \mathbb{R}\setminus\{-1 \}\), we have
3 Chebyshev type integral inequalities
In the sequel, we establish our main Chebyshev type integral inequalities involving the generalized Riemann-Liouville k-fractional integral operator (1.10), with the help of the Pólya-Szegö fractional integral inequality in Lemma 2.1 as follows.
Theorem 3.1
Let f and g be two positive integrable functions on \([a, \infty)\), \(a\geq0\). Assume that there exist four positive integrable functions \(\varphi_{1}\), \(\varphi_{2}\), \(\psi_{1}\), and \(\psi_{2}\) satisfying \((H_{1})\). Then, for \(t>a\), \(k>0\), \(a\geq0\), \(\alpha>0\), and \(r\in\mathbb{R}\setminus\{-1 \}\), the following inequality is fulfilled:
where
Proof
Let f and g be two positive integrable functions on \([a,\infty)\). For \(\tau, \rho\in(a,t)\) with \(t> a\), we define \(A(\tau,\rho)\) as
or, equivalently,
Multiplying both sides of (3.4) by \(\frac{ (1+r )^{2(1-\frac{\alpha}{k})} (t^{r+1}-\tau^{r+1})^{\frac{\alpha }{k}-1}(t^{r+1}-\rho^{r+1})^{\frac{\alpha}{k}-1}}{ (k\Gamma _{k}(\alpha) )^{2}} \) and double integrating with respect to τ and ρ from a to t, we get
By using the Cauchy-Schwartz inequality for double integrals, we have
Therefore, we obtain
By applying Lemma 2.1, for \(\psi_{1}(t)=\psi_{2}(t)=g(t)=1\), we get
which leads to
Similarly, we get
Finally, combining (3.5), (3.7), (3.8), and (3.9), we arrive at the desired result in (3.1). This completes the proof. □
Remark 3.2
If \(\varphi_{1}=m\), \(\varphi_{2}=M\), \(\psi_{1}=n\), and \(\psi_{2}=N\), then we have
Theorem 3.3
Let f and g be two positive integrable functions on \([a, \infty)\), \(a\geq0\). Assume that there exist four positive integrable functions \(\varphi_{1}\), \(\varphi_{2}\), \(\psi_{1}\), and \(\psi_{2}\) satisfying \((H_{1})\). Then, for \(t>a\), \(k>0\), \(a\geq0\), \(\alpha>0\), \(\beta>0\), and \(r\in\mathbb{R}\setminus\{-1 \}\), the following inequality is true:
where
Proof
Multiplying both sides of (3.4) by \(\frac { (1+r )^{2-\frac{\alpha+\beta}{k}} (t^{r+1}-\tau ^{r+1})^{\frac {\alpha}{k}-1}(t^{r+1}-\rho^{r+1})^{\frac{\beta}{k}-1}}{ k^{2}\Gamma_{k}(\alpha)\Gamma_{k}(\beta)}\) and double integrating with respect to τ and ρ from a to t, we obtain
By using the Cauchy-Schwartz inequality for double integrals, we have
Therefore, we get
Applying Lemma 2.1 with \(\psi_{1}(t)=\psi_{2}(t)=g(t)= 1\), we have
This implies that
and
Also, applying the same procedure with \(\phi_{1}(t)=\phi_{2}(t)=f(t)= 1\), we get
and
Finally, considering (3.13) to (3.18), we arrive at the desired result in (3.12). This completes the proof of Theorem 3.3. □
Remark 3.4
We conclude the present investigation by remarking that if we follow Sarikaya and Karaca [18] then our main results become the results recently given by Ntouyas et al. [8]. Similarly, after some parametric changes our results reduce to numerous well-known results presented in the literature.
4 Examples
In this section, we show some approximations of unknown functions by using four linear functions. Let us define the constants \(m_{1}, m_{2}, M_{1}, M_{2}, n_{1}, n_{2}, N_{1}, N_{2}\in\mathbb{R}\) such that
- \((H_{3})\) :
-
\(0< m_{1}\tau+m_{2} \leq f(\tau)\leq M_{1}\tau+M_{2}\), \(0< n_{1}\tau +n_{2}\leq g(\tau)\leq N_{1}\tau+N_{2}\) (\(\tau\in[a,t]\), \(t>a \)).
Proposition 4.1
Suppose that f and g are two positive integrable functions on \([a,\infty)\), \(a\geq0\) satisfying \((H_{3})\). Then, for \(t>a\), \(k>0\), \(a\geq0\), \(\alpha>0\), \(\beta>0\), and \(r\in\mathbb{R}\setminus\{-1 \}\), we have
Proof
Setting \(\varphi_{1}(\tau)=m_{1}\tau+m_{2}\), \(\varphi_{2}(\tau )=M_{1}\tau+ M_{2}\), \(\psi_{1}(\tau)=n_{1}\tau+n_{2}\), and \(\psi_{2}(\tau )=N_{1}\tau +N_{2}\), and applying Lemma 2.1, we obtain (4.1) as desired. □
Corollary 4.1
Let all assumptions of Proposition 4.1 be fulfilled with \(m_{1}=M_{1}=n_{1}={N_{1}=0}\). Then, for \(t>a\), \(k>0\), \(a\geq0\), \(\alpha>0\), \(\beta>0\), and \(r\in\mathbb{R}\setminus\{-1 \}\), the following inequality holds:
Proposition 4.2
Suppose that f and g are two positive integrable functions on \([a,\infty)\), \({a\geq0}\) satisfying \((H_{3})\). Then, for \(t>a\), \(k>0\), \(a\geq0\), \(\alpha>0\), and \(r\in\mathbb{R}\setminus\{-1 \}\), we get the following inequality:
where
Proof
By setting \(\varphi_{1}(\tau)\), \(\varphi_{2}(\tau)\), \(\psi _{1}(\tau)\), and \(\psi_{2}(\tau)\) as in Proposition 4.1 and using Theorem 3.1, we get the inequality (4.3). □
Remark 4.3
If \(m_{1}=M_{1}=n_{1}=N_{1}=0\), then we have
where \(G(f,m,M)(t)\) and \(G(g,n,N)(t)\) are defined by (3.10) and (3.11), respectively.
Proposition 4.4
Assume that f and g are two positive integrable functions on \([a,\infty)\), \(a\geq0\) satisfying \((H_{3})\). Then, for \(t>a\), \(k>0\), \(a\geq0\), \(\alpha>0\), \(\beta>0\), and \(r\in\mathbb{R}\setminus\{-1 \}\), we obtain the following estimate:
where
Proof
By setting the four linear functions as in Proposition 4.1 and using Theorem 3.3, we get the estimate (4.5). □
Corollary 4.2
If \(m_{1}=M_{1}=n_{1}=N_{1}=v=x=0\), then we obtain
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Agarwal, P., Tariboon, J. & Ntouyas, S.K. Some generalized Riemann-Liouville k-fractional integral inequalities. J Inequal Appl 2016, 122 (2016). https://doi.org/10.1186/s13660-016-1067-3
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DOI: https://doi.org/10.1186/s13660-016-1067-3