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Some generalized Riemann-Liouville k-fractional integral inequalities

Journal of Inequalities and Applications20162016:122

https://doi.org/10.1186/s13660-016-1067-3

Received: 3 February 2016

Accepted: 11 April 2016

Published: 19 April 2016

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.

Keywords

integral inequalitiesChebyshev functionalRiemann-Liouville k-fractional integral operatorPólya and Szegö type inequalities

MSC

26D1026A3326D15

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
$$ T(f,g)= \frac{1}{b-a} \int_{a}^{b} f(x) g(x)\,dx- \biggl(\frac{1}{b-a} \int_{a}^{b} f(x)\,dx \biggr) \biggl( \frac{1}{b-a} \int_{a}^{b} g(x)\,dx \biggr), $$
(1.1)
where f and g are two integrable functions which are synchronous on \([a, b]\), i.e.,
$$ \bigl(f(x)-f(y) \bigr) \bigl(g(x)-g(y) \bigr)\geq0, $$
(1.2)
for any \(x, y \in[a, b]\).
The well-known Grüss inequality [13] is defined by
$$ \bigl\vert T(f, g)\bigr\vert \leq\frac{(M-m)(N-n)}{4}, $$
(1.3)
where f and g are two integrable functions which are synchronous on \([a, b]\) and satisfy the following inequalities:
$$ m \leq f(x)\leq M\quad \text{and}\quad n \leq g(y)\leq N, $$
(1.4)
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:
$$ \frac{\int_{a}^{b}f^{2}(x)\,dx \int_{a}^{b}g^{2}(x)\,dx}{ (\int _{a}^{b}f(x)\,dx \int_{a}^{b}g(x)\,dx )^{2}}\leq \frac{1}{4} \biggl(\sqrt{ \frac{MN}{mn}}+\sqrt{\frac{mn}{MN}} \biggr)^{2}. $$
(1.5)
Dragomir and Diamond [15] by using the Pólya and Szegö inequality, proved that
$$ \bigl\vert T(f, g)\bigr\vert \leq\frac{ (M-m ) (N-n )}{4(b-a)^{2}\sqrt{mMnN}} \int_{a}^{b}f(x)\,dx \int_{a}^{b}g(x)\,dx, $$
(1.6)
where f and g are two positive integrable functions which are synchronous on \([a, b]\), and
$$ 0< m \leq f(x)\leq M< \infty,\qquad 0< n \leq g(y)\leq N< \infty, $$
(1.7)
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 [1618], 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]
$$ \Gamma_{k}(x)=\lim_{n \to\infty} \frac{n!k^{n}(nk)^{\frac {x}{k}-1}}{(x)_{n,k}}, $$
(1.8)
where \((x)_{n,k}\), is the Pochhammer k-symbol defined by
$$(x)_{n,k}=x(x+k) (x+2k)\cdots \bigl(x+(n-1)k \bigr)\quad (n\geq1). $$

Definition 1.2

The k-gamma function is defined by
$$\Gamma_{k}(x)= \int_{0}^{\infty}t^{x-1}e^{-\frac{t^{k}}{k}}\,dt,\quad \Re(x)>0. $$
It is well known that the Mellin transform of the exponential function \(e^{-\frac{t^{k}}{k}}\) is the k-gamma function. Clearly
$$ \Gamma(x)=\lim_{k\to1}\Gamma_{k}(x),\quad \Gamma_{k}(x)=k^{\frac {x}{k}-1}\Gamma \biggl(\frac{x}{k} \biggr) \quad \text{and}\quad \Gamma_{k}(x+k)=x\Gamma_{k}(x). $$

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])
$$ R^{\alpha}_{a,k} \bigl\{ f(t) \bigr\} = \frac{1}{k\Gamma_{k}(\alpha)} \int_{a}^{t}(t-\tau)^{\frac{\alpha}{k}-1}f(\tau)\,d\tau \quad \bigl(t\in [a,b] \bigr). $$
(1.9)
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])
$$ R^{\alpha,r}_{a,k} \bigl\{ f(t) \bigr\} = \frac{ (1+r )^{1-\frac{\alpha}{k}}}{k\Gamma _{k}(\alpha)} \int_{a}^{t}\bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1}f(\tau)\,d\tau \quad \bigl(t\in[a,b] \bigr), $$
(1.10)
where \(\Gamma_{k}\) is the Euler gamma k-function.
The generalized Riemann-Liouville k-fractional integral (1.10) has the properties
$$ R^{\alpha,r}_{a,k} \bigl\{ R^{\beta,r}_{a,k}f(t) \bigr\} =R^{\alpha +\beta,r}_{a,k} \bigl\{ f(t) \bigr\} =R^{\beta,r}_{a,k} \bigl\{ R^{\alpha,r}_{a,k}f(t) \bigr\} $$
(1.11)
and
$$ R^{\alpha,r}_{a,k} \{1 \}= \frac{(t^{r+1}-a^{r+1})^{\frac {\alpha}{k}}}{(r+1)^{\frac{\alpha}{k}}\Gamma_{k}(\alpha+k)},\quad \alpha>0. $$
(1.12)

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:
$$ \frac{ R^{\alpha,r}_{a,k}\{\psi_{1}\psi_{2}f^{2}\}(t) R^{\alpha,r}_{a,k}\{ \varphi_{1}\varphi_{2}g^{2}\}(t)}{ ( R^{\alpha,r}_{a,k}\{(\varphi _{1}\psi _{1}+\varphi_{2}\psi_{2})fg\}(t) )^{2}} \leq\frac{1}{4}. $$
(2.1)

Proof

From \((H_{1})\), for \(\tau\in[a,t]\), \(t>a\), we have
$$ \frac{f(\tau)}{g(\tau)}\leq\frac{\varphi_{2}(\tau)}{\psi_{1}(\tau)}, $$
(2.2)
which yields
$$ \biggl(\frac{\varphi_{2}(\tau)}{\psi_{1}(\tau)}-\frac{f(\tau )}{g(\tau)} \biggr)\geq0. $$
(2.3)
Analogously, we have
$$ \frac{\varphi_{1}(\tau)}{\psi_{2}(\tau)}\leq\frac{f(\tau)}{g(\tau)}, $$
(2.4)
from which one has
$$ \biggl(\frac{f(\tau)}{g(\tau)}-\frac{\varphi_{1}(\tau)}{\psi _{2}(\tau)} \biggr)\geq0. $$
(2.5)
Multiplying (2.3) and (2.5), we obtain
$$ \biggl(\frac{\varphi_{2}(\tau)}{\psi_{1}(\tau)}-\frac{f(\tau )}{g(\tau)} \biggr) \biggl(\frac{f(\tau)}{g(\tau)}- \frac{\varphi_{1}(\tau)}{\psi _{2}(\tau )} \biggr)\geq0, $$
or
$$ \biggl(\frac{\varphi_{2}(\tau)}{\psi_{1}(\tau)}+\frac{\varphi_{1}(\tau )}{\psi _{2}(\tau)} \biggr) \frac{f(\tau)}{g(\tau)}\geq\frac{f^{2}(\tau )}{g^{2}(\tau)} +\frac{\varphi_{1}(\tau)\varphi_{2}(\tau)}{\psi_{1}(\tau)\psi_{2}(\tau)}. $$
(2.6)
The inequality (2.6) can be written as
$$ \bigl(\varphi_{1}(\tau)\psi_{1}(\tau)+ \varphi_{2}(\tau)\psi_{2}(\tau ) \bigr)f(\tau)g(\tau)\geq \psi_{1}(\tau)\psi_{2}(\tau)f^{2}(\tau)+ \varphi_{1}(\tau)\varphi_{2}(\tau )g^{2}(\tau). $$
(2.7)
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
$$ R^{\alpha,r}_{a,k}\bigl\{ (\varphi_{1} \psi_{1}+\varphi_{2}\psi_{2} )fg\bigr\} (t)\geq R^{\alpha,r}_{a,k}\bigl\{ \psi_{1}\psi_{2}f^{2} \bigr\} (t)+ R^{\alpha,r}_{a,k}\bigl\{ \varphi _{1} \varphi_{2}g^{2}\bigr\} (t). $$
Applying the AM-GM inequality, i.e., \(a+b\geq2\sqrt{ab}\), \(a,b\in\mathbb{R}^{+}\), we have
$$ R^{\alpha,r}_{a,k}\bigl\{ (\varphi_{1} \psi_{1}+\varphi_{2}\psi_{2} )fg\bigr\} (t)\geq 2 \sqrt{ R^{\alpha,r}_{a,k}\bigl\{ \psi_{1} \psi_{2}f^{2}\bigr\} (t) R^{\alpha,r}_{a,k}\bigl\{ \varphi_{1}\varphi_{2}g^{2}\bigr\} (t)}, $$
which leads to
$$ R^{\alpha,r}_{a,k}\bigl\{ \psi_{1}\psi_{2}f^{2} \bigr\} (t) R^{\alpha,r}_{a,k}\bigl\{ \varphi _{1} \varphi_{2}g^{2}\bigr\} (t)\leq \frac{1}{4} \bigl( R^{\alpha,r}_{a,k}\bigl\{ (\varphi_{1}\psi _{1}+\varphi _{2}\psi_{2} )fg\bigr\} (t) \bigr)^{2}. $$
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:
$$ \frac{ R^{\alpha,r}_{a,k}\{\varphi_{1}\varphi_{2}\}(t)R^{\beta,r}_{a,k}\{ \psi_{1}\psi_{2}\}(t) R^{\alpha,r}_{a,k}\{f^{2}\}(t)R^{\beta,r}_{a,k}\{g^{2}\}(t)}{ ( R^{\alpha,r}_{a,k}\{\varphi_{1}f\}(t)R^{\beta,r}_{a,k}\{\psi_{1}g\}(t)+ R^{\alpha,r}_{a,k}\{\varphi_{2}f\}(t)R^{\beta,r}_{a,k}\{\psi_{2}g\} (t) )^{2}} \leq\frac{1}{4}. $$
(2.8)

Proof

To prove (2.8), using the condition \((H_{1})\), we obtain
$$ \biggl(\frac{\varphi_{2}(\tau)}{\psi_{1}(\rho)}-\frac{f(\tau )}{g(\rho)} \biggr)\geq0 $$
(2.9)
and
$$ \biggl(\frac{f(\tau)}{g(\rho)}-\frac{\varphi_{1}(\tau)}{\psi _{2}(\rho)} \biggr)\geq0, $$
(2.10)
which imply that
$$ \biggl(\frac{\varphi_{1}(\tau)}{\psi_{2}(\rho)}+\frac{\varphi_{2}(\tau )}{\psi _{1}(\rho)} \biggr) \frac{f(\tau)}{g(\rho)}\geq\frac{f^{2}(\tau )}{g^{2}(\rho)} +\frac{\varphi_{1}(\tau)\varphi_{2}(\tau)}{\psi_{1}(\rho)\psi_{2}(\rho)}. $$
(2.11)
Multiplying both sides of (2.11) by \(\psi_{1}(\rho)\psi_{2}(\rho )g^{2}(\rho)\), we have
$$ \begin{aligned}[b] &\varphi_{1}(\tau)f(\tau)\psi_{1}(\rho)g( \rho)+\varphi_{2}(\tau)f(\tau )\psi _{2}(\rho)g(\rho)\\ &\quad \geq \psi_{1}(\rho)\psi_{2}(\rho)f^{2}(\tau)+\varphi _{1}(\tau )\varphi_{2}(\tau)g^{2}(\rho).\end{aligned} $$
(2.12)
Multiplying both sides of (2.12) by
$$\frac{ (1+r )^{1-\frac{\alpha}{k}} (1+r )^{1-\frac {\beta}{k}} (t^{r+1}-\tau^{r+1})^{\frac{\alpha}{k}-1} (t^{r+1}-\rho^{r+1})^{\frac{\beta}{k}-1}}{k\Gamma_{k}(\alpha) k\Gamma _{k}(\beta)}, $$
and double integrating with respect to τ and ρ from a to t, we have
$$\begin{aligned} & R^{\alpha,r}_{a,k}\{\varphi_{1}f\}(t)R^{\beta,r}_{a,k} \{\psi_{1}g\}(t)+ R^{\alpha,r}_{a,k}\{ \varphi_{2}f\}(t)R^{\beta,r}_{a,k}\{\psi_{2}g \}(t) \\ &\quad \geq R^{\alpha,r}_{a,k}\bigl\{ f^{2}\bigr\} (t)R^{\beta,r}_{a,k}\{\psi_{1}\psi_{2}\}(t)+ R^{\alpha,r}_{a,k}\{\varphi_{1}\varphi_{2} \}(t)R^{\beta,r}_{a,k}\bigl\{ g^{2}\bigr\} (t). \end{aligned}$$
Applying the AM-GM inequality, we get
$$\begin{aligned} & R^{\alpha,r}_{a,k}\{\varphi_{1}f\}(t)R^{\beta,r}_{a,k} \{\psi_{1}g\}(t)+ R^{\alpha,r}_{a,k}\{ \varphi_{2}f\}(t)R^{\beta,r}_{a,k}\{\psi_{2}g \}(t) \\ &\quad \geq 2\sqrt{ R^{\alpha,r}_{a,k}\bigl\{ f^{2} \bigr\} (t)R^{\beta,r}_{a,k}\{\psi_{1}\psi _{2} \} (t) R^{\alpha,r}_{a,k}\{\varphi_{1} \varphi_{2}\}(t)R^{\beta,r}_{a,k}\bigl\{ g^{2} \bigr\} (t)}, \end{aligned}$$
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:
$$ R^{\alpha,r}_{a,k}\bigl\{ f^{2}\bigr\} (t)R^{\beta,r}_{a,k}\bigl\{ g^{2}\bigr\} (t)\leq R^{\alpha,r}_{a,k}\bigl\{ (\varphi_{2}fg)/ \psi_{1}\bigr\} (t)R^{\beta,r}_{a,k}\bigl\{ (\psi _{2}fg)/\varphi_{1}\bigr\} (t). $$
(2.13)

Proof

From (2.2), we have
$$ \begin{aligned}[b] &\frac{ (1+r )^{1-\frac{\alpha}{k}}}{k\Gamma_{k}(\alpha )} \int _{a}^{t}\bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1}f^{2}(\tau)\,d\tau\\ &\quad \leq \frac{ (1+r )^{1-\frac{\alpha}{k}}}{k\Gamma_{k}(\alpha )} \int _{a}^{t}\bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1} \frac{\varphi_{2}(\tau)}{\psi_{1}(\tau)}f(\tau)g(\tau)\,d\tau,\end{aligned} $$
which implies
$$ R^{\alpha,r}_{a,k}\bigl\{ f^{2}\bigr\} (t) \leq R^{\alpha,r}_{a,k}\bigl\{ (\varphi _{2}fg)/ \psi_{1}\bigr\} (t). $$
(2.14)
By (2.4), we get
$$ \begin{aligned}[b] &\frac{ (1+r )^{1-\frac{\beta}{k}}}{k\Gamma_{k}(\beta )} \int _{a}^{t}\bigl(t^{r+1}- \rho^{r+1}\bigr)^{\frac{\beta}{k}-1}g^{2}(\rho)\,d\rho\\ &\quad \leq \frac{ (1+r )^{1-\frac{\beta}{k}}}{k\Gamma_{k}(\beta )} \int _{a}^{t}\bigl(t^{r+1}- \rho^{r+1}\bigr)^{\frac{\beta}{k}-1} \frac{\psi_{2}(\rho)}{\varphi_{1}(\rho)}f(\rho)g(\rho)\,d\rho,\end{aligned} $$
from which one has
$$ R_{0,t}^{\beta}\bigl\{ g^{2}\bigr\} (t) \leq R^{\beta,r}_{a,k}\bigl\{ (\psi_{2}fg)/\varphi _{1}\bigr\} (t). $$
(2.15)
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
$$ \frac{ ( R^{\alpha,r}_{a,k} \{f^{2} \}(t) ) (R^{\beta,r}_{a,k} \{g^{2} \}(t) )}{ R^{\alpha,r}_{a,k} \{fg \}(t)R^{\beta,r}_{a,k} \{fg \}(t)} \leq \frac{MN}{mn}. $$
(2.16)

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:
$$\begin{aligned} & \biggl\vert \frac{(t^{r+1}-a^{r+1})^{\frac{\alpha}{k}}}{(r+1)^{\frac {\alpha }{k}}\Gamma_{k}(\alpha+k)} R^{\alpha,r}_{a,k} \{fg \}(t) -R^{\alpha,r}_{a,k} \{f \}(t)R^{\alpha,r}_{a,k} \{ g \}(t)\biggr\vert \\ &\quad \leq \bigl\vert G(f,\varphi_{1},\varphi_{2}) (t)G(g, \psi_{1},\psi_{2}) (t)\bigr\vert ^{\frac{1}{2}}, \end{aligned}$$
(3.1)
where
$$\begin{aligned} G(u,v,w) (t)=\frac{(t^{r+1}-a^{r+1})^{\frac{\alpha }{k}}}{4(r+1)^{\frac {\alpha}{k}}\Gamma_{k}(\alpha+k)} \frac{ (R^{\alpha,r}_{a,k}\{(v+w)u\}(t) )^{2}}{R^{\alpha,r}_{a,k}\{vw\}(t)}- \bigl(R^{\alpha,r}_{a,k}\{u\}(t) \bigr)^{2}. \end{aligned}$$
(3.2)

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
$$ A(\tau, \rho)= \bigl(f(\tau)-f(\rho) \bigr) \bigl(g(\tau)-g(\rho ) \bigr), $$
(3.3)
or, equivalently,
$$ A(\tau, \rho)= f(\tau)g(\tau)+f(\rho)g(\rho) - f(\tau)g(\rho )-f( \rho )g(\tau). $$
(3.4)
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
$$\begin{aligned} & \frac{ (1+r )^{2(1-\frac{\alpha}{k})} }{ (k\Gamma _{k}(\alpha) )^{2}} \int_{a}^{t} \int_{a}^{t} \bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1}\bigl(t^{r+1}-\rho ^{r+1}\bigr)^{\frac {\alpha}{k}-1} A(\tau,\rho )\,d\tau \,d\rho \\ &\quad = 2\frac{(t^{r+1}-a^{r+1})^{\frac{\alpha}{k}}}{(r+1)^{\frac{\alpha }{k}}\Gamma_{k}(\alpha+k)}R^{\alpha,r}_{a,k} \{f g \}(t)- 2 \bigl(R^{\alpha,r}_{a,k} \{g \}(t) \bigr) \bigl(R^{\alpha,r}_{a,k} \{f \} (t) \bigr). \end{aligned}$$
(3.5)
By using the Cauchy-Schwartz inequality for double integrals, we have
$$ \begin{aligned}[b] & \biggl\vert \frac{ (1+r )^{2(1-\frac{\alpha}{k})} }{ (k\Gamma_{k}(\alpha) )^{2}} \int_{a}^{t} \int_{a}^{t} \bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1}\bigl(t^{r+1}-\rho ^{r+1}\bigr)^{\frac {\alpha}{k}-1} A(\tau,\rho )\,d\tau \,d\rho\biggr\vert \\ & \quad \leq \biggl[\frac{ (1+r )^{2(1-\frac{\alpha}{k})} }{ (k\Gamma_{k}(\alpha) )^{2}} \int_{a}^{t} \int_{a}^{t} \bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1}\bigl(t^{r+1}-\rho ^{r+1}\bigr)^{\frac {\alpha}{k}-1} f^{2}(\tau )\,d\tau \,d\rho \\ & \qquad {}+\frac{ (1+r )^{2(1-\frac{\alpha}{k})} }{ (k\Gamma _{k}(\alpha) )^{2}} \int_{a}^{t} \int_{a}^{t} \bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1}\bigl(t^{r+1}-\rho ^{r+1}\bigr)^{\frac {\alpha}{k}-1} f^{2}(\rho )\,d\tau \,d\rho \\ &\qquad {} -2\frac{ (1+r )^{2(1-\frac{\alpha}{k})} }{ (k\Gamma _{k}(\alpha) )^{2}} \int_{a}^{t} \int_{a}^{t} \bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1}\bigl(t^{r+1}-\rho ^{r+1}\bigr)^{\frac {\alpha}{k}-1} f(\tau) f(\rho)\,d\tau \,d\rho \biggr]^{\frac{1}{2}} \\ & \qquad {}\times \biggl[\frac{ (1+r )^{2(1-\frac{\alpha}{k})} }{ (k\Gamma_{k}(\alpha) )^{2}} \int_{a}^{t} \int_{a}^{t} \bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1}\bigl(t^{r+1}-\rho ^{r+1}\bigr)^{\frac {\alpha}{k}-1} g^{2}(\tau )\,d\tau \,d\rho \\ & \qquad {}+\frac{ (1+r )^{2(1-\frac{\alpha}{k})} }{ (k\Gamma _{k}(\alpha) )^{2}} \int_{a}^{t} \int_{a}^{t} \bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1}\bigl(t^{r+1}-\rho ^{r+1}\bigr)^{\frac {\alpha}{k}-1} g^{2}(\rho )\,d\tau \,d\rho \\ & \qquad {}-2\frac{ (1+r )^{2(1-\frac{\alpha}{k})} }{ (k\Gamma _{k}(\alpha) )^{2}} \int_{a}^{t} \int_{a}^{t} \bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1}\bigl(t^{r+1}-\rho ^{r+1}\bigr)^{\frac {\alpha}{k}-1} g(\tau) g(\rho)\,d\tau \,d\rho \biggr]^{\frac{1}{2}}. \end{aligned} $$
(3.6)
Therefore, we obtain
$$ \begin{aligned}[b] & \biggl\vert \frac{ (1+r )^{2(1-\frac{\alpha}{k})} }{ (k\Gamma_{k}(\alpha) )^{2}} \int_{a}^{t} \int_{a}^{t} \bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1}\bigl(t^{r+1}-\rho ^{r+1}\bigr)^{\frac {\alpha}{k}-1} A(\tau,\rho )\,d\tau \,d\rho\biggr\vert \\ &\quad \leq2 \biggl[\frac{(t^{r+1}-a^{r+1})^{\frac{\alpha }{k}}}{(r+1)^{\frac {\alpha}{k}}\Gamma_{k}(\alpha+k)}R^{\alpha,r}_{a,k} \bigl\{ f^{2} \bigr\} (t)- \bigl(R^{\alpha,r}_{a,k} \{f \}(t) \bigr)^{2} \biggr]^{\frac{1}{2}} \\ &\qquad {} \times \biggl[\frac{(t^{r+1}-a^{r+1})^{\frac{\alpha}{k}}}{(r+1)^{\frac {\alpha }{k}}\Gamma_{k}(\alpha+k)}R^{\alpha,r}_{a,k} \bigl\{ g^{2} \bigr\} (t)- \bigl(R^{\alpha,r}_{a,k} \{g \}(t) \bigr)^{2} \biggr]^{\frac{1}{2}}. \end{aligned} $$
(3.7)
By applying Lemma 2.1, for \(\psi_{1}(t)=\psi_{2}(t)=g(t)=1\), we get
$$ R^{\alpha,r}_{a,k}\bigl\{ f^{2}\bigr\} (t)\leq \frac{1}{4} \frac{ (R^{\alpha,r}_{a,k}\{(\varphi_{1}+\varphi_{2})f\}(t) )^{2}}{R^{\alpha,r}_{a,k}\{\varphi_{1}\varphi_{2}\}(t)}, $$
which leads to
$$\begin{aligned} \begin{aligned}[b] &\frac{(t^{r+1}-a^{r+1})^{\frac{\alpha}{k}}}{(r+1)^{\frac{\alpha }{k}}\Gamma_{k}(\alpha+k)}R^{\alpha,r}_{a,k}\bigl\{ f^{2}\bigr\} (t)- \bigl(R^{\alpha,r}_{a,k}\{f\}(t) \bigr)^{2} \\ &\quad \leq \frac{(t^{r+1}-a^{r+1})^{\frac{\alpha}{k}}}{4(r+1)^{\frac{\alpha }{k}}\Gamma_{k}(\alpha+k)} \frac{ (R^{\alpha,r}_{a,k}\{(\varphi_{1}+\varphi_{2})f\}(t) )^{2}}{R^{\alpha,r}_{a,k}\{\varphi_{1}\varphi_{2}\}(t)}- \bigl(R^{\alpha,r}_{a,k} \{f\}(t) \bigr)^{2} \\ &\quad =G(f,\varphi_{1},\varphi_{2}) (t).\end{aligned} \end{aligned}$$
(3.8)
Similarly, we get
$$\begin{aligned} &\frac{(t^{r+1}-a^{r+1})^{\frac{\alpha}{k}}}{(r+1)^{\frac{\alpha }{k}}\Gamma_{k}(\alpha+k)}R^{\alpha,r}_{a,k}\bigl\{ g^{2}\bigr\} (t)- \bigl(R^{\alpha,r}_{a,k}\{g\}(t) \bigr)^{2} \\ &\quad \leq \frac{(t^{r+1}-a^{r+1})^{\frac{\alpha}{k}}}{4(r+1)^{\frac{\alpha }{k}}\Gamma_{k}(\alpha+k)} \frac{ (R^{\alpha,r}_{a,k}\{(\psi_{1}+\psi_{2})g\}(t) )^{2}}{R^{\alpha,r}_{a,k}\{\psi_{1}\psi_{2}\}(t)}- \bigl(R^{\alpha,r}_{a,k} \{ g\}(t) \bigr)^{2} \\ &\quad =G(g,\psi_{1},\psi_{2}) (t). \end{aligned}$$
(3.9)
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
$$\begin{aligned}& G(f,m,M) (t) = \frac{(M-m)^{2}}{4mM} \bigl(R^{\alpha,r}_{a,k}\{f\} (t) \bigr)^{2}, \end{aligned}$$
(3.10)
$$\begin{aligned}& G(g,n,N) (t) = \frac{(N-n)^{2}}{4nN} \bigl(R^{\alpha,r}_{a,k}\{g\} (t) \bigr)^{2}. \end{aligned}$$
(3.11)

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:
$$\begin{aligned} &\biggl\vert \frac{(t^{r+1}-a^{r+1})^{\frac{\alpha}{k}}}{(r+1)^{\frac {\alpha }{k}}\Gamma_{k}(\alpha+k)}R^{\beta,r}_{a,k} \{f g \}(t)+\frac{(t^{r+1}-a^{r+1})^{\frac{\beta }{k}}}{(r+1)^{\frac {\beta}{k}}\Gamma_{k}(\beta+k)}R^{\alpha,r}_{a,k} \{f g \}(t) \\ &\qquad {} -R^{\alpha,r}_{a,k} \{f \}(t) R^{\beta,r}_{a,k} \{ g \}(t)- R^{\alpha,r}_{a,k} \{g \}(t) R^{\beta,r}_{a,k} \{ f \} (t)\biggr\vert \\ &\quad \leq \bigl\vert G_{1}(f,\varphi_{1}, \varphi_{2}) (t)+G_{2}(f,\varphi_{1},\varphi _{2}) (t)\bigr\vert ^{\frac{1}{2}} \\ &\qquad {} \times\bigl\vert G_{1}(g,\psi_{1},\psi_{2}) (t)+G_{1}(g,\psi_{1},\psi _{2}) (t)\bigr\vert ^{\frac{1}{2}}, \end{aligned}$$
(3.12)
where
$$\begin{aligned}& G_{1}(u,v,w) (t) = \frac{(t^{r+1}-a^{r+1})^{\frac{\beta }{k}}}{4(r+1)^{\frac {\beta}{k}}\Gamma_{k}(\beta+k)}\frac{ (R^{\alpha,r}_{a,k}\{ (v+w)u\} (t) )^{2}}{ R^{\alpha,r}_{a,k}\{vw\}(t)} -R^{\alpha,r}_{a,k}\{u\}(t)R^{\beta,r}_{a,k}\{u\}(t), \\& G_{2}(u,v,w) (t) = \frac{(t^{r+1}-a^{r+1})^{\frac{\alpha }{k}}}{4(r+1)^{\frac{\alpha}{k}}\Gamma_{k}(\alpha+k)}\frac{ (R^{\beta,r}_{a,k}\{(v+w)u\}(t) )^{2}}{ R^{\beta,r}_{a,k}\{vw\}(t)} -R^{\alpha,r}_{a,k}\{u\}(t)R^{\beta,r}_{a,k}\{u\}(t). \end{aligned}$$

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
$$\begin{aligned} &\frac{ (1+r )^{2-\frac{\alpha+\beta}{k}} }{k^{2}\Gamma _{k}(\alpha) \Gamma_{k}(\beta)} \int_{a}^{t} \int_{a}^{t} \bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1}\bigl(t^{r+1}-\rho ^{r+1}\bigr)^{\frac {\beta}{k}-1} A(\tau,\rho)\,d\tau \,d\rho \\ &\quad = \frac{(t^{r+1}-a^{r+1})^{\frac{\alpha}{k}}}{4(r+1)^{\frac {\alpha }{k}}\Gamma_{k}(\alpha+k)}R^{\beta,r}_{a,k} \{f g \}(t)+ \frac{(t^{r+1}-a^{r+1})^{\frac{\beta }{k}}}{(r+1)^{\frac {\beta}{k}}\Gamma_{k}(\beta+k)}R^{\alpha,r}_{a,k} \{f g \}(t) \\ &\qquad {} - R^{\alpha,r}_{a,k} \{f \}(t) R^{\beta,r}_{a,k} \{ g \} (t)-R^{\beta,r}_{a,k} \{f \}(t) R^{\alpha,r}_{a,k} \{ g \}(t). \end{aligned}$$
(3.13)
By using the Cauchy-Schwartz inequality for double integrals, we have
$$\begin{aligned} & \biggl\vert \frac{ (1+r )^{2-\frac{\alpha+\beta}{k}} }{k^{2}\Gamma_{k}(\alpha) \Gamma_{k}(\beta)} \int_{a}^{t} \int_{a}^{t} \bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1}\bigl(t^{r+1}-\rho ^{r+1}\bigr)^{\frac {\beta}{k}-1} A(\tau,\rho)\,d\tau \,d\rho\biggr\vert \\ &\quad \leq \biggl[\frac{ (1+r )^{2-\frac{\alpha+\beta}{k}} }{k^{2}\Gamma_{k}(\alpha) \Gamma_{k}(\beta)} \int_{a}^{t} \int_{a}^{t} \bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1}\bigl(t^{r+1}-\rho ^{r+1}\bigr)^{\frac {\beta}{k}-1} f^{2}(\tau)\,d\tau \,d\rho \\ & \qquad {} +\frac{ (1+r )^{2-\frac{\alpha+\beta}{k}} }{k^{2}\Gamma_{k}(\alpha) \Gamma_{k}(\beta)} \int_{a}^{t} \int_{a}^{t} \bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1}\bigl(t^{r+1}-\rho ^{r+1}\bigr)^{\frac {\beta}{k}-1} f^{2}(\rho)\,d\tau \,d\rho \\ &\qquad {} -2 \frac{ (1+r )^{2-\frac{\alpha+\beta}{k}} }{k^{2}\Gamma_{k}(\alpha) \Gamma_{k}(\beta)} \int_{a}^{t} \int_{a}^{t} \bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1}\bigl(t^{r+1}-\rho ^{r+1}\bigr)^{\frac {\beta}{k}-1} f(\rho)\,d\tau \,d\rho \biggr]^{\frac{1}{2}} \\ &\qquad {} \times \biggl[\frac{ (1+r )^{2-\frac{\alpha+\beta}{k}} }{k^{2}\Gamma_{k}(\alpha) \Gamma_{k}(\beta)} \int_{a}^{t} \int_{a}^{t} \bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1}\bigl(t^{r+1}-\rho ^{r+1}\bigr)^{\frac {\beta}{k}-1} g^{2}(\tau)\,d\tau \,d\rho \\ &\qquad {} +\frac{ (1+r )^{2-\frac{\alpha+\beta}{k}} }{k^{2}\Gamma_{k}(\alpha) \Gamma_{k}(\beta)} \int_{a}^{t} \int_{a}^{t} \bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1}\bigl(t^{r+1}-\rho ^{r+1}\bigr)^{\frac {\beta}{k}-1} g^{2}(\rho)\,d\tau \,d\rho \\ &\qquad {} -2 \frac{ (1+r )^{2-\frac{\alpha+\beta}{k}} }{k^{2}\Gamma_{k}(\alpha) \Gamma_{k}(\beta)} \int_{a}^{t} \int_{a}^{t} \bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1}\bigl(t^{r+1}-\rho ^{r+1}\bigr)^{\frac {\beta}{k}-1} g(\tau) g(\rho)\,d\tau \,d\rho \biggr]^{\frac{1}{2}}. \end{aligned}$$
Therefore, we get
$$\begin{aligned} & \biggl\vert \frac{ (1+r )^{2-\frac{\alpha+\beta}{k}} }{k^{2}\Gamma_{k}(\alpha) \Gamma_{k}(\beta)} \int_{a}^{t} \int_{a}^{t} \bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1}\bigl(t^{r+1}-\rho ^{r+1}\bigr)^{\frac {\beta}{k}-1} A(\tau,\rho)\,d\tau \,d\rho\biggr\vert \\ &\quad \leq \biggl[\frac{(t^{r+1}-a^{r+1})^{\frac{\alpha }{k}}}{(r+1)^{\frac {\alpha}{k}}\Gamma_{k}(\alpha+k)} R^{\beta,r}_{a,k} \bigl\{ f^{2}(t) \bigr\} +\frac {(t^{r+1}-a^{r+1})^{\frac {\beta}{k}}}{(r+1)^{\frac{\beta}{k}}\Gamma_{k}(\beta+k)} R^{\alpha,r}_{a,k} \bigl\{ f^{2}(t) \bigr\} \\ &\qquad {}- 2R^{\beta,r}_{a,k} \bigl\{ f(t) \bigr\} R^{\alpha,r}_{a,k} \bigl\{ f(t) \bigr\} \biggr]^{\frac{1}{2}} \\ &\qquad {} \times \biggl[\frac{(t^{r+1}-a^{r+1})^{\frac{\alpha }{k}}}{(r+1)^{\frac {\alpha}{k}}\Gamma_{k}(\alpha+k)} R^{\beta,r}_{a,k} \bigl\{ g^{2}(t) \bigr\} +\frac {(t^{r+1}-a^{r+1})^{\frac {\beta}{k}}}{(r+1)^{\frac{\beta}{k}}\Gamma_{k}(\beta+k)} R^{\alpha,r}_{a,k} \bigl\{ g^{2}(t) \bigr\} \\ &\qquad {} - 2R^{\beta,r}_{a,k} \bigl\{ g(t) \bigr\} R^{\alpha,r}_{a,k} \bigl\{ g(t) \bigr\} \biggr]^{\frac{1}{2}}. \end{aligned}$$
(3.14)
Applying Lemma 2.1 with \(\psi_{1}(t)=\psi_{2}(t)=g(t)= 1\), we have
$$ \frac{(t^{r+1}-a^{r+1})^{\frac{\beta}{k}}}{(r+1)^{\frac{\beta }{k}}\Gamma _{k}(\beta+k)}R^{\alpha,r}_{a,k}\bigl\{ f^{2}\bigr\} (t)\leq \frac{(t^{r+1}-a^{r+1})^{\frac{\beta}{k}}}{4(r+1)^{\frac{\beta }{k}}\Gamma_{k}(\beta+k)}\frac{ (R^{\alpha,r}_{a,k}\{(\varphi _{1}+\varphi_{2})f\}(t) )^{2}}{ R^{\alpha,r}_{a,k}\{\varphi_{1}\varphi_{2}\}(t)}. $$
This implies that
$$\begin{aligned} &\frac{(t^{r+1}-a^{r+1})^{\frac{\beta}{k}}}{(r+1)^{\frac{\beta }{k}}\Gamma_{k}(\beta+k)}R^{\alpha,r}_{a,k}\bigl\{ f^{2}\bigr\} (t)-R^{\alpha,r}_{a,k}\{f \}(t)R^{\beta,r}_{a,k}\{f\}(t) \\ &\quad \leq \frac{(t^{r+1}-a^{r+1})^{\frac{\beta}{k}}}{4(r+1)^{\frac{\beta }{k}}\Gamma_{k}(\beta+k)}\frac{ (R^{\alpha,r}_{a,k}\{(\varphi _{1}+\varphi_{2})f\}(t) )^{2}}{ R^{\alpha,r}_{a,k}\{\varphi_{1}\varphi_{2}\}(t)} -R^{\alpha,r}_{a,k}\{f \}(t)R^{\beta,r}_{a,k}\{f\}(t) \\ &\quad =G_{1}(f,\varphi_{1},\varphi_{2}) (t) \end{aligned}$$
(3.15)
and
$$\begin{aligned} &\frac{(t^{r+1}-a^{r+1})^{\frac{\alpha}{k}}}{(r+1)^{\frac{\alpha }{k}}\Gamma_{k}(\alpha+k)}R^{\beta,r}_{a,k}\bigl\{ f^{2}\bigr\} (t)-R^{\alpha,r}_{a,k}\{f \}(t)R^{\beta,r}_{a,k}\{f\}(t) \\ &\quad \leq \frac{(t^{r+1}-a^{r+1})^{\frac{\alpha}{k}}}{4(r+1)^{\frac{\alpha }{k}}\Gamma_{k}(\alpha+k)}\frac{ (R^{\beta,r}_{a,k}\{(\varphi _{1}+\varphi_{2})f\}(t) )^{2}}{ R_{0,t}^{\beta}\{\varphi_{1}\varphi_{2}\}(t)} -R^{\alpha,r}_{a,k}\{f \}(t)R^{\beta,r}_{a,k}\{f\}(t) \\ &\quad =G_{2}(f,\varphi_{1},\varphi_{2}) (t). \end{aligned}$$
(3.16)
Also, applying the same procedure with \(\phi_{1}(t)=\phi_{2}(t)=f(t)= 1\), we get
$$\begin{aligned} &\frac{(t^{r+1}-a^{r+1})^{\frac{\beta}{k}}}{(r+1)^{\frac{\beta }{k}}\Gamma_{k}(\beta+k)}R^{\alpha,r}_{a,k}\bigl\{ g^{2}\bigr\} (t)-R^{\alpha,r}_{a,k}\{g \}(t)R^{\beta,r}_{a,k}\{g\}(t) \\ &\quad \leq G_{1}(g,\psi_{1},\psi_{2}) (t) \end{aligned}$$
(3.17)
and
$$\begin{aligned} &\frac{(t^{r+1}-a^{r+1})^{\frac{\alpha}{k}}}{(r+1)^{\frac{\alpha }{k}}\Gamma_{k}(\alpha+k)}R^{\beta,r}_{a,k}\bigl\{ g^{2}\bigr\} (t)-R^{\alpha,r}_{a,k}\{g \}(t)R^{\beta,r}_{a,k}\{g\}(t) \\ &\quad \leq G_{2}(g,\psi_{1},\psi_{2}) (t). \end{aligned}$$
(3.18)
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
$$\begin{aligned} & \bigl(n_{1}N_{1}R_{a,k}^{\alpha,r} \bigl\{ \tau^{2}f^{2}\bigr\} (t)+(n_{1}N_{2}+n_{2}N_{1})R_{a,k}^{\alpha,r} \bigl\{ \tau f^{2}\bigr\} (t)+n_{2}N_{2}R_{a,k}^{\alpha,r} \bigl\{ f^{2}\bigr\} (t) \bigr) \\ &\qquad {}\times \bigl(m_{1}M_{1}R_{a,k}^{\alpha,r} \bigl\{ \tau^{2}g^{2}\bigr\} (t)+(m_{1}M_{2}+m_{2}M_{1})R_{a,k}^{\alpha,r} \bigl\{ \tau g^{2}\bigr\} (t)+m_{2}M_{2}R_{a,k}^{\alpha,r} \bigl\{ g^{2}\bigr\} (t) \bigr) \\ &\quad \leq\frac{1}{4} \bigl((m_{1}n_{1}+M_{1}N_{1})R_{a,k}^{\alpha,r} \bigl\{ \tau^{2}fg\bigr\} (t)+(m_{1}n_{2}+m_{2}n_{1}+M_{1}N_{2}+M_{2}N_{1})R_{a,k}^{\alpha,r} \{\tau fg\} (t) \\ &\qquad {}+(m_{2}n_{2}+M_{2}N_{2})R_{a,k}^{\alpha,r} \{fg\}(t) \bigr)^{2}. \end{aligned}$$
(4.1)

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:
$$ \frac{R_{a,k}^{\alpha,r}\{f^{2}\}(t)R_{a,k}^{\alpha,r}\{g^{2}\} (t)}{ (R_{a,k}^{\alpha,r}\{fg\}(t) )^{2}}\leq\frac{1}{4} \biggl(\sqrt { \frac {m_{2}n_{2}}{M_{2}N_{2}}}+ \sqrt{\frac{M_{2}N_{2}}{m_{2}n_{2}}} \biggr)^{2}. $$
(4.2)

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:
$$\begin{aligned} &\biggl\vert \frac{(t^{r+1}-a^{r+1})^{\frac{\alpha}{k}}}{(r+1)^{\frac {\alpha }{k}}\Gamma_{k}(\alpha+k)} R^{\alpha,r}_{a,k} \{fg \}(t) -R^{\alpha,r}_{a,k} \{f \}(t)R^{\alpha,r}_{a,k} \{ g \}(t)\biggr\vert \\ &\quad \leq \bigl\vert G^{*}(f,m_{1},m_{2},M_{1},M_{2}) (t)G^{*}(g,n_{1},n_{2},N_{1},N_{2}) (t) \bigr\vert ^{\frac{1}{2}}, \end{aligned}$$
(4.3)
where
$$\begin{aligned} & G^{*}(u,v,w,x,y) (t) \\ &\quad =\frac{(t^{r+1}-a^{r+1})^{\frac{\alpha}{k}}}{4(r+1)^{\frac{\alpha }{k}}\Gamma_{k}(\alpha+k)}\cdot \frac{ ((v+x)R^{\alpha,r}_{a,k}\{\tau u\}(t)+(w+y)R^{\alpha,r}_{a,k}\{ u\}(t) )^{2}}{vxR^{\alpha,r}_{a,k}\{\tau^{2}\} (t)+(vy+wx)R^{\alpha,r}_{a,k}\{\tau\}(t)+wyR^{\alpha,r}_{a,k}\{1\} (t)} \\ &\qquad {}- \bigl(R^{\alpha,r}_{a,k}\{u\}(t) \bigr)^{2}. \end{aligned}$$
(4.4)

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
$$\begin{aligned}& G^{*}(f,0,m_{2},0,M_{2}) (t) = G(f,m,M) (t), \\& G^{*}(g,0,n_{2},0,N_{2}) (t) = G(g,n,N) (t), \end{aligned}$$
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:
$$\begin{aligned} & \biggl\vert \frac{(t^{r+1}-a^{r+1})^{\frac{\alpha}{k}}}{(r+1)^{\frac {\alpha }{k}}\Gamma_{k}(\alpha+k)}R^{\beta,r}_{a,k} \{f g \}(t)+\frac{(t^{r+1}-a^{r+1})^{\frac{\beta }{k}}}{(r+1)^{\frac {\beta}{k}}\Gamma_{k}(\beta+k)}R^{\alpha,r}_{a,k} \{f g \}(t) \\ &\qquad {} -R^{\alpha,r}_{a,k} \{f \}(t) R^{\beta,r}_{a,k} \{ g \}(t)- R^{\alpha,r}_{a,k} \{g \}(t) R^{\beta,r}_{a,k} \{ f \} (t)\biggr\vert \\ &\quad \leq \bigl\vert G_{1}^{*}(f,m_{1},m_{2},M_{1},M_{2}) (t)+G_{2}^{*}(f,m_{1},m_{2},M_{1},M_{2}) (t)\bigr\vert ^{\frac{1}{2}} \\ &\qquad {} \times\bigl\vert G_{1}^{*}(g,n_{1},n_{2},N_{1},N_{2}) (t)+G_{2}^{*}(g,n_{1},n_{2},N_{1},N_{2}) (t)\bigr\vert ^{\frac{1}{2}}, \end{aligned}$$
(4.5)
where
$$\begin{aligned}& G_{1}^{*}(u,v,w,x,y) (t) \\& \quad =\frac{(t^{r+1}-a^{r+1})^{\frac{\beta}{k}}}{4(r+1)^{\frac{\beta }{k}}\Gamma_{k}(\beta+k)}\cdot\frac{ ((v+x)R^{\alpha,r}_{a,k}\{\tau u\}(t)+(w+y)R^{\alpha,r}_{a,k}\{ u\}(t) )^{2}}{ vxR^{\alpha,r}_{a,k}\{\tau^{2}\}(t)+(vy+wx)R^{\alpha,r}_{a,k}\{\tau\} (t)+wyR^{\alpha,r}_{a,k}\{1\}(t)} \\& \qquad {}-R^{\alpha,r}_{a,k}\{u\}(t)R^{\beta,r}_{a,k}\{u \}(t), \\& G_{2}^{*}(u,v,w,x,y) (t) \\& \quad =\frac{(t^{r+1}-a^{r+1})^{\frac{\alpha}{k}}}{4(r+1)^{\frac{\alpha }{k}}\Gamma_{k}(\alpha+k)}\cdot\frac{ ((v+x)R^{\beta,r}_{a,k}\{\tau u\}(t)+(w+y)R^{\beta,r}_{a,k}\{ u\}(t) )^{2}}{ vxR^{\beta,r}_{a,k}\{\tau^{2}\}(t)+(vy+wx)R^{\beta,r}_{a,k}\{\tau\} (t)+wyR^{\beta,r}_{a,k}\{1\}(t)} \\& \qquad {} -R^{\alpha,r}_{a,k}\{u\}(t)R^{\beta,r}_{a,k}\{u \}(t). \end{aligned}$$

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
$$\begin{aligned}& G_{1}^{*}(u,0,w,0,y) (t) \\& \quad =\frac{1}{4} \biggl(\sqrt{\frac{w}{y}}+\sqrt{\frac{y}{w}} \biggr)^{2}\frac {(t^{r+1}-a^{r+1})^{\frac{\beta-\alpha}{k}}\Gamma_{k}(\alpha +k)}{(r+1)^{\frac{\beta-\alpha}{k}} \Gamma_{k}(\beta+k)}\cdot \bigl(R^{\alpha,r}_{a,k} \{ u\}(t) \bigr)^{2} \\& \qquad {} -R^{\alpha,r}_{a,k}\{u\}(t)R^{\beta,r}_{a,k}\{u \}(t), \\& G_{2}^{*}(u,0,w,0,y) (t) \\& \quad =\frac{1}{4} \biggl(\sqrt{\frac{w}{y}}+\sqrt{\frac{y}{w}} \biggr)^{2}\frac {(t^{r+1}-a^{r+1})^{\frac{\alpha-\beta}{k}}\Gamma_{k}(\beta +k)}{(r+1)^{\frac{\alpha-\beta}{k}}\Gamma_{k}(\alpha+k)} \cdot \bigl(R^{\beta,r}_{a,k} \{ u\}(t) \bigr)^{2} \\& \qquad {} -R^{\alpha,r}_{a,k}\{u\}(t)R^{\beta,r}_{a,k}\{u \}(t). \end{aligned}$$

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics, Anand International College of Engineering, Jaipur, India
(2)
Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, Thailand
(3)
Department of Mathematics, University of Ioannina, Ioannina, Greece
(4)
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

References

  1. Chebyshev, PL: Sur les expressions approximatives des integrales definies par les autres prises entre les mêmes limites. Proc. Math. Soc. Charkov 2, 93-98 (1882) Google Scholar
  2. Anastassiou, GA: Advances on Fractional Inequalities. Springer Briefs in Mathematics. Springer, New York (2011) View ArticleMATHGoogle Scholar
  3. Belarbi, S, Dahmani, Z: On some new fractional integral inequalities. J. Inequal. Pure Appl. Math. 10(3), Article ID 86 (2009) MathSciNetMATHGoogle Scholar
  4. Dahmani, Z, Mechouar, O, Brahami, S: Certain inequalities related to the Chebyshev’s functional involving a type Riemann-Liouville operator. Bull. Math. Anal. Appl. 3(4), 38-44 (2011) MathSciNetMATHGoogle Scholar
  5. Dragomir, SS: Some integral inequalities of Grüss type. Indian J. Pure Appl. Math. 31(4), 397-415 (2000) MathSciNetMATHGoogle Scholar
  6. Kalla, SL, Rao, A: On Grüss type inequality for hypergeometric fractional integrals. Matematiche 66(1), 57-64 (2011) MathSciNetMATHGoogle Scholar
  7. Lakshmikantham, V, Vatsala, AS: Theory of fractional differential inequalities and applications. Commun. Appl. Anal. 11, 395-402 (2007) MathSciNetMATHGoogle Scholar
  8. Ntouyas, SK, Agarwal, P, Tariboon, J: On Pólya-Szegö and Chebyshev types inequalities involving the Riemann-Liouville fractional integral operators. J. Math. Inequal. 10(2), 491-504 (2016) MATHGoogle Scholar
  9. Öǧünmez, H, Özkan, UM: Fractional quantum integral inequalities. J. Inequal. Appl. 2011, Article ID 787939 (2011) MathSciNetView ArticleMATHGoogle Scholar
  10. Sudsutad, W, Ntouyas, SK, Tariboon, J: Fractional integral inequalities via Hadamard’s fractional integral. Abstr. Appl. Anal. 2014, Article ID 563096 (2014) MathSciNetView ArticleMATHGoogle Scholar
  11. Sulaiman, WT: Some new fractional integral inequalities. J. Math. Anal. 2(2), 23-28 (2011) MathSciNetMATHGoogle Scholar
  12. Wang, G, Agarwal, P, Chand, M: Certain Grüss type inequalities involving the generalized fractional integral operator. J. Inequal. Appl. 2014, Article ID 147 (2014) MathSciNetView ArticleMATHGoogle Scholar
  13. Grüss, G: Über das maximum des absoluten Betrages von \(\frac {1}{b-a}\int_{a}^{b}f(x) g(x)\,dx-\frac{1}{(b-a)^{2}}\int_{a}^{b}f(x)\,dx\int_{a}^{b}g(x)\,dx\). Math. Z. 39, 215-226 (1935) MathSciNetView ArticleMATHGoogle Scholar
  14. Pólya, G, Szegö, G: Aufgaben und Lehrsatze aus der Analysis, Bd. 1. Die Grundlehren der mathmatischen Wissenschaften, Bd. 19. Springer, Berlin (1925) View ArticleMATHGoogle Scholar
  15. Dragomir, SS, Diamond, NT: Integral inequalities of Grüss type via Pólya-Szegö and Shisha-Mond results. East Asian Math. J. 19(1), 27-39 (2003) MATHGoogle Scholar
  16. Agarwal, P, Jain, S: Some k-Riemann-Liouville type fractional integral inequalities via Pólya-Szegö inequality (submitted) Google Scholar
  17. Romero, LG, Luque, LL, Dorrego, GA, Cerutti, RA: On the k-Riemann-Liouville fractional derivative. Int. J. Contemp. Math. Sci. 8(1), 41-51 (2013) MathSciNetMATHGoogle Scholar
  18. Sarikaya, MZ, Karaca, A: On the k-Riemann-Liouville fractional integral and applications. Int. J. Stat. Math. 1(3), 33-43 (2014) Google Scholar
  19. Sarikaya, MZ, Dahmani, Z, Kiris, ME, Ahmad, F: \((k; s)\)-Riemann-Liouville fractional integral and applications. Hacet. J. Math. Stat. 45(1), 77-89 (2016) Google Scholar
  20. Diaz, R, Pariguan, E: On hypergeometric functions and Pochhammer k-symbol. Divulg. Mat. 15, 179-192 (2007) MathSciNetMATHGoogle Scholar
  21. Mubeen, S, Habibullah, GM: k-fractional integrals and application. Int. J. Contemp. Math. Sci. 7, 89-94 (2012) MathSciNetMATHGoogle Scholar
  22. Set, E, Tomar, M, Sarikaya, MZ: On generalized Grüss type inequalities for k-fractional integrals. Appl. Math. Comput. 269, 29-34 (2015) MathSciNetView ArticleGoogle Scholar

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