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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:

## 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 inequalities
• Chebyshev functional
• Riemann-Liouville k-fractional integral operator
• Pólya and Szegö type inequalities

• 26D10
• 26A33
• 26D15

## 1 Introduction and motivation

The celebrated functionals were introduced by the Chebyshev in his famous paper  and were subsequently rediscovered in various inequalities (for the celebrated functionals) by numerous authors, including Anastassiou , Belarbi and Dahmani , Dahmani et al. , Dragomir , Kalla and Rao , Lakshmikantham and Vatsala , Ntouyas et al. , Öǧünmez and Özkan , Sudsutad et al. , Sulaiman ; and, for very recent work, see also Wang et al. . 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  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  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  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 , and ). Here, we begin with the following.

### Definition 1.1

Let $$k>0$$, then the generalized k-gamma and k-beta functions defined by 
$$\Gamma_{k}(x)=\lim_{n \to\infty} \frac{n!k^{n}(nk)^{\frac {x}{k}-1}}{(x)_{n,k}},$$
(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 (; see also )
$$R^{\alpha}_{a,k} \bigl\{ f(t) \bigr\} = \frac{1}{k\Gamma_{k}(\alpha)} \int_{a}^{t}(t-\tau)^{\frac{\alpha}{k}-1}f(\tau)\,d\tau \quad \bigl(t\in [a,b] \bigr).$$
(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 ()
$$R^{\alpha,r}_{a,k} \bigl\{ f(t) \bigr\} = \frac{ (1+r )^{1-\frac{\alpha}{k}}}{k\Gamma _{k}(\alpha)} \int_{a}^{t}\bigl(t^{r+1}- \tau^{r+1}\bigr)^{\frac{\alpha}{k}-1}f(\tau)\,d\tau \quad \bigl(t\in[a,b] \bigr),$$
(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
$$R^{\alpha,r}_{a,k}\bigl\{ (\varphi_{1} \psi_{1}+\varphi_{2}\psi_{2} )fg\bigr\} (t)\geq 2 \sqrt{ R^{\alpha,r}_{a,k}\bigl\{ \psi_{1} \psi_{2}f^{2}\bigr\} (t) R^{\alpha,r}_{a,k}\bigl\{ \varphi_{1}\varphi_{2}g^{2}\bigr\} (t)},$$
$$R^{\alpha,r}_{a,k}\bigl\{ \psi_{1}\psi_{2}f^{2} \bigr\} (t) R^{\alpha,r}_{a,k}\bigl\{ \varphi _{1} \varphi_{2}g^{2}\bigr\} (t)\leq \frac{1}{4} \bigl( R^{\alpha,r}_{a,k}\bigl\{ (\varphi_{1}\psi _{1}+\varphi _{2}\psi_{2} )fg\bigr\} (t) \bigr)^{2}.$$
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
$$R^{\alpha,r}_{a,k}\bigl\{ f^{2}\bigr\} (t)\leq \frac{1}{4} \frac{ (R^{\alpha,r}_{a,k}\{(\varphi_{1}+\varphi_{2})f\}(t) )^{2}}{R^{\alpha,r}_{a,k}\{\varphi_{1}\varphi_{2}\}(t)},$$
\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  then our main results become the results recently given by Ntouyas et al. . 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}

## Authors’ Affiliations

(1)
Department of Mathematics, Anand International College of Engineering, Jaipur, 303012, India
(2)
Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand
(3)
Department of Mathematics, University of Ioannina, Ioannina, 451 10, Greece
(4)
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

## References 