Some generalized Riemann-Liouville k-fractional integral inequalities

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.

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 [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]

$$ \Gamma_{k}(x)=\lim_{n \to\infty} \frac{n!k^{n}(nk)^{\frac {x}{k}-1}}{(x)_{n,k}}, $$

(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

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