Ostrowski–Trapezoid–Grüss-Type on (q, ω)-Hahn Difference Operator

Abstract

We use two parameters for functions whose second (q, ω)-derivatives are bounded in order to prove several recent extensions of the Ostrowski inequality and its companion inequalities on (q, ω)-Hahn difference operator. Furthermore, we procure some q-integral and continuous inequalities as special cases of the main results as well these generalizations. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

1. Introduction

In 1938, the following integral inequality was proved by Ostrowski [1].

Theorem 1.

Let the function $\mathsf{\Xi }:[\tau ,\nu ]\to \mathbb{R}$ be continuous on $[\tau ,\nu ]$ and differentiable on $(\tau ,\nu )$, then for all $\varsigma \in [\tau ,\nu ]$, we obtain

$$|\mathsf{\Xi }\left(\varsigma \right)-\frac{1}{\nu -\tau }{\int }_{\tau }^{\nu }\mathsf{\Xi }\left(\xi \right)d\xi |\le \underset{\tau <\xi <\nu }{sup}\left|{\mathsf{\Xi }}^{\prime }\left(\xi \right)\right|(\nu -\tau )\left[\frac{{\left(\varsigma -\frac{\tau +\nu }{2}\right)}^2}{{(\nu -\tau )}^2}+\frac{1}{4}\right].$$

(1)

Clearly, an upper bound is estimated by inequality (1) on account of the absolute deviation between the value of $\mathsf{\Xi }$ at a point $\varsigma$ in $[\tau ,\nu ]$ and its integral mean over $[\tau ,\nu ]$.

The following inequality was proved by Grüss [2] with a view to make the absolute deviation of the integral mean of the product of two functions be estimated from the product of the integral means.

Theorem 2.

Assume that Ξ and ϕ are continuous functions on $[\tau ,\nu ]$ such that

$$m_1\le \mathsf{\Xi }\left(\xi \right)\le M_{1}and{m}_2\le \varphi \left(\xi \right)\le M_2,forall\xi \in [\tau ,\nu ].$$

Then the following inequality

$$|\frac{1}{\nu -\tau }{\int }_{\tau }^{\nu }\mathsf{\Xi }\left(\xi \right)\varphi \left(\xi \right)d\xi -\frac{1}{{(\nu -\tau )}^2}{\int }_{\tau }^{\nu }\mathsf{\Xi }\left(\xi \right)d\xi {\int }_{\tau }^{\nu }\varphi \left(\xi \right)d\xi |\le \frac{1}{4}(M_1-m_1)(M_2-m_2)$$

(2)

holds.

In the literature, inequality (2) is well known as the Grüss inequality.

The next inequality, which is known in the literature as the trapezoid inequality [3], is one of the companion inequalities of the Ostrowski inequality.

Theorem 3.

Assume that Ξ is a twice differentiable function on $[\tau ,\nu ]$, then

$$|\frac{\mathsf{\Xi }\left(\tau \right)+\mathsf{\Xi }\left(\nu \right)}{2}\left(\nu -\tau \right)-{\int }_{\tau }^{\nu }\mathsf{\Xi }\left(\xi \right)d\xi |\le \underset{\tau <\xi <\nu }{sup}\left|{\mathsf{\Xi }}^{″}\left(\xi \right)\right|\frac{{(\nu -\tau )}^3}{12}.$$

In [4], Pachpatte procured the next trapezoid and Grüss-type inequality.

Theorem 4.

Suppose that $\mathsf{\Xi }:[\tau ,\nu ]\to \mathbb{R}$ is continuous and differentiable on $(\tau ,\nu )$, and its first derivative ${\mathsf{\Xi }}^{\prime }:(\tau ,\nu )\to \mathbb{R}$ is bounded on $(\tau ,\nu )$, then

$$|\frac{{\mathsf{\Xi }}^2\left(\nu \right)-{\mathsf{\Xi }}^2\left(\tau \right)}{2}-\frac{\mathsf{\Xi }\left(\nu \right)-\mathsf{\Xi }\left(\tau \right)}{b-\tau }{\int }_{\tau }^{\nu }\mathsf{\Xi }\left(\xi \right)d\xi |\le \frac{M^2{(\nu -\tau )}^2}{3},$$

where $M=\underset{\tau <\xi <\nu }{sup}{\mathsf{\Xi }}^{\prime }\left(\xi \right)$.

Additionally, in the same paper [4], Pachpatte proved the following inequality.

Theorem 5.

Assume that $\mathsf{\Xi }:[\tau ,\nu ]\to \mathbb{R}$ is continuous and differentiable on $(\tau ,\nu )$, and its first derivative ${\mathsf{\Xi }}^{\prime }:(\tau ,\nu )\to \mathbb{R}$ is bounded on $(\tau ,\nu )$, then

$$|\frac{{\mathsf{\Xi }}^2\left(\nu \right)-{\mathsf{\Xi }}^2\left(\tau \right)}{2}-\frac{\mathsf{\Xi }\left(\nu \right)-\mathsf{\Xi }\left(\tau \right)}{b-\tau }{\int }_{\tau }^{\nu }\mathsf{\Xi }\left(\xi \right)d\xi |\le \frac{M^2{(\nu -\tau )}^2}{3},$$

where $M=\underset{\tau <\xi <\nu }{sup}{\mathsf{\Xi }}^{\prime }\left(\xi \right)$.

In several fields, especially in numerical analysis, Ostrowski’s inequality has a very significant role, where we could use it in the estimation of the error in the approximation of integrals.

During the past few decades, many generalizations and amendments of the Ostrowski inequality and its associated inequalities have been done. The articles [2,3,5,6,7,8,9,10,11,12,13,14,15,16], the books [3,17] and the references cited therein are a few examples we refer to. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

The purpose of the present paper is as follows: first, we institute a new Ostrowski-type inequality on (q, ω)-Hahn difference operator for functions whose second (q, ω)-derivatives are bounded. Then, we confirm new generalized Trapezoid- and Grüss-type inequalities on (q, ω)-Hahn difference operator. A number of continuous and q-integral inequalities are obtained as special cases of our result.

This paper is ordered as follows: In Section 2, we set the essential definitions and concepts associated to the calculus on (q, ω)-Hahn difference operator substantially. In Section 3, our major results are stated and proved. In Section 4, the conclusion is stated.

2. Preliminaries

This section briefly introduces the calculus of Hahn-difference operators as established in [18]. Let $q\in (0,1),$$\omega >0$ be fixed, ${\omega }_0:=\omega /\left(1-q\right)$, and I be an interval of $\mathbb{R}$ containing ${\omega }_0$. Let $h\left(\zeta \right):=q\zeta +\omega ,\zeta \in I.$ Both $h\left(\zeta \right)$ and $h^{-1}\left(\zeta \right)$. The numbers ${\omega }_0$ play the role played by $\zeta =0$ in the q-setting. In fact,

$${\omega }_0=\underset{k\to \infty }{lim}\left(\underset{k-\mathrm{times}}{\underbrace{h\circ h\circ \cdots \circ h}}\right)\left(\zeta \right).$$

The transformation $h\left(\zeta \right)$ has the inverse $h^{-1}\left(\zeta \right)=(\zeta -\omega )/q,\zeta \in I.$ The (q, ω)-Hahn difference operator introduced by Hahn in [19] can be defined as follows

Definition 1.

Let f be a function defined on I. The Hahn difference operator is defined by

$$D_{q,\omega }f\left(\zeta \right):=\left\{\begin{array}{cc}{\displaystyle \frac{f(q\zeta +\omega )-f\left(\zeta \right)}{(q\zeta +\omega )-\zeta },}\hfill & \zeta \ne {\omega }_0,\hfill \\ \hfill & \hfill & \hfill \\ f^{\prime }\left({\omega }_0\right),\hfill & \zeta ={\omega }_0,\hfill \end{array}\right.$$

(3)

provided that f is differentiable at ${\omega }_0.$ In this case, we call D_q,ωf, the (q, ω)-derivative of f. We say that f is (q, ω)-differentiable, i.e., throughout I, if $D_{q,\omega }f\left({\omega }_0\right)$ exists.

Note that

$$\underset{\omega \uparrow 0}{lim}{D}_{q,\omega }f\left(\zeta \right)=D_{q}f\left(\zeta \right),$$

$$\underset{q\downarrow 1,\omega \uparrow 0}{lim}{D}_{q,\omega }f\left(\zeta \right)=f^{\prime }\left(\zeta \right)$$

taking into account $\downarrow ,\uparrow$ that mean limits from left and right at finite points, respectively. One can easily check that if $f,g$ are (q, ω)-differentiable at $\zeta \in I,$ then

$$\begin{array}{ccc}\hfill D_{q,\omega }(\alpha f+\beta g)\left(\zeta \right)& =& \alpha D_{q,\omega }f\left(\zeta \right)+\beta D_{q,\omega }g\left(\zeta \right),\alpha ,\beta \in \mathbb{C},\hfill \\ \hfill D_{q,\omega }\left(fg\right)\left(\zeta \right)& =& D_{q,\omega }\left(f\left(\zeta \right)\right)g\left(\zeta \right)+f(q\zeta +\omega )D_{q,\omega }g\left(\zeta \right),\hfill \\ \hfill {\displaystyle D_{q,\omega }\left(\frac{f}{g}\right)\left(\zeta \right)}& =& \frac{D_{q,\omega }\left(f\left(\zeta \right)\right)g\left(\zeta \right)-f\left(\zeta \right)D_{q,\omega }g\left(\zeta \right)}{g\left(\zeta \right)g(q\zeta +\omega )},\hfill \end{array}$$

(4)

provided in the last identity $g\left(\zeta \right)g(qt+\omega )\ne 0,$ cf. [18]. The right inverse for $D_{q,\omega }$ is defined in [18] in terms of Jackson–Nörlund sums as follows cf. [20]. Let $a,b\in I$ the $(q,\omega )$-integral of f from a to b is defined to be

$${\displaystyle {\int }_a^{b}f\left(\zeta \right)d_{q,\omega }\zeta :={\int }_{{\omega }_0}^{b}f\left(\zeta \right)d_{q,\omega }\zeta -{\int }_{{\omega }_0}^{a}f\left(\zeta \right)d_{q,\omega }\zeta ,}$$

(5)

$${\displaystyle {\int }_{{\omega }_0}^{x}f\left(\zeta \right)d_{q,\omega }\zeta :=\left(x(1-q)-\omega \right)\sum _{k=0}^{\infty }{q}^{k}f(x{q}^k+\omega {\left[k\right]}_q),x\in I,}$$

(6)

provided that the series converges at $x=a$ and $x=b.$ Here ${\left[k\right]}_q$ is the q-number ${\left[k\right]}_q:={\displaystyle \frac{1-q^k}{1-q}},k\in {\mathbb{N}}_0.$ In this case, f is called $(q,\omega )$-integrable on $[a,b]$ and the sum to the right hand side of (6) will be called the Jackson–Nörlund sum—see [20] for the relationship between Nörlund sums and the difference operators. The fundamental theorem of $(q,\omega )$-calculus given in [18] states that if $f:I⟶\mathbb{R}$ is continuous at ${\omega }_0$,

$$F\left(\zeta \right):={\int }_{{\omega }_0}^{\zeta }f\left(x\right)d_{q,\omega }x.\zeta \in I,$$

then F is continuous at ${\omega }_0.$ Furthermore, $D_{q,\omega }F\left(\zeta \right)$ exists for every $\zeta \in I$ and

$$D_{q,\omega }F\left(\zeta \right)=f\left(\zeta \right).$$

Conversely,

$${\int }_a^b{D}_{q,\omega }f\left(\zeta \right)d_{q,\omega }\zeta =f\left(b\right)-f\left(a\right)\mathrm{for} \mathrm{all}a,b\in I.$$

Consequently, the $(q,\omega )$- integration by parts for continuous function $f,g$ is given in [18] by

$${\displaystyle {\int }_a^{b}f\left(\zeta \right)D_{q,\omega }g\left(\zeta \right)d_{q,\omega }\zeta =f\left(\zeta \right)g\left(\zeta \right){|}_a^b-{\int }_a^b{D}_{q,\omega }\left(f\left(\zeta \right)\right)g(qt+\omega )d_{q,\omega }\zeta ,a,b\in I.}$$

(7)

Lemma 1

([18]). Let $s\in I$, $s>{\omega }_0,$ f and g be $(q,\omega )$-integrable on $I,$ then for $a,b\in {\{s{q}^k+\omega {\left[k\right]}_q\}}_{k=0}^{\infty }$ we have

$$\left|{\int }_a^{b}f\left(\zeta \right)d_{q,\omega }\zeta \right|⩽{\int }_a^b\left|f\left(\zeta \right)\right|d_{q,\omega }\zeta .$$

(8)

Consequently, if $g\left(\zeta \right)⩾0$ for all $\zeta \in {\{s{q}^k+\omega {\left[k\right]}_q\}}_{k=0}^{\infty },$ then for all $a,b\in {\{s{q}^k+\omega {\left[k\right]}_q\}}_{k=0}^{\infty }$ the inequalities

$${\int }_{{\omega }_0}^{b}g\left(\zeta \right)d_{q,\omega }\zeta ⩾0\mathit{and}{\int }_a^{b}g\left(\zeta \right)d_{q,\omega }\zeta ⩾0$$

(9)

holds.

3. Main Results

3.1. An Ostrowski-Type Inequality on (q, ω)-Hahn Difference Operator

Theorem 6.

Let $\zeta \in I,$$\zeta >{\omega }_0$ with τ, ν, ς, $\xi \in {\{t{q}^k+\omega {\left[k\right]}_q\}}_{k=0}^{\infty }$ and $\tau <\nu$. Further, assume that $\mathsf{\Xi }:[\tau ,\nu ]\to \mathbb{R}$ is a twice $(q,\omega )$-differentiable function. Then, for all $\varsigma \in [\tau ,\nu ]$ and $\eta ,\gamma \in \mathbb{R}$, we get

$$\begin{array}{ccc}\hfill & & |\mathsf{\Xi }\left(\varsigma \right)-\frac{1}{\eta +\gamma }\left[\frac{\eta }{\varsigma -\tau }{\int }_{\tau }^{\varsigma }\mathsf{\Xi }(q\xi +\omega )d_{q,\omega }\xi +\frac{\gamma }{\nu -\varsigma }{\int }_{\varsigma }^{\nu }\mathsf{\Xi }(q\xi +\omega )d_{q,\omega }\xi \right]\hfill \\ \hfill & & -\frac{1}{\eta +\gamma }[{\int }_{\tau }^{\nu }{\int }_{\tau }^{\xi }\frac{\eta }{\xi -\tau }\mathsf{\Omega }(\varsigma ,\xi )D_{q,\omega }\mathsf{\Xi }(qs+\omega )d_{q,\omega }s{d}_{q,\omega }\xi \hfill \\ \hfill & & +{\int }_{\tau }^{\nu }{\int }_{\xi }^{\nu }\frac{\gamma }{\nu -\varsigma }\mathsf{\Omega }(\varsigma ,\xi )D_{q,\omega }\mathsf{\Xi }(qs+\omega )d_{q,\omega }s{d}_{q,\omega }\xi \left]\right|\hfill \\ & & ⩽K{\int }_{\tau }^{\nu }{\int }_{\tau }^{\nu }\left|\mathsf{\Omega }(\varsigma ,\xi )\mathsf{\Omega }(\xi ,s)\right|d_{q,\omega }s{d}_{q,\omega }\xi ,\hfill \end{array}$$

(10)

where

$$\mathsf{\Omega }(\varsigma ,\xi )=\left\{\begin{array}{cc}{\displaystyle \frac{\eta }{\eta +\gamma }\left(\frac{\xi -\tau }{\varsigma -\tau }\right),}\hfill & \tau \le \xi <\varsigma ,\hfill \\ {\displaystyle \frac{-\gamma }{\eta +\gamma }\left(\frac{\nu -\xi }{\nu -\varsigma }\right),}\hfill & \varsigma \le \xi \le \nu \hfill \end{array}\right.$$

and

$$K=\underset{\tau <\xi <\nu }{sup}\left|D_{q,\omega }^2\mathsf{\Xi }\left(\xi \right)\right|<\infty .$$

Proof. 

If we use integration by parts Formula (7), we obtain

$${\int }_{\tau }^{\varsigma }\frac{\eta }{\eta +\gamma }\left(\frac{\xi -\tau }{\varsigma -\tau }\right)D_{q,\omega }\mathsf{\Xi }\left(\xi \right)d_{q,\omega }\xi =\frac{\eta }{\eta +\gamma }\mathsf{\Xi }\left(\varsigma \right)-\frac{\eta }{(\eta +\gamma )(\varsigma -\tau )}{\int }_{\tau }^{\varsigma }\mathsf{\Xi }(q\xi +\omega )d_{q,\omega }\xi ,$$

(11)

and

$${\int }_{\varsigma }^{\nu }\frac{-\gamma }{\eta +\gamma }\left(\frac{\nu -\xi }{\nu -\varsigma }\right)D_{q,\omega }\mathsf{\Xi }\left(\xi \right)d_{q,\omega }\xi =\frac{\gamma }{\eta +\gamma }\mathsf{\Xi }\left(\varsigma \right)-\frac{\gamma }{(\eta +\gamma )(\nu -\varsigma )}{\int }_{\varsigma }^{\nu }\mathsf{\Xi }(q\xi +\omega )d_{q,\omega }\xi .$$

(12)

Adding (11) and (12), we obtain

$${\int }_{\tau }^{\nu }\mathsf{\Omega }(\varsigma ,\xi )D_{q,\omega }\mathsf{\Xi }\left(\xi \right)d_{q,\omega }\xi =\mathsf{\Xi }\left(\varsigma \right)-\frac{1}{\eta +\gamma }\left[\frac{\eta }{\varsigma -\tau }{\int }_{\tau }^{\varsigma }\mathsf{\Xi }(q\xi +\omega )d_{q,\omega }\xi +\frac{\gamma }{\nu -\varsigma }{\int }_{\varsigma }^{\nu }\mathsf{\Xi }(q\xi +\omega )d_{q,\omega }\xi \right].$$

(13)

Similarly, we have

$$\begin{array}{ccc}\hfill & & {\int }_{\tau }^{\nu }\mathsf{\Omega }(\xi ,s)D_{q,\omega }^2\mathsf{\Xi }\left(s\right)d_{q,\omega }s\hfill \\ \hfill & & =D_{q,\omega }\mathsf{\Xi }\left(\xi \right)-\frac{1}{\eta +\gamma }\left[\frac{\eta }{\xi -\tau }{\int }_{\tau }^{\xi }{D}_{q,\omega }\mathsf{\Xi }(qs+\omega )d_{q,\omega }s+\frac{\gamma }{\nu -\xi }{\int }_{\xi }^{\nu }{D}_{q,\omega }\mathsf{\Xi }(qs+\omega )d_{q,\omega }s\right].\hfill \end{array}$$

(14)

Substituting (14) into (13) leads to

$$\begin{array}{ccc}\hfill & & {\int }_{\tau }^{\nu }{\int }_{\tau }^{\nu }\mathsf{\Omega }(\varsigma ,\xi )\mathsf{\Omega }(\xi ,s)D_{q,\omega }^2\mathsf{\Xi }\left(s\right)d_{q,\omega }s{d}_{q,\omega }\xi \hfill \\ \hfill & & +\frac{1}{\eta +\gamma }[{\int }_{\tau }^{\nu }{\int }_{\tau }^{\xi }\frac{\eta }{\xi -\tau }\mathsf{\Omega }(\varsigma ,\xi )D_{q,\omega }\mathsf{\Xi }(qs+\omega )d_{q,\omega }s{d}_{q,\omega }\xi \hfill \\ & & +{\int }_{\tau }^{\nu }{\int }_{\xi }^{\nu }\frac{\gamma }{\nu -\xi }\mathsf{\Omega }(\varsigma ,\xi )D_{q,\omega }\mathsf{\Xi }(qs+\omega )d_{q,\omega }s{d}_{q,\omega }\xi ]\hfill \\ \hfill & & =\mathsf{\Xi }\left(\varsigma \right)-\frac{1}{\eta +\gamma }\left[\frac{\eta }{\varsigma -\tau }{\int }_{\tau }^{\varsigma }\mathsf{\Xi }(q\xi +\omega )d_{q,\omega }\xi +\frac{\gamma }{\nu -\varsigma }{\int }_{\varsigma }^{\nu }\mathsf{\Xi }(q\xi +\omega )d_{q,\omega }\xi \right].\hfill \end{array}$$

(15)

Inequality (10) follows directly from (15) and the properties of modulus. So, the proof is completed by this. □

Corollary 1.

Tn Theorem 6, if one takes $\omega ⟶0^{+}$, then, inequality (10) becomes

$$\begin{array}{ccc}\hfill & & |\mathsf{\Xi }\left(\varsigma \right)-\frac{1}{\eta +\gamma }\left[\frac{\eta }{\varsigma -\tau }{\int }_{\tau }^{\varsigma }\mathsf{\Xi }\left(q\xi \right)d_q\xi +\frac{\gamma }{\nu -\varsigma }{\int }_{\varsigma }^{\nu }\mathsf{\Xi }\left(q\xi \right)d_q\xi \right]\hfill \\ \hfill & & -\frac{1}{\eta +\gamma }[{\int }_{\tau }^{\nu }{\int }_{\tau }^{\xi }\frac{\eta }{\xi -\tau }\mathsf{\Omega }(\varsigma ,\xi )D_q\mathsf{\Xi }\left(qs\right)d_{q}s{d}_q\xi \hfill \\ \hfill & & +{\int }_{\tau }^{\nu }{\int }_{\xi }^{\nu }\frac{\gamma }{\nu -\varsigma }\mathsf{\Omega }(\varsigma ,\xi )D_q\mathsf{\Xi }\left(qs\right)d_{q}s{d}_q\xi \left]\right|\hfill \\ & & ⩽K{\int }_{\tau }^{\nu }{\int }_{\tau }^{\nu }\left|\mathsf{\Omega }(\varsigma ,\xi )\mathsf{\Omega }(\xi ,s)\right|d_{q}s{d}_q\xi ,\hfill \end{array}$$

where

$$\mathsf{\Omega }(\varsigma ,\xi )=\left\{\begin{array}{cc}{\displaystyle \frac{\eta }{\eta +\gamma }\left(\frac{\xi -\tau }{\varsigma -\tau }\right),}\hfill & \tau \le \xi <\varsigma ,\hfill \\ {\displaystyle \frac{-\gamma }{\eta +\gamma }\left(\frac{\nu -\xi }{\nu -\varsigma }\right),}\hfill & \varsigma \le \xi \le \nu \hfill \end{array}\right.$$

and

$$K=\underset{\tau <\xi <\nu }{sup}\left|D_q^2\mathsf{\Xi }\left(\xi \right)\right|<\infty .$$

Corollary 2.

If we take $q⟶1^{-}$ and $\omega ⟶0^{+}$ in Theorem 6, then, inequality (10) becomes

$$\begin{array}{ccc}\hfill & & |\mathsf{\Xi }\left(\varsigma \right)-\frac{1}{\eta +\gamma }\left[\frac{\eta }{\varsigma -\tau }{\int }_{\tau }^{\varsigma }\mathsf{\Xi }\left(\xi \right)d\xi +\frac{\gamma }{\nu -\varsigma }{\int }_{\varsigma }^{\nu }\mathsf{\Xi }\left(\xi \right)d\xi \right]\hfill \\ & & -\frac{1}{\eta +\gamma }[{\int }_{\tau }^{\nu }{\int }_{\tau }^{\xi }\frac{\eta }{\xi -\tau }\mathsf{\Omega }(\varsigma ,\xi ){\mathsf{\Xi }}^{\prime }\left(s\right)dsd\xi \hfill \\ & & +{\int }_{\tau }^{\nu }{\int }_{\xi }^{\nu }\frac{\gamma }{\nu -\varsigma }\mathsf{\Omega }(\varsigma ,\xi ){\mathsf{\Xi }}^{\prime }\left(s\right)dsd\xi \left]\right|\hfill \\ & & ⩽K{\int }_{\tau }^{\nu }{\int }_{\tau }^{\nu }\left|\mathsf{\Omega }(\varsigma ,\xi )\mathsf{\Omega }(\xi ,s)\right|dsd\xi ,\hfill \end{array}$$

where

$$\mathsf{\Omega }(\varsigma ,\xi )=\left\{\begin{array}{cc}{\displaystyle \frac{\eta }{\eta +\gamma }\left(\frac{\xi -\tau }{\varsigma -\tau }\right),}\hfill & \tau \le \xi <\varsigma ,\hfill \\ {\displaystyle \frac{-\gamma }{\eta +\gamma }\left(\frac{\nu -\xi }{\nu -\varsigma }\right),}\hfill & \varsigma \le \xi \le \nu \hfill \end{array}\right.$$

and

$$K=\underset{\tau <\xi <\nu }{sup}\left|{\mathsf{\Xi }}^{″}\left(\xi \right)\right|<\infty .$$

3.2. A Trapezoid-Type Inequality on (q, ω)-Hahn Difference Operator

Theorem 7.

Under the same assumptions as in Theorem 6, we have

$$\begin{array}{ccc}\hfill & & |{\mathsf{\Xi }}^2\left(\nu \right)-{\mathsf{\Xi }}^2\left(\tau \right)-\frac{1}{\eta +\gamma }{\int }_{\tau }^{\nu }[\frac{\eta }{\varsigma -\tau }{\int }_{\tau }^{\varsigma }\left[\mathsf{\Xi }(q\xi +\omega )+\mathsf{\Xi }(q^2\xi +\omega {\left[2\right]}_q)\right]d_{q,\omega }\xi \hfill \\ & & +\frac{\gamma }{\nu -\varsigma }{\int }_{\varsigma }^{\nu }\left[\mathsf{\Xi }(q\xi +\omega )+\mathsf{\Xi }(q^2\xi +\omega {\left[2\right]}_q)\right]d_{q,\omega }\xi \left]d_{q,\omega }\varsigma \right|\hfill \\ & & \le M\left(M+P\right){\int }_{\tau }^{\nu }{\int }_{\tau }^{\nu }\left|\mathsf{\Omega }(\varsigma ,\xi )\right|d_{q,\omega }\xi d_{q,\omega }\varsigma ,\hfill \end{array}$$

(16)

where

$$\mathsf{\Omega }(\varsigma ,\xi )=\left\{\begin{array}{cc}{\displaystyle \frac{\eta }{\eta +\gamma }\left(\frac{\xi -\tau }{\varsigma -\tau }\right),}\hfill & \tau \le \xi <\varsigma ,\hfill \\ {\displaystyle \frac{-\gamma }{\eta +\gamma }\left(\frac{\nu -\xi }{\nu -\varsigma }\right),}\hfill & \varsigma \le \xi \le \nu \hfill \end{array}\right.$$

and

$$M=\underset{\tau <\xi <\nu }{sup}\left|D_{q,\omega }\mathsf{\Xi }\left(\xi \right)\right|andP=\underset{\tau <\xi <\nu }{sup}\left|D_{q,\omega }\mathsf{\Xi }(q\xi +\omega )\right|.$$

Proof. 

From (13) we have

$$\mathsf{\Xi }\left(\varsigma \right)={\int }_{\tau }^{\nu }\mathsf{\Omega }(\varsigma ,\xi )D_{q,\omega }\mathsf{\Xi }\left(\xi \right)d_{q,\omega }\xi +\frac{1}{\eta +\gamma }\left[\frac{\eta }{\varsigma -\tau }{\int }_{\tau }^{\varsigma }\mathsf{\Xi }(q\xi +\omega )d_{q,\omega }\xi +\frac{\gamma }{\nu -\varsigma }{\int }_{\varsigma }^{\nu }\mathsf{\Xi }(q\xi +\omega )d_{q,\omega }\xi \right]$$

(17)

and similarly

$$\begin{array}{ccc}\hfill \mathsf{\Xi }(q\varsigma +\omega )& & ={\int }_{\tau }^{\nu }\mathsf{\Omega }(\varsigma ,\xi )D_{q,\omega }\mathsf{\Xi }(q\xi +\omega )d_{q,\omega }\xi +\frac{1}{\eta +\gamma }[\frac{\eta }{\varsigma -\tau }{\int }_{\tau }^{\varsigma }\mathsf{\Xi }(q^2\xi +\omega {\left[2\right]}_q)d_{q,\omega }\xi \hfill \\ & & +\frac{\gamma }{\nu -\varsigma }{\int }_{\varsigma }^{\nu }\mathsf{\Xi }(q^2\xi +\omega {\left[2\right]}_q)d_{q,\omega }\xi ].\hfill \end{array}$$

(18)

Now, adding (17) and (18) produces

$$\begin{array}{ccc}\hfill \mathsf{\Xi }\left(\varsigma \right)+\mathsf{\Xi }(q\varsigma +\omega )& =& {\int }_{\tau }^{\nu }\mathsf{\Omega }(\varsigma ,\xi )\left[D_{q,\omega }\mathsf{\Xi }\left(\xi \right)+D_{q,\omega }\mathsf{\Xi }(q\xi +\omega )\right]d_{q,\omega }\xi \hfill \\ & & +\frac{1}{\eta +\gamma }[\frac{\eta }{\varsigma -\tau }{\int }_{\tau }^{\varsigma }\left[\mathsf{\Xi }(q\xi +\omega )+\mathsf{\Xi }(q^2\xi +\omega {\left[2\right]}_q)\right]d_{q,\omega }\xi \hfill \\ & & +\frac{\gamma }{\nu -\varsigma }{\int }_{\varsigma }^{\nu }\left[\mathsf{\Xi }(q\xi +\omega )+\mathsf{\Xi }(q^2\xi +\omega {\left[2\right]}_q)\right]d_{q,\omega }\xi ].\hfill \end{array}$$

If we multiply the latest identity by $D_{q,\omega }\mathsf{\Xi }\left(\varsigma \right)$, and by employing (7) and integrating the resulting identity with respect to $\varsigma$ from $\tau$ to $\nu$ we obtain

$$\begin{array}{ccc}\hfill {\mathsf{\Xi }}^2\left(\nu \right)-{\mathsf{\Xi }}^2\left(\tau \right)& =& {\int }_{\tau }^{\nu }{\int }_{\tau }^{\nu }{D}_{q,\omega }\mathsf{\Xi }\left(\varsigma \right)\mathsf{\Omega }(\varsigma ,\xi )\left[D_{q,\omega }\mathsf{\Xi }\left(\xi \right)+D_{q,\omega }\mathsf{\Xi }(q\xi +\omega )\right]d_{q,\omega }\xi d_{q,\omega }\varsigma \hfill \\ & & +\frac{1}{\eta +\gamma }{\int }_{\tau }^{\nu }{D}_{q,\omega }\mathsf{\Xi }\left(\varsigma \right)[\frac{\eta }{\varsigma -\tau }{\int }_{\tau }^{\varsigma }\left[\mathsf{\Xi }(q\xi +\omega )+\mathsf{\Xi }(q^2\xi +\omega {\left[2\right]}_q)\right]d_{q,\omega }\xi \hfill \\ & & +\frac{\gamma }{\nu -\varsigma }{\int }_{\varsigma }^{\nu }\left[\mathsf{\Xi }(q\xi +\omega )+\mathsf{\Xi }(q^2\xi +\omega {\left[2\right]}_q)\right]d_{q,\omega }\xi ]d_{q,\omega }\varsigma .\hfill \end{array}$$

Equivalently

$$\begin{array}{ccc}& & {\mathsf{\Xi }}^2\left(\nu \right)-{\mathsf{\Xi }}^2\left(\tau \right)-\frac{1}{\eta +\gamma }{\int }_{\tau }^{\nu }{D}_{q,\omega }\mathsf{\Xi }\left(\varsigma \right)[\frac{\eta }{\varsigma -\tau }{\int }_{\tau }^{\varsigma }\left[\mathsf{\Xi }(q\xi +\omega )+\mathsf{\Xi }(q^2\xi +\omega {\left[2\right]}_q)\right]d_{q,\omega }\xi \hfill \\ & & +\frac{\gamma }{b-\varsigma }{\int }_{\varsigma }^{\nu }\left[\mathsf{\Xi }(q\xi +\omega )+\mathsf{\Xi }(q^2\xi +\omega {\left[2\right]}_q)\right]d_{q,\omega }\xi ]d_{q,\omega }\varsigma \hfill \\ & & ={\int }_{\tau }^{\nu }{\int }_{\tau }^{\nu }{D}_{q,\omega }\mathsf{\Xi }\left(\varsigma \right)\mathsf{\Omega }(\varsigma ,\xi )\left[D_{q,\omega }\mathsf{\Xi }\left(\xi \right)+D_{q,\omega }\mathsf{\Xi }(q\xi +\omega )\right]d_{q,\omega }\xi d_{q,\omega }\varsigma .\hfill \end{array}$$

If we take the absolute value on both sides, we obtain

$$\begin{array}{ccc}& & |{\mathsf{\Xi }}^2\left(\nu \right)-{\mathsf{\Xi }}^2\left(\tau \right)-\frac{1}{\eta +\gamma }{\int }_{\tau }^{\nu }{D}_{q,\omega }\mathsf{\Xi }\left(\varsigma \right)[\frac{\eta }{\varsigma -\tau }{\int }_{\tau }^{\varsigma }\left[\mathsf{\Xi }(q\xi +\omega )+\mathsf{\Xi }(q^2\xi +\omega {\left[2\right]}_q)\right]d_{q,\omega }\xi \hfill \\ & & +\frac{\gamma }{\nu -\varsigma }{\int }_{\varsigma }^{\nu }\left[\mathsf{\Xi }(q\xi +\omega )+\mathsf{\Xi }(q^2\xi +\omega {\left[2\right]}_q)\right]d_{q,\omega }\xi \left]d_{q,\omega }\varsigma \right|\hfill \\ & & =\left|{\int }_{\tau }^{\nu }{\int }_{\tau }^{\nu }{D}_{q,\omega }\mathsf{\Xi }\left(\varsigma \right)\mathsf{\Omega }(\varsigma ,\xi )\left[D_{q,\omega }\mathsf{\Xi }\left(\xi \right)+D_{q,\omega }\mathsf{\Xi }(q\xi +\omega )\right]d_{q,\omega }\xi d_{q,\omega }\varsigma \right|\hfill \\ & & \le {\int }_{\tau }^{\nu }{\int }_{\tau }^{\nu }\left|D_{q,\omega }\mathsf{\Xi }\left(\varsigma \right)\right|\left|\mathsf{\Omega }(\varsigma ,\xi )\right|\left[\left|D_{q,\omega }\mathsf{\Xi }\left(\xi \right)\right|+\left|D_{q,\omega }\mathsf{\Xi }(q\xi +\omega )\right|\right]d_{q,\omega }\xi d_{q,\omega }\varsigma \hfill \\ & & \le M\left(M+P\right){\int }_{\tau }^{\nu }{\int }_{\tau }^{\nu }\left|\mathsf{\Omega }(\varsigma ,\xi )\right|d_{q,\omega }\xi d_{q,\omega }\varsigma .\hfill \end{array}$$

This shows the validity of (16). □

Corollary 3.

If we take $\omega ⟶0^{+}$ in Theorem 7, then, inequality (16) becomes

$$\begin{array}{ccc}\hfill & & |{\mathsf{\Xi }}^2\left(\nu \right)-{\mathsf{\Xi }}^2\left(\tau \right)-\frac{1}{\eta +\gamma }{\int }_{\tau }^{\nu }[\frac{\eta }{\varsigma -\tau }{\int }_{\tau }^{\varsigma }\left[\mathsf{\Xi }\left(q\xi \right)+\mathsf{\Xi }\left(q^2\xi \right)\right]d_q\xi \hfill \\ & & +\frac{\gamma }{\nu -\varsigma }{\int }_{\varsigma }^{\nu }\left[\mathsf{\Xi }\left(q\xi \right)+\mathsf{\Xi }\left(q^2\xi \right)\right]d_q\xi \left]d_q\varsigma \right|\hfill \\ & & \le M\left(M+P\right){\int }_{\tau }^{\nu }{\int }_{\tau }^{\nu }\left|\mathsf{\Omega }(\varsigma ,\xi )\right|d_q\xi d_q\varsigma ,\hfill \end{array}$$

where

$$\mathsf{\Omega }(\varsigma ,\xi )=\left\{\begin{array}{cc}{\displaystyle \frac{\eta }{\eta +\gamma }\left(\frac{\xi -\tau }{\varsigma -\tau }\right),}\hfill & \tau \le \xi <\varsigma ,\hfill \\ {\displaystyle \frac{-\gamma }{\eta +\gamma }\left(\frac{\nu -\xi }{\nu -\varsigma }\right),}\hfill & \varsigma \le \xi \le \nu \hfill \end{array}\right.$$

and

$$M=\underset{\tau <\xi <\nu }{sup}\left|D_q\mathsf{\Xi }\left(\xi \right)\right|andP=\underset{\tau <\xi <\nu }{sup}\left|D_q\mathsf{\Xi }\left(q\xi \right)\right|.$$

Corollary 4.

If we take $q⟶1^{-}$ and $\omega ⟶0^{+}$ in Theorem 7, then, inequality (16) becomes

$$\begin{array}{ccc}& & |\frac{{\mathsf{\Xi }}^2\left(\nu \right)-{\mathsf{\Xi }}^2\left(\tau \right)}{2}-\frac{1}{\eta +\gamma }{\int }_{\tau }^{\nu }{\mathsf{\Xi }}^{\prime }\left(\varsigma \right)\left[\frac{\eta }{\varsigma -\tau }{\int }_{\tau }^{\varsigma }\mathsf{\Xi }\left(\xi \right)d\xi +\frac{\gamma }{\nu -\varsigma }{\int }_{\varsigma }^{\nu }\mathsf{\Xi }\left(\xi \right)d\xi \right]d\varsigma |\hfill \\ & & \le M^2{\int }_{\tau }^{\nu }{\int }_{\tau }^{\nu }\left|\mathsf{\Omega }(\varsigma ,\xi )\right|dtdx,\hfill \end{array}$$

where

$$\mathsf{\Omega }(\varsigma ,\xi )=\left\{\begin{array}{cc}{\displaystyle \frac{\eta }{\eta +\gamma }\left(\frac{\xi -\tau }{\varsigma -\tau }\right),}\hfill & \tau \le \xi <\varsigma ,\hfill \\ {\displaystyle \frac{-\gamma }{\eta +\gamma }\left(\frac{\nu -\xi }{\nu -\varsigma }\right),}\hfill & \varsigma \le \xi \le \nu \hfill \end{array}\right.$$

and

$$M=\underset{\tau <\xi <\nu }{sup}\left|{\mathsf{\Xi }}^{\prime }\left(\xi \right)\right|.$$

3.3. A Grüss-Type Inequality on (q, ω)-Hahn Difference Operator

Theorem 8.

Let $\zeta \in I,$$\zeta >{\omega }_0$ with τ, ν, ς, $\xi \in {\{t{q}^k+\omega {\left[k\right]}_q\}}_{k=0}^{\infty }$ and $\tau <\nu$. Moreover, assume that Ξ, $\varphi :[\tau ,\nu ]\to \mathbb{R}$ are $(q,\omega )$- differentiable functions. Then, for all $\varsigma \in [\tau ,\nu ]$ and $\eta ,\gamma \in \mathbb{R}$, we obtain

$$\begin{array}{ccc}\hfill & & |2{\int }_{\tau }^{\nu }\mathsf{\Xi }\left(\varsigma \right)\varphi \left(\varsigma \right)d_{q,\omega }\varsigma -\frac{1}{\eta +\gamma }[\frac{\eta }{\varsigma -\tau }{\int }_{\tau }^{\nu }{\int }_{\tau }^{\varsigma }\left(\mathsf{\Xi }(q\xi +\omega )\varphi \left(\varsigma \right)+\varphi (q\xi +\omega )\mathsf{\Xi }\left(\varsigma \right)\right)d_{q,\omega }\xi d_{q,\omega }\varsigma \hfill \\ \hfill & & +\frac{\gamma }{\nu -\varsigma }{\int }_{\tau }^{\nu }{\int }_{\varsigma }^{\nu }\left(\mathsf{\Xi }(q\xi +\omega )\varphi \left(\varsigma \right)+\varphi (q\xi +\omega )\mathsf{\Xi }\left(\varsigma \right)\right)d_{q,\omega }\xi d_{q,\omega }\varsigma \left]\right|\hfill \\ & & \le {\int }_{\tau }^{\nu }{\int }_{\tau }^{\nu }\left|\mathsf{\Omega }(\varsigma ,\xi )\right|\left[M\left|\varphi \right(\varsigma \left)\right|+N\left|\mathsf{\Xi }\right(\varsigma \left)\right|\right]d_{q,\omega }\xi d_{q,\omega }\varsigma ,\hfill \end{array}$$

(19)

where

$$\mathsf{\Omega }(\varsigma ,\xi )=\left\{\begin{array}{cc}{\displaystyle \frac{\eta }{\eta +\gamma }\left(\frac{\xi -\tau }{\varsigma -\tau }\right),}\hfill & \tau \le \xi <\varsigma ,\hfill \\ {\displaystyle \frac{-\gamma }{\eta +\gamma }\left(\frac{\nu -\xi }{\nu -\varsigma }\right),}\hfill & \varsigma \le \xi \le \nu \hfill \end{array}\right.$$

and

$$M=\underset{\tau <\xi <\nu }{sup}|D_{q,\omega }\mathsf{\Xi }\left(\xi \right)|<\infty andN=\underset{\tau <\xi <\nu }{sup}\left|D_{q,\omega }\varphi \left(\xi \right)\right|<\infty .$$

Proof. 

From (13) we have

$$\mathsf{\Xi }\left(\varsigma \right)={\int }_{\tau }^{\nu }\mathsf{\Omega }(\varsigma ,\xi )D_{q,\omega }\mathsf{\Xi }\left(\xi \right)d_{q,\omega }\xi +\frac{1}{\eta +\gamma }\left[\frac{\eta }{\varsigma -\tau }{\int }_{\tau }^{\varsigma }\mathsf{\Xi }(q\xi +\omega )d_{q,\omega }\xi +\frac{\gamma }{\nu -\varsigma }{\int }_{\varsigma }^{\nu }\mathsf{\Xi }(q\xi +\omega )d_{q,\omega }\xi \right]$$

(20)

and similarly

$$\varphi \left(\varsigma \right)={\int }_{\tau }^{\nu }\mathsf{\Omega }(\varsigma ,\xi )D_{q,\omega }\varphi \left(\xi \right)d_{q,\omega }\xi +\frac{1}{\eta +\gamma }\left[\frac{\eta }{\varsigma -\tau }{\int }_{\tau }^{\varsigma }\varphi (q\xi +\omega )d_{q,\omega }\xi +\frac{\gamma }{\nu -\varsigma }{\int }_{\varsigma }^{\nu }\varphi (q\xi +\omega )d_{q,\omega }\xi \right].$$

(21)

Multiplying (20) by $\varphi \left(\varsigma \right)$ and (21) by $\mathsf{\Xi }\left(\varsigma \right)$, adding them and integrating the resulting identity with respect to $\varsigma$ from $\tau$ to $\nu$ yield

$$\begin{array}{ccc}\hfill 2{\int }_{\tau }^{\nu }\mathsf{\Xi }\left(\varsigma \right)\varphi \left(\varsigma \right)d_{q,\omega }\varsigma & =& {\int }_{\tau }^{\nu }{\int }_{\tau }^{\nu }\mathsf{\Omega }(\varsigma ,\xi )\left[D_{q,\omega }\mathsf{\Xi }\left(\xi \right)\varphi \left(\varsigma \right)+D_{q,\omega }\varphi \left(\xi \right)\mathsf{\Xi }\left(\varsigma \right)\right]d_{q,\omega }\xi d_{q,\omega }\varsigma \hfill \\ & & +\frac{1}{\eta +\gamma }[\frac{\eta }{\varsigma -\tau }{\int }_{\tau }^{\nu }{\int }_{\tau }^{\varsigma }\left(\mathsf{\Xi }(q\xi +\omega )\varphi \left(\varsigma \right)+\varphi (q\xi +\omega )\mathsf{\Xi }\left(\varsigma \right)\right)d_{q,\omega }\xi d_{q,\omega }\varsigma \hfill \\ & & +\frac{\gamma }{\nu -\varsigma }{\int }_{\tau }^{\nu }{\int }_{\varsigma }^{\nu }\left(\mathsf{\Xi }(q\xi +\omega )\varphi \left(\varsigma \right)+\varphi (q\xi +\omega )\mathsf{\Xi }\left(\varsigma \right)\right)d_{q,\omega }\xi d_{q,\omega }\varsigma ].\hfill \end{array}$$

By using modulus properties, we obtain

$$\begin{array}{ccc}& & |2{\int }_{\tau }^{\nu }\mathsf{\Xi }\left(\varsigma \right)\varphi \left(\varsigma \right)d_{q,\omega }\varsigma -\frac{1}{\eta +\gamma }[\frac{\eta }{\varsigma -\tau }{\int }_{\tau }^{\nu }{\int }_{\tau }^{\varsigma }\left(\mathsf{\Xi }(q\xi +\omega )\varphi \left(\varsigma \right)+\varphi (q\xi +\omega )\mathsf{\Xi }\left(\varsigma \right)\right)d_{q,\omega }\xi d_{q,\omega }\varsigma \hfill \\ & & +\frac{\gamma }{\nu -\varsigma }{\int }_{\tau }^{\nu }{\int }_{\varsigma }^{\nu }\left(\mathsf{\Xi }(q\xi +\omega )\varphi \left(\varsigma \right)+\varphi (q\xi +\omega )\mathsf{\Xi }\left(\varsigma \right)\right)d_{q,\omega }\xi d_{q,\omega }\varsigma \left]\right|\hfill \\ & & =\left|{\int }_{\tau }^{\nu }{\int }_{\tau }^{\nu }\mathsf{\Omega }(\varsigma ,\xi )\left[D_{q,\omega }\mathsf{\Xi }\left(\xi \right)\varphi \left(\varsigma \right)+D_{q,\omega }\varphi \left(\xi \right)\mathsf{\Xi }\left(\varsigma \right)\right]d_{q,\omega }\xi d_{q,\omega }\varsigma \right|\hfill \\ & & \le {\int }_{\tau }^{\nu }{\int }_{\tau }^{\nu }\left|\mathsf{\Omega }(\varsigma ,\xi )\right|\left[\left|D_{q,\omega }\mathsf{\Xi }\left(\xi \right)\left|\right|\varphi \left(\varsigma \right)|+|D_{q,\omega }\varphi \left(\xi \right)\right|\left|\mathsf{\Xi }\left(\varsigma \right)\right|\right]d_{q,\omega }\xi d_{q,\omega }\varsigma \hfill \\ & & \le {\int }_{\tau }^{\nu }{\int }_{\tau }^{\nu }\left|\mathsf{\Omega }(\varsigma ,\xi )\right|\left[M\left|\varphi \right(\varsigma \left)\right|+N\left|\mathsf{\Xi }\right(\varsigma \left)\right|\right]d_{q,\omega }\xi d_{q,\omega }\varsigma .\hfill \end{array}$$

The proof is terminated by this. □

Corollary 5.

If we take $\omega ⟶0^{+}$ in Theorem 8, then, inequality (19) becomes

$$\begin{array}{ccc}\hfill & & |2{\int }_{\tau }^{\nu }\mathsf{\Xi }\left(\varsigma \right)\varphi \left(\varsigma \right)d_q\varsigma -\frac{1}{\eta +\gamma }[\frac{\eta }{\varsigma -\tau }{\int }_{\tau }^{\nu }{\int }_{\tau }^{\varsigma }\left(\mathsf{\Xi }\left(q\xi \right)\varphi \left(\varsigma \right)+\varphi \left(q\xi \right)\mathsf{\Xi }\left(\varsigma \right)\right)d_q\xi d_q\varsigma \hfill \\ \hfill & & +\frac{\gamma }{\nu -\varsigma }{\int }_{\tau }^{\nu }{\int }_{\varsigma }^{\nu }\left(\mathsf{\Xi }\left(q\xi \right)\varphi \left(\varsigma \right)+\varphi \left(q\xi \right)\mathsf{\Xi }\left(\varsigma \right)\right)d_q\xi d_q\varsigma \left]\right|\hfill \\ & & \le {\int }_{\tau }^{\nu }{\int }_{\tau }^{\nu }\left|\mathsf{\Omega }(\varsigma ,\xi )\right|\left[M\left|\varphi \right(\varsigma \left)\right|+N\left|\mathsf{\Xi }\right(\varsigma \left)\right|\right]d_q\xi d_q\varsigma \hfill \end{array}$$

where

$$\mathsf{\Omega }(\varsigma ,\xi )=\left\{\begin{array}{cc}{\displaystyle \frac{\eta }{\eta +\gamma }\left(\frac{\xi -\tau }{\varsigma -\tau }\right),}\hfill & \tau \le \xi <\varsigma ,\hfill \\ {\displaystyle \frac{-\gamma }{\eta +\gamma }\left(\frac{\nu -\xi }{\nu -\varsigma }\right),}\hfill & \varsigma \le \xi \le \nu \hfill \end{array}\right.$$

and

$$M=\underset{\tau <\xi <\nu }{sup}|D_q\mathsf{\Xi }\left(\xi \right)|<\infty andN=\underset{\tau <\xi <\nu }{sup}\left|D_q\varphi \left(\xi \right)\right|<\infty .$$

Corollary 6.

If we take $q⟶1^{-}$ and $\omega ⟶0^{+}$ in Theorem 8, then, inequality (19) becomes

$$\begin{array}{ccc}& & |2{\int }_{\tau }^{\nu }\mathsf{\Xi }\left(\varsigma \right)\varphi \left(\varsigma \right)d\varsigma -\frac{1}{\eta +\gamma }[\frac{\eta }{\varsigma -\tau }{\int }_{\tau }^{\nu }{\int }_{\tau }^{\varsigma }\left(\mathsf{\Xi }\left(\xi \right)\varphi \left(\varsigma \right)+\varphi \left(\xi \right)\mathsf{\Xi }\left(\varsigma \right)\right)d\xi d\varsigma \hfill \\ & & +\frac{\gamma }{\nu -\varsigma }{\int }_{\tau }^{\nu }{\int }_{\varsigma }^{\nu }\left(\mathsf{\Xi }\left(\xi \right)\varphi \left(\varsigma \right)+\varphi \left(\xi \right)\mathsf{\Xi }\left(\varsigma \right)\right)d\xi d\varsigma \left]\right|\hfill \\ & & \le {\int }_{\tau }^{\nu }{\int }_{\tau }^{\nu }\left|\mathsf{\Omega }(\varsigma ,\xi )\right|\left[M\left|\varphi \right(\varsigma \left)\right|+N\left|\mathsf{\Xi }\right(\varsigma \left)\right|\right]d\xi d\varsigma ,\hfill \end{array}$$

where

$$\mathsf{\Omega }(\varsigma ,\xi )=\left\{\begin{array}{cc}{\displaystyle \frac{\eta }{\eta +\gamma }\left(\frac{\xi -\tau }{\varsigma -\tau }\right),}\hfill & \tau \le \xi <\varsigma ,\hfill \\ {\displaystyle \frac{-\gamma }{\eta +\gamma }\left(\frac{\nu -\xi }{\nu -\varsigma }\right),}\hfill & \varsigma \le \xi \le \nu \hfill \end{array}\right.$$

and

$$M=\underset{\tau <\xi <\nu }{sup}|{\mathsf{\Xi }}^{\prime }\left(\xi \right)|<\infty andN=\underset{\tau <\xi <\nu }{sup}\left|{\varphi }^{\prime }\left(\xi \right)\right|<\infty .$$

4. Conclusions

In this manuscript, a number of new investigations into the Ostrowski inequality and its attendant inequalities on $(q,\omega )$-Hahn difference operator were discussed by using two parameters. Furthermore, definite conditions, which have not been studied before, existed in these inequalities. For instance, in Theorem 6, the second $(q,\omega )$-derivative of the function $\mathsf{\Xi }$ is bounded, and we deal with it; however, all of the existing literature deals with functions whose first derivatives are bounded. Besides that, our inequalities were extended to be continuous and q calculus with a view to procure a number of new inequalities as special cases.