Two-Sided Ostrowski-Type Inequalities on Time Scales Under Integrable Derivative Bounds

Abstract

In this paper, we introduce new two-sided Ostrowski-type inequalities on arbitrary time scales. Using the delta derivative and delta integral operators, we obtain explicit bounds for the deviation of a function value from the mean delta integral, under the assumption that the delta derivative is bounded between two integrable functions. With additional monotonicity conditions on the bound functions, further refinement of the obtained estimates is possible. The results recover the classical continuous Ostrowski inequality, as well as its discrete and quantum counterparts on $\mathbb{Z}$ and $q^{\mathbb{Z}}$, as special cases.

1. Introduction

The classical Ostrowski inequality, established by Ostrowski in [1], provides an upper bound for the deviation of a differentiable function from its integral mean. More precisely, if $f:[u_1,u_2]\to \mathbb{R}$ is continuous on $[u_1,u_2]$ and differentiable on $(u_1,u_2)$ with $|f^{\prime }\left(\xi \right)|\le M$, then for any $\varkappa \in [u_1,u_2]$ one has

$$\left|f\left(\varkappa \right)-{\displaystyle \frac{1}{u_2-u_1}}{\int }_{u_1}^{u_2}f\left(\xi \right)d\xi \right|\le \left[{\displaystyle \frac{1}{4}}+{\displaystyle \frac{{\left(\varkappa -\frac{u_1+u_2}{2}\right)}^2}{{(u_2-u_1)}^2}}\right](u_2-u_1)M.$$

(1)

Due to its relevance in numerical integration and error estimation, this inequality has been extended in several directions. Cerone and Dragomir [2,3] obtained generalizations for n-times differentiable mappings. Liu [4] and Alomari [5] studied companion inequalities and applications. Further refinements under convexity assumptions, including s-convexity, were investigated by Alomari et al. [6] and Set et al. [7].

The theory of time scales, introduced by Hilger and developed by Bohner and Peterson [8], provides a unified framework for continuous and discrete analysis. Ostrowski-type inequalities on time scales were first obtained by Bohner and Matthews [9], and further extended by Karpuz and Özkan [10], Tuna and Daghan [11], and Liu et al. [12,13,14] who considered weighted and multi-point versions.

Additional structural improvements were studied by Sarikaya [15], who introduced weighted Ostrowski and Čebyšev type inequalities on time scales, and further refined in [16]. Generalized Ostrowski–Grüss inequalities were recently studied by Farid et al. [17], and related k-point generalizations were considered by Nwaeze et al. [18] and Khan et al. [19]. Cerone and Dragomir [2] also discussed inequalities for functions satisfying convexity-type conditions. Earlier companion results of Alomari [5] and Liu [4] remain fundamental for understanding these developments.

Based on these results, this paper presents new two-sided Ostrowski-type inequalities on arbitrary time scales. The main assumption is that the delta derivative satisfies

$$m\left(\xi \right)\le f^{\Delta }\left(\xi \right)\le M\left(\xi \right),\xi \in {[u_1,u_2]}_{\mathbb{T}}^{\kappa },$$

where m and M are delta integrable functions. With additional monotonicity conditions on these limiting functions, it leads to sharper and more computable estimates. The results are particularly applicable to uniform time scales such as $h\mathbb{Z}$ and quantum time scales such as $q^{\mathbb{Z}}$.

The main novelty of the present paper lies in replacing the classical constant derivative bounds by variable monotone functions on arbitrary time scales. As a consequence, the obtained inequalities yield locally adaptive error estimates that reflect the graininess structure of the underlying time scale.

2. Preliminaries and Basic Definitions

For a comprehensive introduction to time scales calculus, we refer to Bohner and Peterson [8]. A time scale $\mathbb{T}$ is a nonempty closed subset of $\mathbb{R}$. The forward and backward jump operators $\sigma ,\rho :\mathbb{T}\to \mathbb{T}$ are defined by

$$\sigma \left(\xi \right)=inf\{s\in \mathbb{T}:s>\xi \},\rho \left(\xi \right)=sup\{s\in \mathbb{T}:s<\xi \}.$$

The forward graininess function is

$$\mu \left(\xi \right)=\sigma \left(\xi \right)-\xi .$$

2.1. Delta Differentiation and Integration

A function $f:\mathbb{T}\to \mathbb{R}$ is said to be delta differentiable at $\xi \in {\mathbb{T}}^{\kappa }$ if there exists a number $f^{\Delta }\left(\xi \right)$ such that, for every $\epsilon >0$, there exists a neighborhood U of $\xi$ satisfying

$$|f\left(\sigma \left(\xi \right)\right)-f\left(s\right)-f^{\Delta }\left(\xi \right)(\sigma \left(\xi \right)-s)|\le \epsilon |\sigma \left(\xi \right)-s|$$

for all $s\in U$.

The fundamental identity

$$f\left(\sigma \left(\xi \right)\right)=f\left(\xi \right)+\mu \left(\xi \right)f^{\Delta }\left(\xi \right)$$

holds whenever f is delta differentiable at $\xi$.

The Cauchy delta integral is defined by

$${\int }_{u_1}^{u_2}f\left(\xi \right)\Delta \xi =F\left(u_2\right)-F\left(u_1\right),$$

whenever $F^{\Delta }\left(\xi \right)=f\left(\xi \right)$ on ${[u_1,u_2]}_{\mathbb{T}}^{\kappa }$.

In particular:

• If $\mathbb{T}=\mathbb{R}$,

  $${\int }_{u_1}^{u_2}f\left(\xi \right)\Delta \xi ={\int }_{u_1}^{u_2}f\left(\xi \right)d\xi .$$
• If $\mathbb{T}=h\mathbb{Z}$,

  $${\int }_{u_1}^{u_2}f\left(\xi \right)\Delta \xi =h\sum _{k=0}^{{\displaystyle \frac{u_2-u_1}{h}}-1}f(u_1+kh).$$
• If $\mathbb{T}=q^{\mathbb{Z}}$,

  $${\int }_{u_1}^{u_2}f\left(\xi \right)\Delta \xi =(q-1)\sum _{\xi \in {[u_1,u_2)}_{\mathbb{T}}}\xi f\left(\xi \right).$$

2.2. Integration by Parts on Time Scales

If f and g are delta differentiable on ${[u_1,u_2]}_{\mathbb{T}}^{\kappa }$, then

$${\int }_{u_1}^{u_2}f\left(\xi \right)g^{\Delta }\left(\xi \right)\Delta \xi ={\left[f\left(\xi \right)g\left(\xi \right)\right]}_{u_1}^{u_2}-{\int }_{u_1}^{u_2}{f}^{\Delta }\left(\xi \right)g\left(\sigma \left(\xi \right)\right)\Delta \xi ,$$

or equivalently,

$${\int }_{u_1}^{u_2}f\left(\sigma \left(\xi \right)\right)g^{\Delta }\left(\xi \right)\Delta \xi ={\left[f\left(\xi \right)g\left(\xi \right)\right]}_{u_1}^{u_2}-{\int }_{u_1}^{u_2}{f}^{\Delta }\left(\xi \right)g\left(\xi \right)\Delta \xi .$$

2.3. Basic Properties of the Delta Derivative

For delta differentiable functions $f,g$ and constants $u_1,u_2\in \mathbb{R}$:

1.

${(u_{1}f+u_{2}g)}^{\Delta }\left(\xi \right)=u_1{f}^{\Delta }\left(\xi \right)+u_2{g}^{\Delta }\left(\xi \right)$.

2.

${\left(fg\right)}^{\Delta }\left(\xi \right)=f^{\Delta }\left(\xi \right)g\left(\xi \right)+f\left(\sigma \left(\xi \right)\right)g^{\Delta }\left(\xi \right)$.

3.

If $f\left(\xi \right)={\xi }^2$, then

$$f^{\Delta }\left(\xi \right)=\xi +\sigma \left(\xi \right).$$

In particular, on $\mathbb{T}=h\mathbb{Z}$,

$$f^{\Delta }\left(\xi \right)=2\xi +h.$$

4.

On $\mathbb{T}=q^{\mathbb{Z}}$, for $f\left(\xi \right)={\xi }^n$,

$$f^{\Delta }\left(\xi \right)={\displaystyle \frac{q^n-1}{q-1}}{\xi }^{n-1}.$$

3. Two-Sided Ostrowski-Type Inequalities on Time Scales

In this section, we develop a unified approach to Ostrowski-type inequalities on arbitrary time scales. The results are obtained by estimating the delta derivative of the basis function through appropriate integrable comparison functions. This approach provides sharp error bounds for the Ostrowski functional in a general time scale setting and will allow us to recover various continuous, discrete, and quantum versions known as special cases.

Theorem 1. 

Let $\mathbb{T}$ be a time scale and let $f:{[u_1,u_2]}_{\mathbb{T}}\to \mathbb{R}$ be delta differentiable on ${[u_1,u_2]}_{\mathbb{T}}$. Assume that $f^{\Delta }$ is delta integrable on ${[u_1,u_2]}_{\mathbb{T}}$ and that there exist delta integrable functions $m,M:{[u_1,u_2]}_{\mathbb{T}}\to \mathbb{R}$ such that

$$m\left(\xi \right)\le f^{\Delta }\left(\xi \right)\le M\left(\xi \right),\xi \in {[u_1,u_2]}_{\mathbb{T}}.$$

Then, for every $\varkappa \in {[u_1,u_2]}_{\mathbb{T}}$, the following two-sided Ostrowski-type inequality holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{u_2-u_1}}\left[{\int }_{u_1}^{\varkappa }(\xi -u_1)m\left(\xi \right)\Delta \xi +{\int }_{\varkappa }^{u_2}(u_2-\xi )m\left(\xi \right)\Delta \xi \right]\hfill \\ \hfill & \hfill \\ \hfill & \le f\left(\varkappa \right)-{\displaystyle \frac{1}{u_2-u_1}}{\int }_{u_1}^{u_2}f\left(\xi \right)\Delta \xi \hfill \\ \hfill & \hfill \\ \hfill & \le {\displaystyle \frac{1}{u_2-u_1}}\left[{\int }_{u_1}^{\varkappa }(\xi -u_1)M\left(\xi \right)\Delta \xi +{\int }_{\varkappa }^{u_2}(u_2-\xi )M\left(\xi \right)\Delta \xi \right].\hfill \end{array}$$

(2)

Proof. 

Fix $\varkappa \in {[u_1,u_2]}_{\mathbb{T}}$. For any $\xi \in {[u_1,u_2]}_{\mathbb{T}}$, by the Fundamental Theorem of Calculus on time scales,

$$f\left(\varkappa \right)-f\left(\xi \right)={\int }_{\xi }^{\varkappa }{f}^{\Delta }\left(\tau \right)\Delta \tau .$$

Integrating both sides with respect to $\xi$ over ${[u_1,u_2]}_{\mathbb{T}}$, we obtain

$${\int }_{u_1}^{u_2}\left[f\left(\varkappa \right)-f\left(\xi \right)\right]\Delta \xi ={\int }_{u_1}^{u_2}{\int }_{\xi }^{\varkappa }{f}^{\Delta }\left(\tau \right)\Delta \tau \Delta \xi .$$

The left-hand side simplifies to

$$f\left(\varkappa \right)(u_2-u_1)-{\int }_{u_1}^{u_2}f\left(\xi \right)\Delta \xi .$$

Since $f^{\Delta }$ is delta integrable, we may change the order of delta integration. Splitting the region according to $\xi \le \varkappa$ and $\xi \ge \varkappa$, we obtain

$${\int }_{u_1}^{\varkappa }{f}^{\Delta }\left(\tau \right)\left({\int }_{u_1}^{\tau }\Delta \xi \right)\Delta \tau +{\int }_{\varkappa }^{u_2}{f}^{\Delta }\left(\tau \right)\left({\int }_{\tau }^{u_2}\Delta \xi \right)\Delta \tau .$$

Using the identity valid on any time scale,

$${\int }_a^{b}1\Delta \xi =b-a,$$

we have

$${\int }_{u_1}^{\tau }\Delta \xi =\tau -u_1,{\int }_{\tau }^{u_2}\Delta \xi =u_2-\tau .$$

Hence,

$${\int }_{u_1}^{u_2}{\int }_{\xi }^{\varkappa }{f}^{\Delta }\left(\tau \right)\Delta \tau \Delta \xi ={\int }_{u_1}^{\varkappa }(\tau -u_1)f^{\Delta }\left(\tau \right)\Delta \tau +{\int }_{\varkappa }^{u_2}(u_2-\tau )f^{\Delta }\left(\tau \right)\Delta \tau .$$

Dividing by $u_2-u_1$, we obtain the identity

$$\begin{array}{ccc}& & f\left(\varkappa \right)-{\displaystyle \frac{1}{u_2-u_1}}{\int }_{u_1}^{u_2}f\left(\xi \right)\Delta \xi \hfill \\ & & \\ & =& {\displaystyle \frac{1}{u_2-u_1}}\left[{\int }_{u_1}^{\varkappa }(\xi -u_1)f^{\Delta }\left(\xi \right)\Delta \xi +{\int }_{\varkappa }^{u_2}(u_2-\xi )f^{\Delta }\left(\xi \right)\Delta \xi \right].\hfill \end{array}$$

Since

$$m\left(\xi \right)\le f^{\Delta }\left(\xi \right)\le M\left(\xi \right)$$

and the weights $(\xi -u_1)$ and $(u_2-\xi )$ are nonnegative on their respective intervals, we obtain

$${\int }_{u_1}^{\varkappa }(\xi -u_1)m\left(\xi \right)\Delta \xi \le {\int }_{u_1}^{\varkappa }(\xi -u_1)f^{\Delta }\left(\xi \right)\Delta \xi \le {\int }_{u_1}^{\varkappa }(\xi -u_1)M\left(\xi \right)\Delta \xi ,$$

and

$${\int }_{\varkappa }^{u_2}(u_2-\xi )m\left(\xi \right)\Delta \xi \le {\int }_{\varkappa }^{u_2}(u_2-\xi )f^{\Delta }\left(\xi \right)\Delta \xi \le {\int }_{\varkappa }^{u_2}(u_2-\xi )M\left(\xi \right)\Delta \xi .$$

Adding these inequalities and multiplying by ${\displaystyle \frac{1}{u_2-u_1}}$ yields (2). □

The bounds obtained in Theorem 1 can be further improved by introducing additional structural assumptions to the limiting functions. In particular, monotonicity leads to sharper and more explicit estimates.

Proposition 1. 

Let $\mathbb{T}$ be a time scale and let $u_1,u_2\in \mathbb{T}$ with $u_1<u_2$. Assume that $f:{[u_1,u_2]}_{\mathbb{T}}\to \mathbb{R}$ is delta differentiable on ${[u_1,u_2]}_{\mathbb{T}}^{\kappa }$ and that $f^{\Delta }$ is delta integrable on ${[u_1,u_2]}_{\mathbb{T}}$. Suppose that there exist delta integrable functions $m,M:{[u_1,u_2]}_{\mathbb{T}}\to \mathbb{R}$ such that

$$m\left(\xi \right)\le f^{\Delta }\left(\xi \right)\le M\left(\xi \right),\xi \in {[u_1,u_2]}_{\mathbb{T}}^{\kappa }.$$

If m and M are monotone on ${[u_1,u_2]}_{\mathbb{T}}$, then the bounds in Theorem 1 admit the estimates

$$\begin{array}{cc}\hfill f\left(\varkappa \right)-{\displaystyle \frac{1}{u_2-u_1}}{\int }_{u_1}^{u_2}f\left(\xi \right)\Delta \xi \ge & {\displaystyle \frac{m\left(u_1\right)}{u_2-u_1}}{\int }_{u_1}^{\varkappa }(\xi -u_1)\Delta \xi +{\displaystyle \frac{m\left(\varkappa \right)}{u_2-u_1}}{\int }_{\varkappa }^{u_2}(u_2-\xi )\Delta \xi ,\hfill \\ \hfill f\left(\varkappa \right)-{\displaystyle \frac{1}{u_2-u_1}}{\int }_{u_1}^{u_2}f\left(\xi \right)\Delta \xi \le & {\displaystyle \frac{M\left(\varkappa \right)}{u_2-u_1}}{\int }_{u_1}^{\varkappa }(\xi -u_1)\Delta \xi +{\displaystyle \frac{M\left(u_2\right)}{u_2-u_1}}{\int }_{\varkappa }^{u_2}(u_2-\xi )\Delta \xi .\hfill \end{array}$$

Proof. 

From Theorem 1 we have

$$\begin{array}{cc}\hfill f\left(\varkappa \right)-{\displaystyle \frac{1}{u_2-u_1}}{\int }_{u_1}^{u_2}f\left(\xi \right)\Delta \xi =& {\displaystyle \frac{1}{u_2-u_1}}\left[{\int }_{u_1}^{\varkappa }(\xi -u_1)f^{\Delta }\left(\xi \right)\Delta \xi +{\int }_{\varkappa }^{u_2}(u_2-\xi )f^{\Delta }\left(\xi \right)\Delta \xi \right].\hfill \end{array}$$

Assume that m is monotone increasing on ${[u_1,u_2]}_{\mathbb{T}}$. Then

$$m\left(\xi \right)\ge m\left(u_1\right),\xi \in {[u_1,\varkappa ]}_{\mathbb{T}},m\left(\xi \right)\ge m\left(\varkappa \right),\xi \in {[\varkappa ,u_2]}_{\mathbb{T}}.$$

Using $m\left(\xi \right)\le f^{\Delta }\left(\xi \right)$ and the positivity of the weights $(\xi -u_1)$ and $(u_2-\xi )$, we obtain

$$\begin{array}{cc}\hfill {\int }_{u_1}^{\varkappa }(\xi -u_1)f^{\Delta }\left(\xi \right)\Delta \xi \ge & m\left(u_1\right){\int }_{u_1}^{\varkappa }(\xi -u_1)\Delta \xi ,\hfill \\ \hfill {\int }_{\varkappa }^{u_2}(u_2-\xi )f^{\Delta }\left(\xi \right)\Delta \xi \ge & m\left(\varkappa \right){\int }_{\varkappa }^{u_2}(u_2-\xi )\Delta \xi .\hfill \end{array}$$

Substituting these estimates into the above identity yields the lower bound. For the upper bound, assume that M is monotone increasing on ${[u_1,u_2]}_{\mathbb{T}}$. Then

$$M\left(\xi \right)\le M\left(\varkappa \right),\xi \in {[u_1,\varkappa ]}_{\mathbb{T}},M\left(\xi \right)\le M\left(u_2\right),\xi \in {[\varkappa ,u_2]}_{\mathbb{T}}.$$

Using $f^{\Delta }\left(\xi \right)\le M\left(\xi \right)$ and the positivity of the weights $(\xi -u_1)$ and $(u_2-\xi )$, we obtain the corresponding upper estimate, which yields the stated bound. □

Remark 1. 

It is worth noting that the terms

$${\int }_{u_1}^{\varkappa }(\xi -u_1)\Delta \xi \mathit{and}{\int }_{\varkappa }^{u_2}(u_2-\xi )\Delta \xi$$

naturally depend on the graininess function $\mu \left(\xi \right)$ of the time scale. Indeed, on any time scale $\mathbb{T}$ the identity

$${\int }_a^b(\xi -a)\Delta \xi ={\displaystyle \frac{{(b-a)}^2}{2}}-{\displaystyle \frac{1}{2}}{\int }_a^b\mu \left(\xi \right)\Delta \xi$$

holds. Therefore, the lower and upper bounds obtained in Proposition 1 inherently incorporate the local structure of the time scale through the graininess function. This provides a refined error estimate compared with the classical continuous case where $\mu \left(\xi \right)=0$.

Remark 2. 

The bounds obtained in Proposition 1 are sharp for the class of functions whose delta derivatives are monotone. Indeed, if $f^{\Delta }\left(\xi \right)$ itself is a monotone function on ${[u_1,u_2]}_{\mathbb{T}}$, we may choose

$$m\left(\xi \right)=M\left(\xi \right)=f^{\Delta }\left(\xi \right).$$

Substituting this choice into the bounds of Proposition 1 reduces the inequalities to equalities and exactly reproduces the error functional

$$E\left(\varkappa \right)=f\left(\varkappa \right)-{\displaystyle \frac{1}{u_2-u_1}}{\int }_{u_1}^{u_2}f\left(\xi \right)\Delta \xi .$$

Hence the obtained bounds are best possible for this class of functions.

Remark 3. 

The estimates obtained in Theorem 1 and Proposition 1 are closely related to recent results in the literature. In particular, results similar in spirit were considered in [16], where inequalities on time scales were derived under derivative bounds given by constants.

The present results differ from [16] in two main aspects. First, the bounds for $f^{\Delta }$ are given by monotone functions m and M rather than constants, which allows a more flexible description of the behavior of the derivative. Second, the obtained estimates explicitly reflect the structure of the time scale through the delta integrals

$${\int }_{u_1}^{\sigma }(\xi -u_1)\Delta \xi ,{\int }_{\sigma }^{u_2}(u_2-\xi )\Delta \xi ,$$

whose values depend on the graininess function $\mu \left(\xi \right)$.

Consequently, the bounds obtained here may provide sharper estimates, especially on nonuniform or discrete time scales where the graininess function varies.

Comparison with Classical Ostrowski-Type Inequalities

Table 1. Comparison of features between classical and proposed Ostrowski-type inequalities.

Feature	Classical [1]	[9]	This Paper
Time scale	$\mathbb{R}$	Arbitrary $\mathbb{T}$	Arbitrary $\mathbb{T}$
Derivative bound	Constant M	Constant M	Functions $m\left(\xi \right),M\left(\xi \right)$
Error sensitivity	Global bound	Global bound	Local bound
Graininess $\mu \left(\xi \right)$	Not present	Implicit	Explicitly involved
Type of estimate	One-sided	One-sided	Two-sided

highlights the main differences between the classical Ostrowski inequality, its time scale extension, and the result obtained in Proposition 1.

The comparison shows that the present results provide more flexible bounds since the derivative constraints are allowed to vary along the time scale.

Example 1. 

Let $\mathbb{T}=\mathbb{R}$ and consider the function $f\left(\xi \right)={\xi }^2$ on $[0,1]$. Then

$$f^{\prime }\left(\xi \right)=2\xi .$$

Taking

$$m\left(\xi \right)=2\xi ,M\left(\xi \right)=2\xi ,$$

which are monotone functions on $[0,1]$, the bounds obtained in Proposition 1 reduce to equalities. For $\varkappa \in [0,1]$ we compute

$$E\left(\varkappa \right)={\varkappa }^2-{\int }_0^1{\xi }^{2}d\xi ={\varkappa }^2-{\displaystyle \frac{1}{3}}.$$

This example illustrates that the bounds in Proposition 1 are exact for functions with monotone derivatives.

Example 2. 

Let $\mathbb{T}=h\mathbb{Z}$ with step size $h>0$ and consider $f\left(t\right)=t^2$ on $[0,1]\cap h\mathbb{Z}$. The delta derivative becomes

$$f^{\Delta }\left(t\right)={\displaystyle \frac{{(t+h)}^2-t^2}{h}}=2t+h.$$

Thus the derivative bounds can be chosen as

$$m\left(t\right)=2t,M\left(t\right)=2t+h.$$

Applying Proposition 1 yields bounds that explicitly depend on the graininess h of the time scale.

Example 3. 

Consider the exponential function $f\left(\zeta \right)=e^{\zeta }$ on the time scale $\mathbb{T}=\mathbb{Z}$ or $\mathbb{T}=\mathbb{R}$. The delta derivative is given by

$$f^{\Delta }\left(\zeta \right)=\left\{\begin{array}{cc}{e}^{\zeta },\hfill & \mathbb{T}=\mathbb{R},\hfill \\ e^{\zeta +1}-e^{\zeta }=(e-1)e^{\zeta },\hfill & \mathbb{T}=\mathbb{Z}.\hfill \end{array}\right.$$

Applying the two-sided Ostrowski-type inequality from Proposition 1, we have

$$\left|f\left(\zeta \right)-{\displaystyle \frac{1}{b-a}}{\int }_a^{b}f\left(\sigma \left(t\right)\right)\Delta t\right|\le {\displaystyle \frac{{(b-a)}^2+{(2\zeta -a-b)}^2}{2(b-a)}}\underset{t\in {[a,b]}_{\mathbb{T}}}{sup}\left|f^{\Delta }\left(t\right)\right|.$$

For the interval $[a,b]=[0,4]$ and $\mathbb{T}=\mathbb{Z}$, the delta derivative values are

$$f^{\Delta }\left(t\right)=(e-1)e^t,t=0,1,2,3,4,$$

and the actual error $E\left(\zeta \right)$ is bounded between

$$L\left(\zeta \right)=-{\displaystyle \frac{{(b-a)}^2+{(2\zeta -a-b)}^2}{2(b-a)}}\underset{t\in {[a,b]}_{\mathbb{T}}}{sup}|f^{\Delta }\left(t\right)|,U\left(\zeta \right)={\displaystyle \frac{{(b-a)}^2+{(2\zeta -a-b)}^2}{2(b-a)}}\underset{t\in {[a,b]}_{\mathbb{T}}}{sup}\left|f^{\Delta }\left(t\right)\right|.$$

The numerical comparison between the actual error and the theoretical bounds can be plotted as in Figure 1.

4. Error Analysis

Now, let us show how Theorem 1 is reduced to classical discrete Ostrowski inequalities by considering certain choices of time scales as follows:

Corollary 1. 

Let $n_1,n_2\in \mathbb{Z}$ with $n_1<n_2$, and let $f:\{n_1,n_1+1,\dots ,n_2\}\to \mathbb{R}$ be a real sequence. Assume that there exist real constants L and U such that

$$L\le \Delta f\left(k\right):=f(k+1)-f\left(k\right)\le U,k=n_1,n_1+1,\dots ,n_2-1.$$

Then, for any $s\in \{n_1,n_1+1,\dots ,n_2\}$, the following inequality holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{L}{2(n_2-n_1)}}\left[(s-n_1-1)(s-n_1)+(n_2-s)(n_2-s+1)\right]\hfill \\ \hfill & \hfill \\ \hfill & \le f\left(s\right)-{\displaystyle \frac{1}{n_2-n_1}}\sum _{k=n_1}^{n_2-1}f\left(k\right)\hfill \\ \hfill & \hfill \\ \hfill & \le {\displaystyle \frac{U}{2(n_2-n_1)}}\left[(s-n_1-1)(s-n_1)+(n_2-s)(n_2-s+1)\right].\hfill \end{array}$$

(3)

Proof. 

This result follows as a direct specialization of Theorem 1 to the time scale $\mathbb{T}=\mathbb{Z}$.

On the integer time scale, the delta derivative reduces to the forward difference operator,

$$f^{\Delta }\left(k\right)=\Delta f\left(k\right)=f(k+1)-f\left(k\right),$$

and the delta integral becomes a finite sum,

$${\int }_a^{b}g\left(k\right)\Delta k=\sum _{k=a}^{b-1}g\left(k\right).$$

Substituting these identities into inequality (2) of Theorem 1 yields

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{n_2-n_1}}\left[\sum _{k=n_1}^{s-1}(k-n_1)m_k+\sum _{k=s}^{n_2-1}(n_2-k)m_k\right]\hfill \\ \hfill & \\ \hfill & \le f\left(s\right)-{\displaystyle \frac{1}{n_2-n_1}}\sum _{k=n_1}^{n_2-1}f\left(k\right)\hfill \\ \hfill & \\ \hfill & \le {\displaystyle \frac{1}{n_2-n_1}}\left[\sum _{k=n_1}^{s-1}(k-n_1)M_k+\sum _{k=s}^{n_2-1}(n_2-k)M_k\right].\hfill \end{array}$$

Under the additional assumption that the bounds are constant, namely $m_k=L$ and $M_k=U$ for all k, the weighted sums can be evaluated explicitly using standard formulas for arithmetic series:

$$\sum _{k=n_1}^{s-1}(k-n_1)={\displaystyle \frac{(s-n_1-1)(s-n_1)}{2}},$$

and

$$\sum _{k=s}^{n_2-1}(n_2-k)={\displaystyle \frac{(n_2-s)(n_2-s+1)}{2}}.$$

Substituting these expressions into the previous inequality immediately yields (3), which completes the proof. □

Remark 4. 

If $n_2-n_1$ is even and $s={\displaystyle \frac{n_1+n_2}{2}}$ is the midpoint of the interval, then inequality (3) reduces to

$$\left|f\left(s\right)-{\displaystyle \frac{1}{n_2-n_1}}\sum _{k=n_1}^{n_2-1}f\left(k\right)\right|\le {\displaystyle \frac{n_2-n_1}{4}}max\left\{\right|L|,|U\left|\right\},$$

which represents the discrete analogue of the classical Ostrowski inequality for differentiable functions.

Corollary 2. 

Let $q>1$ and let $\mathbb{T}=q^{\mathbb{Z}}$ be the quantum time scale. Assume that $u_1,u_2\in \mathbb{T}$ with $u_1<u_2$, and let $f:{[u_1,u_2]}_{\mathbb{T}}\to \mathbb{R}$ be continuous on ${[u_1,u_2]}_{\mathbb{T}}$ and delta differentiable on ${[u_1,u_2]}_{\mathbb{T}}^{\kappa }$. Suppose that $f^{\Delta }$ is delta integrable and that there exist delta integrable functions $m,M:{[u_1,u_2]}_{\mathbb{T}}\to \mathbb{R}$ such that

$$m\left(\xi \right)\le f^{\Delta }\left(\xi \right)\le M\left(\xi \right),\xi \in {[u_1,u_2]}_{\mathbb{T}}^{\kappa }.$$

Then, for every $\varkappa \in {[u_1,u_2]}_{\mathbb{T}}$, the following quantum Ostrowski-type inequality holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{u_2-u_1}}\left[(q-1)\sum _{\xi \in {[u_1,\varkappa )}_{\mathbb{T}}}\xi (\xi -u_1)m\left(\xi \right)+(q-1)\sum _{\xi \in {[\varkappa ,u_2)}_{\mathbb{T}}}\xi (u_2-\xi )m\left(\xi \right)\right]\hfill \\ \hfill & \hfill \\ \hfill & \le f\left(\varkappa \right)-{\displaystyle \frac{q-1}{u_2-u_1}}\sum _{\xi \in {[u_1,u_2)}_{\mathbb{T}}}\xi f\left(\xi \right)\hfill \\ \hfill & \hfill \\ \hfill & \le {\displaystyle \frac{1}{u_2-u_1}}\left[(q-1)\sum _{\xi \in {[u_1,\varkappa )}_{\mathbb{T}}}\xi (\xi -u_1)M\left(\xi \right)+(q-1)\sum _{\xi \in {[\varkappa ,u_2)}_{\mathbb{T}}}\xi (u_2-\xi )M\left(\xi \right)\right].\hfill \end{array}$$

(4)

Proof. 

The result follows directly from Theorem 1 by choosing the time scale $\mathbb{T}=q^{\mathbb{Z}}$.

On the quantum time scale, the delta derivative is

$$f^{\Delta }\left(\xi \right)={\displaystyle \frac{f\left(q\xi \right)-f\left(\xi \right)}{(q-1)\xi }},$$

and the delta integral reduces to the weighted sum

$${\int }_a^{b}g\left(\xi \right)\Delta \xi =(q-1)\sum _{\xi \in {[a,b)}_{\mathbb{T}}}\xi g\left(\xi \right).$$

Applying these representations to the integrals appearing in (2), we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{u_2-u_1}}\left[(q-1)\sum _{\xi \in {[u_1,\varkappa )}_{\mathbb{T}}}\xi (\xi -u_1)m\left(\xi \right)+(q-1)\sum _{\xi \in {[\varkappa ,u_2)}_{\mathbb{T}}}\xi (u_2-\xi )m\left(\xi \right)\right]\hfill \\ \hfill & \\ \hfill & \le f\left(\varkappa \right)-{\displaystyle \frac{q-1}{u_2-u_1}}\sum _{\xi \in {[u_1,u_2)}_{\mathbb{T}}}\xi f\left(\xi \right)\hfill \\ \hfill & \\ \hfill & \le {\displaystyle \frac{1}{u_2-u_1}}\left[(q-1)\sum _{\xi \in {[u_1,\varkappa )}_{\mathbb{T}}}\xi (\xi -u_1)M\left(\xi \right)+(q-1)\sum _{\xi \in {[\varkappa ,u_2)}_{\mathbb{T}}}\xi (u_2-\xi )M\left(\xi \right)\right],\hfill \end{array}$$

which completes the proof. □

Remark 5. 

Assume that the bounds of the delta derivative are constants, that is, $m\left(\xi \right)\equiv L$ and $M\left(\xi \right)\equiv U$ with $L\le U$. Then inequality (4) yields the explicit quantum Ostrowski-type estimate

$$\begin{array}{ccc}& & \left|f\left(\varkappa \right)-{\displaystyle \frac{q-1}{u_2-u_1}}\sum _{\xi \in {[u_1,u_2)}_{\mathbb{T}}}\xi f\left(\xi \right)\right|\hfill \\ & & \\ & \le & {\displaystyle \frac{U-L}{u_2-u_1}}\left[{\int }_{u_1}^{\varkappa }(\xi -u_1)\Delta \xi +{\int }_{\varkappa }^{u_2}(u_2-\xi )\Delta \xi \right].\hfill \end{array}$$

On the quantum time scale $\mathbb{T}=q^{\mathbb{Z}}$, the delta integrals admit the explicit representations

$${\int }_{u_1}^{\varkappa }(\xi -u_1)\Delta \xi =(q-1)\sum _{\xi \in {[u_1,\varkappa )}_{\mathbb{T}}}\xi (\xi -u_1),$$

$${\int }_{\varkappa }^{u_2}(u_2-\xi )\Delta \xi =(q-1)\sum _{\xi \in {[\varkappa ,u_2)}_{\mathbb{T}}}\xi (u_2-\xi ).$$

Moreover, as $q\to 1^{+}$, the quantum delta derivative and delta integral converge to the classical derivative and the Riemann integral, and the above estimate reduces to the classical Ostrowski inequality on real intervals.

Corollary 3. 

Let $h>0$ and let $\mathbb{T}=h\mathbb{Z}$. Assume that $u_1,u_2\in \mathbb{T}$ with $u_1<u_2$, and let $f:{[u_1,u_2]}_{\mathbb{T}}\to \mathbb{R}$. Suppose that there exist functions $m,M:{[u_1,u_2]}_{\mathbb{T}}^{\kappa }\to \mathbb{R}$ such that

$$m\left(\xi \right)\le f^{\Delta }\left(\xi \right)={\displaystyle \frac{f(\xi +h)-f\left(\xi \right)}{h}}\le M\left(\xi \right),\xi \in {[u_1,u_2]}_{\mathbb{T}}^{\kappa }.$$

Then, for every $\varkappa \in {[u_1,u_2]}_{\mathbb{T}}$, the following discrete Ostrowski-type inequality holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{h}{u_2-u_1}}\left[\sum _{\xi \in {[u_1,\varkappa )}_{\mathbb{T}}}(\xi -u_1)m\left(\xi \right)+\sum _{\xi \in {[\varkappa ,u_2)}_{\mathbb{T}}}(u_2-\xi )m\left(\xi \right)\right]\hfill \\ \hfill & \hfill \\ \hfill & \le f\left(\varkappa \right)-{\displaystyle \frac{h}{u_2-u_1}}\sum _{\xi \in {[u_1,u_2)}_{\mathbb{T}}}f\left(\xi \right)\hfill \\ \hfill & \hfill \\ \hfill & \le {\displaystyle \frac{h}{u_2-u_1}}\left[\sum _{\xi \in {[u_1,\varkappa )}_{\mathbb{T}}}(\xi -u_1)M\left(\xi \right)+\sum _{\xi \in {[\varkappa ,u_2)}_{\mathbb{T}}}(u_2-\xi )M\left(\xi \right)\right].\hfill \end{array}$$

(5)

Remark 6. 

This corollary follows from Theorem 1 by taking the uniform time scale $\mathbb{T}=h\mathbb{Z}$. Let $u_1,u_2\in \mathbb{T}$ with $u_1<u_2$. On this time scale, the forward jump operator satisfies $\sigma \left(\xi \right)=\xi +h$, the graininess function is constant $\mu \left(\xi \right)=h$, and the delta derivative reduces to

$$f^{\Delta }\left(\xi \right)={\displaystyle \frac{f(\xi +h)-f\left(\xi \right)}{h}},\xi \in {[u_1,u_2-h]}_{\mathbb{T}}.$$

Moreover, for any function $g:{[u_1,u_2]}_{\mathbb{T}}\to \mathbb{R}$, the delta integral over ${[u_1,u_2]}_{\mathbb{T}}$ becomes

$${\int }_{u_1}^{u_2}g\left(\xi \right)\Delta \xi =h\sum _{k=0}^{{\displaystyle \frac{u_2-u_1}{h}}-1}g(u_1+kh).$$

Therefore, inequality (5) represents the discrete version of Theorem 1 on the uniform lattice $\mathbb{T}=h\mathbb{Z}$. In particular, when $h=1$, it reduces to the classical Ostrowski-type inequality for integer sequences.

Corollary 4. 

Let $h>0$ and let $\mathbb{T}=h\mathbb{Z}$. Assume that $u_1,u_2\in \mathbb{T}$ with $u_1<u_2$, and let $f:{[u_1,u_2]}_{\mathbb{T}}\to \mathbb{R}$. Suppose that the forward differences satisfy the constant bounds

$$L\le {\Delta }_{h}f\left(\xi \right):={\displaystyle \frac{f(\xi +h)-f\left(\xi \right)}{h}}\le U,\xi \in {[u_1,u_2-h]}_{\mathbb{T}}.$$

Then, for any $\varkappa \in {[u_1,u_2]}_{\mathbb{T}}$, the following explicit Ostrowski-type inequality holds:

$$\begin{array}{ccc}& & \left|f\left(\varkappa \right)-{\displaystyle \frac{h}{u_2-u_1}}\sum _{k=0}^{{\displaystyle \frac{u_2-u_1}{h}}-1}f(u_1+kh)\right|\hfill \\ & & \\ & \le & {\displaystyle \frac{U-L}{2(u_2-u_1)}}\left[{(\varkappa -u_1)}^2+{(u_2-\varkappa )}^2-h(u_2-u_1)\right].\hfill \end{array}$$

Proof. 

From Corollary 3, setting $m\left(\xi \right)=L$ and $M\left(\xi \right)=U$, we obtain

$$\begin{array}{ccc}& & \left|f\left(\varkappa \right)-{\displaystyle \frac{h}{u_2-u_1}}\sum _{k=0}^{{\displaystyle \frac{u_2-u_1}{h}}-1}f(u_1+kh)\right|\hfill \\ & & \\ & \le & {\displaystyle \frac{U-L}{u_2-u_1}}\left[h\sum _{k=0}^{N_1-1}kh+h\sum _{k=0}^{N_2-1}(u_2-\varkappa -kh)\right],\hfill \end{array}$$

where

$$N_1={\displaystyle \frac{\varkappa -u_1}{h}},N_2={\displaystyle \frac{u_2-\varkappa }{h}}.$$

Using the classical Gauss summation formula

$$\sum _{k=0}^{n-1}k={\displaystyle \frac{n(n-1)}{2}},$$

we compute

$$h\sum _{k=0}^{N_1-1}kh={\displaystyle \frac{{(\varkappa -u_1)}^2-h(\varkappa -u_1)}{2}},$$

and

$$h\sum _{k=0}^{N_2-1}(u_2-\varkappa -kh)={\displaystyle \frac{{(u_2-\varkappa )}^2-h(u_2-\varkappa )}{2}}.$$

Adding these expressions yields

$${\displaystyle \frac{1}{2}}\left[{(\varkappa -u_1)}^2+{(u_2-\varkappa )}^2-h(u_2-u_1)\right].$$

Multiplying by ${\displaystyle \frac{U-L}{u_2-u_1}}$ completes the proof. □

Proposition  2. 

Let $\mathbb{T}$ be a time scale and let $u_1,u_2\in \mathbb{T}$ with $u_1<u_2$. Assume that $f:{[u_1,u_2]}_{\mathbb{T}}\to \mathbb{R}$ is delta differentiable on ${[u_1,u_2]}_{\mathbb{T}}^{\kappa }$ and that

$$m\left(\xi \right)\le f^{\Delta }\left(\xi \right)\le M\left(\xi \right),\xi \in {[u_1,u_2]}_{\mathbb{T}}^{\kappa },$$

where m and M are delta integrable and monotone functions on ${[u_1,u_2]}_{\mathbb{T}}$. Then, for any $\sigma \in {[u_1,u_2]}_{\mathbb{T}}$, the following estimates hold.

(i) 

If m is increasing on ${[u_1,u_2]}_{\mathbb{T}}$, then

$$\begin{array}{ccc}& & {\int }_{u_1}^{\sigma }(\xi -u_1)m\left(\xi \right)\Delta \xi +{\int }_{\sigma }^{u_2}(u_2-\xi )m\left(\xi \right)\Delta \xi \hfill \\ & & \\ & \le & m\left(\sigma \right){\int }_{u_1}^{\sigma }(\xi -u_1)\Delta \xi +m\left(u_2\right){\int }_{\sigma }^{u_2}(u_2-\xi )\Delta \xi .\hfill \end{array}$$

(ii) 

If m is decreasing on ${[u_1,u_2]}_{\mathbb{T}}$, then

$$\begin{array}{ccc}& & {\int }_{u_1}^{\sigma }(\xi -u_1)m\left(\xi \right)\Delta \xi +{\int }_{\sigma }^{u_2}(u_2-\xi )m\left(\xi \right)\Delta \xi \hfill \\ & & \\ & \ge & m\left(u_1\right){\int }_{u_1}^{\sigma }(\xi -u_1)\Delta \xi +m\left(\sigma \right){\int }_{\sigma }^{u_2}(u_2-\xi )\Delta \xi .\hfill \end{array}$$

Analogous inequalities hold for M with the inequality signs reversed where appropriate.

Proof. 

We prove part (i); part (ii) follows analogously by reversing the inequalities.

Assume that m is increasing on ${[u_1,u_2]}_{\mathbb{T}}$. Then, for all $\xi \in {[u_1,\varkappa ]}_{\mathbb{T}}$, we have $m\left(\xi \right)\le m\left(\varkappa \right)$. Since $\xi -u_1\ge 0$ on this interval, it follows that

$$(\xi -u_1)m\left(\xi \right)\le (\xi -u_1)m\left(\varkappa \right).$$

Integrating both sides with respect to the delta integral yields

$${\int }_{u_1}^{\varkappa }(\xi -u_1)m\left(\xi \right)\Delta \xi \le m\left(\varkappa \right){\int }_{u_1}^{\varkappa }(\xi -u_1)\Delta \xi .$$

Similarly, for $\xi \in {[\varkappa ,u_2]}_{\mathbb{T}}$, monotonicity of m implies $m\left(\xi \right)\le m\left(u_2\right)$, and since $u_2-\xi \ge 0$, we obtain

$${\int }_{\varkappa }^{u_2}(u_2-\xi )m\left(\xi \right)\Delta \xi \le m\left(u_2\right){\int }_{\varkappa }^{u_2}(u_2-\xi )\Delta \xi .$$

Adding the two inequalities completes the proof of part (i). □

Remark 7. 

If $\mathbb{T}=\mathbb{R}$, then the delta integral coincides with the classical Riemann integral and the identities

$${\int }_{u_1}^{u_2}(\xi -u_1)d\xi ={\displaystyle \frac{{(u_2-u_1)}^2}{2}},{\int }_{u_1}^{u_2}(u_2-\xi )d\xi ={\displaystyle \frac{{(u_2-u_1)}^2}{2}}$$

hold exactly. In this case, Proposition 2 reduces to the classical monotone bounds used in Ostrowski-type inequalities. For general time scales, the integrals

$${\int }_{u_1}^{u_2}(\xi -u_1)\Delta \xi \mathrm{and}{\int }_{u_1}^{u_2}(u_2-\xi )\Delta \xi$$

depend on the graininess function $\mu \left(\xi \right)$ and are, therefore, left in integral form.

Remark 8. 

Proposition 2 significantly enhances the applicability of the main theorem on time scales. In discrete settings, such as $\mathbb{T}=\mathbb{Z}$ or $\mathbb{T}=h\mathbb{Z}$, the error bounds can be evaluated by using only finitely many terms of a sequence, avoiding the explicit computation of delta integrals. In quantum calculus, where the q-integral may become technically involved, the monotonicity of the bounding functions allows one to estimate the bounds using only endpoint values. This feature makes the result particularly effective for numerical and computational purposes.

Example 4. 

Let $\mathbb{T}=h\mathbb{Z}$ with $h>0$, and consider the interval ${[u_1,u_2]}_{\mathbb{T}}={[0,Nh]}_{\mathbb{T}}$ for some $N\in \mathbb{N}$. Let $f\left(\xi \right)={\xi }^2$, so that

$$f^{\Delta }\left(\xi \right)={\displaystyle \frac{{(\xi +h)}^2-{\xi }^2}{h}}=2\xi +h.$$

Clearly, $f^{\Delta }$ is increasing on ${[0,Nh]}_{\mathbb{T}}$. Set

$$m\left(\xi \right)=2\xi +h,M\left(\xi \right)=2\xi +h.$$

For $\varkappa =kh$ with $k\in \{1,2,\dots ,N-1\}$, Proposition 2 yields

$$\begin{array}{ccc}& & {\int }_0^{kh}(\xi -0)m\left(\xi \right)\Delta \xi +{\int }_{kh}^{Nh}(Nh-\xi )m\left(\xi \right)\Delta \xi \hfill \\ & & \\ & \le & m\left(kh\right){\int }_0^{kh}(\xi -0)\Delta \xi +m\left(Nh\right){\int }_{kh}^{Nh}(Nh-\xi )\Delta \xi .\hfill \end{array}$$

Since

$${\int }_0^{kh}\xi \Delta \xi =h\sum _{j=0}^{k-1}\left(jh\right)=h^2\sum _{j=0}^{k-1}j,$$

and

$${\int }_{kh}^{Nh}(Nh-\xi )\Delta \xi =h\sum _{j=k}^{N-1}(Nh-jh)=h^2\sum _{j=k}^{N-1}(N-j),$$

the bound is obtained explicitly in terms of finite sums, without any approximation.

Example 5. 

Let $\mathbb{T}=q^{\mathbb{Z}}$ with $q>1$, and consider the interval ${[u_1,u_2]}_{\mathbb{T}}={[q^{n_1},q^{n_2}]}_{\mathbb{T}}$, where $n_1<n_2$ are integers. Let

$$f\left(\xi \right)={\xi }^{k+1},k\ge 1.$$

Then the delta derivative on $q^{\mathbb{Z}}$ is given by

$$f^{\Delta }\left(\xi \right)={\displaystyle \frac{f\left(q\xi \right)-f\left(\xi \right)}{(q-1)\xi }}={\displaystyle \frac{q^{k+1}-1}{q-1}}{\xi }^k.$$

Hence $f^{\Delta }$ is increasing on ${[u_1,u_2]}_{\mathbb{T}}$. Define

$$m\left(\xi \right)=M\left(\xi \right)={\displaystyle \frac{q^{k+1}-1}{q-1}}{\xi }^k.$$

For $\varkappa =q^n$ with $n_1<n<n_2$, Theorem 1 gives

$$\begin{array}{ccc}& & f\left(\varkappa \right)-{\displaystyle \frac{1}{u_2-u_1}}{\int }_{u_1}^{u_2}f\left(\xi \right)\Delta \xi \hfill \\ & & \\ & =& {\displaystyle \frac{1}{u_2-u_1}}\left[{\int }_{u_1}^{\varkappa }(\xi -u_1)f^{\Delta }\left(\xi \right)\Delta \xi +{\int }_{\varkappa }^{u_2}(u_2-\xi )f^{\Delta }\left(\xi \right)\Delta \xi \right].\hfill \end{array}$$

Since $m=f^{\Delta }$ is increasing on ${[u_1,u_2]}_{\mathbb{T}}$, Proposition 2(i) implies

$$\begin{array}{ccc}& & {\int }_{u_1}^{\varkappa }(\xi -u_1)m\left(\xi \right)\Delta \xi +{\int }_{\varkappa }^{u_2}(u_2-\xi )m\left(\xi \right)\Delta \xi \hfill \\ & & \\ & \le & m\left(\varkappa \right){\int }_{u_1}^{\varkappa }(\xi -u_1)\Delta \xi +m\left(u_2\right){\int }_{\varkappa }^{u_2}(u_2-\xi )\Delta \xi .\hfill \end{array}$$

Consequently, we obtain the estimate

$$\begin{array}{cc}\hfill & f\left(x\right)-{\displaystyle \frac{1}{u_2-u_1}}{\int }_{u_1}^{u_2}f\left(\lambda \right)\Delta \lambda \hfill \\ \hfill & \le {\displaystyle \frac{1}{u_2-u_1}}\left[m\left(x\right){\int }_{u_1}^x(\lambda -u_1)\Delta \lambda +m\left(u_2\right){\int }_x^{u_2}(u_2-\lambda )\Delta \lambda \right],\hfill \end{array}$$

which provides a sharper Ostrowski-type bound expressed only in terms of the endpoint values $m\left(\varkappa \right)$ and $m\left(u_2\right)$.

In this section, we analyze the deviation between the function value $f\left(\varkappa \right)$ and its delta integral mean over the interval ${[u_1,u_2]}_{\mathbb{T}}$. Throughout, we follow the notation of Theorem 1.

Define the error functional by

$$E\left(\varkappa \right):=f\left(\varkappa \right)-{\displaystyle \frac{1}{u_2-u_1}}{\int }_{u_1}^{u_2}f\left(\xi \right)\Delta \xi ,\varkappa \in {[u_1,u_2]}_{\mathbb{T}}.$$

4.1. Error Bound Under Derivative Constraints

Assume that

$$m\left(\xi \right)\le f^{\Delta }\left(\xi \right)\le M\left(\xi \right),\xi \in {[u_1,u_2]}_{\mathbb{T}},$$

as in Theorem 1. If

$$m\left(\xi \right)\equiv L,M\left(\xi \right)\equiv U,$$

with constants $L\le U$, then Theorem 1 yields

$$\left|E\left(\varkappa \right)\right|\le {\displaystyle \frac{U-L}{u_2-u_1}}\left[{\int }_{u_1}^{\varkappa }(\xi -u_1)\Delta \xi +{\int }_{\varkappa }^{u_2}(u_2-\xi )\Delta \xi \right].$$

4.2. Minimizing the Error Bound

Set

$$\mathcal{E}\left(\varkappa \right)={\int }_{u_1}^{\varkappa }(\xi -u_1)\Delta \xi +{\int }_{\varkappa }^{u_2}(u_2-\xi )\Delta \xi .$$

In the continuous case $\mathbb{T}=\mathbb{R}$,

$$\mathcal{E}\left(\varkappa \right)={\displaystyle \frac{{(\varkappa -u_1)}^2}{2}}+{\displaystyle \frac{{(u_2-\varkappa )}^2}{2}},$$

which is minimized at

$$\varkappa ={\displaystyle \frac{u_1+u_2}{2}}.$$

For uniform discrete time scales $\mathbb{T}=h\mathbb{Z}$, the minimum is attained at a grid point closest to ${\displaystyle \frac{u_1+u_2}{2}}$.

4.3. Endpoint Behavior on Uniform Time Scales

For $\mathbb{T}=h\mathbb{Z}$,

$$\mathcal{E}\left(\varkappa \right)\le {\displaystyle \frac{{(u_2-u_1)}^2}{2}},$$

and, therefore,

$$\left|E\left(\varkappa \right)\right|\le {\displaystyle \frac{U-L}{2}}(u_2-u_1).$$

5. Numerical Illustration of the Error Bound

In this section, we illustrate the sharpness of Theorem 1 on the uniform discrete time scale $\mathbb{T}=h\mathbb{Z}$. The purpose is to compare the actual error with the theoretical bound.

Uniform Discrete Time Scale

Let $u_1=0$, $u_2=4$, and $h=1$. Consider the function

$$f\left(\xi \right)={\xi }^2$$

defined on ${[u_1,u_2]}_{h\mathbb{Z}}=\{0,1,2,3,4\}$.

Delta integral mean. Since

$${\int }_0^{4}f\left(\xi \right)\Delta \xi =\sum _{k=0}^3{k}^2=14,$$

the delta integral mean is

$$\overline{f}={\displaystyle \frac{1}{u_2-u_1}}{\int }_0^{4}f\left(\xi \right)\Delta \xi ={\displaystyle \frac{14}{4}}=3.5.$$

Actual error. The error functional is

$$E\left(\varkappa \right)=f\left(\varkappa \right)-\overline{f}={\varkappa }^2-3.5.$$

$$\begin{array}{ccc}\hfill \varkappa \hfill & f\left(\varkappa \right)& \hfill \left|E\right(\varkappa \left)\right|\hfill \\ \hfill 0\hfill & 0& \hfill 3.5\hfill \\ \hfill 1\hfill & 1& \hfill 2.5\hfill \\ \hfill 2\hfill & 4& \hfill 0.5\hfill \\ \hfill 3\hfill & 9& \hfill 5.5\hfill \\ \hfill 4\hfill & 16& \hfill 12.5\hfill \end{array}$$

The comparison between the actual errors and the theoretical bounds across different time scales is illustrated in Figure 2.

Theoretical bound from Theorem 1.

On ${[0,4]}_{h\mathbb{Z}}$,

$$f^{\Delta }\left(\xi \right)=2\xi +1,$$

hence

$$L=1,U=7,U-L=6.$$

The bound given by Theorem 1 becomes

$$\left|E\left(\varkappa \right)\right|\le {\displaystyle \frac{6}{4}}\left[{\int }_0^{\varkappa }\xi \Delta \xi +{\int }_{\varkappa }^4(4-\xi )\Delta \xi \right].$$

For example, at $\varkappa =4$ we obtain

$$\left|E\left(4\right)\right|\le {\displaystyle \frac{6}{4}}\sum _{k=0}^3(4-k)={\displaystyle \frac{6}{4}}·10=15.$$

Since the actual error is $\left|E\right(4\left)\right|=12.5$, the estimate is verified.

6. Conclusions

In this paper, we established two-sided Ostrowski-type inequalities on arbitrary time scales by assuming upper and lower bounds for the delta derivative of the function. The obtained inequalities provide explicit estimates for the deviation $f\left(\varkappa \right)-{\displaystyle \frac{1}{u_2-u_1}}{\int }_{u_1}^{u_2}f\left(\xi \right)\Delta \xi$ and include, as particular cases, the classical continuous, uniform discrete, and quantum settings.

Furthermore, the established results provide a rigorous “safety corridor” for error estimation in the numerical analysis of dynamic equations on time scales. In practical scenarios such as population dynamics or control theory, where growth rates or parameters often vary monotonically, our bounds offer a reliable tool to determine the maximum possible deviation from average behavior without requiring an exact solution to the dynamic equation. This connection between abstract time scale theory and applied dynamic systems ensures the robustness of numerical simulations across non-uniform time domains.

Under additional monotonicity assumptions regarding the limiting functions, improved integral estimates were obtained that remained fully consistent with the structure of the underlying time scale. These improvements are particularly important when the granularity function is not constant, since the error limit then clearly depends on the time scale geometry. Numerical samples on uniform discrete and quantum time scales validate the theoretical predictions. The approach developed here can be extended to weighted Ostrowski-type inequalities or inequalities involving higher-order delta derivatives.