Weighted Milne-type inequalities through Riemann-Liouville fractional integrals and diverse function classes

1.   Introduction

Exploring error upper limits through numerical integration formulas and various methodologies has been an area of study for numerous mathematicians. The investigation into error bounds for these formulas involves the analysis of mathematical inequalities via different types of functions, involving convex, Lipschitzian, and bounded functions. This paper specifically focuses on establishing bounds for functions where either the derivatives or second derivatives exhibit convex behavior.

First, let's present a few numerical integration methods along with their respective upper error bounds:

(ⅰ) The expression below represents Simpson's quadrature formula, commonly known as Simpson's $1/3$ rule:

(ⅱ) The characterization of Simpson's second formula, also known as the Newton-Cotes quadratic formula or Simpson's $3/8$ rule (see ^[1]), is as follows:

Formulas (1.1) and (1.2) are applicable to any function $\mathcal{P}$ that has a continuous fourth derivative over the interval $[\eta, \epsilon]$.

The classical Simpson inequality ^[2] is expressed as follows:

Theorem 1.1. Assume $\mathcal{P}:\left[ \eta, \epsilon\right] \rightarrow \mathbb{R}$ is a function with continuous fourth derivative on $\left(\eta, \epsilon\right),$ and $\left \Vert \mathcal{P}^{\left(4\right) }\right \Vert _{\infty } = \underset{x\in \left(\eta, \epsilon\right) }{\sup }\left \vert \mathcal{P}^{\left(4\right) }(x)\right \vert < \infty.$ Then, the following inequality holds:

Sarikaya et al. initially established the Simpson type inequality using convex functions in ^[3]. Regarding Riemann-Liouville fractional integrals, there exist three distinct types of Simpson inequalities characterized by the representation of fractional integrals. The proofs for these inequalities are documented in the papers ^[4,5,6]. Additionally, some papers are devoted to Simpson-type inequalities specifically tailored for twice-differentiable functions, as demonstrated ^{[7,8,9,10,11]}.

The classical Newton inequality is expressed as follows:

Theorem 1.2. (See ^[1]) Suppose $\mathcal{P}:\left[ \eta, \epsilon\right] \rightarrow \mathbb{R}$ is a function with four continuous derivatives on the open interval $\left(\eta, \epsilon\right),$ and $\left \Vert \mathcal{P}^{\left(4\right) }\right \Vert _{\infty } = \underset{x\in \left(\eta, \epsilon\right) }{\sup }\left \vert \mathcal{P}^{\left(4\right) }(x)\right \vert < \infty.$ In such a case, the subsequent inequality is satisfied:

In the works ^[12,13,14], several Newton-type inequalities were presented, utilizing convex functions for local fractional integrals. In the paper ^[15], the authors were the first to establish Newton-type inequalities for Riemann-Liouville fractional integrals. Subsequent to this development, several papers have been devoted to deriving Newton-type inequalities for Riemann-Liouville fractional integrals ^[16,17]. Gao and Shi demonstrated Newton-type inequalities for functions with two derivatives ^[18].

Djenaoui and Meftah were the first to establish Milne-type inequalities employing the concept of convexity ^[19]. Subsequently, Budak et al. extended these inequalities to Riemann-Liouville fractional integrals ^[20]. In the same work, the authors introduced several Milne-type inequalities applicable to diverse function classes, including bounded functions, Lipschitz functions, and functions of bounded variation. Furthermore, in recent publications such as ^[11,21], new fractional versions of Milne-type inequalities have been derived using differentiable convex functions and various function classes like bounded functions, Lipschitz functions, and functions of bounded variation. For additional Milne-type inequalities, please see ^{[22,23,24,25]}.

The objective of this paper is to derive some fractional Milne-type inequalities for functions whose second derivatives exhibit convexity. First, we introduce a fundamental integral equality enabled by a nonnegative weighted function, essential for deriving the main results. Second, we systematically establish weighted Milne-type inequalities across various function classes, encompassing differentiable convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation. These results not only advance the understanding of Milne-type inequalities but also provide insights for potential future research in the considered topics. Moreover, the findings of this study incorporate numerous results previously presented in various studies as special cases, thus consolidating and extending the existing knowledge in the field.

To achieve this goal, we will rely on the following definition of Riemann-Liouville fractional integrals.

The commonly known Riemann-Liouville fractional integrals are defined as follows:

Definition 1.1. (See ^[26,27]) The Riemann-Liouville integrals $\mathcal{I}_{\eta+}^{\delta }\mathcal{P}$ and $\mathcal{I}_{\epsilon-}^{\delta }\mathcal{P}$ of order $\delta > 0$ are expressed as

and

respectively. Here, $\mathcal{P}$ belongs to $L_{1}[\eta, \epsilon]$, and $\Gamma (\delta)$ denotes the Gamma function, which is defined as

It is noteworthy that the fractional integral converges to the classical integral when $\delta$ approaches $1$.

This paper is structured into seven parts, starting with an introduction and preliminary concepts. Section 2 focuses on proving a pivotal integral equality, crucial for establishing the main results, utilizing a nonnegative weighted function. Section 3 presents weighted Milne-type inequalities for various function classes employing Riemann-Liouville fractional integrals. Also, Section 3 addresses weighted Milne-type inequalities for differentiable convex functions. In Section 4, several weighted Milne-type inequalities for bounded functions through fractional integrals are provided. Section 5 establishes weighted fractional Milne-type inequalities for Lipschitzian functions. Additionally, Section 6 proves weighted fractional Milne-type inequalities for functions of bounded variation. Finally, Section 7 offers insights into Milne-type inequalities and discuss potential avenues for future research in this domain.

2.   An essential equality

Throughout the paper, we make the assumption that $w:[\eta, \epsilon]\rightarrow \mathbb{R}$ is both nonnegative and continuous over the interval $[\eta, \epsilon]$, exhibiting symmetry with respect to $\frac{\eta+\epsilon}{2}$ (i.e., $w(x) = w\left(\eta+\epsilon-x\right)$ for all $x\in \left[ \eta, \epsilon\right]$). Now, let us introduce the functions $W_{1}$ and $W_{2}$ as follows:

and

Given the symmetry of the function $w$ with respect to $\frac{\eta+\epsilon}{2}$, we can establish the following equalities, which will be consistently employed in the subsequent sections:

Especially, we also have

Lemma 2.1. Let $f:[\eta, \epsilon]\rightarrow \mathbb{R}$ be an absolutely continuous function on the interval $(\eta, \epsilon)$ such that $\mathcal{P}^{\prime }\in L_{1}\left[ \eta, \epsilon\right]$. Hence, the following equality holds:

Here, $\Gamma$ represents the Euler Gamma function, and

Proof.

Applying the principles of integration by parts, we can easily derive:

Similar to the previous process, we have

By combining the equalities (2.6) and (2.7), we can easily derive

Hence, the proof of Lemma 2.1 is concluded by multiplying both sides of (2.8) by $\frac{1}{4}$. □

3.   Weighted fractional Milne-type inequalities for convex functions

In this segment, we derive weighted fractional Milne-type inequalities for differentiable convex functions by applying the modulus of the established equality (2.5). Additionally, we formulate certain weighted fractional Milne-type inequalities using Hölder and power-mean inequalities.

Theorem 3.1. Assuming that all the conditions of Lemma 2.1 are satisfied and $\left \vert \mathcal{P}^{\prime }\right \vert$ represents a convex function on the interval $\left[ \eta, \epsilon\right]$, we can establish the subsequent weighted fractional Milne-type inequality.

Proof. By Lemma 2.1, we have

Since the function $\left \vert \mathcal{P}^{\prime }\right \vert$ is convex, we have

and

Then, it follows that

which completes the proof. □

Remark 3.1. By selecting $w(t) = 1$ for every $t$ in the interval $\left[ \eta, \epsilon\right]$ in Theorem 3.1, we obtain the ensuing fractional Milne-type inequality, a result previously demonstrated by Budak et al. ^[20]:

Corollary 3.1. Upon setting $\delta = 1$ in Theorem 3.1, we derive the subsequent weighted Milne-type inequality.

Proof. Due to the symmetry of the function $w$ with respect to $\frac{\eta+\epsilon}{2}$, the proof (3.2) is based on the fact that

Conversely, we also obtain

Corollary 3.2. Under the stipulations of Theorem 3.1, the subsequent weighted fractional Milne-type inequality holds.

Corollary 3.3. By selecting $\delta = 1$ in Corollary 3.2, we arrive at the following weighted Milne-type inequality:

Remark 3.2. By choosing $w(t) = 1$ for every $t$ in the interval $\left[ \eta, \epsilon\right]$ in Corollary 3.3, we obtain the following Milne-type inequality:

which is obtained by Budak et al. ^[20].

Theorem 3.2. Assuming the fulfillment of all conditions stated in Lemma 2.1, if $\left \vert \mathcal{P}^{\prime }\right \vert ^{q}$ exhibits convexity on the interval $[\eta, \epsilon]$, where $q > 1$, then the following Milne-type inequality is established:

where $\frac{1}{p}+\frac{1}{q} = 1.$

Proof. Applying the well-known Hölder inequality to (3.1), we obtain

Due to the convexity of the function $\left \vert \mathcal{P}^{\prime }\right \vert ^{q}$, we get

Hence, the proof of Theorem 3.2 is now concluded. □

Remark 3.3. If we set $w(t) = 1$ for every $t\in\left[ \eta, \epsilon\right]$ in Theorem 3.2, the theorem simplifies to [20,Theorem 2].

Corollary 3.4. Setting $\delta = 1$ in Theorem 3.2 yields the subsequent weighted Milne-type inequality:

Corollary 3.5. Under the conditions specified in Theorem 3.2, the following weighted fractional Milne-type inequality is established:

Proof. The proof of (3.5) is obvious from the fact that

Corollary 3.6. If we choose $\delta = 1$ in Corollary 3.5, then we have the following weighted Milne-type inequality:

Remark 3.4. By selecting $w(t) = 1$ for every $t\in\left[ \eta, \epsilon\right]$ in Corollary 3.6, we obtain the following Milne-type inequality:

which is given by Budak et al. ^[20].

Theorem 3.3. Assuming all conditions specified in Lemma 2.1 are met, if $\left \vert \mathcal{P}^{\prime }\right \vert ^{q}$ is convex on $[\eta, \epsilon]$ with $q\geq 1$, then the following Milne-type inequality holds:

Here,

and

Proof. Utilizing the power mean inequality in (3.1), it consequently follows that:

Since the function $\left \vert \mathcal{P}^{\prime }\right \vert$ is convex, we have

and

Then, it follows that

This completes the proof. □

Remark 3.5. By selecting $w(t) = 1$ for every $t\in\left[ \eta, \epsilon\right]$ in Theorem 3.3, the theorem simplifies to [20,Theorem 3].

Corollary 3.7. Under the assumptions of Theorem 3.3, the following weighted fractional Milne-type inequality holds:

Corollary 3.8. Setting $\delta = 1$ in Corollary 3.7 results in the subsequent weighted Milne-type inequality:

Remark 3.6. By choosing $w(t) = 1$ for all $t\in \left[ \eta, \epsilon\right]$ in Corollary 3.8, we obtain the following Milne-type inequality:

which is given by Budak et al. ^[20].

4.   Weighted fractional Milne-type inequalities for bounded functions

Within this portion, we present a set of weighted fractional Milne-type inequalities designed for functions that are bounded.

Theorem 4.1. Let us assume that the conditions outlined in Lemma 2.1 are satisfied. If there exist real numbers $m$ and $M$ such that $m \leq \mathcal{P}^{\prime}(t) \leq M$ for $t \in [\eta, \epsilon]$, then it results in

Proof. By using Lemma 2.1, we have

By taking the absolute value of (4.1), we derive

Given that $m \leq \mathcal{P}^{\prime}(t) \leq M$ for $t \in [\eta, \epsilon]$, we can deduce

and

If we consider (4.2) and (4.3), then we get

This completes the proof. □

Remark 4.1. Selecting $w(t) = 1$ for all $t\in \left[ \eta, \epsilon\right]$ in Theorem 4.1, we arrive at the following fractional Milne-type inequality, as demonstrated by Budak et al. ^[20]:

Corollary 4.1. When $\delta = 1$ is chosen in Theorem 4.1, the resulting weighted Milne-type inequality is as follows:

Proof. The proof is obvious from the equality (3.3). □

Corollary 4.2. Under the assumptions outlined in Theorem 4.1, we arrive at the following weighted fractional Milne-type inequality:

Corollary 4.3. When $\delta = 1$ is selected in Corollary 4.2, we obtain the following weighted Milne-type inequality:

Remark 4.2. When $w(t) = 1$ for all $t\in \left[ \eta, \epsilon\right]$ in Corollary 4.3, the ensuing Milne-type inequality is as follows:

which is given by Budak et al. ^[20].

Corollary 4.4. Given the conditions stipulated in Theorem 4.1, if there exists $M\in \mathbb{R}^{+}$ such that $\left \vert \mathcal{P}(t)\right \vert \leq M$ for all $t\in \left[\eta, \epsilon\right]$, then we obtain

5.   Weighted fractional Milne-type inequalities for Lipschitzian functions

Next, we present several weighted fractional Milne-type inequalities specifically for Lipschitzian functions.

Theorem 5.1. Suppose the conditions of Lemma 2.1 are satisfied. If $\mathcal{P}^{\prime}$ is an $L$-Lipschitzian function on $\left[ \eta, \epsilon\right]$, then the ensuing inequality holds:

Proof. Using the fact that $\mathcal{P}^{\prime }$ is $L$-Lipschitzian function, by Lemma 2.1, we have

This completes the proof. □

Remark 5.1. When we set $w(t) = 1$ for all $t\in \left[ \eta, \epsilon\right]$ in Theorem 5.1, the resulting fractional Milne-type inequality, proven by Budak et al. ^[20], is as follows:

Corollary 5.1. If we select $\delta = 1$ in Theorem 5.1, we obtain the subsequent weighted Milne-type inequality:

Proof. As the function $w$ is symmetric with respect to $\frac{\eta+\epsilon}{2}$, the proof of (5.1) follows straightforwardly from the fact that

Corollary 5.2. Under the conditions specified in Theorem 5.1, the following weighted fractional Milne-type inequality holds:

Corollary 5.3. If we choose $\delta = 1$ in Corollary 5.2, then we have the following weighted Milne-type inequality:

Remark 5.2. If we choose $w(t) = 1$ for all $t\in \left[ \eta, \epsilon\right]$ in Corollary 5.3, then we have the following Milne-type inequality:

which is given by Budak et al. ^[20].

6.   Weighted fractional Milne-type inequalities for functions of bounded variation

Now, we present some weighted fractional Milne-type inequalities that are applicable to functions of bounded variation.

Theorem 6.1. Let us assume that $\mathcal{P}:\left[ \eta, \epsilon\right] \rightarrow \mathbb{R}$ is a function of bounded variation on $\left[ \eta, \epsilon\right]$. Then, we deduce

Here, $\bigvee \limits_{\eta}^{\epsilon}\left(f\right)$ denotes the total variation of $\mathcal{P}$ on $\left[ \eta, \epsilon\right]$.

Proof. Define the function $K_{\delta }(x)$ by

With the help of the integrating by parts, we obtain

Thus, we have

It is a known fact that if $g$ and $\mathcal{P}$ are functions mapping from $\left[ \eta, \epsilon\right]$ to $\mathbb{R}$ such that $g$ is continuous on $\left[ \eta, \epsilon\right]$ and $\mathcal{P}$ is of bounded variation on $\left[ \eta, \epsilon\right]$, then $\int_{\eta}^{\epsilon} g(t), d\mathcal{P}(t)$ exists and

By using (6.1), we have

By applying (6.2), we have

If we substitute inequalities (6.4) into (6.3), then we get the desired result. □

Remark 6.1. When we set $w(t) = 1$ for all $t\in \left[ \eta, \epsilon\right]$ in Theorem 6.1, the resultant fractional Milne-type inequality, established by Budak et al. ^[20], is as follows:

Corollary 6.1. When $\delta = 1$ is chosen in Theorem 6.1, the resulting weighted Milne-type inequality is as follows:

Corollary 6.2. Under the assumptions stated in Theorem 6.1, we derive the following weighted fractional Milne-type inequality:

Corollary 6.3. If we choose $\delta = 1$ in Corollary 6.2, then we have the following weighted Milne-type inequality:

Remark 6.2. If we choose $w(t) = 1$ for all $t\in \left[ \eta, \epsilon\right]$ in Corollary 6.3, then we have the following Milne-type inequality:

which is given by Alomari ^[28].

7.   Concluding remarks

In conclusion, this research paper has investigated weighted Milne-type inequalities utilizing Riemann-Liouville fractional integrals across diverse function classes. A key contribution has been the establishment of a fundamental integral equality, facilitated by the use of a nonnegative weighted function, which has been pivotal for deriving the main results. The paper has systematically proven weighted Milne-type inequalities for various function classes, including differentiable convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation. The obtained results have not only contributed to the understanding of Milne-type inequalities but have also offered insights that have paved the way for potential future research in the considered topics. Moreover, Remaks 3.1–3.6, 4.1, 4.2, 5.1, 5.2, 6.1, and 6.2 show that the acquired results may be compared with a variety of earlier findings in the literature, several of which are included as special cases.

Author contributions

Areej A Almoneef, Abd-Allah Hyder and Hüseyin Budak: Conceptualization, Formal analysis, Investigation, Methodology, Writing-original draft, Writing-review & editing. All authors of this article have contributed equally. All authors have read and approved the final version of the manuscript for publication.

Use of AI tools declarationt

We have used the AI, especially ChatGPT 3.5, in rephrasing some statements to provide a clear meaning for these statements and avoid the similarity in the used words.

Acknowledgments

The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through the Research Groups Program under grant (RGP.2/82/45). The authors would like to acknowledge the Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R337).

Conflict of interest

The authors declare no conflicts of interest regarding the publication of this article.