Error Estimation of Weddle’s Rule for Generalized Convex Functions with Applications to Numerical Integration and Computational Analysis

Abstract

This paper presents new integral inequalities for differentiable generalized convex functions in the second sense, with a focus on improving the accuracy of Weddle’s formula for numerical integration. The study is motivated by the following three key factors: the generalization of convexity through s-convex functions, the enhancement of the approximation quality, particularly as $s\to 0^{+}$, and the effectiveness of Weddle’s formula in cases where Simpson’s 1/3 rule fails. An integral identity is derived for differentiable functions, which is then used to establish sharp error bounds for Weddle’s formula under s-convexity. Numerical examples and comparative tables demonstrate that the proposed inequalities yield significantly tighter bounds than those based on classical convexity. Applications to numerical quadrature highlight the practical utility of the results in computational mathematics.

1. Introduction

Convexity is a cornerstone concept in mathematical analysis with wide-ranging applications in optimization theory and differential calculus. Formally, a real-valued function $\phi :[\omega ,\varpi ]\to \mathbb{R}$ is said to be convex if for any two points $x_1,x_2\in [\omega ,\varpi ]$ and any interpolation parameter $\lambda \in [0,1]$, the following inequality holds:

$$\phi (\omega \rho +(1-\rho )\varpi )\le \rho \phi \left(\omega \right)+(1-\rho )\phi \left(\varpi \right),$$

(1)

for each values of $\rho$ between 0 and 1. Hermite–Hadamard type inequalities are significant mathematical inequalities that involve convex mappings. They are stated as follows: Let $\phi :\left[\omega ,\varpi \right]\subseteq \mathbb{R}\to {\mathbb{R}}^{+}$ be a convex function defined on the interval $\left[\omega ,\varpi \right]$ of real numbers. The following inequalities are called Hermite–Hadamard inequalities:

$$\phi \left({\displaystyle \frac{\omega +\varpi }{2}}\right)\le {\displaystyle \frac{1}{\varpi -\omega }}{\int }_{\omega }^{\varpi }\phi \left(u\right)du\le {\displaystyle \frac{\phi \left(\omega \right)+\phi \left(\varpi \right)}{2}}.$$

(2)

For a concave function, the inequalities mentioned above also hold but in the opposite direction. In the past two decades, researchers have explored several new upper bounds for both the left and right sides of the inequality (2).

The first use of the term “numerical integration” is documented to have been used in 1915 when David Gibb used it in the title of a book titled “A Course in Interpolation and Numerical Integration for the Mathematical Laboratory” [1]. Over the last few decades, numerical integration has also been suitably applied to scientific computing, engineering, and data analysis. To solve increasingly more complicated problems, adaptive quadrature algorithms, numerical integration with error estimation, and high dimensionality integration techniques are used. Quadrature is an old terminology in mathematics, especially in numerical integration, whereby the term means to calculate area. It should be noted that many quadrature rules can be obtained by constructing different interpolating polynomials. In modern computational mathematics, numerical integration has emerged as a critical tool across scientific computing, engineering, and data science. Contemporary advancements include adaptive quadrature algorithms, error-controlled integration schemes, and techniques for high-dimensional problems, all designed to address increasingly sophisticated applications. The term quadrature, historically rooted in numerical integration, refers to the approximation of definite integrals. A broad family of quadrature rules can be constructed through polynomial interpolation as follows:

• Midpoint rule: Derived from a constant (degree-0) interpolant.
• Trapezoidal rule: Based on linear (degree-1) polynomial interpolation.
• Simpson’s rule: Uses quadratic (degree-2). interpolation, named after Thomas Simpson (1710–1761).
• Boole’s rule: Employing quartic (degree-4) interpolation.
• Weddle’s rule: Utilizing sextic (degree-6) polynomial interpolation.

These methods form the foundation of classical numerical integration, with each successive rule generally offering improved accuracy for sufficiently smooth integrands. The theoretical framework and convergence properties of these quadrature formulas are well-established in the numerical analysis literature. We have the following well-known results related to abovementioned formulas:

• The trapezoidal formula:

$${\displaystyle \frac{\varpi -\omega }{2}}\left[\phi \left(\omega \right)+\phi \left(\varpi \right)\right]\approx {\int }_{\omega }^{\varpi }\phi \left(u\right)du.$$

• The midpoint formula:

$$\left(\varpi -\omega \right)\left[\phi \left({\displaystyle \frac{\varpi +\omega }{2}}\right)\right]\approx {\int }_{\omega }^{\varpi }\phi \left(u\right)du.$$

• Simpson’s 1/3 formula:

$${\displaystyle \frac{\varpi -\omega }{6}}\left[\phi \left(\omega \right)+4\phi \left({\displaystyle \frac{\varpi +\omega }{2}}\right)+\phi \left(\varpi \right)\right]\approx {\int }_{\omega }^{\varpi }\phi \left(u\right)du.$$

• The following Boole formula:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\varpi -\omega }{90}}\left[7\left[\phi \left(\omega \right)+\phi \left(\varpi \right)\right]+32\left[\phi \left({\displaystyle \frac{\varpi +3\omega }{4}}\right)+\phi \left({\displaystyle \frac{3\varpi +\omega }{4}}\right)\right]\right.\hfill \\ \hfill & \left.+12\phi \left({\displaystyle \frac{\omega +\varpi }{2}}\right)\right]\approx {\int }_{\omega }^{\varpi }\phi \left(u\right)du.\hfill \end{array}$$

• The six-point Weddle’s formula for numerical integration is defined as follows:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\varpi -\omega }{20}}\left[\phi \left(\omega \right)+5\phi \left({\displaystyle \frac{\varpi +5\omega }{6}}\right)+\phi \left({\displaystyle \frac{\varpi +2\omega }{3}}\right)+6\phi \left({\displaystyle \frac{\omega +\varpi }{2}}\right)\right.\hfill \\ \hfill & \left.+\phi \left({\displaystyle \frac{\omega +2\varpi }{3}}\right)+5\phi \left({\displaystyle \frac{5\varpi +\omega }{6}}\right)+\phi \left(\varpi \right)\right]\approx {\int }_{\omega }^{\varpi }\phi \left(u\right)du.\hfill \end{array}$$

The preceding formulas are part of the Newton–Cotes family of formulas, which work by dividing the area under a curve into segments and approximating each segment with a polynomial function. Weddle’s formula was developed by British mathematician John Weddle in the late 19th century and extends Simpson’s rule which uses the quadratic polynomials by employing the cubic polynomials for greater accuracy. Weddle’s rule approximates better than other numerical methods when a polynomial has a degree of six or more. Weddle’s rule can be applied to various types of integrals, including both smooth and oscillatory functions. Its versatility and adaptability make it suitable for multiple scientific and engineering applications, from physics and biology to finance and economics. One can visit [2,3,4,5] to learn more about numerical integration and its applications.

For Weddle’s formula, the error term typically requires the function to be differentiable six times. However, if the function is convex and differentiable at least once, we can still determine an error bound for Weddle’s formula. This is a significant achievement in inequality theory because the class of functions that are differentiable once is much broader than the class of functions that have six derivatives.

This article is motivated by the need to address the issue of estimating error terms for numerical integration methods, such as the trapezoidal rule, the midpoint rule, Simpson’s rule, etc. In 1998, Dragomir and Agarwal introduced an error term for the trapezoidal rule that could be estimated using only the first derivative of the convex function [6]. Subsequently, Kirmaci inspired by the work of Dragomir and Agarwal, proved some error bounds for the midpoint formula for differentiable convex functions [7]. After these two publications, many researchers started their work to establish the error bounds for numerical quadrature formulas with different approaches. In their respective works [8,9], Alomari and Dragomir provided error bounds for Simpson’s formula for both convex and general convex functions, along with their applications. On the other hand, researchers established error bounds for Simpson’s formula for two-variable functions in [10,11], respectively. Additionally, Du [12] utilized the general form of convexity to establish error bounds for Simpson’s formula. Hermite–Hadamard, Ostrowski’s, midpoint, Simpson’s, and trapezoidal inequalities can all be utilized for determining the error bounds of numerical integration formulas; see [13,14,15,16,17,18,19,20].

In [21], Breckner was the first mathematician to introduce a generalized convex function in 1979. The number of associations with s-convexity in the first sense was negotiated in [22]. Direct proof of Breckner’s result was esteemed in 2001 by Pycia [23]. Due to the importance of convexity and s-convexity in the investigation of optimality to resolve mathematical programming, many researchers seriously focused on s-convex functions. For example, H. Hudzik et al. presented two kinds of s-convexity $s\in (0,1)$ in [22]. They demonstrated that the s-convexity in the second sense is fundamentally stronger than the s-convexity in the first sense. We use the s-convexity of a function in the second sense generally known as the s-convex function. Since $s\in (0,1)$, this class of functions is more important than the convex functions. We observe that the results obtained via s-convexity are significantly sharper than those from standard convexity. Moreover, since s-convexity generalizes classical convexity, all results for convex functions can be recovered by setting $s=1$ in the s-convex case. The formal definition of this class is given as follows:

Definition 1.

A function $\phi :[0,\infty )\to \mathbb{R}$ is called s-convex in the second sense if for some fixed $s\in (0,1]$, the inequality

$$\phi \left(\omega \rho +(1-\rho )\varpi \right)\le {\rho }^s\phi \left(\omega \right)+{(1-\rho )}^s\phi \left(\varpi \right),$$

(3)

holds for all $\omega ,\varpi \in [0,\infty )$ and $\rho \in [0,1]$. When $s=1$, this reduces to the ordinary convexity condition on $[0,\infty )$.

In [24], the following Hermite–Hadamard inequality in terms of s-convex functions in the second sense holds:

Theorem 1.

If $\phi :\left[0,\infty \right)\to \left[0,\infty \right)$ is an s-convex function in the second sense where $0<s<1$ and letting ${\varkappa }_1,{\varkappa }_2\in \left[0,\infty \right)$, ${\varkappa }_1<{\varkappa }_2.$ If $\phi \in L^1\left[{\varkappa }_1,{\varkappa }_2\right]$, then the following inequalities hold:

$$2^{s-1}\phi \left({\displaystyle \frac{\omega +\varpi }{2}}\right)\le {\displaystyle \frac{1}{\varpi -\omega }}{\int }_{\omega }^{\varpi }\phi \left(u\right)du\le {\displaystyle \frac{\phi \left(\omega \right)+\phi \left(\varpi \right)}{s+1}}.$$

(4)

Refer to [25,26,27] for more recent results and insights regarding Hadamard’s inequality.

Motivated by current research developments, we employ the differentiable s-convexity property to establish novel integral inequalities. Compared to conventional convex functions, s-convexity offers significantly greater flexibility in bound variation, which proves crucial for enhancing the robustness and precision of Weddle formula-type inequalities. This framework consequently provides a more versatile and refined analytical tool than classical convexity approaches. The construction of suitable kernels presented particular challenges due to the distinctive characteristics of s-convex functions. Our derived inequalities enable the effective approximation of error bounds in Weddle’s formula without requiring the computation of higher-order derivatives, which may be nonexistent or computationally intensive to obtain. The results presented in this work provide optimal approximation bounds while operating entirely within classical calculus.

The paper is structured as follows: Section 2 develops the principal theoretical contributions, establishing a Weddle-type quadrature formula for differentiable s-convex functions in the second sense. Section 3 presents numerical implementations and the computational verification of these theoretical results through carefully constructed examples. The practical application of these findings to quadrature formulas is examined in Section 4. Finally, Section 5 offers concluding observations and suggests potential avenues for future research in this direction.

2. Main Results

This section presents a new foundational identity that enables the derivation of innovative inequalities related to Weddle’s formula. The following lemma plays a pivotal role in establishing our main results concerning Weddle-type inequalities for s-convex functions.

Lemma 1.

Let $\phi :I\subset \mathbb{R}\to \mathbb{R}$ be a differentiable mapping on $I^{\circ },\omega ,\varpi \in I^{\circ }$ with $\omega <\varpi ,$ and ${\phi }^{\prime }\in L^1\left[\omega ,\varpi \right]$ then the following equality holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{20}}\left[\phi \left(\omega \right)+5\phi \left({\displaystyle \frac{\varpi +5\omega }{6}}\right)+\phi \left({\displaystyle \frac{\varpi +2\omega }{3}}\right)+6\phi \left({\displaystyle \frac{\varpi +\omega }{2}}\right)+\phi \left({\displaystyle \frac{2\varpi +\omega }{3}}\right)\right.\hfill \\ \hfill & \left.+5\phi \left({\displaystyle \frac{5\varpi +\omega }{6}}\right)+\phi \left(\varpi \right)\right]-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_{\omega }^{\varpi }\phi \left(u\right)du\hfill \\ \hfill & =\left(\varpi -\omega \right){\int }_0^1\Lambda \left(\rho \right){\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)d\rho ,\hfill \end{array}$$

(5)

where

$$\Lambda \left(\rho \right)=\left\{\begin{array}{c}\left(\rho -{\displaystyle \frac{1}{20}}\right);\rho \in [0,{\displaystyle \frac{1}{6}}),\\ \\ \left(\rho -{\displaystyle \frac{6}{20}}\right);\rho \in [{\displaystyle \frac{1}{6}},{\displaystyle \frac{1}{3}}),\\ \\ \left(\rho -{\displaystyle \frac{7}{20}}\right);\rho \in [{\displaystyle \frac{1}{3}}.{\displaystyle \frac{1}{2}}),\\ \\ \left(\rho -{\displaystyle \frac{13}{20}}\right);\rho \in [{\displaystyle \frac{1}{2}},{\displaystyle \frac{2}{3}}),\\ \\ \left(\rho -{\displaystyle \frac{14}{20}}\right);\rho \in [{\displaystyle \frac{2}{3}},{\displaystyle \frac{5}{6}}),\\ \\ \left(\rho -{\displaystyle \frac{19}{20}}\right);\rho \in \left[{\displaystyle \frac{5}{6}},1\right].\end{array}\right.$$

Proof. 

From definition of $\Lambda \left(\rho \right),$ we have

$$\begin{array}{cc}\hfill I& ={\int }_0^1\Lambda \left(\rho \right){\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)d\rho \hfill \\ \hfill & ={\int }_0^{{\displaystyle \frac{1}{6}}}\left(\rho -{\displaystyle \frac{1}{20}}\right){\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)d\rho +{\int }_{{\displaystyle \frac{1}{6}}}^{{\displaystyle \frac{1}{3}}}\left(\rho -{\displaystyle \frac{6}{20}}\right){\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)d\rho \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{3}}}^{{\displaystyle \frac{1}{2}}}\left(\rho -{\displaystyle \frac{7}{20}}\right){\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)d\rho +{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{2}{3}}}\left(\rho -{\displaystyle \frac{13}{20}}\right){\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)d\rho \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{2}{3}}}^{{\displaystyle \frac{5}{6}}}\left(\rho -{\displaystyle \frac{14}{20}}\right){\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)d\rho +{\int }_{{\displaystyle \frac{5}{6}}}^1\left(\rho -{\displaystyle \frac{19}{20}}\right){\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)d\rho \hfill \\ \hfill & =I_1+I_2+I_3+I_4+I_5+I_6.\hfill \end{array}$$

(6)

Using integration by parts, we have

$$\begin{array}{cc}\hfill I_1& ={\int }_0^{{\displaystyle \frac{1}{6}}}\left(\rho -{\displaystyle \frac{1}{20}}\right){\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)d\rho \hfill \\ \hfill & ={\displaystyle \frac{1}{\varpi -\omega }}{\left.\left(\rho -{\displaystyle \frac{1}{20}}\right)\phi \left(\rho \varpi +\left(1-\rho \right)\omega \right)\right|}_0^{{\displaystyle \frac{1}{6}}}-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_0^{{\displaystyle \frac{1}{6}}}\phi \left(\rho \varpi +\left(1-\rho \right)\omega \right)d\rho \hfill \\ \hfill & ={\displaystyle \frac{7}{60(\varpi -\omega )}}\phi \left({\displaystyle \frac{\varpi +5\omega }{6}}\right)+{\displaystyle \frac{1}{20(\varpi -\omega )}}\phi \left(\omega \right)-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_0^{{\displaystyle \frac{1}{6}}}\phi \left(\rho \varpi +\left(1-\rho \right)\omega \right)d\rho .\hfill \end{array}$$

(7)

Similarly, we have

$$\begin{array}{cc}\hfill I_2& ={\int }_{{\displaystyle \frac{1}{6}}}^{{\displaystyle \frac{1}{3}}}\left(\rho -{\displaystyle \frac{6}{20}}\right){\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)d\rho \hfill \\ \hfill & ={\displaystyle \frac{1}{\varpi -\omega }}\left[{\displaystyle \frac{2}{60}}\phi \left({\displaystyle \frac{\varpi +2\omega }{3}}\right)+{\displaystyle \frac{8}{60}}\phi \left({\displaystyle \frac{\varpi +5\omega }{6}}\right)\right]-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_{{\displaystyle \frac{1}{6}}}^{{\displaystyle \frac{1}{3}}}\phi \left(\rho \varpi +\left(1-\rho \right)\omega \right)d\rho ,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill I_3& ={\int }_{{\displaystyle \frac{1}{3}}}^{{\displaystyle \frac{1}{2}}}\left(\rho -{\displaystyle \frac{7}{20}}\right){\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)d\rho \hfill \\ \hfill & ={\displaystyle \frac{1}{\varpi -\omega }}\left[{\displaystyle \frac{3}{20}}\phi \left({\displaystyle \frac{\varpi +\omega }{2}}\right)+{\displaystyle \frac{1}{60}}\phi \left({\displaystyle \frac{\varpi +2\omega }{3}}\right)\right]-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_{{\displaystyle \frac{1}{3}}}^{{\displaystyle \frac{1}{2}}}\phi \left(\rho \varpi +\left(1-\rho \right)\omega \right)d\rho ,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill I_4& ={\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{2}{3}}}\left(\rho -{\displaystyle \frac{13}{20}}\right){\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)d\rho \hfill \\ \hfill & ={\displaystyle \frac{1}{\varpi -\omega }}\left[{\displaystyle \frac{1}{60}}\phi \left({\displaystyle \frac{2\varpi +\omega }{3}}\right)+{\displaystyle \frac{3}{20}}\phi \left({\displaystyle \frac{\varpi +\omega }{2}}\right)\right]-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{2}{3}}}\phi \left(\rho \varpi +\left(1-\rho \right)\omega \right)d\rho ,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill I_5& ={\int }_{{\displaystyle \frac{2}{3}}}^{{\displaystyle \frac{5}{6}}}\left(\rho -{\displaystyle \frac{14}{20}}\right){\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)d\rho \hfill \\ \hfill & ={\displaystyle \frac{1}{\varpi -\omega }}\left[{\displaystyle \frac{8}{60}}\phi \left({\displaystyle \frac{5\varpi +\omega }{6}}\right)+{\displaystyle \frac{2}{60}}\phi \left({\displaystyle \frac{2\varpi +\omega }{3}}\right)\right]-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_{{\displaystyle \frac{2}{3}}}^{{\displaystyle \frac{5}{6}}}\phi \left(\rho \varpi +\left(1-\rho \right)\omega \right)d\rho ,\hfill \end{array}$$

(11)

and

$$\begin{array}{cc}\hfill I_6& ={\int }_{{\displaystyle \frac{5}{6}}}^1\left(\rho -{\displaystyle \frac{19}{20}}\right){\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)d\rho \hfill \\ \hfill & ={\displaystyle \frac{1}{\varpi -\omega }}\left[{\displaystyle \frac{1}{20}}\phi \left(\varpi \right)+{\displaystyle \frac{7}{60}}\phi \left({\displaystyle \frac{5\varpi +\omega }{6}}\right)\right]-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_{{\displaystyle \frac{5}{6}}}^1\phi \left(\rho \varpi +\left(1-\rho \right)\omega \right)d\rho .\hfill \end{array}$$

(12)

By replacing the values from (7) to (12) in (6), multiplying by $\left(\varpi -\omega \right)$, and using the change of variables $u=\rho \varpi +\left(1-\rho \right)\omega ,$ we obtain the desired result. This completes the proof. □

Theorem 2.

Assume that $\phi :I\subset [0,\infty )\to \mathbb{R}$ is a differentiable on $I^{\circ }$ such that ${\phi }^{\prime }\in L_1\left[\omega ,\varpi \right]$, where $\omega ,\varpi \in I^{\circ }.$ If $\left|{\phi }^{\prime }\right|$ is s-convex on $\left[\omega ,\varpi \right]$ for some fixed $s\in \left(0,1\right]$, then we have the following inequality:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{20}}\left[\phi \left(\omega \right)+5\phi \left({\displaystyle \frac{\varpi +5\omega }{6}}\right)+\phi \left({\displaystyle \frac{\varpi +2\omega }{3}}\right)+6\phi \left({\displaystyle \frac{\varpi +\omega }{2}}\right)+\phi \left({\displaystyle \frac{2\varpi +\omega }{3}}\right)\right.\right.\hfill \\ \hfill & \left.\left.+5\phi \left({\displaystyle \frac{5\varpi +\omega }{6}}\right)+\phi \left(\varpi \right)\right]-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_{\omega }^{\varpi }\phi \left(u\right)du\right|\hfill \\ \hfill & \le {\displaystyle \frac{\left(\varpi -\omega \right)}{(s+1)(s+2)}}\left[2^{-2s-3}·{15}^{-s-2}\left(-2^s·5^{s+1}\left(-2^{s+1}-2^{s+1}·3^{s+2}+4^{s+1}-5^{s+1}+1\right)s\right.\right.\hfill \\ \hfill & -11·{10}^{s+1}-3^{s+4}·{20}^{s+1}-19·{20}^{s+1}-41·{40}^{s+1}-49·{50}^{s+1}+3^{s+2}+{18}^{s+2}+{21}^{s+2}\hfill \\ \hfill & \left.\left.-{30}^{s+2}+{39}^{s+2}+{42}^{s+2}+{57}^{s+2}\right)\right]\left[\left|{\phi }^{\prime }\left(\varpi \right)\right|+\left|{\phi }^{\prime }\left(\omega \right)\right|\right].\hfill \end{array}$$

(13)

Proof. 

Taking the absolute of Lemma 1, we have the following:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{20}}\left[\phi \left(\omega \right)+5\phi \left({\displaystyle \frac{\varpi +5\omega }{6}}\right)+\phi \left({\displaystyle \frac{\varpi +2\omega }{3}}\right)+6\phi \left({\displaystyle \frac{\varpi +\omega }{2}}\right)+\phi \left({\displaystyle \frac{2\varpi +\omega }{3}}\right)\right.\right.\hfill \\ \hfill & \left.\left.+5\phi \left({\displaystyle \frac{5\varpi +\omega }{6}}\right)+\phi \left(\varpi \right)\right]-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_{\omega }^{\varpi }\phi \left(u\right)du\right|\hfill \\ \hfill & \le \left(\varpi -\omega \right){\int }_0^1\left|\Lambda \left(\rho \right)\right|\left|{\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)\right|d\rho \hfill \\ \hfill & =\left(\varpi -\omega \right)\left[{\int }_0^{{\displaystyle \frac{1}{6}}}\left|\rho -{\displaystyle \frac{1}{20}}\right|\left|{\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)\right|d\rho \right.+{\int }_{{\displaystyle \frac{1}{6}}}^{{\displaystyle \frac{1}{3}}}\left|\rho -{\displaystyle \frac{6}{20}}\right|\left|{\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)\right|d\rho \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{3}}}^{{\displaystyle \frac{1}{2}}}\left|\rho -{\displaystyle \frac{7}{20}}\right|\left|{\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)\right|d\rho +{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{2}{3}}}\left|\rho -{\displaystyle \frac{13}{20}}\right|\left|{\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)\right|d\rho \hfill \\ \hfill & \left.+{\int }_{{\displaystyle \frac{2}{3}}}^{{\displaystyle \frac{5}{6}}}\left|\rho -{\displaystyle \frac{14}{20}}\right|\left|{\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)\right|d\rho +{\int }_{{\displaystyle \frac{5}{6}}}^1\left|\rho -{\displaystyle \frac{19}{20}}\right|\left|{\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)\right|d\rho \right].\hfill \end{array}$$

Since $\left|{\phi }^{\prime }\right|$ is s-convex on $\left[\omega ,\varpi \right],$ we get

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{20}}\left[\phi \left(\omega \right)+5\phi \left({\displaystyle \frac{\varpi +5\omega }{6}}\right)+\phi \left({\displaystyle \frac{\varpi +2\omega }{3}}\right)+6\phi \left({\displaystyle \frac{\varpi +\omega }{2}}\right)+\phi \left({\displaystyle \frac{2\varpi +\omega }{3}}\right)\right.\right.\hfill \\ \hfill & \left.\left.+5\phi \left({\displaystyle \frac{5\varpi +\omega }{6}}\right)+\phi \left(\varpi \right)\right]-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_{\omega }^{\varpi }\phi \left(\vartheta \right)d\vartheta \right|\hfill \\ \hfill & \le \left(\varpi -\omega \right)\left[{\int }_0^{{\displaystyle \frac{1}{6}}}\left|\rho -{\displaystyle \frac{1}{20}}\right|\left[{\rho }^s\left|{\phi }^{\prime }\left(\varpi \right)\right|d\rho +{\left(1-\rho \right)}^s\left|{\phi }^{\prime }\left(\omega \right)\right|\right]d\rho \right.\hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{6}}}^{{\displaystyle \frac{1}{3}}}\left|\rho -{\displaystyle \frac{6}{20}}\right|\left[{\rho }^s\left|{\phi }^{\prime }\left(\varpi \right)\right|d\rho +{\left(1-\rho \right)}^s\left|{\phi }^{\prime }\left(\omega \right)\right|\right]d\rho \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{3}}}^{{\displaystyle \frac{1}{2}}}\left|\rho -{\displaystyle \frac{7}{20}}\right|\left[{\rho }^s\left|{\phi }^{\prime }\left(\varpi \right)\right|d\rho +{\left(1-\rho \right)}^s\left|{\phi }^{\prime }\left(\omega \right)\right|\right]d\rho \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{2}{3}}}\left|\rho -{\displaystyle \frac{13}{20}}\right|\left[{\rho }^s\left|{\phi }^{\prime }\left(\varpi \right)\right|d\rho +{\left(1-\rho \right)}^s\left|{\phi }^{\prime }\left(\omega \right)\right|\right]d\rho \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{2}{3}}}^{{\displaystyle \frac{5}{6}}}\left|\rho -{\displaystyle \frac{14}{20}}\right|\left[{\rho }^s\left|{\phi }^{\prime }\left(\varpi \right)\right|d\rho +{\left(1-\rho \right)}^s\left|{\phi }^{\prime }\left(\omega \right)\right|\right]d\rho \hfill \\ \hfill & \left.+{\int }_{{\displaystyle \frac{5}{6}}}^1\left|\rho -{\displaystyle \frac{19}{20}}\right|\left[{\rho }^s\left|{\phi }^{\prime }\left(\varpi \right)\right|d\rho +{\left(1-\rho \right)}^s\left|{\phi }^{\prime }\left(\omega \right)\right|\right]d\rho \right]\hfill \\ \hfill & =\left(\varpi -\omega \right)\left[\left({\int }_0^{{\displaystyle \frac{1}{6}}}\left|\rho -{\displaystyle \frac{1}{20}}\right|{\rho }^{s}d\rho +{\int }_{{\displaystyle \frac{1}{6}}}^{{\displaystyle \frac{1}{3}}}\left|\rho -{\displaystyle \frac{6}{20}}\right|{\rho }^{s}d\rho +{\int }_{{\displaystyle \frac{1}{3}}}^{{\displaystyle \frac{1}{2}}}\left|\rho -{\displaystyle \frac{7}{20}}\right|{\rho }^{s}d\rho \right.\right.\hfill \\ \hfill & \left.+{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{2}{3}}}\left|\rho -{\displaystyle \frac{13}{20}}\right|{\rho }^{s}d\rho +{\int }_{{\displaystyle \frac{2}{3}}}^{{\displaystyle \frac{5}{6}}}\left|\rho -{\displaystyle \frac{14}{20}}\right|{\rho }^{s}d\rho +{\int }_{{\displaystyle \frac{5}{6}}}^1\left|\rho -{\displaystyle \frac{19}{20}}\right|{\rho }^{s}d\rho \right)\left|{\phi }^{\prime }\left(\varpi \right)\right|\hfill \\ \hfill & +\left({\int }_0^{{\displaystyle \frac{1}{6}}}\left|\rho -{\displaystyle \frac{1}{20}}\right|{\left(1-\rho \right)}^{s}d\rho +{\int }_{{\displaystyle \frac{1}{6}}}^{{\displaystyle \frac{1}{3}}}\left|\rho -{\displaystyle \frac{6}{20}}\right|{\left(1-\rho \right)}^{s}d\rho +{\int }_{{\displaystyle \frac{1}{3}}}^{{\displaystyle \frac{1}{2}}}\left|\rho -{\displaystyle \frac{7}{20}}\right|{\left(1-\rho \right)}^{s}d\rho \right.\hfill \\ \hfill & \left.\left.+{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{2}{3}}}\left|\rho -{\displaystyle \frac{13}{20}}\right|{\left(1-\rho \right)}^{s}d\rho +{\int }_{{\displaystyle \frac{2}{3}}}^{{\displaystyle \frac{5}{6}}}\left|\rho -{\displaystyle \frac{14}{20}}\right|{\left(1-\rho \right)}^{s}d\rho +{\int }_{{\displaystyle \frac{5}{6}}}^1\left|\rho -{\displaystyle \frac{19}{20}}\right|{\left(1-\rho \right)}^{s}d\rho \right)\left|{\phi }^{\prime }\left(\omega \right)\right|\right].\hfill \end{array}$$

Here, we used the equalities

$$\begin{array}{cc}\hfill & {\int }_0^{{\displaystyle \frac{1}{6}}}\left|\rho -{\displaystyle \frac{1}{20}}\right|{\rho }^{s}d\rho \hfill \\ \hfill & ={\displaystyle \frac{3^{-s-2}{400}^{-s-1}\left(7·2^{3s+1}·5^{2s+1}s+2^{2s+1}·3^{s+2}·5^s+2^{3s+3}·5^{2s+1}\right)}{(s+1)(s+2)}},\hfill \\ \hfill & {\int }_{{\displaystyle \frac{1}{6}}}^{{\displaystyle \frac{1}{3}}}\left|\rho -{\displaystyle \frac{6}{20}}\right|{\rho }^{s}d\rho \hfill \\ \hfill & ={\displaystyle \frac{{30}^{-s-2}\left(-4·5^{s+1}·s+{10}^{s+1}·s-2^{s+4}{5}^{s+1}-13·5^{s+1}+2·3^{2s+4}\right)}{(s+1)(s+2)}},\hfill \\ \hfill & {\int }_{{\displaystyle \frac{1}{3}}}^{{\displaystyle \frac{1}{2}}}\left|\rho -{\displaystyle \frac{7}{20}}\right|{\rho }^{s}d\rho \hfill \\ \hfill & ={\displaystyle \frac{2^{-2s-3}·{15}^{-s-2}\left({21}^{s+2}-2^s·5^{s+1}\left(2^{s+1}(s+22)+3^{s+2}(4-3s)\right)\right)}{(s+1)(s+2)}},\hfill \\ \hfill & {\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{2}{3}}}\left|\rho -{\displaystyle \frac{13}{20}}\right|{\rho }^{s}d\rho \hfill \\ \hfill & ={\displaystyle \frac{2^{-2s-3}·{15}^{-s-2}\left(2^s·5^{s+1}\left(4^{s+1}(s-38)-3^{s+2}(3s+16)\right)+{39}^{s+2}\right)}{(s+1)(s+2)}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\int }_{{\displaystyle \frac{2}{3}}}^{{\displaystyle \frac{5}{6}}}\left|\rho -{\displaystyle \frac{14}{20}}\right|{\rho }^{s}d\rho \hfill \\ \hfill & ={\displaystyle \frac{{30}^{-s-2}\left(20\left(5^{2s+1}-{20}^s\right)s-11·2^{2s+3}{5}^{s+1}-17·{25}^{s+1}+2·{21}^{s+2}\right)}{(s+1)(s+2)}},\hfill \\ \hfill & {\int }_{{\displaystyle \frac{5}{6}}}^1\left|\rho -{\displaystyle \frac{19}{20}}\right|{\rho }^{s}d\rho \hfill \\ \hfill & =-{\displaystyle \frac{-90(s-18)-9·{19}^{s+2}·{20}^{-s}+5^{s+2}·6^{-s}(7s+64)}{1800(s+1)(s+2)}},\hfill \end{array}$$

Similarly, the following integrals should be calculated as the previous elementary integrals

$$\begin{array}{cc}\hfill & {\int }_0^{{\displaystyle \frac{1}{6}}}\left|\rho -{\displaystyle \frac{1}{20}}\right|{\left(1-\rho \right)}^{s}d\rho ,{\int }_{{\displaystyle \frac{1}{6}}}^{{\displaystyle \frac{1}{3}}}\left|\rho -{\displaystyle \frac{6}{20}}\right|{\left(1-\rho \right)}^{s}d\rho ,{\int }_{{\displaystyle \frac{1}{3}}}^{{\displaystyle \frac{1}{2}}}\left|\rho -{\displaystyle \frac{7}{20}}\right|{\left(1-\rho \right)}^{s}d\rho ,\hfill \\ \hfill & {\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{2}{3}}}\left|\rho -{\displaystyle \frac{13}{20}}\right|{\left(1-\rho \right)}^{s}d\rho ,{\int }_{{\displaystyle \frac{2}{3}}}^{{\displaystyle \frac{5}{6}}}\left|\rho -{\displaystyle \frac{14}{20}}\right|{\left(1-\rho \right)}^{s}d\rho ,{\int }_{{\displaystyle \frac{5}{6}}}^1\left|\rho -{\displaystyle \frac{19}{20}}\right|{\left(1-\rho \right)}^{s}d\rho .\hfill \end{array}$$

Hence, the proof of Theorem 2 has been completed. □

Corollary 1.

By setting $s=1$ in Theorem 2, we have the following inequality:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{20}}\left[\phi \left(\omega \right)+5\phi \left({\displaystyle \frac{\varpi +5\omega }{6}}\right)+\phi \left({\displaystyle \frac{\varpi +2\omega }{3}}\right)+6\phi \left({\displaystyle \frac{\varpi +\omega }{2}}\right)+\phi \left({\displaystyle \frac{2\varpi +\omega }{3}}\right)\right.\right.\hfill \\ \hfill & \left.\left.+5\phi \left({\displaystyle \frac{5\varpi +\omega }{6}}\right)+\phi \left(\varpi \right)\right]-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_{\omega }^{\varpi }\phi \left(u\right)du\right|\le {\displaystyle \frac{13\left(\varpi -\omega \right)}{450}}\left[\left|{\phi }^{\prime }\left(\varpi \right)\right|+\left|{\phi }^{\prime }\left(\omega \right)\right|\right].\hfill \end{array}$$

(14)

Theorem 3.

Assume that $\phi :I\subset [0,\infty )\to \mathbb{R}$ is a differentiable on $I^{\circ }$ such that ${\phi }^{\prime }\in L^1\left[\omega ,\varpi \right]$, where $\omega ,\varpi \in I^{\circ }.$ If ${\left|{\phi }^{\prime }\right|}^q$ is s-convex on $\left[\omega ,\varpi \right]$ for some fixed $s\in \left(0,1\right]$ and $p,q>1,$ then we have the following inequality:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{20}}\left[\phi \left(\omega \right)+5\phi \left({\displaystyle \frac{\varpi +5\omega }{6}}\right)+\phi \left({\displaystyle \frac{\varpi +2\omega }{3}}\right)+6\phi \left({\displaystyle \frac{\varpi +\omega }{2}}\right)+\phi \left({\displaystyle \frac{2\varpi +\omega }{3}}\right)\right.\right.\hfill \\ \hfill & \left.\left.+5\phi \left({\displaystyle \frac{5\varpi +\omega }{6}}\right)+\phi \left(\varpi \right)\right]-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_{\omega }^{\varpi }\phi \left(u\right)du\right|\hfill \\ \hfill & \le \left(\varpi -\omega \right){\left({\displaystyle \frac{2^{-1-2p}·{15}^{-1-p}\left(1+2^{1+p}+3^{1+p}+7^{1+p}+8^{1+p}+9^{1+p}\right)}{1+p}}\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \times {\left[{\displaystyle \frac{{\left|{\phi }^{\prime }\left(\varpi \right)\right|}^q+{\left|{\phi }^{\prime }\left(\omega \right)\right|}^q}{s+1}}\right]}^{{\displaystyle \frac{1}{q}}},\hfill \end{array}$$

(15)

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

Proof. 

From Lemma 1, and using Hölder integral inequality, we have the following:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{20}}\left[\phi \left(\omega \right)+5\phi \left({\displaystyle \frac{\varpi +5\omega }{6}}\right)+\phi \left({\displaystyle \frac{\varpi +2\omega }{3}}\right)+6\phi \left({\displaystyle \frac{\varpi +\omega }{2}}\right)+\phi \left({\displaystyle \frac{2\varpi +\omega }{3}}\right)\right.\right.\hfill \\ \hfill & \left.\left.+5\phi \left({\displaystyle \frac{5\varpi +\omega }{6}}\right)+\phi \left(\varpi \right)\right]-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_{\omega }^{\varpi }\phi \left(u\right)du\right|\hfill \\ \hfill & \le \left(\varpi -\omega \right){\left({\int }_0^1{\left|\Lambda \left(\rho \right)\right|}^{p}d\rho \right)}^{{\displaystyle \frac{1}{p}}}\times {\left({\int }_0^1{\left|{\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)\right|}^{q}d\rho \right)}^{{\displaystyle \frac{1}{q}}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\left(\varpi -\omega \right)\left[{\left({\int }_0^{{\displaystyle \frac{1}{6}}}{\left|\rho -{\displaystyle \frac{1}{20}}\right|}^{p}d\rho \right)}^{{\displaystyle \frac{1}{p}}}+{\left({\int }_{{\displaystyle \frac{1}{6}}}^{{\displaystyle \frac{1}{3}}}{\left|\rho -{\displaystyle \frac{6}{20}}\right|}^{p}d\rho \right)}^{{\displaystyle \frac{1}{p}}}\right.\hfill \\ \hfill & +{\left({\int }_{{\displaystyle \frac{1}{3}}}^{{\displaystyle \frac{1}{2}}}{\left|\rho -{\displaystyle \frac{7}{20}}\right|}^{p}d\rho \right)}^{{\displaystyle \frac{1}{p}}}+{\left({\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{2}{3}}}{\left|\rho -{\displaystyle \frac{13}{20}}\right|}^{p}d\rho \right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \left.+{\left({\int }_{{\displaystyle \frac{2}{3}}}^{{\displaystyle \frac{5}{6}}}{\left|\rho -{\displaystyle \frac{14}{20}}\right|}^{p}d\rho \right)}^{{\displaystyle \frac{1}{p}}}+{\left({\int }_{{\displaystyle \frac{5}{6}}}^1{\left|\rho -{\displaystyle \frac{19}{20}}\right|}^{p}d\rho \right)}^{{\displaystyle \frac{1}{p}}}\right]\hfill \\ \hfill & \times {\left({\int }_0^1{\left|{\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)\right|}^{q}d\rho \right)}^{{\displaystyle \frac{1}{q}}},\hfill \end{array}$$

since ${\left|{\phi }^{\prime }\right|}^q$ is s-convex on $\left[\omega ,\varpi \right],$ we get

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{20}}\left[\phi \left(\omega \right)+5\phi \left({\displaystyle \frac{\varpi +5\omega }{6}}\right)+\phi \left({\displaystyle \frac{\varpi +2\omega }{3}}\right)+6\phi \left({\displaystyle \frac{\varpi +\omega }{2}}\right)+\phi \left({\displaystyle \frac{2\varpi +\omega }{3}}\right)\right.\right.\hfill \\ \hfill & \left.\left.+5\phi \left({\displaystyle \frac{5\varpi +\omega }{6}}\right)+\phi \left(\varpi \right)\right]-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_{\omega }^{\varpi }\phi \left(u\right)du\right|\hfill \\ \hfill & \le \left(\varpi -\omega \right)\left[{\left({\int }_0^{{\displaystyle \frac{1}{6}}}{\left|\rho -{\displaystyle \frac{1}{20}}\right|}^{p}d\rho \right)}^{{\displaystyle \frac{1}{p}}}+{\left({\int }_{{\displaystyle \frac{1}{6}}}^{{\displaystyle \frac{1}{3}}}{\left|\rho -{\displaystyle \frac{6}{20}}\right|}^{p}d\rho \right)}^{{\displaystyle \frac{1}{p}}}\right.\hfill \\ \hfill & +{\left({\int }_{{\displaystyle \frac{1}{3}}}^{{\displaystyle \frac{1}{2}}}{\left|\rho -{\displaystyle \frac{7}{20}}\right|}^{p}d\rho \right)}^{{\displaystyle \frac{1}{p}}}+{\left({\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{2}{3}}}{\left|\rho -{\displaystyle \frac{13}{20}}\right|}^{p}d\rho \right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \left.+{\left({\int }_{{\displaystyle \frac{2}{3}}}^{{\displaystyle \frac{5}{6}}}{\left|\rho -{\displaystyle \frac{14}{20}}\right|}^{p}d\rho \right)}^{{\displaystyle \frac{1}{p}}}+{\left({\int }_{{\displaystyle \frac{5}{6}}}^1{\left|\rho -{\displaystyle \frac{19}{20}}\right|}^{p}d\rho \right)}^{{\displaystyle \frac{1}{p}}}\right]\hfill \\ \hfill & \times {\left({\int }_0^1{\rho }^s{\left|{\phi }^{\prime }\left(\varpi \right)\right|}^q+{\left(1-\rho \right)}^s{\left|{\phi }^{\prime }\left(\omega \right)\right|}^{q}d\rho \right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & =\left(\varpi -\omega \right){\left({\displaystyle \frac{2^{-1-2p}·{15}^{-1-p}\left(1+2^{1+p}+3^{1+p}+7^{1+p}+8^{1+p}+9^{1+p}\right)}{1+p}}\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \times {\left[{\displaystyle \frac{{\left|{\phi }^{\prime }\left(\varpi \right)\right|}^q+{\left|{\phi }^{\prime }\left(\omega \right)\right|}^q}{s+1}}\right]}^{{\displaystyle \frac{1}{q}}}.\hfill \end{array}$$

Thus, Theorem 3 is completed. □

Corollary 2.

By setting $s=1$ in Theorem 2, we have the following inequality:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{20}}\left[\phi \left(\omega \right)+5\phi \left({\displaystyle \frac{\varpi +5\omega }{6}}\right)+\phi \left({\displaystyle \frac{\varpi +2\omega }{3}}\right)+6\phi \left({\displaystyle \frac{\varpi +\omega }{2}}\right)+\phi \left({\displaystyle \frac{2\varpi +\omega }{3}}\right)\right.\right.\hfill \\ \hfill & \left.\left.+5\phi \left({\displaystyle \frac{5\varpi +\omega }{6}}\right)+\phi \left(\varpi \right)\right]-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_{\omega }^{\varpi }\phi \left(u\right)du\right|\hfill \\ \hfill & \le \left(\varpi -\omega \right){\left[{\displaystyle \frac{2^{-1-2p}·{15}^{-1-p}\left(1+2^{1+p}+3^{1+p}+7^{1+p}+8^{1+p}+9^{1+p}\right)}{1+p}}\right]}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \times {\left[{\displaystyle \frac{{\left|{\phi }^{\prime }\left(\varpi \right)\right|}^q+{\left|{\phi }^{\prime }\left(\omega \right)\right|}^q}{2}}\right]}^{{\displaystyle \frac{1}{q}}}.\hfill \end{array}$$

(16)

Theorem 4.

Assume that $\phi :I\subset [0,\infty )\to \mathbb{R}$ is a differentiable on $I^{\circ }$ such that ${\phi }^{\prime }\in L^1\left[\omega ,\varpi \right]$, where $\omega ,\varpi \in I^{\circ }.$ If ${\left|{\phi }^{\prime }\right|}^q$ is s-convex on $\left[\omega ,\varpi \right]$ for some fixed $s\in \left(0,1\right]$ and $q\ge 1,$ then the following inequality holds:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{20}}\left[\phi \left(\omega \right)+5\phi \left({\displaystyle \frac{\varpi +5\omega }{6}}\right)+\phi \left({\displaystyle \frac{\varpi +2\omega }{3}}\right)+6\phi \left({\displaystyle \frac{\varpi +\omega }{2}}\right)+\phi \left({\displaystyle \frac{2\varpi +\omega }{3}}\right)\right.\right.\hfill \\ \hfill & \left.\left.+5\phi \left({\displaystyle \frac{5\varpi +\omega }{6}}\right)+\phi \left(\varpi \right)\right]-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_{\omega }^{\varpi }\phi \left(u\right)du\right|\hfill \\ \hfill & \le {\displaystyle \frac{\left(\varpi -\omega \right)}{{\left((s+1)(s+2)\right)}^{\frac{1}{q}}}}{\left({\displaystyle \frac{13}{225}}\right)}^{1-{\displaystyle \frac{1}{q}}}\left[2^{-2s-3}·{15}^{-s-2}\left(-2^s·5^{s+1}\left(-2^{s+1}-2^{s+1}·3^{s+2}+4^{s+1}-5^{s+1}+1\right)s\right.\right.\hfill \\ \hfill & -11·{10}^{s+1}-3^{s+4}·{20}^{s+1}-19·{20}^{s+1}-41·{40}^{s+1}-49·{50}^{s+1}+3^{s+2}+{18}^{s+2}+{21}^{s+2}-{30}^{s+2}\hfill \\ \hfill & \left.+{39}^{s+2}+{42}^{s+2}+{57}^{s+2}\right){\left.\left({\left|{\phi }^{\prime }\left(\varpi \right)\right|}^q+{\left|{\phi }^{\prime }\left(\omega \right)\right|}^q\right)\right]}^{{\displaystyle \frac{1}{q}}}.\hfill \end{array}$$

(17)

Proof. 

From Lemma 1, and using the Power mean integral inequality, we have the following:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{20}}\left[\phi \left(\omega \right)+5\phi \left({\displaystyle \frac{\varpi +5\omega }{6}}\right)+\phi \left({\displaystyle \frac{\varpi +2\omega }{3}}\right)+6\phi \left({\displaystyle \frac{\varpi +\omega }{2}}\right)+\phi \left({\displaystyle \frac{2\varpi +\omega }{3}}\right)\right.\right.\hfill \\ \hfill & \left.\left.+5\phi \left({\displaystyle \frac{5\varpi +\omega }{6}}\right)+\phi \left(\varpi \right)\right]-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_{\omega }^{\varpi }\phi \left(u\right)du\right|\hfill \\ \hfill & \le \left(\varpi -\omega \right){\left({\int }_0^1\left|\Lambda \left(\rho \right)\right|d\rho \right)}^{1-{\displaystyle \frac{1}{q}}}\times {\left({\int }_0^1\left|\Lambda \left(\rho \right)\right|{\left|{\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)\right|}^{q}d\rho \right)}^{{\displaystyle \frac{1}{q}}},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\int }_0^1\left|\Lambda \left(\rho \right)\right|d\rho & =\left[{\int }_0^{{\displaystyle \frac{1}{6}}}\left|\rho -{\displaystyle \frac{1}{20}}\right|d\rho +{\int }_{{\displaystyle \frac{1}{6}}}^{{\displaystyle \frac{1}{3}}}\left|\rho -{\displaystyle \frac{6}{20}}\right|d\rho +{\int }_{{\displaystyle \frac{1}{3}}}^{{\displaystyle \frac{1}{2}}}\left|\rho -{\displaystyle \frac{7}{20}}\right|d\rho \right.\hfill \\ \hfill & \left.+{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{2}{3}}}\left|\rho -{\displaystyle \frac{13}{20}}\right|d\rho +{\int }_{{\displaystyle \frac{2}{3}}}^{{\displaystyle \frac{5}{6}}}\left|\rho -{\displaystyle \frac{14}{20}}\right|d\rho +{\int }_{{\displaystyle \frac{5}{6}}}^1\left|\rho -{\displaystyle \frac{19}{20}}\right|d\rho \right]={\displaystyle \frac{13}{225}}.\hfill \end{array}$$

(19)

Since ${\left|{\phi }^{\prime }\right|}^q$ is s-convex on $\left[\omega ,\varpi \right],$ we get

$$\begin{array}{cc}\hfill & {\int }_0^1\left|\Lambda \left(\rho \right)\right|{\left|{\phi }^{\prime }\left(\rho \varpi +\left(1-\rho \right)\omega \right)\right|}^{q}d\rho \hfill \\ \hfill & \le \left[{\int }_0^{{\displaystyle \frac{1}{6}}}\left|\rho -{\displaystyle \frac{1}{20}}\right|\left({\rho }^s{\left|{\phi }^{\prime }\left(\varpi \right)\right|}^q+{\left(1-\rho \right)}^s{\left|{\phi }^{\prime }\left(\omega \right)\right|}^q\right)d\rho \right.\hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{6}}}^{{\displaystyle \frac{1}{3}}}\left|\rho -{\displaystyle \frac{6}{20}}\right|\left({\rho }^s{\left|{\phi }^{\prime }\left(\varpi \right)\right|}^q+{\left(1-\rho \right)}^s{\left|{\phi }^{\prime }\left(\omega \right)\right|}^q\right)d\rho \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{3}}}^{{\displaystyle \frac{1}{2}}}\left|\rho -{\displaystyle \frac{7}{20}}\right|\left({\rho }^s{\left|{\phi }^{\prime }\left(\varpi \right)\right|}^q+{\left(1-\rho \right)}^s{\left|{\phi }^{\prime }\left(\omega \right)\right|}^q\right)d\rho \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{2}{3}}}\left|\rho -{\displaystyle \frac{13}{20}}\right|\left({\rho }^s{\left|{\phi }^{\prime }\left(\varpi \right)\right|}^q+{\left(1-\rho \right)}^s{\left|{\phi }^{\prime }\left(\omega \right)\right|}^q\right)d\rho \hfill \\ \hfill & +{\int }_{{\displaystyle \frac{2}{3}}}^{{\displaystyle \frac{5}{6}}}\left|\rho -{\displaystyle \frac{14}{20}}\right|\left({\rho }^s{\left|{\phi }^{\prime }\left(\varpi \right)\right|}^q+{\left(1-\rho \right)}^s{\left|{\phi }^{\prime }\left(\omega \right)\right|}^q\right)d\rho \hfill \\ \hfill & \left.+{\int }_{{\displaystyle \frac{5}{6}}}^1\left|\rho -{\displaystyle \frac{19}{20}}\right|\left({\rho }^s{\left|{\phi }^{\prime }\left(\varpi \right)\right|}^q+{\left(1-\rho \right)}^s{\left|{\phi }^{\prime }\left(\omega \right)\right|}^q\right)d\rho \right]\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\left[\left({\int }_0^{{\displaystyle \frac{1}{6}}}\left|\rho -{\displaystyle \frac{1}{20}}\right|{\rho }^{s}d\rho +{\int }_{{\displaystyle \frac{1}{6}}}^{{\displaystyle \frac{1}{3}}}\left|\rho -{\displaystyle \frac{6}{20}}\right|{\rho }^{s}d\rho +{\int }_{{\displaystyle \frac{1}{3}}}^{{\displaystyle \frac{1}{2}}}\left|\rho -{\displaystyle \frac{7}{20}}\right|{\rho }^{s}d\rho \right.\right.\hfill \\ \hfill & \left.+{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{2}{3}}}\left|\rho -{\displaystyle \frac{13}{20}}\right|{\rho }^{s}d\rho +{\int }_{{\displaystyle \frac{2}{3}}}^{{\displaystyle \frac{5}{6}}}\left|\rho -{\displaystyle \frac{14}{20}}\right|{\rho }^{s}d\rho +{\int }_{{\displaystyle \frac{5}{6}}}^1\left|\rho -{\displaystyle \frac{19}{20}}\right|{\rho }^{s}d\rho \right){\left|{\phi }^{\prime }\left(\varpi \right)\right|}^q\hfill \\ \hfill & +\left({\int }_0^{{\displaystyle \frac{1}{6}}}\left|\rho -{\displaystyle \frac{1}{20}}\right|{\left(1-\rho \right)}^{s}d\rho +{\int }_{{\displaystyle \frac{1}{6}}}^{{\displaystyle \frac{1}{3}}}\left|\rho -{\displaystyle \frac{6}{20}}\right|{\left(1-\rho \right)}^{s}d\rho +{\int }_{{\displaystyle \frac{1}{3}}}^{{\displaystyle \frac{1}{2}}}\left|\rho -{\displaystyle \frac{7}{20}}\right|{\left(1-\rho \right)}^{s}d\rho \right.\hfill \\ \hfill & \left.+{\int }_{{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{2}{3}}}\left|\rho -{\displaystyle \frac{13}{20}}\right|{\left(1-\rho \right)}^{s}d\rho +{\int }_{{\displaystyle \frac{2}{3}}}^{{\displaystyle \frac{5}{6}}}\left|\rho -{\displaystyle \frac{14}{20}}\right|{\left(1-\rho \right)}^{s}d\rho +{\int }_{{\displaystyle \frac{5}{6}}}^1\left|\rho -{\displaystyle \frac{19}{20}}\right|{\left(1-\rho \right)}^{s}d\rho \right){\left|{\phi }^{\prime }\left(\omega \right)\right|}^q.\hfill \end{array}$$

(20)

Using the values of (19) and (20) in (18), we obtain

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{20}}\left[\phi \left(\omega \right)+5\phi \left({\displaystyle \frac{\varpi +5\omega }{6}}\right)+\phi \left({\displaystyle \frac{\varpi +2\omega }{3}}\right)+6\phi \left({\displaystyle \frac{\varpi +\omega }{2}}\right)+\phi \left({\displaystyle \frac{2\varpi +\omega }{3}}\right)\right.\right.\hfill \\ \hfill & \left.\left.+5\phi \left({\displaystyle \frac{5\varpi +\omega }{6}}\right)+\phi \left(\varpi \right)\right]-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_{\omega }^{\varpi }\phi \left(u\right)du\right|\hfill \\ \hfill & \le {\displaystyle \frac{\left(\varpi -\omega \right)}{{\left((s+1)(s+2)\right)}^{\frac{1}{q}}}}{\left({\displaystyle \frac{13}{225}}\right)}^{1-{\displaystyle \frac{1}{q}}}\left[2^{-2s-3}·{15}^{-s-2}\left(-2^s·5^{s+1}\left(-2^{s+1}-2^{s+1}·3^{s+2}+4^{s+1}-5^{s+1}+1\right)s\right.\right.\hfill \\ \hfill & -11·{10}^{s+1}-3^{s+4}·{20}^{s+1}-19·{20}^{s+1}-41·{40}^{s+1}-49·{50}^{s+1}+3^{s+2}+{18}^{s+2}+{21}^{s+2}-{30}^{s+2}\hfill \\ \hfill & \left.+{39}^{s+2}+{42}^{s+2}+{57}^{s+2}\right){\left.\left({\left|{\phi }^{\prime }\left(\varpi \right)\right|}^q+{\left|{\phi }^{\prime }\left(\omega \right)\right|}^q\right)\right]}^{{\displaystyle \frac{1}{q}}}.\hfill \end{array}$$

Thus, the proof of Theorem 4 is completed. □

Remark 1.

If we take $q=1$ in Theorem 4, we then capture the inequality (13).

Corollary 3.

By setting $s=1$ in Theorem 4, we then have the following inequality:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{20}}\left[\phi \left(\omega \right)+5\phi \left({\displaystyle \frac{\varpi +5\omega }{6}}\right)+\phi \left({\displaystyle \frac{\varpi +2\omega }{3}}\right)+6\phi \left({\displaystyle \frac{\varpi +\omega }{2}}\right)+\phi \left({\displaystyle \frac{2\varpi +\omega }{3}}\right)\right.\right.\hfill \\ \hfill & \left.\left.+5\phi \left({\displaystyle \frac{5\varpi +\omega }{6}}\right)+\phi \left(\varpi \right)\right]-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_{\omega }^{\varpi }\phi \left(u\right)du\right|\hfill \\ \hfill & \le \left(\varpi -\omega \right){\left({\displaystyle \frac{13}{225}}\right)}^{1-{\displaystyle \frac{1}{q}}}{\left[{\displaystyle \frac{13}{450}}\left({\left|{\phi }^{\prime }\left(\varpi \right)\right|}^q+{\left|{\phi }^{\prime }\left(\omega \right)\right|}^q\right)\right]}^{{\displaystyle \frac{1}{q}}}.\hfill \end{array}$$

3. Numerical Examples

In this section, we provide numerical examples and computational analysis for newly established Weddle formula-type inequalities for s-convex functions. These results show that the inequalities presented in this work are numerically correct.

Example 1.

Let $\phi :\left[\omega ,\varpi \right]=\left[1,2\right]\to \mathbb{R}$ be a function defined by $\phi \left(\rho \right)=Exp\left[s\rho \right]$. Then by applying inequality (13) to the function $\phi \left(\rho \right)=Exp\left[s\rho \right],$ we obtain that the left-hand side of (13) for $s={\displaystyle \frac{1}{3}}$ is

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{20}}\left[\phi \left(\omega \right)+5\phi \left({\displaystyle \frac{\varpi +5\omega }{6}}\right)+\phi \left({\displaystyle \frac{\varpi +2\omega }{3}}\right)+6\phi \left({\displaystyle \frac{\varpi +\omega }{2}}\right)+\phi \left({\displaystyle \frac{2\varpi +\omega }{3}}\right)\right.\right.\hfill \\ \hfill & \left.\left.+5\phi \left({\displaystyle \frac{5\varpi +\omega }{6}}\right)+\phi \left(\varpi \right)\right]-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_{\omega }^{\varpi }\phi \left(u\right)du\right|=5.7913\times {10}^{-11}.\hfill \end{array}$$

(21)

and the right-hand side of (13) for $s={\displaystyle \frac{1}{3}}$ is

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\left(\varpi -\omega \right)}{(s+1)(s+2)}}[2^{-2s-3}·{15}^{-s-2}(-2^s·5^{s+1}\left(-2^{s+1}-2^{s+1}·3^{s+2}+4^{s+1}-5^{s+1}+1\right)s\hfill \\ \hfill & -11·{10}^{s+1}-3^{s+4}·{20}^{s+1}-19·{20}^{s+1}-41·{40}^{s+1}-49·{50}^{s+1}+3^{s+2}+{18}^{s+2}\hfill \\ \hfill & +{21}^{s+2}-{30}^{s+2}+{39}^{s+2}+{42}^{s+2}+{57}^{s+2}\left)\right]\left[\left|{\phi }^{\prime }\left(\varpi \right)\right|+\left|{\phi }^{\prime }\left(\omega \right)\right|\right]=0.0489.\hfill \end{array}$$

(22)

From (21) and (22), it is clear that the left-hand side is less than the right-hand side

$$5.7913\times {10}^{-11}<0.0489.$$

This demonstrates that the inequality (13) is numerically valid.

The graph of the inequality of Example 1 is depicted in Figure 1 for $s\in (0,1]$, which demonstrates the validity of Theorem 2.

Example 2.

Let $\phi :\left[\omega ,\varpi \right]=\left[1,2\right]\to \mathbb{R}$ be a function defined by $\phi \left(\tau \right)=Exp\left[s\rho \right]$ and $p=q=2$. Then by applying inequality (15) to the function $\phi \left(\tau \right)=Exp\left[s\rho \right],$ the right-hand side of (15) for $s={\displaystyle \frac{1}{3}}$ is

$$\begin{array}{cc}\hfill & \left(\varpi -\omega \right){\left({\displaystyle \frac{2^{-1-2p}·{15}^{-1-p}\left(1+2^{1+p}+3^{1+p}+7^{1+p}+8^{1+p}+9^{1+p}\right)}{1+p}}\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \times {\left({\displaystyle \frac{{\left|{\phi }^{\prime }\left(\varpi \right)\right|}^q+{\left|{\phi }^{\prime }\left(\omega \right)\right|}^q}{s+1}}\right)}^{{\displaystyle \frac{1}{q}}}=0.0562.\hfill \end{array}$$

(23)

From (21) and (23), it is clear that the left-hand side is less than the right-hand side

$$5.7913\times {10}^{-11}<0.0562.$$

This demonstrates that the inequality (15) is numerically valid.

The graph of the inequality of Example 2 is depicted in Figure 2 for $s\in (0,1]$, which demonstrates the validity of Theorem 3.

Example 3.

Let $\phi :\left[\omega ,\varpi \right]=\left[1,2\right]\to \mathbb{R}$ be a function defined by $\phi \left(\tau \right)=Exp\left[2s\rho \right]$ and $q=2$. Then by applying inequality (17) to the function $\phi \left(\tau \right)=Exp\left[2s\rho \right],$ we obtain that the left-hand side of (17) for $s={\displaystyle \frac{1}{3}}$ is

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{20}}\left[\phi \left(\omega \right)+5\phi \left({\displaystyle \frac{\varpi +5\omega }{6}}\right)+\phi \left({\displaystyle \frac{\varpi +2\omega }{3}}\right)+6\phi \left({\displaystyle \frac{\varpi +\omega }{2}}\right)+\phi \left({\displaystyle \frac{2\varpi +\omega }{3}}\right)\right.\right.\hfill \\ \hfill & \left.\left.+5\phi \left({\displaystyle \frac{5\varpi +\omega }{6}}\right)+\phi \left(\varpi \right)\right]-{\displaystyle \frac{1}{\varpi -\omega }}{\int }_{\omega }^{\varpi }\phi \left(u\right)du\right|=6.1759\times {10}^{-9}.\hfill \end{array}$$

(24)

and the right-hand side of (17) for $s={\displaystyle \frac{1}{3}}$ is

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\left(\varpi -\omega \right)}{{\left((s+1)(s+2)\right)}^{\frac{1}{q}}}}{\left({\displaystyle \frac{13}{225}}\right)}^{1-{\displaystyle \frac{1}{q}}}\left[2^{-2s-3}·{15}^{-s-2}\left(-2^s·5^{s+1}\left(-2^{s+1}-2^{s+1}·3^{s+2}+4^{s+1}-5^{s+1}+1\right)s\right.\right.\hfill \\ \hfill & -11·{10}^{s+1}-3^{s+4}·{20}^{s+1}-19·{20}^{s+1}-41·{40}^{s+1}-49·{50}^{s+1}+3^{s+2}+{18}^{s+2}+{21}^{s+2}-{30}^{s+2}\hfill \\ \hfill & \left.+{39}^{s+2}+{42}^{s+2}+{57}^{s+2}\right){\left.\left({\left|{\phi }^{\prime }\left(\varpi \right)\right|}^q+{\left|{\phi }^{\prime }\left(\omega \right)\right|}^q\right)\right]}^{{\displaystyle \frac{1}{q}}}=0.0529.\hfill \end{array}$$

(25)

From (24) and (25), it is clear that the left-hand side is less than the right-hand side

$$6.1759\times {10}^{-9}<0.0529.$$

This demonstrates that the inequality (17) is numerically valid.

The graph of the inequality of Example 3 is depicted in Figure 3 for $s\in (0,1]$, which demonstrates the validity of Theorem 4.

Remark 2.

One can easily observe from

Table 1. A comparative analysis of the left-hand and right-hand inequalities in the discretization of s as presented in Theorem 2.

s	Left Inequality	Right Inequality
0.1	3.0864 × 10−14	0.0123
0.2	2.2071 × 10−12	0.0264
0.3	2.9257 × 10−11	0.0428
0.4	1.9141 × 10−10	0.0621
0.5	8.5077 × 10−10	0.0851
0.6	2.9618 × 10−9	0.1126
0.7	8.7132 × 10−9	0.1456
0.8	2.2664 × 10−8	0.1854
0.9	5.3668 × 10−8	0.2336
1	1.1803 × 10−7	0.2920

and

Table 2. A comparative analysis of the left-hand and right-hand inequalities in the discretization of s as presented in Theorem 4.

s	Left Inequality	Right Inequality
0.1	2.20712 × 10−12	0.00744
0.2	1.9141 × 10−10	0.02035
0.3	2.96184 × 10−9	0.04247
0.4	2.26636 × 10−8	0.07996
0.5	1.18029 × 10−7	0.14283
0.6	4.82308 × 10−7	0.24714
0.7	1.66828 × 10−6	0.41854
0.8	5.11063 × 10−6	0.69778
0.9	0.0000142757	1.14924
1	0.0000370908	1.87429

that when $s\to 1^{-}$ we get the worst approximation and when $s\to 0^{+}$ we get very good lower and upper bounds for the inequality of Theorem 2. So, we concluded that s-convex gives a better approximation as compared to the convex function.

4. Application to Numerical Integration Formulas

In this section, we present applications of the newly established results to the following numerical quadrature formulas:

Assume that ${\mathrm{{\rm Y}}}_n$ is the partition given by

$${\mathrm{{\rm Y}}}_n:\omega ={\vartheta }_0<{\vartheta }_1<{\vartheta }_2<\cdots <{\vartheta }_{n-1}<{\vartheta }_n=\varpi ,$$

$$h_j={\displaystyle \frac{{\vartheta }_{j+1}-{\vartheta }_j}{n}},j=1,2,3,\dots ,n-1,$$

where n must be divisible by $6$. It is well-known that if the mapping $\phi :\left[\omega ,\varpi \right]\to \mathbb{R}$ is differentiable such that ${\phi }^{\left(6\right)}\left(\vartheta \right)$ exists on $\left(\omega ,\varpi \right)$ and $M={max}_{\vartheta \in \left(\omega ,\varpi \right)}\left|{\phi }^{\left(6\right)}\left(\vartheta \right)\right|<\infty$, then

$${\int }_{\omega }^{\varpi }\phi \left(\vartheta \right)d\vartheta =S_w\left({\mathrm{{\rm Y}}}_n,\phi \right)+{\mathbb{R}}_w\left({\mathrm{{\rm Y}}}_n,\phi \right),$$

(26)

where

$$\begin{array}{cc}\hfill S_w\left({\mathrm{{\rm Y}}}_n,\phi \right)& ={\displaystyle \frac{1}{20}}\sum _{j=0}^{n-1}\left({\vartheta }_{j+1}-{\vartheta }_j\right)\left[\phi \left({\vartheta }_j\right)+5\phi \left({\vartheta }_j+h\right)+\phi \left({\vartheta }_j+2h\right)\right.\hfill \\ \hfill & \left.+6\phi \left({\vartheta }_j+3h\right)+\phi \left({\vartheta }_j+4h\right)+5\phi \left({\vartheta }_j+5h\right)+\phi \left({\vartheta }_{j+1}\right)\right],\hfill \end{array}$$

and the remainder term satisfies the approximation

$$\left|{\mathbb{R}}_w\left({\mathrm{{\rm Y}}}_n,\phi \right)\right|\le {\displaystyle \frac{13}{450}}\sum _{j=0}^{n-1}{\displaystyle \frac{{\left({\vartheta }_{j+1}-{\vartheta }_j\right)}^2}{n}}\left[\left|{\phi }^{\prime }\left({\vartheta }_j\right)\right|+\left|{\phi }^{\prime }\left({\vartheta }_{j+1}\right)\right|\right].$$

(27)

It is clear that if the mapping $\phi$ is not 6th times differentiable or the 6th derivative is not bounded on $\left(\omega ,\varpi \right)$, then (26) cannot be applied. In the following we give many different estimations for the remainder term ${\mathbb{R}}_w\left({\mathrm{{\rm Y}}}_n,\phi \right)$ in terms of the first derivative. We prove the error bounds of Weddle’s rule in the following propositions.

Proposition 1.

Suppose that $\phi :\left[\omega ,\varpi \right]\to \mathbb{R}$ is a differentiable mapping on $\left(\omega ,\varpi \right)$. Then we have the following:

$$\left|{\mathbb{R}}_w\left({\mathrm{{\rm Y}}}_n,\phi \right)\right|\le {\displaystyle \frac{13}{450}}\sum _{j=0}^{n-1}{\displaystyle \frac{{\left({\vartheta }_{j+1}-{\vartheta }_j\right)}^2}{6}}\left[\left|{\phi }^{\prime }\left({\vartheta }_j\right)\right|+\left|{\phi }^{\prime }\left({\vartheta }_{j+1}\right)\right|\right],$$

(28)

for all $j=1,2,3,\dots ,n-1.$

Proof. 

Applying Corollary 1 on the subinterval $\left[{\vartheta }_j,{\vartheta }_{j+1}\right]$, we have

$$\begin{array}{c}\hfill \omega ={\vartheta }_j,\varpi ={\vartheta }_{j+1},h_i={\displaystyle \frac{{\vartheta }_{j+1}-{\vartheta }_j}{6}},\end{array}$$

where $j=1,2,3,\dots ,n-1.$ Then we have the following estimation

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{20}}\left({\vartheta }_{j+1}-{\vartheta }_j\right)\left[\phi \left({\vartheta }_j\right)+5\phi \left({\vartheta }_j+h\right)+\phi \left({\vartheta }_j+2h\right)\right.\right.\hfill \\ \hfill & \left.\left.+6\phi \left({\vartheta }_j+3h\right)+\phi \left({\vartheta }_j+4h\right)+5\phi \left({\vartheta }_j+5h\right)+\phi \left({\vartheta }_{j+1}\right)\right]-{\int }_{{\vartheta }_j}^{{\vartheta }_{j+1}}\phi \left(u\right)du\right|\hfill \\ \hfill & \le {\displaystyle \frac{13}{450}}\left({\displaystyle \frac{{\left({\vartheta }_{j+1}-{\vartheta }_j\right)}^2}{6}}\right)\left[\left|{\phi }^{\prime }\left({\vartheta }_j\right)\right|+\left|{\phi }^{\prime }\left({\vartheta }_{j+1}\right)\right|\right],\mathrm{for}\mathrm{all}j=1,2,3,\dots ,n-1.\hfill \end{array}$$

After summing and by the triangular inequality, we have

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{20}}\sum _{j=0}^{n-1}\left({\vartheta }_{j+1}-{\vartheta }_j\right)\left[\phi \left({\vartheta }_j\right)+5\phi \left({\vartheta }_j+h\right)+\phi \left({\vartheta }_j+2h\right)\right.\right.\hfill \\ \hfill & \left.\left.+6\phi \left({\vartheta }_j+3h\right)+\phi \left({\vartheta }_j+4h\right)+5\phi \left({\vartheta }_j+5h\right)+\phi \left({\vartheta }_{j+1}\right)\right]-{\int }_{\omega }^{\varpi }\phi \left(u\right)du\right|\hfill \\ \hfill & \le {\displaystyle \frac{13}{450}}\sum _{j=0}^{n-1}{\displaystyle \frac{{\left({\vartheta }_{j+1}-{\vartheta }_j\right)}^2}{6}}\left[\left|{\phi }^{\prime }\left({\vartheta }_j\right)\right|+\left|{\phi }^{\prime }\left({\vartheta }_{j+1}\right)\right|\right].\hfill \end{array}$$

The proof of Proposition 1 is completed. □

Proposition 2.

Suppose that $\phi :\left[\omega ,\varpi \right]\to \mathbb{R}$ is a differentiable mapping on $\left(\omega ,\varpi \right)$. Then we have the following:

$$\begin{array}{cc}\hfill \left|{\mathbb{R}}_w\left({\mathrm{{\rm Y}}}_n,\phi \right)\right|& \le {\left({\displaystyle \frac{2^{-1-2p}·{15}^{-1-p}\left(1+2^{1+p}+3^{1+p}+7^{1+p}+8^{1+p}+9^{1+p}\right)}{1+p}}\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \times \sum _{j=0}^{n-1}{\displaystyle \frac{{\left({\vartheta }_{i+1}-{\vartheta }_i\right)}^2}{6}}{\left({\displaystyle \frac{{\left|{\phi }^{\prime }\left({\vartheta }_i\right)\right|}^q+{\left|{\phi }^{\prime }\left({\vartheta }_{i+1}\right)\right|}^q}{2}}\right)}^{{\displaystyle \frac{1}{q}}},\hfill \end{array}$$

for all $j=1,2,3,\dots ,n-1.$

Proof. 

By applying Corollary 2, the proof is similar to Proposition 1. □

Proposition 3.

Suppose that $\phi :\left[\omega ,\varpi \right]\to \mathbb{R}$ is a differentiable mapping on $\left(\omega ,\varpi \right)$. Then we have the following:

$$\begin{array}{c}\hfill \left|{\mathbb{R}}_w\left({\mathrm{{\rm Y}}}_n,\phi \right)\right|\le {\displaystyle \frac{13}{225}}\sum _{j=0}^{n-1}{\displaystyle \frac{{\left({\vartheta }_{i+1}-{\vartheta }_i\right)}^2}{6}}{\left({\displaystyle \frac{{\left|{\phi }^{\prime }\left({\vartheta }_i\right)\right|}^q+{\left|{\phi }^{\prime }\left({\vartheta }_{i+1}\right)\right|}^q}{2}}\right)}^{{\displaystyle \frac{1}{q}}},\end{array}$$

for all $j=1,2,3,\dots ,n-1.$

Proof. 

By applying Corollary 3, the proof is similar to Proposition 1. □

Remark 3.

The error approximations in classical methods are based on Taylor expansion for the Weddle formula and involve the 6th derivative ${\left|\left|{\phi }^{\left(6\right)}\right|\right|}_{\infty }$. If the ${\phi }^{\left(6\right)}$ derivative does not exists or is very large at some points in $\left[\omega ,\varpi \right]$, the classical approximation cannot be applied and thus (28) provides an alternative approximation for Weddle’s formula.

5. Conclusions

This study introduces novel Weddle rule-type inequalities for single-time differentiable s-convex functions in the second sense. By using an integral identity, we derive new error bounds for Weddle’s quadrature formula, significantly improving upon the existing results. Our findings demonstrate that s-convex functions yield sharper approximations compared to classical convex functions, as evidenced by the comparative analysis in Table 1 and Table 2. To underscore the practical utility of our results, we present applications, supported by numerical examples and computational analyses, which validate the theoretical advancements. The methodologies developed here extend to diverse classes of functions, offering broader implications for research in convexity, s-convexity, fractional integrals, coordinated convexity, q-calculus, and higher-order differential equations. This work not only advances the field of integral inequalities but also opens new avenues for exploration in numerical analysis and applied mathematics. The results are pivotal for future studies seeking to refine error estimations and expand the applicability of quadrature rules in computational frameworks.