On s-Convexity of Dual Simpson Type Integral Inequalities

Abstract

Integral inequalities are a powerful tool for estimating errors of quadrature formulas. In this study, some symmetric dual Simpson type integral inequalities for the classes of s-convex, bounded and Lipschitzian functions are proposed. The obtained results are based on a new identity and the use of some standard techniques such as Hölder as well as power mean inequalities. We give at the end some applications to the estimation of quadrature rules and to particular means.

1. Introduction

The concept of convexity and its variants plays a fundamental and important role in the development of various fields of science and engineering in a direct or indirect way. This concept has a closed relationship in the development of the theory of inequalities, which is an important tool in the study of some qualitative properties of solutions for differential and integral equations as well as in numerical analysis for estimating the errors in quadrature formulas. Noting that the most used methods for evaluating the integrals by a numerical approach is that of Newton–Cotes, which comprises a group of formulas involving a certain numbers of equally spaced points.

Definition 1

([1]). A function $\mathsf{\Lambda }:I\to \mathbb{R}$ is said to be convex, if

$$\mathsf{\Lambda }(ie+(1-i\left)k\right)\le i\mathsf{\Lambda }\left(e\right)+(1-i)\mathsf{\Lambda }\left(k\right)$$

holds for all $e,k\in I$ and all $i\in [0,1]$.

Bakula et al. [2], studied the following general form of three point Newton–Cotes formula via weighted Montgomery identities:

$$\stackrel{f}{\underset{e}{\int }}w\left(j\right)\aleph \left(j\right)dj=C\left(\chi \right)\left(\aleph \left(\chi \right)+\aleph \left(e+f-\chi \right)\right)+\left(1-2C\left(\chi \right)\right)\aleph \left(\frac{e+f}{2}\right)+\mathcal{R}\left(w,\aleph ,\chi \right),$$

(1)

where $e<f$ and $\chi \in \left[e,\frac{e+f}{2}\right),\mathcal{R}\left(w,\aleph ,\chi \right)$ is the remainder term and $C\left(\chi \right)$ is an arbitrary real function defined on $\left[e,\frac{e+f}{2}\right)$ and gives the following results:

$$\begin{array}{c}\left|\stackrel{f}{\underset{e}{\int }}w\left(j\right)\aleph \left(j\right)dj-C\left(\chi \right)\left(\aleph \left(\chi \right)+\aleph \left(e+f-\chi \right)\right)+\left(1-2C\left(\chi \right)\right)\aleph \left(\frac{e+f}{2}\right)-g_n\left(\chi \right)\right|\hfill \\ \le \frac{2B\left(\alpha +1,n-1\right)L}{\left(\alpha +n\right)\left(n-2\right)!}\left(\left|C\left(x\right)\right|\left({\left(\chi -e\right)}^{\alpha +n}+{\left(f-\chi \right)}^{\alpha +n}\right)+\left|1-2C\left(\chi \right)\right|{\left(\frac{f-e}{2}\right)}^{\alpha +n}\right),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill g_n\left(\chi \right)=& C\left(\chi \right)\left(\underset{i=0}{\sum ^{n-1}}\frac{{\aleph }^{\left(i+1\right)}\left(f\right)}{i!}\left(\stackrel{f}{\underset{x}{\int }}\left(1-W\left(j\right)\right){\left(j-f\right)}^{i}dj+\stackrel{f}{\underset{e+f-\chi }{\int }}\left(1-W\left(j\right)\right){\left(j-f\right)}^{i}dj\right)\right.\hfill \\ \hfill & -\left.\underset{i=0}{\sum ^{n-1}}\frac{{\aleph }^{\left(i+1\right)}\left(e\right)}{i!}\left(\stackrel{\chi }{\underset{e}{\int }}\left(1-W\left(j\right)\right){\left(j-e\right)}^{i}dj+\stackrel{e+f-\chi }{\underset{a}{\int }}\left(1-W\left(j\right)\right){\left(j-e\right)}^{i}dj\right)\right)\hfill \\ \hfill & +\left(1-2C\left(\chi \right)\right)\left(\underset{i=0}{\sum ^{n-1}}\frac{{\aleph }^{\left(i+1\right)}\left(f\right)}{i!}\stackrel{f}{\underset{\frac{e+f}{2}}{\int }}\left(1-W\left(j\right)\right){\left(j-f\right)}^{i}dj\right.\hfill \\ \hfill & -\left.\underset{i=0}{\sum ^{n-1}}\frac{{\aleph }^{\left(i+1\right)}\left(f\right)}{i!}\stackrel{\frac{e+f}{2}}{\underset{e}{\int }}\left(1-W\left(j\right)\right){\left(j-e\right)}^{i}dj\right),\hfill \end{array}$$

With $W\left(\chi \right)=\stackrel{\chi }{\underset{e}{\int }}w\left(j\right)dj$ for all $x\in \left[e,f\right]$ and $W\left(f\right)=1$. Additionally,

$$\begin{array}{cc}\hfill & \left|\stackrel{b}{\underset{a}{\int }}w\left(j\right)\aleph \left(j\right)dj-C\left(\chi \right)\left(\aleph \left(\chi \right)+\aleph \left(e+f-\chi \right)\right)+\left(1-2C\left(\chi \right)\right)\aleph \left(\frac{e+f}{2}\right)-r_n\left(\chi \right)\right|\hfill \\ \hfill \le & \frac{2B\left(\alpha +1,n-1\right)L}{\left(f-e\right)\left(\alpha +n+1\right)\left(n-2\right)!}\hfill \\ \hfill & \times \left(\left|C\left(\chi \right)\right|\left({\left(\chi -e\right)}^{\alpha +n+1}+{\left(f-\chi \right)}^{\alpha +n+1}\right)+\left|1-2C\left(\chi \right)\right|{\left(\frac{f-e}{2}\right)}^{\alpha +n+1}\right),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill r_n\left(\chi \right)=& C\left(\chi \right)\underset{i=0}{\sum ^{n-1}}\frac{\left({\left(-1\right)}^i{\aleph }^{\left(i+1\right)}\left(f\right)-f^{\left(i+1\right)}\left(e\right)\right)\left({\left(x-e\right)}^{i+2}+{\left(f-x\right)}^{i+2}\right)}{i!\left(i+2\right)\left(f-e\right)}\hfill \\ \hfill & +\left(1-2C\left(\chi \right)\right)\underset{i=0}{\sum ^{n-1}}\frac{\left({\left(-1\right)}^i{\aleph }^{\left(i+1\right)}\left(f\right)-{\aleph }^{\left(i+1\right)}\left(e\right)\right){\left(f-e\right)}^{i+2}}{i!\left(i+2\right)2^{i+2}}.\hfill \end{array}$$

Obviously, if $C\left(\frac{3e+f}{4}\right)=\frac{2}{3}$, then identity (1) gives the weighted version of the dual Simpson inequality. Moreover, if we choose $w\left(j\right)=\frac{1}{f-e}$, we obtain the classical dual Simpson type inequality for functions whose $n^{th}$ derivatives are $\alpha$-L-Hölderians.

In [3], Pečarić and Vukelić used the Euler-type identities and gave some estimates of the general dual Simpson quadrature formula for functions as well as first derivatives are of bounded variation on $[0,1]$, L-Lipschitzian and R-integrable as follows.

In the case where ℵ is L-Lipschitzian on $[0,1]$, we have

$$\left|\stackrel{1}{\underset{0}{\int }}\aleph \left(j\right)dj-\frac{1}{2u-v}\left(u\aleph \left(\frac{1}{4}\right)-v\aleph \left(\frac{1}{2}\right)+u\aleph \left(\frac{3}{4}\right)\right)\right|\le \frac{2u+v}{8\left(2u-v\right)}L.$$

(2)

If ${\aleph }^{\prime }$ is L-Lipschitzian on $[0,1]$, then

$$\begin{array}{cc}\hfill & \left|\stackrel{1}{\underset{0}{\int }}\aleph \left(j\right)dj-\frac{1}{2u-v}\left(u\aleph \left(\frac{1}{4}\right)-v\aleph \left(\frac{1}{2}\right)+u\aleph \left(\frac{3}{4}\right)\right)\right|\hfill \\ \hfill \le & \frac{2u^2\left(3v+\sqrt{2uv}\right)+uv\left(5v-\sqrt{2uv}\right)+2v^2\left(v+3\sqrt{2uv}\right)}{48\left(2u-v\right)\left(v+\sqrt{2uv}\right)\left(2u+v+2\sqrt{2uv}\right)}L.\hfill \end{array}$$

(3)

If ℵ is a continuous function of bounded variation on $[0,1]$, then

$$\left|\stackrel{1}{\underset{0}{\int }}\aleph \left(j\right)dj-\frac{1}{2u-v}\left(u\aleph \left(\frac{1}{4}\right)-v\aleph \left(\frac{1}{2}\right)+u\aleph \left(\frac{3}{4}\right)\right)\right|\le \frac{2u+v}{4\left(2u-v\right)}\stackrel{1}{\underset{0}{\vee }}\left(f\right).$$

(4)

If ${\aleph }^{\prime }$ is a continuous function of bounded variation $[0,1]$, then

$$\begin{array}{cc}\hfill & \left|\stackrel{1}{\underset{0}{\int }}\aleph \left(j\right)dj-\frac{1}{2u-v}\left(u\aleph \left(\frac{1}{4}\right)-v\aleph \left(\frac{1}{2}\right)+u\aleph \left(\frac{3}{4}\right)\right)\right|\hfill \\ \hfill \le & \frac{2u+3v+\left|2u-5v\right|}{64\left(2u-v\right)}\stackrel{1}{\underset{0}{\vee }}\left({\aleph }^{\prime }\right).\hfill \end{array}$$

(5)

By taking $u=2$ and $v=1$, inequalities (2)–(5) will be reduced to the classical dual Simpson inequality, of which the general form is as follows:

$$\left|\frac{1}{3}\left(2\aleph \left(\frac{3e+f}{4}\right)-\aleph \left(\frac{e+f}{2}\right)+2\aleph \left(\frac{e+3f}{4}\right)\right)-\frac{1}{f-e}\stackrel{f}{\underset{e}{\int }}\aleph \left(j\right)dj\right|\le \frac{7{\left(f-e\right)}^4}{23040}{∥{\aleph }^{\left(4\right)}∥}_{\infty },$$

(6)

where ℵ is a four-times continuously differentiable function on $\left(e,f\right)$, and ${∥{\aleph }^{\left(4\right)}∥}_{\infty }={\displaystyle \underset{x\in \left(e,f\right)}{sup}}\left|{\aleph }^{\left(4\right)}\left(x\right)\right|$, (see [4,5,6]).

In [7], Dragomir gave the following Simpson inequality for mapping of bounded variation:

$$\left|\frac{1}{6}\left(\aleph \left(e\right)+4\aleph \left(\frac{e+f}{2}\right)+\aleph \left(f\right)\right)-\frac{1}{f-e}\stackrel{f}{\underset{e}{\int }}\aleph \left(j\right)dj\right|\le \frac{1}{3}\left(f-e\right)\underset{e}{\stackrel{f}{\vee }}\left(\aleph \right),$$

where $\underset{e}{\stackrel{f}{\vee }}\left({\aleph }^{\left(n\right)}\right)$ is the total variation of function ℵ.

Pečarić and Varošanec [8] discussed the Simpson inequality for derivatives of bounded variation

$$\left|\frac{1}{6}\left(\aleph \left(e\right)+4\aleph \left(\frac{e+f}{2}\right)+\aleph \left(f\right)\right)-\frac{1}{f-e}\stackrel{f}{\underset{e}{\int }}\aleph \left(j\right)dj\right|\le c_n{\left(f-e\right)}^n\underset{e}{\stackrel{f}{\vee }}\left({\aleph }^{\left(n\right)}\right),$$

where $n\in \left\{0,1,2,3\right\}$ with $c_0=\frac{1}{3},c_1=\frac{1}{24},c_2=\frac{1}{324},c_3=\frac{1}{1152}$ and $\underset{e}{\stackrel{f}{\vee }}\left({\aleph }^{\left(n\right)}\right)$ is the total variation of function ${\aleph }^{\left(n\right)}$.

Regarding some papers dealing with three-point Newton–Cotes, we refer readers to [9,10,11,12,13,14] and references therein.

In this paper, by adopting a novel approach, we establish some dual Simpson-type inequalities for functions whose first derivatives are s-convex. The cases where the first derivatives are bounded as well as Lipschitzian functions are also discussed. Applications of the results are given.

2. Main Results

We recall that a non-negative function $\mathsf{\Lambda }:I\subset \left[0,\infty \right)\to \mathbb{R}$ is said to be s-convex in the second sense for some fixed $s\in \left(0,1\right]$, if

$$\mathsf{\Lambda }(ie+(1-i)k)\le i^s\mathsf{\Lambda }\left(e\right)+{(1-i)}^s\mathsf{\Lambda }\left(k\right),$$

holds for all $e,k\in I$ and $i\in [0,1]$ (see [15]).

Now, we prove the following identity, which is basic to establish our main results.

Lemma 1.

Let $\aleph :\left[\vartheta ,\kappa \right]\to \mathbb{R}$ be a differentiable function on $\left[\vartheta ,\kappa \right]$, with $\vartheta <\kappa$ and ${\aleph }^{\prime }\in L^1\left[\vartheta ,\kappa \right]$, then the following equality holds

$$\begin{array}{cc}\hfill & \frac{1}{3}\left(2\aleph \left(\frac{3\vartheta +\kappa }{4}\right)-\aleph \left(\frac{\vartheta +\kappa }{2}\right)+2\aleph \left(\frac{\vartheta +3\kappa }{4}\right)\right)-\frac{1}{\kappa -\vartheta }\underset{\vartheta }{\stackrel{\kappa }{\int }}\aleph \left(j\right)dj\hfill \\ \hfill =& \frac{\kappa -\vartheta }{16}\left(\underset{0}{\stackrel{1}{\int }}i{\aleph }^{\prime }\left(\left(1-i\right)\vartheta +i\frac{3\vartheta +\kappa }{4}\right)di+\underset{0}{\stackrel{1}{\int }}\left(i-\frac{5}{3}\right){\aleph }^{\prime }\left(\left(1-i\right)\frac{3\vartheta +\kappa }{4}+i\frac{\vartheta +\kappa }{2}\right)di\right.\hfill \\ \hfill & +\left.\underset{0}{\stackrel{1}{\int }}\left(i+\frac{2}{3}\right){\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +\kappa }{2}+i\frac{\vartheta +3\kappa }{4}\right)di+\underset{0}{\stackrel{1}{\int }}\left(i-1\right){\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +3\kappa }{4}+i\kappa \right)di\right).\hfill \end{array}$$

Proof. 

Let

$$\begin{array}{cc}\hfill I_1& =\underset{0}{\stackrel{1}{\int }}i{\aleph }^{\prime }\left(\left(1-i\right)\vartheta +i\frac{3\vartheta +\kappa }{4}\right)di,\hfill \\ \hfill I_2& =\underset{0}{\stackrel{1}{\int }}\left(i-\frac{5}{3}\right){\aleph }^{\prime }\left(\left(1-i\right)\frac{3\vartheta +\kappa }{4}+i\frac{\vartheta +\kappa }{2}\right)di,\hfill \\ \hfill I_3& =\underset{0}{\stackrel{1}{\int }}\left(i+\frac{2}{3}\right){\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +\kappa }{2}+i\frac{\vartheta +3\kappa }{4}\right)di,\hfill \\ \hfill I_4& =\underset{0}{\stackrel{1}{\int }}\left(i-1\right){\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +3\kappa }{4}+i\kappa \right)di.\hfill \end{array}$$

Integrating by parts $I_1$, we obtain

$$\begin{array}{cc}\hfill I_1& ={\left.\frac{4}{\kappa -\vartheta }i\aleph \left(\left(1-i\right)\vartheta +i\frac{3\vartheta +\kappa }{4}\right)\right|}_{i=0}^{i=1}\hfill \\ \hfill & -\frac{4}{\kappa -\vartheta }\underset{0}{\stackrel{1}{\int }}\aleph \left(\left(1-i\right)\vartheta +i\frac{3\vartheta +\kappa }{4}\right)di\hfill \\ \hfill & =\frac{4}{\kappa -\vartheta }\aleph \left(\frac{3\vartheta +\kappa }{4}\right)-\frac{4}{\kappa -\vartheta }\underset{0}{\stackrel{1}{\int }}\aleph \left(\left(1-i\right)\vartheta +i\frac{3\vartheta +\kappa }{4}\right)di\hfill \\ \hfill & =\frac{4}{\kappa -\vartheta }\aleph \left(\frac{3\vartheta +\kappa }{4}\right)-\frac{16}{{\left(\kappa -\vartheta \right)}^2}\underset{\vartheta }{\stackrel{\frac{3\vartheta +\kappa }{4}}{\int }}\aleph \left(j\right)dj.\hfill \end{array}$$

(7)

Similarly, we obtain

$$I_2=-\frac{8}{3\left(\kappa -\vartheta \right)}\aleph \left(\frac{\vartheta +\kappa }{2}\right)+\frac{20}{3\left(\kappa -\vartheta \right)}\aleph \left(\frac{3\vartheta +\kappa }{4}\right)-\frac{16}{{\left(\kappa -\vartheta \right)}^2}\underset{\frac{3\vartheta +\kappa }{4}}{\stackrel{\frac{\vartheta +\kappa }{2}}{\int }}\aleph \left(j\right)dj,$$

(8)

$$I_3=\frac{20}{3\left(\kappa -\vartheta \right)}\aleph \left(\frac{\vartheta +3\kappa }{4}\right)-\frac{8}{3\left(\kappa -\vartheta \right)}\aleph \left(\frac{\vartheta +\kappa }{2}\right)-\frac{16}{{\left(\kappa -\vartheta \right)}^2}\underset{\frac{\vartheta +\kappa }{2}}{\stackrel{\frac{\vartheta +3\kappa }{4}}{\int }}\aleph \left(j\right)dj,$$

(9)

and

$$I_4=\frac{4}{\kappa -\vartheta }\aleph \left(\frac{\vartheta +3\kappa }{4}\right)-\frac{16}{{\left(\kappa -\vartheta \right)}^2}\underset{\frac{\vartheta +3\kappa }{4}}{\stackrel{\kappa }{\int }}\aleph \left(j\right)dj.$$

(10)

Adding (7)–(10), multiplying the result by $\frac{\kappa -\vartheta }{16}$, we obtain the desired result. □

Theorem 1.

Let ℵ be as in Lemma 1 with $0\le \vartheta <\kappa$. If $\left|{\aleph }^{\prime }\right|$ is s-convex in the second sense for some fixed $s\in \left(0,1\right]$, then we have

$$\begin{array}{cc}\hfill & \left|\frac{1}{3}\left(2\aleph \left(\frac{3\vartheta +\kappa }{4}\right)-\aleph \left(\frac{\vartheta +\kappa }{2}\right)+2\aleph \left(\frac{\vartheta +3\kappa }{4}\right)\right)-\frac{1}{\kappa -\vartheta }\underset{\vartheta }{\stackrel{\kappa }{\int }}\aleph \left(j\right)dj\right|\hfill \\ \hfill \le & \frac{\kappa -\vartheta }{16}\left(\frac{1}{\left(s+1\right)\left(s+2\right)}\left(\left|{\aleph }^{\prime }\left(\vartheta \right)\right|+\left|{\aleph }^{\prime }\left(\kappa \right)\right|\right)+\frac{4s+14}{3\left(s+1\right)\left(s+2\right)}\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|\right.\hfill \\ \hfill & +\left.\frac{8s+10}{3\left(s+1\right)\left(s+2\right)}\left(\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|+\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|\right)\right).\hfill \end{array}$$

Proof. 

From Lemma 1, properties of modulus, and s-convexity in the second sense of $\left|{\aleph }^{\prime }\right|$, we have

$$\begin{array}{cc}\hfill & \left|\frac{1}{3}\left(2\aleph \left(\frac{3\vartheta +\kappa }{4}\right)-\aleph \left(\frac{\vartheta +\kappa }{2}\right)+2\aleph \left(\frac{\vartheta +3\kappa }{4}\right)\right)-\frac{1}{\kappa -\vartheta }\underset{\vartheta }{\stackrel{\kappa }{\int }}\aleph \left(j\right)dj\right|\hfill \\ \hfill \le & \frac{\kappa -\vartheta }{16}\left(\underset{0}{\stackrel{1}{\int }}t\left|{\aleph }^{\prime }\left(\left(1-i\right)\vartheta +i\frac{3\vartheta +\kappa }{4}\right)\right|di+\underset{0}{\stackrel{1}{\int }}\left(\frac{5}{3}-i\right)\left|{\aleph }^{\prime }\left(\left(1-i\right)\frac{3\vartheta +\kappa }{4}+i\frac{\vartheta +\kappa }{2}\right)\right|di\right.\hfill \\ \hfill & +\left.\underset{0}{\stackrel{1}{\int }}\left(i+\frac{2}{3}\right)\left|{\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +\kappa }{2}+i\frac{\vartheta +3\kappa }{4}\right)\right|di+\underset{0}{\stackrel{1}{\int }}\left(1-i\right)\left|{\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +3\kappa }{4}+i\kappa \right)\right|di\right)\hfill \\ \hfill \le & \frac{\kappa -\vartheta }{16}\left(\underset{0}{\stackrel{1}{\int }}i\left({\left(1-i\right)}^s\left|{\aleph }^{\prime }\left(\vartheta \right)\right|+i^s\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|\right)di\right.\hfill \\ \hfill & +\underset{0}{\stackrel{1}{\int }}\left(\frac{5}{3}-i\right)\left({\left(1-i\right)}^s\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|+i^s\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|\right)di\hfill \\ \hfill & +\underset{0}{\stackrel{1}{\int }}\left(i+\frac{2}{3}\right)\left({\left(1-i\right)}^s\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|+i^s\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|\right)di\hfill \\ \hfill & +\left.\underset{0}{\stackrel{1}{\int }}\left(1-i\right)\left({\left(1-i\right)}^s\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|+i^s\left|{\aleph }^{\prime }\left(\kappa \right)\right|\right)di\right)\hfill \\ \hfill =& \frac{\kappa -\vartheta }{16}\left(\left(\underset{0}{\stackrel{1}{\int }}i{(1-i)}^{s}di\right)\left|{\aleph }^{\prime }\left(\vartheta \right)\right|+\left(\underset{0}{\stackrel{1}{\int }}{i}^{s+1}di+\underset{0}{\stackrel{1}{\int }}(\frac{5}{3}-i){\left(1-i\right)}^{s}di\right)\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|\right.\hfill \\ \hfill & +\left(\underset{0}{\stackrel{1}{\int }}(\frac{5}{3}-i)i^{s}di+\underset{0}{\stackrel{1}{\int }}(i+\frac{2}{3}){(1-i)}^{s}di\right)\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|\hfill \\ \hfill & +\left.\left(\underset{0}{\stackrel{1}{\int }}(i+\frac{2}{3})i^{s}di+\underset{0}{\stackrel{1}{\int }}{(1-i)}^{s+1}di\right)\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|+\left(\underset{0}{\stackrel{1}{\int }}(1-i)i^{s}di\right)\left|{\aleph }^{\prime }\left(\kappa \right)\right|\right)\hfill \\ \hfill =& \frac{\kappa -\vartheta }{16}\left(\frac{1}{\left(s+1\right)\left(s+2\right)}\left(\left|{\aleph }^{\prime }\left(\vartheta \right)\right|+\left|{\aleph }^{\prime }\left(\kappa \right)\right|\right)+\frac{4s+14}{3\left(s+1\right)\left(s+2\right)}\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|\right.\hfill \\ \hfill & +\left.\frac{8s+10}{3\left(s+1\right)\left(s+2\right)}\left(\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|+\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|\right)\right),\hfill \end{array}$$

where we have used the fact that

$$\begin{array}{cc}\hfill \underset{0}{\stackrel{1}{\int }}i{\left(1-i\right)}^{s}di& =\underset{0}{\stackrel{1}{\int }}{i}^s\left(1-i\right)di=\frac{1}{\left(s+1\right)\left(s+2\right)},\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill \underset{0}{\stackrel{1}{\int }}{i}^{s+1}di& =\underset{0}{\stackrel{1}{\int }}{\left(1-i\right)}^{s+1}di=\frac{1}{s+2},\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill \underset{0}{\stackrel{1}{\int }}\left(\frac{5}{3}-i\right){\left(1-i\right)}^{s}di& =\underset{0}{\stackrel{1}{\int }}{i}^s\left(i+\frac{2}{3}\right)di=\frac{5s+7}{3\left(s+1\right)\left(s+2\right)},\hfill \end{array}$$

(13)

and

$$\underset{0}{\stackrel{1}{\int }}{i}^s\left(\frac{5}{3}-i\right)di=\underset{0}{\stackrel{1}{\int }}\left(i+\frac{2}{3}\right){\left(1-i\right)}^{s}di=\frac{2s+7}{3\left(s+1\right)\left(s+2\right)}.$$

(14)

The proof is completed. □

Corollary 1.

For $s=1$, Theorem 1 gives

$$\begin{array}{cc}\hfill & \left|\frac{1}{3}\left(2\aleph \left(\frac{3\vartheta +\kappa }{4}\right)-\aleph \left(\frac{\vartheta +\kappa }{2}\right)+2\aleph \left(\frac{\vartheta +3\kappa }{4}\right)\right)-\frac{1}{\kappa -\vartheta }\underset{\vartheta }{\stackrel{\kappa }{\int }}\aleph \left(j\right)dj\right|\hfill \\ \hfill \le & \frac{5\left(\kappa -\vartheta \right)}{24}\left(\frac{\left|{\aleph }^{\prime }\left(\vartheta \right)\right|+6\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|+6\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|+6\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|+\left|{\aleph }^{\prime }\left(\kappa \right)\right|}{20}\right).\hfill \end{array}$$

Theorem 2.

Let ℵ be as in Lemma 1 with $0\le \vartheta <\kappa$. If ${\left|{\aleph }^{\prime }\right|}^q$ is s-convex in the second sense for some fixed $s\in \left(0,1\right]$ where $q>1$ with $\frac{1}{p}+\frac{1}{q}=1$, then we have

$$\begin{array}{cc}\hfill & \left|\frac{1}{3}\left(2\aleph \left(\frac{3\vartheta +\kappa }{4}\right)-\aleph \left(\frac{\vartheta +\kappa }{2}\right)+2\aleph \left(\frac{\vartheta +3\kappa }{4}\right)\right)-\frac{1}{\kappa -\vartheta }\underset{\vartheta }{\stackrel{\kappa }{\int }}\aleph \left(j\right)dj\right|\hfill \\ \hfill \le & \frac{\kappa -\vartheta }{16{\left(p+1\right)}^{\frac{1}{p}}}\left({\left(\frac{{\left|{\aleph }^{\prime }\left(\vartheta \right)\right|}^q+{\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|}^q}{1+s}\right)}^{\frac{1}{q}}+{\left(\frac{{\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|}^q+{\left|{\aleph }^{\prime }\left(\kappa \right)\right|}^q}{1+s}\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +\left.{\left(\frac{5^{p+1}-2^{p+1}}{3^{p+1}}\right)}^{\frac{1}{p}}\left({\left(\frac{{\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|}^q+{\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|}^q}{1+s}\right)}^{\frac{1}{q}}+{\left(\frac{{\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|}^q+{\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|}^q}{1+s}\right)}^{\frac{1}{q}}\right)\right).\hfill \end{array}$$

Proof. 

From Lemma 1, properties of modulus, Hölder’s inequality, and s-convexity in the second sense of ${\left|{\aleph }^{\prime }\right|}^q$, we have

$$\begin{array}{cc}\hfill & \left|\frac{1}{3}\left(2\aleph \left(\frac{3\vartheta +\kappa }{4}\right)-\aleph \left(\frac{\vartheta +\kappa }{2}\right)+2\aleph \left(\frac{\vartheta +3\kappa }{4}\right)\right)-\frac{1}{\kappa -\vartheta }\underset{\vartheta }{\stackrel{\kappa }{\int }}\aleph \left(j\right)dj\right|\hfill \\ \hfill \le & \frac{\kappa -\vartheta }{16}\left({\left(\underset{0}{\stackrel{1}{\int }}{i}^{p}di\right)}^{\frac{1}{p}}{\left(\underset{0}{\stackrel{1}{\int }}{\left|{\aleph }^{\prime }\left(\left(1-i\right)\vartheta +i\frac{3\vartheta +\kappa }{4}\right)\right|}^{q}di\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +{\left(\underset{0}{\stackrel{1}{\int }}{\left(\frac{5}{3}-i\right)}^{p}di\right)}^{\frac{1}{p}}{\left(\underset{0}{\stackrel{1}{\int }}{\left|{\aleph }^{\prime }\left(\left(1-i\right)\frac{3\vartheta +\kappa }{4}+i\frac{\vartheta +\kappa }{2}\right)\right|}^{q}di\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\underset{0}{\stackrel{1}{\int }}{\left(i+\frac{2}{3}\right)}^{p}di\right)}^{\frac{1}{p}}{\left(\underset{0}{\stackrel{1}{\int }}{\left|{\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +\kappa }{2}+i\frac{\vartheta +3\kappa }{4}\right)\right|}^{q}di\right)}^{\frac{1}{q}}\hfill \\ \hfill & +\left.{\left(\underset{0}{\stackrel{1}{\int }}{\left(1-i\right)}^{p}di\right)}^{\frac{1}{p}}{\left(\underset{0}{\stackrel{1}{\int }}{\left|{\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +3\kappa }{4}+i\kappa \right)\right|}^{q}di\right)}^{\frac{1}{q}}\right)\hfill \\ \hfill \le & \frac{\kappa -\vartheta }{16{\left(p+1\right)}^{\frac{1}{p}}}\left({\left(\underset{0}{\stackrel{1}{\int }}\left({\left(1-i\right)}^s{\left|{\aleph }^{\prime }\left(\vartheta \right)\right|}^q+i^s{\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|}^q\right)di\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +{\left(\frac{5^{p+1}-2^{p+1}}{3^{p+1}}\right)}^{\frac{1}{p}}\left({\left(\underset{0}{\stackrel{1}{\int }}\left({\left(1-i\right)}^s{\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|}^q+i^s{\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|}^q\right)di\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +\left.{\left(\underset{0}{\stackrel{1}{\int }}\left({\left(1-i\right)}^s{\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|}^q+i^s{\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|}^q\right)di\right)}^{\frac{1}{q}}\right)\hfill \\ \hfill & +\left.{\left(\underset{0}{\stackrel{1}{\int }}\left({\left(1-i\right)}^s{\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|}^q+i^s{\left|{\aleph }^{\prime }\left(\kappa \right)\right|}^q\right)di\right)}^{\frac{1}{q}}\right)\hfill \\ \hfill =& \frac{\kappa -\vartheta }{16{\left(p+1\right)}^{\frac{1}{p}}}\left({\left(\frac{{\left|{\aleph }^{\prime }\left(\vartheta \right)\right|}^q+{\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|}^q}{1+s}\right)}^{\frac{1}{q}}+{\left(\frac{{\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|}^q+{\left|{\aleph }^{\prime }\left(\kappa \right)\right|}^q}{1+s}\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +\left.{\left(\frac{5^{p+1}-2^{p+1}}{3^{p+1}}\right)}^{\frac{1}{p}}\left({\left(\frac{{\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|}^q+{\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|}^q}{1+s}\right)}^{\frac{1}{q}}+{\left(\frac{{\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|}^q+{\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|}^q}{1+s}\right)}^{\frac{1}{q}}\right)\right).\hfill \end{array}$$

The proof is completed. □

Corollary 2.

For $s=1$, Theorem 2 gives

$$\begin{array}{cc}\hfill & \left|\frac{1}{3}\left(2\aleph \left(\frac{3\vartheta +\kappa }{4}\right)-\aleph \left(\frac{\vartheta +\kappa }{2}\right)+2\aleph \left(\frac{\vartheta +3\kappa }{4}\right)\right)-\frac{1}{\kappa -\vartheta }\underset{\vartheta }{\stackrel{\kappa }{\int }}\aleph \left(j\right)dj\right|\hfill \\ \hfill \le & \frac{\kappa -\vartheta }{16{\left(p+1\right)}^{\frac{1}{p}}}\left({\left(\frac{{\left|{\aleph }^{\prime }\left(\vartheta \right)\right|}^q+{\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|}^q}{2}\right)}^{\frac{1}{q}}+{\left(\frac{{\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|}^q+{\left|{\aleph }^{\prime }\left(\kappa \right)\right|}^q}{2}\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +\left.{\left(\frac{5^{p+1}-2^{p+1}}{3^{p+1}}\right)}^{\frac{1}{p}}\left({\left(\frac{{\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|}^q+{\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|}^q}{2}\right)}^{\frac{1}{q}}+{\left(\frac{{\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|}^q+{\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|}^q}{2}\right)}^{\frac{1}{q}}\right)\right).\hfill \end{array}$$

Theorem 3.

Let ℵ be as in Lemma 1 with $0\le \vartheta <\kappa$. If ${\left|{\aleph }^{\prime }\right|}^q$ is s-convex in the second sense for some fixed $s\in \left(0,1\right]$ where $q\ge 1$, then we have

$$\begin{array}{cc}\hfill & \left|\frac{1}{3}\left(2\aleph \left(\frac{3\vartheta +\kappa }{4}\right)-\aleph \left(\frac{\vartheta +\kappa }{2}\right)+2\aleph \left(\frac{\vartheta +3\kappa }{4}\right)\right)-\frac{1}{\kappa -\vartheta }\underset{\vartheta }{\stackrel{\kappa }{\int }}\aleph \left(j\right)dj\right|\hfill \\ \hfill \le & \frac{\kappa -\vartheta }{16}\left({\left(\frac{1}{2}\right)}^{1-\frac{1}{q}}\left({\left(\frac{{\left|{\aleph }^{\prime }\left(\vartheta \right)\right|}^q+\left(s+1\right){\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|}^q}{\left(s+1\right)\left(s+2\right)}\right)}^{\frac{1}{q}}+{\left(\frac{\left(s+1\right){\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|}^q+{\left|{\aleph }^{\prime }\left(\kappa \right)\right|}^q}{\left(s+1\right)\left(s+2\right)}\right)}^{\frac{1}{q}}\right)\right.\hfill \\ \hfill & +{\left(\frac{7}{6}\right)}^{1-\frac{1}{q}}{\left(\frac{\left(5s+7\right){\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|}^q+\left(2s+7\right){\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|}^q}{3\left(s+1\right)\left(s+2\right)}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +\left.{\left(\frac{7}{6}\right)}^{1-\frac{1}{q}}{\left(\frac{\left(2s+7\right){\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|}^q+\left(5s+7\right){\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|}^q}{3\left(s+1\right)\left(s+2\right)}\right)}^{\frac{1}{q}}\right).\hfill \end{array}$$

Proof. 

From Lemma 1, properties of modulus, power mean inequality, and s-convexity in the second sense of ${\left|{\aleph }^{\prime }\right|}^q$, we have

$$\begin{array}{cc}\hfill & \left|\frac{1}{3}\left(2\aleph \left(\frac{3\vartheta +\kappa }{4}\right)-\aleph \left(\frac{\vartheta +\kappa }{2}\right)+2\aleph \left(\frac{\vartheta +3\kappa }{4}\right)\right)-\frac{1}{\kappa -\vartheta }\underset{\vartheta }{\stackrel{\kappa }{\int }}\aleph \left(j\right)dj\right|\hfill \\ \hfill \le & \frac{\kappa -\vartheta }{16}\left({\left(\underset{0}{\stackrel{1}{\int }}idi\right)}^{1-\frac{1}{q}}{\left(\underset{0}{\stackrel{1}{\int }}i{\left|{\aleph }^{\prime }\left(\left(1-i\right)\vartheta +i\frac{3\vartheta +\kappa }{4}\right)\right|}^{q}di\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +{\left(\underset{0}{\stackrel{1}{\int }}\left(\frac{5}{3}-i\right)di\right)}^{1-\frac{1}{q}}{\left(\underset{0}{\stackrel{1}{\int }}\left(\frac{5}{3}-i\right){\left|{\aleph }^{\prime }\left(\left(1-i\right)\frac{3\vartheta +\kappa }{4}+i\frac{\vartheta +\kappa }{2}\right)\right|}^{q}di\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\underset{0}{\stackrel{1}{\int }}\left(i+\frac{2}{3}\right)di\right)}^{1-\frac{1}{q}}{\left(\underset{0}{\stackrel{1}{\int }}\left(i+\frac{2}{3}\right){\left|{\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +\kappa }{2}+i\frac{\vartheta +3\kappa }{4}\right)\right|}^{q}di\right)}^{\frac{1}{q}}\hfill \\ \hfill & +\left.{\left(\underset{0}{\stackrel{1}{\int }}\left(1-i\right)di\right)}^{1-\frac{1}{q}}{\left(\underset{0}{\stackrel{1}{\int }}\left(1-i\right){\left|{\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +3\kappa }{4}+i\kappa \right)\right|}^{q}di\right)}^{\frac{1}{q}}\right)\hfill \\ \hfill \le & \frac{\kappa -\vartheta }{16}\left({\left(\frac{1}{2}\right)}^{1-\frac{1}{q}}{\left(\underset{0}{\stackrel{1}{\int }}i\left({\left(1-i\right)}^s{\left|{\aleph }^{\prime }\left(\vartheta \right)\right|}^q+i^s{\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|}^q\right)di\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +{\left(\frac{7}{6}\right)}^{1-\frac{1}{q}}{\left(\underset{0}{\stackrel{1}{\int }}\left(\frac{5}{3}-i\right)\left({\left(1-i\right)}^s{\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|}^q+i^s{\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|}^q\right)di\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{7}{6}\right)}^{1-\frac{1}{q}}{\left(\underset{0}{\stackrel{1}{\int }}\left(i+\frac{2}{3}\right)\left({\left(1-i\right)}^s{\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|}^q+i^s{\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|}^q\right)di\right)}^{\frac{1}{q}}\hfill \\ \hfill & +\left.{\left(\frac{1}{2}\right)}^{1-\frac{1}{q}}{\left(\underset{0}{\stackrel{1}{\int }}\left(1-i\right)\left({\left(1-i\right)}^s{\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|}^q+i^s{\left|{\aleph }^{\prime }\left(\kappa \right)\right|}^q\right)di\right)}^{\frac{1}{q}}\right)\hfill \\ \hfill =& \frac{\kappa -\vartheta }{16}\left({\left(\frac{1}{2}\right)}^{1-\frac{1}{q}}{\left({\left|{\aleph }^{\prime }\left(\vartheta \right)\right|}^q\underset{0}{\stackrel{1}{\int }}i{\left(1-i\right)}^{s}di+{\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|}^q\underset{0}{\stackrel{1}{\int }}{i}^{s+1}di\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +{\left(\frac{7}{6}\right)}^{1-\frac{1}{q}}{\left({\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|}^q\underset{0}{\stackrel{1}{\int }}\left(\frac{5}{3}-i\right){\left(1-i\right)}^{s}di+{\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|}^q\underset{0}{\stackrel{1}{\int }}\left(\frac{5}{3}-i\right)i^{s}di\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{7}{6}\right)}^{1-\frac{1}{q}}{\left({\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|}^q\underset{0}{\stackrel{1}{\int }}\left(i+\frac{2}{3}\right){\left(1-i\right)}^{s}di+{\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|}^q\underset{0}{\stackrel{1}{\int }}\left(i+\frac{2}{3}\right)i^{s}di\right)}^{\frac{1}{q}}\hfill \\ \hfill & +\left.{\left(\frac{1}{2}\right)}^{1-\frac{1}{q}}{\left({\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|}^q\underset{0}{\stackrel{1}{\int }}{\left(1-i\right)}^{s+1}di+{\left|{\aleph }^{\prime }\left(b\right)\right|}^q\underset{0}{\stackrel{1}{\int }}\left(1-i\right)i^{s}di\right)}^{\frac{1}{q}}\right)\hfill \\ \hfill =& \frac{\kappa -\vartheta }{16}\left({\left(\frac{1}{2}\right)}^{1-\frac{1}{q}}\left({\left(\frac{{\left|{\aleph }^{\prime }\left(\vartheta \right)\right|}^q+\left(s+1\right){\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|}^q}{\left(s+1\right)\left(s+2\right)}\right)}^{\frac{1}{q}}+{\left(\frac{\left(s+1\right){\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|}^q+{\left|{\aleph }^{\prime }\left(\kappa \right)\right|}^q}{\left(s+1\right)\left(s+2\right)}\right)}^{\frac{1}{q}}\right)\right.\hfill \\ \hfill & +{\left(\frac{7}{6}\right)}^{1-\frac{1}{q}}{\left(\frac{\left(5s+7\right){\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|}^q+\left(2s+7\right){\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|}^q}{3\left(s+1\right)\left(s+2\right)}\right)}^{\frac{1}{q}}\hfill \\ \hfill & +\left.{\left(\frac{7}{6}\right)}^{1-\frac{1}{q}}{\left(\frac{\left(2s+7\right){\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|}^q+\left(5s+7\right){\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|}^q}{3\left(s+1\right)\left(s+2\right)}\right)}^{\frac{1}{q}}\right),\hfill \end{array}$$

where we have used (11)–(14). The proof is achieved. □

Corollary 3.

For $s=1$, Theorem 3 gives

$$\begin{array}{cc}\hfill & \left|\frac{1}{3}\left(2\aleph \left(\frac{3\vartheta +\kappa }{4}\right)-\aleph \left(\frac{\vartheta +\kappa }{2}\right)+2\aleph \left(\frac{\vartheta +3\kappa }{4}\right)\right)-\frac{1}{\kappa -\vartheta }\underset{\vartheta }{\stackrel{\kappa }{\int }}\aleph \left(j\right)dj\right|\hfill \\ \hfill \le & \frac{\kappa -\vartheta }{32}\left({\left(\frac{{\left|{\aleph }^{\prime }\left(\vartheta \right)\right|}^q+2{\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|}^q}{3}\right)}^{\frac{1}{q}}+{\left(\frac{2{\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|}^q+{\left|{\aleph }^{\prime }\left(\kappa \right)\right|}^q}{3}\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +\left.\frac{7}{3}\left({\left(\frac{4{\left|{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|}^q+3{\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|}^q}{7}\right)}^{\frac{1}{q}}+{\left(\frac{3{\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|}^q+4{\left|{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|}^q}{7}\right)}^{\frac{1}{q}}\right)\right).\hfill \end{array}$$

3. Further Results

In the following results, we will discuss the cases where ${\aleph }^{\prime }\left(x\right)$ is bounded as well as ${\aleph }^{\prime }\left(x\right)$ of L-Lipschitzian functions.

Theorem 4.

Let ℵ be as in Lemma 1. If there exist constants $-\infty <m<M<+\infty$ such that $m\le {\aleph }^{\prime }\left(x\right)\le M$ for all $x\in \left[\vartheta ,\kappa \right]$, then we have

$$\left|\frac{1}{3}\left(2\aleph \left(\frac{3\vartheta +\kappa }{4}\right)-\aleph \left(\frac{\vartheta +\kappa }{2}\right)+2\aleph \left(\frac{\vartheta +3\kappa }{4}\right)\right)-\frac{1}{\kappa -\vartheta }\underset{\vartheta }{\stackrel{\kappa }{\int }}\aleph \left(j\right)dj\right|\le \frac{5\left(\kappa -\vartheta \right)\left(M-m\right)}{48}.$$

Proof. 

From Lemma 1, we have

$$\begin{array}{cc}\hfill & \frac{1}{3}\left(2\aleph \left(\frac{3\vartheta +\kappa }{4}\right)-\aleph \left(\frac{\vartheta +\kappa }{2}\right)+2\aleph \left(\frac{\vartheta +3\kappa }{4}\right)\right)-\frac{1}{\kappa -\vartheta }\underset{\vartheta }{\stackrel{\kappa }{\int }}\aleph \left(j\right)dj\hfill \\ \hfill =& \frac{\kappa -\vartheta }{16}\left(\underset{0}{\stackrel{1}{\int }}i{\aleph }^{\prime }\left(\left(1-i\right)\vartheta +i\frac{3\vartheta +\kappa }{4}\right)di+\underset{0}{\stackrel{1}{\int }}\left(i-\frac{5}{3}\right){\aleph }^{\prime }\left(\left(1-i\right)\frac{3\vartheta +\kappa }{4}+i\frac{\vartheta +\kappa }{2}\right)di\right.\hfill \\ \hfill & +\left.\underset{0}{\stackrel{1}{\int }}\left(i+\frac{2}{3}\right){\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +\kappa }{2}+i\frac{\vartheta +3\kappa }{4}\right)di+\underset{0}{\stackrel{1}{\int }}\left(i-1\right){\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +3\kappa }{4}+i\kappa \right)di\right)\hfill \\ \hfill =& \frac{\kappa -\vartheta }{16}\left(\underset{0}{\stackrel{1}{\int }}i\left({\aleph }^{\prime }\left(\left(1-i\right)\vartheta +i\frac{3\vartheta +\kappa }{4}\right)-\frac{m+M}{2}+\frac{m+M}{2}\right)di\right.\hfill \\ \hfill & +\underset{0}{\stackrel{1}{\int }}\left(i-\frac{5}{3}\right)\left({\aleph }^{\prime }\left(\left(1-i\right)\frac{3\vartheta +\kappa }{4}+i\frac{\vartheta +\kappa }{2}\right)-\frac{m+M}{2}+\frac{m+M}{2}\right)di\hfill \\ \hfill & +\underset{0}{\stackrel{1}{\int }}\left(i+\frac{2}{3}\right)\left({\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +\kappa }{2}+i\frac{\vartheta +3\kappa }{4}\right)-\frac{m+M}{2}+\frac{m+M}{2}\right)di\hfill \\ \hfill & +\left.\underset{0}{\stackrel{1}{\int }}\left(i-1\right)\left({\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +3\kappa }{4}+i\kappa \right)-\frac{m+M}{2}+\frac{m+M}{2}\right)di\right)\hfill \\ \hfill =& \frac{\kappa -\vartheta }{16}\left(\underset{0}{\stackrel{1}{\int }}i\left({\aleph }^{\prime }\left(\left(1-i\right)\vartheta +i\frac{3\vartheta +\kappa }{4}\right)-\frac{m+M}{2}\right)di\right.\hfill \\ \hfill & +\underset{0}{\stackrel{1}{\int }}\left(i-\frac{5}{3}\right)\left({\aleph }^{\prime }\left(\left(1-i\right)\frac{3\vartheta +\kappa }{4}+i\frac{\vartheta +\kappa }{2}\right)-\frac{m+M}{2}\right)di\hfill \\ \hfill & +\underset{0}{\stackrel{1}{\int }}\left(i+\frac{2}{3}\right)\left({\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +\kappa }{2}+i\frac{\vartheta +3\kappa }{4}\right)-\frac{m+M}{2}\right)di\hfill \\ \hfill & +\underset{0}{\stackrel{1}{\int }}\left(i-1\right)\left({\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +3\kappa }{4}+i\kappa \right)-\frac{m+M}{2}\right)di,\hfill \end{array}$$

(15)

where we have used the fact that

$$\underset{0}{\stackrel{1}{\int }}idi+\underset{0}{\stackrel{1}{\int }}\left(i-\frac{5}{3}\right)di+\underset{0}{\stackrel{1}{\int }}\left(i+\frac{2}{3}\right)di+\underset{0}{\stackrel{1}{\int }}\left(i-1\right)di=\underset{0}{\stackrel{1}{\int }}\left(4i-2\right)di=0.$$

Applying the absolute value in both sides of (15), we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1}{3}\left(2\aleph \left(\frac{3\vartheta +\kappa }{4}\right)-\aleph \left(\frac{\vartheta +\kappa }{2}\right)+2\aleph \left(\frac{\vartheta +3\kappa }{4}\right)\right)-\frac{1}{\kappa -\vartheta }\underset{\vartheta }{\stackrel{\kappa }{\int }}\aleph \left(j\right)dj\right|\hfill \\ \hfill \le & \frac{\kappa -\vartheta }{16}\left(\underset{0}{\stackrel{1}{\int }}i\left|{\aleph }^{\prime }\left(\left(1-i\right)\vartheta +i\frac{3\vartheta +\kappa }{4}\right)-\frac{m+M}{2}\right|di\right.\hfill \\ \hfill & +\underset{0}{\stackrel{1}{\int }}\left(\frac{5}{3}-i\right)\left|{\aleph }^{\prime }\left(\left(1-i\right)\frac{3\vartheta +\kappa }{4}+i\frac{\vartheta +\kappa }{2}\right)-\frac{m+M}{2}\right|di\hfill \\ \hfill & +\underset{0}{\stackrel{1}{\int }}\left(i+\frac{2}{3}\right)\left|{\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +\kappa }{2}+i\frac{\vartheta +3\kappa }{4}\right)-\frac{m+M}{2}\right|di\hfill \\ \hfill & +\left.\underset{0}{\stackrel{1}{\int }}\left(1-i\right)\left(\left|{\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +3\kappa }{4}+i\kappa \right)-\frac{m+M}{2}\right|\right)di\right).\hfill \end{array}$$

(16)

Since $m\le {\aleph }^{\prime }\left(x\right)\le M$ for all $x\in \left[\vartheta ,\kappa \right]$, we have

$$\begin{array}{cc}\hfill \left|{\aleph }^{\prime }\left(\left(1-i\right)\vartheta +i\frac{3\vartheta +\kappa }{4}\right)-\frac{m+M}{2}\right|& \le \frac{M-m}{2},\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill \left|{\aleph }^{\prime }\left(\left(1-i\right)\frac{3\vartheta +\kappa }{4}+i\frac{\vartheta +\kappa }{2}\right)-\frac{m+M}{2}\right|& \le \frac{M-m}{2},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill \left|{\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +\kappa }{2}+i\frac{\vartheta +3\kappa }{4}\right)-\frac{m+M}{2}\right|& \le \frac{M-m}{2},\hfill \end{array}$$

(19)

and

$$\left|{\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +3\kappa }{4}+i\kappa \right)-\frac{m+M}{2}\right|\le \frac{M-m}{2}.$$

(20)

Using (17)–(20) in (16), we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1}{3}\left(2\aleph \left(\frac{3\vartheta +\kappa }{4}\right)-\aleph \left(\frac{\vartheta +\kappa }{2}\right)+2\aleph \left(\frac{\vartheta +3\kappa }{4}\right)\right)-\frac{1}{\kappa -\vartheta }\underset{\vartheta }{\stackrel{\kappa }{\int }}\aleph \left(j\right)dj\right|\hfill \\ \hfill \le & \frac{\left(\kappa -\vartheta \right)\left(M-m\right)}{32}\left(\underset{0}{\stackrel{1}{\int }}idi+\underset{0}{\stackrel{1}{\int }}\left(\frac{5}{3}-i\right)dt+\underset{0}{\stackrel{1}{\int }}\left(i+\frac{2}{3}\right)dt+\underset{0}{\stackrel{1}{\int }}\left(1-i\right)dt\right)\hfill \\ \hfill =& \frac{5\left(\kappa -\vartheta \right)\left(M-m\right)}{48}.\hfill \end{array}$$

The proof is completed. □

Theorem 5.

Let ℵ be as in Lemma 1. If ${\aleph }^{\prime }$ is L-Lipschitzian function on $\left[\vartheta ,\kappa \right]$, then we have

$$\left|\frac{1}{3}\left(2\aleph \left(\frac{3\vartheta +\kappa }{4}\right)-\aleph \left(\frac{\vartheta +\kappa }{2}\right)+2\aleph \left(\frac{\vartheta +3\kappa }{4}\right)\right)-\frac{1}{\kappa -\vartheta }\underset{\vartheta }{\stackrel{\kappa }{\int }}\aleph \left(j\right)dj\right|\le \frac{13{\left(\kappa -\vartheta \right)}^2}{192}L.$$

Proof. 

From Lemma 1, we have

$$\begin{array}{cc}\hfill & \frac{1}{3}\left(2\aleph \left(\frac{3\vartheta +\kappa }{4}\right)-\aleph \left(\frac{\vartheta +\kappa }{2}\right)+2\aleph \left(\frac{\vartheta +3\kappa }{4}\right)\right)-\frac{1}{\kappa -\vartheta }\underset{\vartheta }{\stackrel{\kappa }{\int }}\aleph \left(j\right)dj\hfill \\ \hfill =& \frac{\kappa -\vartheta }{16}\left(\underset{0}{\stackrel{1}{\int }}i{\aleph }^{\prime }\left(\left(1-i\right)\vartheta +i\frac{3\vartheta +\kappa }{4}\right)di+\underset{0}{\stackrel{1}{\int }}\left(i-\frac{5}{3}\right){\aleph }^{\prime }\left(\left(1-i\right)\frac{3\vartheta +\kappa }{4}+i\frac{\vartheta +\kappa }{2}\right)di\right.\hfill \\ \hfill & +\left.\underset{0}{\stackrel{1}{\int }}\left(i+\frac{2}{3}\right){\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +\kappa }{2}+i\frac{\vartheta +3\kappa }{4}\right)di+\underset{0}{\stackrel{1}{\int }}\left(i-1\right){\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +3\kappa }{4}+i\kappa \right)di\right)\hfill \\ \hfill =& \frac{\kappa -\vartheta }{16}\left(\underset{0}{\stackrel{1}{\int }}i\left({\aleph }^{\prime }\left(\left(1-i\right)\vartheta +i\frac{3\vartheta +\kappa }{4}\right)-{\aleph }^{\prime }\left(\vartheta \right)+{\aleph }^{\prime }\left(\vartheta \right)\right)di\right.\hfill \\ \hfill & +\underset{0}{\stackrel{1}{\int }}\left(i-\frac{5}{3}\right)\left({\aleph }^{\prime }\left(\left(1-i\right)\frac{3\vartheta +\kappa }{4}+i\frac{\vartheta +\kappa }{2}\right)-{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)+{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right)di\hfill \\ \hfill & +\underset{0}{\stackrel{1}{\int }}\left(i+\frac{2}{3}\right)\left({\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +\kappa }{2}+i\frac{\vartheta +3\kappa }{4}\right)-{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)+{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right)di\hfill \\ \hfill & +\left.\underset{0}{\stackrel{1}{\int }}\left(i-1\right)\left({\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +3\kappa }{4}+i\kappa \right)-{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)+{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right)di\right)\hfill \\ \hfill =& \frac{\kappa -\vartheta }{16}\left(\underset{0}{\stackrel{1}{\int }}i\left({\aleph }^{\prime }\left(\left(1-i\right)\vartheta +i\frac{3\vartheta +\kappa }{4}\right)-{\aleph }^{\prime }\left(\vartheta \right)\right)di\right.\hfill \\ \hfill & +\underset{0}{\stackrel{1}{\int }}\left(i-\frac{5}{3}\right)\left({\aleph }^{\prime }\left(\left(1-i\right)\frac{3\vartheta +\kappa }{4}+i\frac{\vartheta +\kappa }{2}\right)-{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right)di\hfill \\ \hfill & +\underset{0}{\stackrel{1}{\int }}\left(i+\frac{2}{3}\right)\left({\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +\kappa }{2}+i\frac{\vartheta +3\kappa }{4}\right)-{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right)di\hfill \\ \hfill & +\underset{0}{\stackrel{1}{\int }}\left(i-1\right)\left({\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +3\kappa }{4}+i\kappa \right)-{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right)di\hfill \\ \hfill & +\left.\frac{1}{2}\left({\aleph }^{\prime }\left(\vartheta \right)-{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right)+\frac{7}{6}\left({\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)-{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right)\right).\hfill \end{array}$$

(21)

Applying the absolute value in both sides of (21), and by using the fact that ${\aleph }^{\prime }$ is L-Lipschitzian on $\left[\vartheta ,\kappa \right]$, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1}{3}\left(2\aleph \left(\frac{3\vartheta +\kappa }{4}\right)-\aleph \left(\frac{\vartheta +\kappa }{2}\right)+2\aleph \left(\frac{\vartheta +3\kappa }{4}\right)\right)-\frac{1}{\kappa -\vartheta }\underset{\vartheta }{\stackrel{\kappa }{\int }}\aleph \left(j\right)dj\right|\hfill \\ \hfill \le & \frac{\kappa -\vartheta }{16}\left(\underset{0}{\stackrel{1}{\int }}i\left|{\aleph }^{\prime }\left(\left(1-i\right)\vartheta +i\frac{3\vartheta +\kappa }{4}\right)-{\aleph }^{\prime }\left(\vartheta \right)\right|di\right.\hfill \\ \hfill & +\underset{0}{\stackrel{1}{\int }}\left(\frac{5}{3}-i\right)\left|{\aleph }^{\prime }\left(\left(1-i\right)\frac{3\vartheta +\kappa }{4}+i\frac{\vartheta +\kappa }{2}\right)-{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|di\hfill \\ \hfill & +\underset{0}{\stackrel{1}{\int }}\left(i+\frac{2}{3}\right)\left|{\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +\kappa }{2}+i\frac{\vartheta +3\kappa }{4}\right)-{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)\right|di\hfill \\ \hfill & +\underset{0}{\stackrel{1}{\int }}\left(1-i\right)\left|{\aleph }^{\prime }\left(\left(1-i\right)\frac{\vartheta +3\kappa }{4}+i\kappa \right)-{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|di\hfill \\ \hfill & +\left.\frac{1}{2}\left|{\aleph }^{\prime }\left(\vartheta \right)-{\aleph }^{\prime }\left(\frac{\vartheta +3\kappa }{4}\right)\right|+\frac{7}{6}\left|{\aleph }^{\prime }\left(\frac{\vartheta +\kappa }{2}\right)-{\aleph }^{\prime }\left(\frac{3\vartheta +\kappa }{4}\right)\right|\right)\hfill \\ \hfill \le & \frac{\kappa -\vartheta }{16}\left(\frac{\left(\kappa -\vartheta \right)L}{4}\underset{0}{\stackrel{1}{\int }}{i}^{2}di+\frac{\left(\kappa -\vartheta \right)L}{4}\underset{0}{\stackrel{1}{\int }}\left(\frac{5}{3}-i\right)idi+\frac{\left(\kappa -\vartheta \right)L}{4}\underset{0}{\stackrel{1}{\int }}\left(i+\frac{2}{3}\right)idi\right.\hfill \\ \hfill & +\left.\frac{\left(\kappa -\vartheta \right)L}{4}\underset{0}{\stackrel{1}{\int }}\left(1-i\right)idi+\frac{L}{2}\left|a-\frac{a+3b}{4}\right|+\frac{7L}{6}\left|\frac{a+b}{2}-\frac{3a+b}{4}\right|\right)\hfill \\ \hfill =& \frac{13{\left(\kappa -\vartheta \right)}^2}{192}L.\hfill \end{array}$$

The proof is completed. □

4. Applications

Dual Simpson’s quadrature formula

Let $\mathsf{\Lambda }$ be the partition of the points $\vartheta =e_0<e_1<\dots <e_n=\kappa$ of the interval $\left[\vartheta ,\kappa \right]$, and consider the quadrature formula

$$\underset{\vartheta }{\stackrel{\kappa }{\int }}\aleph \left(j\right)dj=\lambda \left(\aleph ,\mathsf{\Lambda }\right)+R\left(\aleph ,\mathsf{\Lambda }\right),$$

where

$$\lambda \left(\aleph ,\mathsf{\Lambda }\right)=\underset{i=0}{\sum ^{n-1}}\frac{e_{i+1}-e_i}{3}\left(2\aleph \left(\frac{3e_i+e_{i+1}}{4}\right)-\aleph \left(\frac{e_i+e_{i+1}}{2}\right)+2\aleph \left(\frac{e_i+3e_{i+1}}{4}\right)\right),$$

and $R\left(\aleph ,\mathsf{\Lambda }\right)$ denotes the associated approximation error.

Proposition 1.

Let $n\in \mathbb{N}$ and $\aleph :\left[\vartheta ,\kappa \right]\to \mathbb{R}$ be a differentiable function on $\left(\vartheta ,\kappa \right)$ with $0\le \vartheta <\kappa$ and ${\aleph }^{\prime }$$\in L^1\left[\vartheta ,\kappa \right]$. If $\left|{\aleph }^{\prime }\right|$ is s-convex function with $s\in \left(0,1\right]$, we have

$$\begin{array}{cc}\hfill \left|R\left(\aleph ,{\rm Y}\right)\right|\le & \underset{i=0}{\sum ^{n-1}}\frac{{\left(e_{i+1}-e_i\right)}^2}{16}\left(\frac{1}{\left(s+1\right)\left(s+2\right)}\left(\left|{\aleph }^{\prime }\left(e_i\right)\right|+\left|{\aleph }^{\prime }\left(e_{i+1}\right)\right|\right)+\frac{4s+14}{3\left(s+1\right)\left(s+2\right)}\left|{\aleph }^{\prime }\left(\frac{e_i+e_{i+1}}{2}\right)\right|\right.\hfill \\ \hfill & +\left.\frac{8s+10}{3\left(s+1\right)\left(s+2\right)}\left(\left|{\aleph }^{\prime }\left(\frac{3e_i+e_{i+1}}{4}\right)\right|+\left|{\aleph }^{\prime }\left(\frac{e_i+3e_{i+1}}{4}\right)\right|\right)\right).\hfill \end{array}$$

Proof. 

Using Theorem 1 on $\left[e_i,e_{i+1}\right]$$\left(i=0,1,\dots ,n-1\right)$, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1}{3}\left(2\aleph \left(\frac{3e_i+e_{i+1}}{4}\right)-\aleph \left(\frac{e_i+e_{i+1}}{2}\right)+2\aleph \left(\frac{e_i+3e_{i+1}}{4}\right)\right)-\frac{1}{e_{i+1}-e_i}\underset{e_i}{\stackrel{e_{i+1}}{\int }}\aleph \left(j\right)dj\right|\hfill \\ \hfill \le & \frac{e_{i+1}-e_i}{16}\left(\frac{1}{\left(s+1\right)\left(s+2\right)}\left(\left|{\aleph }^{\prime }\left(e_i\right)\right|+\left|{\aleph }^{\prime }\left(e_{i+1}\right)\right|\right)+\frac{4s+14}{3\left(s+1\right)\left(s+2\right)}\left|{\aleph }^{\prime }\left(\frac{e_i+e_{i+1}}{2}\right)\right|\right.\hfill \\ \hfill & +\left.\frac{8s+10}{3\left(s+1\right)\left(s+2\right)}\left(\left|{\aleph }^{\prime }\left(\frac{3e_i+e_{i+1}}{4}\right)\right|+\left|{\aleph }^{\prime }\left(\frac{e_i+3e_{i+1}}{4}\right)\right|\right)\right).\hfill \end{array}$$

(22)

Multiplying both sides of (22) by $\left(e_{i+1}-e_i\right)$, summing the obtained inequalities for all $i=0,1,\dots ,n-1$ and using the triangular inequality, we obtain the result. □

Application to special means

For arbitrary real numbers $e,e_1,e_2,\dots ,e_n,f$ we have:

The Arithmetic mean: $A\left(e_1,e_2,\dots ,e_n\right)=\frac{e_1+e_2+\dots +e_n}{n}$.

The p-Logarithmic mean: $L_p\left(e,f\right)={\left(\frac{f^{p+1}-e^{p+1}}{\left(p+1\right)\left(f-e\right)}\right)}^{\frac{1}{p}}$, $e,f>0,e\ne f$ and $p\in \mathbb{R}‵\left\{-1,0\right\}$.

Proposition 2.

Let $e,f\in \mathbb{R}$ with $0<e<f$, then we have

$$\left|2A^{\frac{3}{2}}\left(e,e,e,f\right)-A^{\frac{3}{2}}\left(e,f\right)+2A^{\frac{3}{2}}\left(e,f,f,f\right)-3L_{\frac{3}{2}}^{\frac{3}{2}}\left(e,f\right)\right|\le \frac{15\left(f-e\right)\left(\sqrt{f}-\sqrt{e}\right)}{32}.$$

Proof. 

Applying Theorem 4 to the function $\aleph \left(j\right)=j^{\frac{3}{2}}$ on $\left[e,f\right]$. □

5. Conclusions

Many practical studies and engineering problems often lead to calculations of integrals, most of which cannot be solved directly, requiring us to evaluate them by different quadrature rules, hence the need to estimate the error made to better circumvent and manage the problem. Thus, in this work, we have considered the dual Simpson quadrature rule. We have firstly established a novel identity. Based on this identity, we have derived some new dual Simpson type integral inequalities for functions whose first derivatives are -convex. We have also discussed the above-mentioned inequality when the first derivatives lie in the classes of bounded and Lipschitzian functions. We have provided at the end some applications to quadrature formulas and special means. We hope that the obtained results stimulate further research, as well as generalizations in various other types of calculus in this interesting field.