An analytical approach of multi-dimensional Navier-Stokes equation in the framework of natural transform

Manoj Singh; Ahmed Hussein; Msmali; Mohammad Tamsir; Abdullah Ali H. Ahmadini; Manoj Singh; Ahmed Hussein; Msmali; Mohammad Tamsir; Abdullah Ali H. Ahmadini

doi:10.3934/math.2024426

AIMS Mathematics

2024, Volume 9, Issue 4: 8776-8802. doi: 10.3934/math.2024426

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Research article Special Issues

An analytical approach of multi-dimensional Navier-Stokes equation in the framework of natural transform

1.
Department of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Kingdom of Saudi Arabia
2.
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong NSW 2522, Australia

Received: 31 December 2023 Revised: 06 February 2024 Accepted: 21 February 2024 Published: 01 March 2024

This article introduces a new iterative transform method and homotopy perturbation transform method along with a natural transform to analyze the multi-dimensional Navier-Stokes equations. To solve the fractional-derivative, the Caputo-Fabrizio definition of the fractional derivative was employed. Four examples were considered to examine the efficacy and accuracy of the proposed methods. The efficiency and accuracy were also demonstrated by the solution comparison via graphs. The proposed methods' convergence and uniqueness are also discussed. The methods mentioned above are straightforward and support a high rate of convergence.

Keywords:

Citation: Manoj Singh, Ahmed Hussein, Msmali, Mohammad Tamsir, Abdullah Ali H. Ahmadini. An analytical approach of multi-dimensional Navier-Stokes equation in the framework of natural transform[J]. AIMS Mathematics, 2024, 9(4): 8776-8802. doi: 10.3934/math.2024426

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Abstract

1. Introduction

Riemann-Liouville fractional integral given by

$\begin{equation*} I_{a+}^{\alpha}\xi(\wp) = \frac{1}{\Gamma(\alpha)}\int_{a}^{\chi}(\chi-\wp)^{\alpha-1}\xi(\wp)dt. \end{equation*}$

Many different concepts of fractional derivative maybe found in ^[9,10,11]. In ^[12] studied a conformable derivative:

$\begin{equation*} \wp_{\alpha}f(\wp) = \lim\limits_{\epsilon\rightarrow 0}\frac{f(\wp+\epsilon \wp^{1-\alpha})-f(\wp)}{\epsilon}. \end{equation*}$

The time scale conformable derivatives was introduced by Benkhettou et al. ^[17].

Further, in recent years, numerous mathematicians claimed that non-integer order derivatives and integrals are well suited to describing the properties of many actual materials, such as polymers. Fractional derivatives are a wonderful tool for describing memory and learning. a variety of materials and procedures inherited properties is one of the most significant benefits of fractional ownership. For more concepts and definition on time scales see ^{[13,14,15,16,17,18,19,33,34,35]}.

Continuous version of Steffensen's inequality ^[7] is written as: For $0\leq g(\wp)\leq1$ on $\in[a, b]$ . Then

$\begin{equation} \int_{b-\lambda}^{b}f(\wp)dt\leq\int_{a}^{b}f(\wp)g(\wp)dt\leq\int_{a}^{a+\lambda}f(\wp)dt, \end{equation}$

(1.1)

where $\lambda = \int_{a}^{b}g(\wp)dt.$

Supposing $f$ is nondecreasing gets the reverse of (1.1).

Also, the discrete inequality of Steffensen ^[6] is: For $\lambda_2\leq \sum_{\ell = 1}^{n}g(\ell)\leq \lambda_1$ . Then

$\begin{equation} \sum\limits_{\ell = n-\lambda_2+1}^{n} f(\ell) \leq \sum\limits_{\ell = 1}^{n}f(\ell)g(\ell) \leq \sum\limits_{\ell = 1}^{\lambda_1} f(\ell). \end{equation}$

(1.2)

Recently, a large number of dynamic inequalities on time scales have been studied by a small number of writers who were inspired by a few applications (see ^{[1,2,3,4,8,28,29,30,31,32,36,37,40,41,42,44,48,49,50,51,52,53]}).

In ^[5] Jakšetić et al. proved that, if $\hat{\mu}([c, d]) = \int_{[a, b]}g(\wp)d \hat{\mu}(\wp)$ , where $[c, d]\subseteq[a, b]$ . Then

$\begin{equation*} \int_{[a,b]}f(\wp)g(\wp)d\hat{\mu}(\wp)\leq \int_{[c,d]}f(\wp)g(\wp)d\hat{\mu}(\wp)+\int_{[a,c]}\big(f(\wp)-f(d)\big)g(\wp)d\hat{\mu}(\wp), \end{equation*}$

and

$\begin{equation*} \int_{[c,d]}f(\wp)d\hat{\mu}(\wp)-\int_{[d,b]}\big(f(c)-f(\wp)\big)g(\wp)d\hat{\mu}(\wp)\leq \int_{[a,b]}f(\wp)g(\wp)d\hat{\mu}(\wp). \end{equation*}$

Anderson, in ^[3], studied the inequality:

$\begin{equation} \int_{b-\lambda}^{b}\phi(\wp)\nabla \wp\leq \int_{a}^{b}\phi(\wp)\psi(\wp)\nabla \wp\leq\int_{a}^{a+\lambda}\phi(\wp)\nabla \wp, \end{equation}$

(1.3)

In ^[47] the authors have proved, for

$\begin{equation*} \int_{m}^{m+\lambda_1}\zeta(\wp)d\wp = \int_{m}^{k}\zeta(\wp)g(\wp)d\wp, \end{equation*}$

and

$\begin{equation*} \int_{n-\lambda_2}^{n}\zeta(\wp)d\wp = \int_{k}^{n}\zeta(\wp)g(\wp)d\wp. \end{equation*}$

If there exists a constant $A$ such that $r(\wp)/\zeta(\wp)-At$ is monotonic on the intervals $[m, k]$ , $[k, n]$ , and

$\begin{equation*} \int_{m}^{n}tq(\wp)g(\wp)d\wp = \int_{m}^{m+\lambda_1}tq(\wp)d\wp+\int_{n-\lambda_2}^{n}tq(\wp)d\wp, \end{equation*}$

then

$\begin{equation*} \int_{m}^{n}r(\wp)g(\wp)d\wp\leq\int_{m}^{m+\lambda_1}r(\wp)d\wp+\int_{n-\lambda_2}^{n}r(\wp)d\wp. \end{equation*}$

In particularly, Anderson ^[3] proved

$\begin{equation*} \int_{n-\lambda}^{n}r(\wp)\nabla \wp\leq\int_{m}^{n}r(\wp)g(\wp)\nabla \wp\leq \int_{m}^{m+\lambda}r(\wp)\nabla \wp. \end{equation*}$

where $m, n\in\mathbb{T}_\kappa$ with $m < n$ , $r$ , $g:[m, n]_{\mathbb{T}}\rightarrow \mathbb{R}$ are $\nabla$ -integrable functions such that $r$ is of one sign and nonincreasing and $0\leq g(\wp)\leq1$ on $[m, n]_{\mathbb{T}}$ and $\lambda = \int_{m}^{n}g(\wp)\nabla \wp$ , $n-\lambda, m+\lambda \in \mathbb{T}$ .

We prove the next two needed results:

Theorem 1.1. Assume $q > 0$ with $0\leq g(\wp)\leq \zeta(\wp)$ $\forall\wp\in[m, n]_{\mathbb{T}}$ and $\lambda$ is given from $\int_{m}^{n}g(\wp)\Delta_{\alpha} \wp = \int_{m}^{m+\lambda}\zeta(\wp)\Delta_{\alpha} \wp$ , then

$\begin{equation} \int_{m}^{n}r(\wp)g(\wp)\Delta_{\alpha} \wp\leq \int_{m}^{m+\lambda}r(\wp)\zeta(\wp)\Delta_{\alpha} \wp. \end{equation}$

(1.4)

Also, provided with $0\leq g(\wp)\leq \zeta(\wp)$ and $\int_{n-\lambda}^{n}\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{n}g(\wp)\Delta_{\alpha} \wp$ , we have

$\begin{equation} \int_{n-\lambda}^{n}r(\wp)\zeta(\wp)\Delta_{\alpha} \wp\leq \int_{m}^{n}r(\wp)g(\wp)\Delta_{\alpha} \wp. \end{equation}$

(1.5)

We get the reverse inequalities of (1.4) and (1.5) when assuming $r/\zeta$ is nondecreasing.

Theorem 1.2. Assume $\psi$ is integrable on time scales interval $[m, n]$ , with $\zeta(\wp)-\psi(\wp)\geq g(\wp)\geq\psi(\wp)\geq 0 \; \forall\wp\in[m, n]_{\mathbb{T}}$ and $\int_{m}^{m+\lambda}\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{n}g(\wp)\Delta_{\alpha} \wp = \int_{n-\lambda}^{n}\zeta(\wp)\Delta_{\alpha} \wp$ and $g$ , $r$ and $\zeta$ are $\Delta_{\alpha}$ -integrable functions, $\zeta(\wp)\geq g(\wp)\geq0$ , we have

$\begin{eqnarray} &&\int_{n-\lambda}^{n}r(\wp)\zeta(\wp)\Delta_{\alpha} \wp+ \int_{m}^{n}\Big|\big(r(\wp)-r(n-\lambda)\big)\psi(\wp)\Big|\Delta_{\alpha} \wp\\ &&\leq \int_{m}^{n}r(\wp)g(\wp)\Delta_{\alpha} \wp\\ &&\leq \int_{m}^{m+\lambda}r(\wp)\zeta(\wp)\Delta_{\alpha} \wp-\int_{m}^{n}\Big|\big(r(\wp)-r(m+\lambda)\big)\psi(\wp)\bigg|\Delta_{\alpha} \wp, \end{eqnarray}$

(1.6)

and

$\begin{eqnarray} \int_{n-\lambda}^{n}r(\wp)\zeta(\wp)\Delta_{\alpha} \wp&\leq& \int_{n-\lambda}^{n}\Big[r(\wp)\zeta(\wp)-\big(r(\wp)-r(n-\lambda)\big)\Big]\big[\zeta(\wp)-g(\wp)\big]\Delta_{\alpha} \wp\\ &\leq&\int_{m}^{n}r(\wp)g(\wp)\Delta_{\alpha} \wp\\ &\leq& \int_{m}^{m+\lambda}\Big[r(\wp)\zeta(\wp)-\big(r(\wp)-r(m+\lambda)\big)\Big]\big[\zeta(\wp)-g(\wp)\big]\Delta_{\alpha} \wp\\ &\leq&\int_{m}^{m+\lambda}r(\wp)\zeta(\wp)\Delta_{\alpha} \wp. \end{eqnarray}$

(1.7)

Proof. The proof techniques of Theorems 1.6 and 1.7 are like to that in ^[4] and is removed.

Several authors proved conformable Hardy's inequality ^[20,21], conformable Hermite-Hadamard's inequality ^[22,23,24], conformable inequality of Opial's ^[26,27] and conformable inequality of Steffensen's ^[25]. In ^[45] Anderson proved the followong results:

Theorem 1.3. ^[45] Suppose $\alpha\in(0, 1]$ and $r_1$ , $r_2\in \mathbb{R}$ such that $0\leq r_1\leq r_2$ . Suppose $\prod: [r_1, r_2]\rightarrow [0, \infty)$ and $\Gamma: [r_1, r_2]\rightarrow [0, 1]$ are $\alpha$ -fractional integrable functions on $[r_1, r_2]$ with $\Pi$ is decreasing, we get

$\begin{equation*} \int_{r_2-\aleph}^{r_2}\Pi(\zeta)d_{\alpha}\zeta\leq \int_{r_1}^{r_2}\Pi(\zeta)\Gamma(\zeta)d_{\alpha}\zeta\leq \int_{r_1}^{r_1+\aleph}\Pi(\zeta)d_{\alpha}\zeta, \end{equation*}$

where $\aleph = \frac{\alpha(r_2-r_1)}{r_2^{\alpha}-r_1^{\alpha}}\int_{r_1}^{r_2}\Gamma(\zeta)d_{\alpha}\zeta\in[0, r_2-r_1]$ .

In ^[46] the authors gave an extension for Theorem 1.8:

Theorem 1.4. Assume $\alpha\in(0, 1]$ and $r_1$ , $r_2\in \mathbb{R}$ such that $0\leq r_1\leq r_2$ . Suppose $\prod, \Gamma, \Sigma: [r_1, r_2]\rightarrow [0, \infty)$ are integrable on $[r_1, r_2]$ with the decreasing function $\Pi$ and $0\leq \Gamma\leq \Sigma$ , we get

$\begin{equation*} \int_{r_2-\aleph}^{r_2}\Sigma(\zeta)\Pi(\zeta)d_{\alpha}\zeta\leq \int_{r_1}^{r_2}\Pi(\zeta)\Gamma(\zeta)d_{\alpha}\zeta\leq \int_{r_1}^{r_1+\aleph}\Sigma(\zeta)\Pi(\zeta)d_{\alpha}\zeta, \end{equation*}$

where $\aleph = \frac{(r_2-r_1)}{\int_{r_1}^{r_2}\Sigma(\zeta)d_{\alpha}\zeta}\int_{r_1}^{r_2}\Gamma(\zeta) d_{\alpha}\zeta\in[0, r_2-r_1]$ .

In this paper, we prove and explore several novel speculations of the Steffensen inequality obtained in ^[47] through the conformable integral containing time scale concept. We furthermore recover certain known results as special cases of our results.

2. Main results

Lemma 2.1. Assume $\zeta > 0$ is $rd$ -continuous function on $[m, n]\cap\mathbb{T}$ , $g$ , $r$ be $rd$ -continuous on $[m, n]\cap\mathbb{T}$ such that $r/\zeta$ nonincreasing function and $0\leq g(\wp)\leq 1$ $\forall\wp\in[m, n]\cap\mathbb{T}$ . Then

$(\Lambda_{1})$

$\begin{equation} \qquad \int_{m}^{n}r(\wp)g(\wp)\Delta_{\alpha} \wp\leq \int_{m}^{m+\lambda}r(\wp)\Delta_{\alpha} \wp, \end{equation}$

(2.1)

where $\lambda$ is given by

$\begin{equation*} \int_{m}^{n}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp = \int_{m}^{m+\lambda}\zeta(\wp)\Delta_{\alpha} \wp. \end{equation*}$

$(\Lambda_{2})$

$\begin{equation} \qquad\int_{n-\lambda}^{n}r(\wp)\Delta_{\alpha} \wp\leq \int_{m}^{n}r(\wp)g(\wp)\Delta_{\alpha} \wp, \end{equation}$

(2.2)

such that

$\begin{equation*} \int_{n-\lambda}^{n}\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{n}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp. \end{equation*}$

(2.1) and (2.2) are reversed when $r/\zeta$ is nondecreasing.

Proof. Putting $g(\wp)\mapsto \zeta(\wp)g(\wp)$ and $r(\wp)\mapsto r(\wp)/\zeta(\wp)$ in (1.4), (1.5) to get $(\Lambda_{1})$ and $(\Lambda_{2})$ simultaneously.

Lemma 2.2. Under the same hypotheses of Lemma 2.1. with $\psi$ be integrable functions on $[m, n]\cap\mathbb{T}$ and $0\leq \psi(\wp)\leq g(\wp)\leq 1-\psi(\wp)$ for all $\wp\in[m, n]_{\mathbb{T}}$ . Then

$\begin{eqnarray*} &&\int_{n-\lambda}^{n}r(\wp)\Delta_{\alpha} \wp+ \int_{m}^{n}\bigg|\bigg(\frac{r(\wp)}{\zeta(\wp)}-\frac{r(n-\lambda)}{\zeta(n-\lambda)}\bigg)\zeta(\wp)\psi(\wp)\bigg|\Delta_{\alpha} \wp\\ &&\leq \int_{m}^{n}r(\wp)g(\wp)\Delta_{\alpha} \wp\\ &&\leq \int_{m}^{m+\lambda}r(\wp)\Delta_{\alpha} \wp-\int_{m}^{n}\bigg|\bigg(\frac{r(\wp)}{\zeta(\wp)}-\frac{r(m+\lambda)}{\zeta(m+\lambda)}\bigg)\zeta(\wp)\psi(\wp)\bigg|\Delta_{\alpha} \wp, \end{eqnarray*}$

where $\lambda$ is obtained from

$\begin{equation*} \int_{m}^{m+\lambda}h(\wp)\Delta_{\alpha} \wp = \int_{m}^{n}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp = \int_{n-\lambda}^{n}\zeta(\wp)\Delta_{\alpha} \wp. \end{equation*}$

Proof. Putting $g(\wp)\mapsto \zeta(\wp)g(\wp)$ , $r(\wp)\mapsto r(\wp)/h(\wp)$ and $\psi(\wp)\mapsto \zeta(\wp)\psi(\wp)$ in (1.6).

Lemma 2.3. Under the same conditions of Lemma 2.1. Then

$\begin{eqnarray*} \int_{n-\lambda}^{n}r(\wp)\Delta_{\alpha} \wp&\leq& \int_{n-\lambda}^{n}\bigg(r(\wp)-\bigg[\frac{r(\wp)}{\zeta(\wp)}-\frac{r(n-\lambda)}{\zeta(n-\lambda)}\bigg]\zeta(\wp)\big[1-g(\wp)\big]\bigg)\Delta_{\alpha} \wp\\ &\leq&\int_{m}^{n}r(\wp)g(\wp)\Delta_{\alpha} \wp\\ &\leq& \int_{m}^{m+\lambda}\bigg(r(\wp)-\bigg[\frac{r(\wp)}{\zeta(\wp)}-\frac{r(a+\lambda)}{\zeta(m+\lambda)}\bigg]\zeta(\wp)\big[1-g(\wp)\big]\bigg)\Delta_{\alpha} \wp\\ &\leq&\int_{m}^{m+\lambda}r(\wp)\Delta_{\alpha} \wp, \end{eqnarray*}$

where $\lambda$ is obtained from

$\begin{equation*} \int_{m}^{m+\lambda}\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{n}g(\wp)\Delta_{\alpha} \wp = \int_{n-\lambda}^{n}\zeta(\wp)\Delta_{\alpha} \wp. \end{equation*}$

Proof. Taking $g(\wp)\mapsto \zeta(\wp)g(\wp)$ and $r(\wp)\mapsto r(\wp)/\zeta(\wp)$ in (1.7).

Theorem 2.1. Under the same conditions of Lemma 2.3 such that $k\in(m, n)$ and $\lambda_1$ , $\lambda_2$ are given from

$(\Lambda_{3})$

$\begin{equation*} \int_{m}^{m+\lambda_1}\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{k}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp, \end{equation*}$

$\begin{equation*} \int_{n-\lambda_2}^{n}\zeta(\wp)\Delta_{\alpha} \wp = \int_{k}^{n}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp. \end{equation*}$

If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

$\begin{equation} \int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp = \int_{m}^{m+\lambda_1}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp+\int_{n-\lambda_2}^{n}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp, \end{equation}$

(2.3)

then

$\begin{equation} \int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\leq\int_{m}^{m+\lambda_1}r^{\sigma}(\wp)\Delta_{\alpha} \wp+\int_{n-\lambda_2}^{n}r^{\sigma}(\wp)\Delta_{\alpha} \wp. \end{equation}$

(2.4)

(2.4) is reversed if $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ and (2.3).

$(\Lambda_{4})$

$\begin{equation*} \int_{k-\lambda_1}^{k}\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{k}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp, \end{equation*}$

$\begin{equation*} \int_{k}^{k+\lambda_2}\zeta(\wp)\Delta_{\alpha} \wp = \int_{k}^{n}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp. \end{equation*}$

If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

$\begin{equation} \int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp = \int_{k-\lambda_1}^{k+\lambda_2}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp, \end{equation}$

(2.5)

then

$\begin{equation} \int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\geq\int_{k-\lambda_1}^{k+\lambda_2}r^{\sigma}(\wp)\Delta_{\alpha} \wp. \end{equation}$

(2.6)

If $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ and (2.5) satisfied, then we reverse (2.6).

$(\Lambda_{5})$ If $\lambda_1$ , $\lambda_2$ be the same as in $(\Lambda_{3})$ and $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ so that

$\begin{eqnarray} &&\int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp\\ && = \int_{m}^{m+\lambda_1}\bigg(\phi(\wp)\zeta(\wp)-[\phi(\wp)-m-\lambda_1]\zeta(\wp)[1-g(\wp)]\bigg)\Delta_{\alpha} \wp \\ &&\quad +\int_{n-\lambda_2}^{n}\bigg(\phi(\wp)\zeta(\wp)-[\phi(\wp)-n+\lambda_2]\zeta(\wp)[1-g(\wp)]\bigg)\Delta_{\alpha} \wp, \end{eqnarray}$

(2.7)

then

$\begin{eqnarray} &&\int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\\ &&\leq\int_{m}^{m+\lambda_1}\bigg(r^{\sigma}(\wp) -\Big|\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(m+\lambda_1)}{\zeta(m+\lambda_1)}\Big|\zeta(\wp)[1-g(\wp)]\bigg)\Delta_{\alpha} \wp\\ &&\quad +\int_{n-\lambda_2}^{n}\bigg(r^{\sigma}(\wp)-\Big|\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(n-\lambda_2)}{\zeta(n-\lambda_2)}\Big|\zeta(\wp)[1-g(\wp)]\bigg)\Delta_{\alpha} \wp. \end{eqnarray}$

(2.8)

If $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ and (2.7) satisfied, the inequality in (2.8) is reversed.

$(\Lambda_{6})$ If $\lambda_1$ , $\lambda_2$ be defined as in $(\Lambda_{4})$ and $r^{\sigma}/\zeta\in\mathbb{AH}_1^k [m, n]$ and

$\begin{eqnarray} &&\int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp\\ && = \int_{k-\lambda_1}^{k}\Big(\phi(\wp)\zeta(\wp)-[\phi(\wp)-k+\lambda_1]\zeta(\wp)[1-g(\wp)]\Big)\Delta_{\alpha} \wp\\ && = \int_{m}^{m+\lambda_1}\Big(\phi(\wp)\zeta(\wp)-[\phi(\wp)-k+\lambda_2]\zeta(\wp)[1-g(\wp)]\Big)\Delta_{\alpha} \wp, \end{eqnarray}$

(2.9)

then

$\begin{eqnarray} &&\int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\\ &&\geq\int_{k-\lambda_1}^{k}\bigg(r^{\sigma}(\wp)-\Big[\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(k-\lambda_1)}{\zeta(k-\lambda_1)}\Big]\zeta(\wp)[1-g(\wp)]\bigg)\Delta_{\alpha} \wp\\ &&\quad +\int_{k}^{k+\lambda_2}\bigg(r^{\sigma}(\wp)-\Big[\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(k+\lambda_2)}{\zeta(k+\lambda_2)}\Big]\zeta(\wp)[1-g(\wp)]\bigg)\Delta_{\alpha} \wp. \end{eqnarray}$

(2.10)

If $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ and (2.9) satisfied, we reverse (2.10).

Proof. $(\Lambda_{3})$ Consider $r^{\sigma}/\zeta\in\mathbb{AH}_1^k [m, n]$ , and $R_{1}(\ell) = r^{\sigma}(\ell)-A\phi(\ell)\zeta(\ell)$ , since $A$ is given in Definition 2.1. Since $R_{1}/\zeta:[m, k]\cap{\mathbb{T}}\rightarrow \mathbb{R}$ , using Lemma 2.1 $(\Lambda_{1})$ , we deduce

$\begin{eqnarray} 0&\leq&\int_{m}^{m+\lambda_1}R_{1}(\wp)\Delta_{\alpha} \wp-\int_{m}^{k}R_{1}(\wp)g(\wp)\Delta_{\alpha} \wp\\ & = &\int_{m}^{m+\lambda_1}r^{\sigma}(\wp)\Delta_{\alpha} \wp-\int_{m}^{k}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\\ &&-A\bigg(\int_{m}^{m+\lambda_1}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp-\int_{m}^{k}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp\bigg). \end{eqnarray}$

(2.11)

As $R_{1}/\zeta:[k, n]\cap{\mathbb{T}}\rightarrow \mathbb{R}$ is nondecreasing, using Lemma 2.1 $(\Lambda_{2})$ , we obtain

$\begin{eqnarray} 0&\geq&\int_{k}^{n}R_{1}(\wp)g(\wp)\Delta_{\alpha} \wp-\int_{n-\lambda_2}^{n}R_{1}(\wp)\Delta_{\alpha} \wp\\ & = &\int_{k}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp-\int_{n-\lambda_2}^{n}r^{\sigma}(\wp)\Delta_{\alpha} \wp\\ &&-A\bigg(\int_{k}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp-\int_{n-\lambda_2}^{n}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp\bigg). \end{eqnarray}$

(2.12)

(2.11) and (2.12) imply that

$\begin{eqnarray*} &&\int_{m}^{m+\lambda_1}r^{\sigma}(\wp)\Delta_{\alpha} \wp+\int_{n-\lambda_2}^{n}r^{\sigma}(\wp)\Delta_{\alpha} \wp-\int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\\ &&\geq A\bigg(\int_{m}^{m+\lambda_1}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp+\int_{n-\lambda_2}^{n}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp -\int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp\bigg) \end{eqnarray*}$

Hence, if (2.3) is hold, then (2.4) holds. For $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ , we get the some steps.

$(\Lambda_{4})$ Let $r^{\sigma}/\zeta\in\mathbb{AH}_1^k [m, n]$ , also $R_{1}(x) = r^{\sigma}(x)-A\phi(x)\zeta(x)$ , where $A$ as in Definition 2.1. $R_{1}/\zeta:[m, k]\cap{\mathbb{T}}\rightarrow \mathbb{R}$ is nonincreasing, so from Lemma 2.1 $(\Lambda_{1})$ we obtain

$\begin{eqnarray} 0&\leq&\int_{m}^{k}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp-\int_{k-\lambda_1}^{k}r^{\sigma}(\wp)\Delta_{\alpha} \wp\\ &&-A\bigg(\int_{m}^{k}\phi(\wp)h(\wp)g(\wp)\Delta_{\alpha} \wp-\int_{c-\lambda_1}^{k}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp\bigg). \end{eqnarray}$

(2.13)

Using Lemma 2.1 $(\Lambda_{1})$ we have

$\begin{eqnarray} 0&\geq&\int_{k}^{k+\lambda_2}r^{\sigma}(\wp)\Delta_{\alpha} \wp-\int_{k}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\\ &&-A\bigg(\int_{k}^{k+\lambda_2}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp-\int_{k}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp\bigg). \end{eqnarray}$

(2.14)

Thus, from (2.13), (2.14), we get

$\begin{eqnarray*} &&\int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp-\int_{k-\lambda_1}^{k+\lambda_2}r^{\sigma}(\wp)\Delta_{\alpha} \wp\\ &&\geq A\bigg(\int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp-\int_{k-\lambda_1}^{k+\lambda_2}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp\bigg) \end{eqnarray*}$

Therefore, if $\int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp = \int_{k-\lambda_1}^{k+\lambda_2}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp$ is satisfied, then (2.8) holds. Follow the same steps for $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ .

Using Lemma 2.3 and repeat the steps of Theorem 2.1 $(\Lambda_{3})$ and Theorem 2.1 $(\Lambda_{4})$ in the proof of $(\Lambda_{5})$ and $(\Lambda_{6})$ respectively.

Corollary 2.1. The inequalities (2.4), (2.6), (2.8) and (2.10) of Theorem 2.1 letting $\mathbb{T} = \mathbb{R}$ takes

$\begin{equation} (i)\qquad \int_{m}^{n}f^{\sigma}(\wp)g(\wp)d_{\alpha} \wp\leq\int_{m}^{m+\lambda_1}r^{\sigma}(\wp)d_{\alpha} \wp+\int_{n-\lambda_2}^{n}r^{\sigma}(\wp)d_{\alpha} \wp. \end{equation}$

(2.15)

$\begin{equation} (ii)\qquad \int_{m}^{n}r^{\sigma}(\wp)g(\wp)d_{\alpha} \wp\geq\int_{k-\lambda_1}^{k+\lambda_2}r^{\sigma}(\wp)d_{\alpha} \wp. \end{equation}$

(2.16)

$\begin{eqnarray} (iii)\qquad &&\int_{m}^{n}r^{\sigma}(\wp)g(\wp)d_{\alpha} \wp\\ &&\leq\int_{m}^{m+\lambda_1}\bigg(r^{\sigma}(\wp)-\Big[\frac{r^{\sigma}(\wp)}{\zeta(\wp)} -\frac{r^{\sigma}(m+\lambda_1)}{\zeta(m+\lambda_1)}\Big]\zeta(\wp)[1-g(\wp)]\bigg)d_{\alpha} \wp\\ &&\quad +\int_{n-\lambda_2}^{n}\bigg(r^{\sigma}(\wp)-\Big[\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(n-\lambda_2)}{\zeta(n-\lambda_2)}\Big]\zeta(\wp)[1-g(\wp)]\bigg)d_{\alpha} \wp. \end{eqnarray}$

(2.17)

$\begin{eqnarray} (iv)\qquad &&\int_{m}^{n}r^{\sigma}(\wp)g(\wp)d_{\alpha} \wp\\ &&\geq\int_{k-\lambda_1}^{k}\bigg(r^{\sigma}(\wp)-\Big[\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(k-\lambda_1)}{\zeta(k-\lambda_1)}\Big]\zeta(\wp)[1-g(\wp)]\bigg)d_{\alpha} \wp\\ &&\quad +\int_{k}^{k+\lambda_2}\bigg(r^{\sigma}(\wp)-\Big[\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(k+\lambda_2)}{\zeta(k+\lambda_2)}\Big]\zeta(\wp)[1-g(\wp)]\bigg)d_{\alpha} \wp. \end{eqnarray}$

(2.18)

Corollary 2.2. We get [,Theorems 8,10,21 and 22], if we put $\alpha = 1$ and $\phi(\wp) = \wp$ in Corollary 2.1 $[(i), (ii), (iii), (iv)]$ simultaneously.

Corollary 2.3. In Corollary 2.1 taking $\mathbb{T} = \mathbb{Z}$ , the results (2.15)–(2.18) will be equivalent to

$\begin{equation*} (i)\qquad \sum\limits_{\wp = m}^{n-1}r(\wp+1)g(\wp)\wp^{\alpha-1}\leq\sum\limits_{\wp = m}^{m+\lambda_1-1}r(\wp+1)+\sum\limits_{\wp = n-\lambda_2}^{n-1}r(\wp+1)\wp^{\alpha-1}. \end{equation*}$

$\begin{equation*} (ii)\qquad \sum\limits_{\wp = m}^{n-1}r(\wp+1)g(\wp)\wp^{\alpha-1}\geq\sum\limits_{\wp = k-\lambda_1}^{k+\lambda_2-1}r(\wp+1)\wp^{\alpha-1}. \end{equation*}$

$\begin{eqnarray*} (iii)\qquad \nonumber&&\sum\limits_{\wp = m}^{n-1}r(\wp+1)g(\wp)\wp^{\alpha-1}\\ \nonumber&&\leq\sum\limits_{\wp = m}^{m+\lambda_1-1}\bigg(r(\wp+1)-\Big[\frac{r(\wp+1)}{\zeta(\wp)}-\frac{r(a+\lambda_1+1)}{\zeta(m+\lambda_1)}\Big]\zeta(\wp)[1-g(\wp)]\bigg)\wp^{\alpha-1}\\ &&\quad +\sum\limits_{\wp = n-\lambda_2}^{n-1}\bigg(r(\wp+1)-\Big[\frac{r(\wp+1)}{\zeta(\wp)}-\frac{r(n-\lambda_2+1)}{\zeta(n-\lambda_2)}\Big]\zeta(\wp)[1-g(\wp)]\bigg)\wp^{\alpha-1}. \end{eqnarray*}$

$\begin{eqnarray*} (iv)\qquad \nonumber&&\sum\limits_{\wp = m}^{n-1}r(\wp+1)g(\wp))\wp^{\alpha-1}\\ \nonumber&&\geq\sum\limits_{\wp = k-\lambda_1}^{k-1}\bigg(r(\wp+1)-\Big[\frac{r(\wp+1)}{\zeta(\wp)}-\frac{r(k-\lambda_1+1)}{\zeta(k-\lambda_1)}\Big]\zeta(\wp)[1-g(\wp)]\bigg))\wp^{\alpha-1}\\ &&\quad +\sum\limits_{\wp = k}^{k+\lambda_2-1}\bigg(r(\wp+1)-\Big[\frac{r(\wp+1)}{\zeta(\wp)}-\frac{r(k+\lambda_2+1)}{\zeta(k+\lambda_2)}\Big]\zeta(\wp)[1-g(\wp)]\bigg))\wp^{\alpha-1}. \end{eqnarray*}$

Theorem 2.2. Under the assumptions in Lemma 2.1 with $0\leq g(\wp)\leq \zeta(\wp)$ and $\lambda_1$ , $\lambda_2$ be defined as

$(\Lambda_{7})$

$\begin{equation*} \int_{m}^{m+\lambda_1}\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{k}g(\wp)\Delta_{\alpha} \wp, \end{equation*}$

$\begin{equation*} \int_{n-\lambda_2}^{n}\zeta(\wp)\Delta_{\alpha} \wp = \int_{k}^{n}g(\wp)\Delta_{\alpha} \wp. \end{equation*}$

If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

$\begin{equation} \int_{m}^{n}\phi(\wp)g(\wp)\Delta_{\alpha} \wp = \int_{m}^{m+\lambda_1}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp+\int_{n-\lambda_2}^{n}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp, \end{equation}$

(2.19)

then

$\begin{equation} \int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\leq\int_{m}^{m+\lambda_1}r^{\sigma}(\wp)\zeta(\wp)\Delta_{\alpha} \wp+\int_{n-\lambda_2}^{n}r^{\sigma}(\wp)\zeta(\wp)\Delta_{\alpha} \wp. \end{equation}$

(2.20)

$(\Lambda_{8})$

$\begin{equation*} \int_{k-\lambda_1}^{k}\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{k}g(\wp)\Delta_{\alpha} \wp, \end{equation*}$

$\begin{equation*} \int_{k}^{k+\lambda_2}\zeta(\wp)\Delta_{\alpha} \wp = \int_{k}^{n}g(\wp)\Delta_{\alpha} \wp. \end{equation*}$

If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

$\begin{equation} \int_{m}^{n}\phi(\wp)g(\wp)\Delta_{\alpha} \wp = \int_{k-\lambda_1}^{k+\lambda_2}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp, \end{equation}$

(2.21)

then

$\begin{equation} \int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\geq\int_{k-\lambda_1}^{k+\lambda_2}r^{\sigma}(\wp)\zeta(\wp)\Delta_{\alpha} \wp. \end{equation}$

(2.22)

If $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ and (2.19), (2.21) satisfied, we get the reverse of (2.20) and (2.22).

Proof. By using Theorem 2.1 $[(\Lambda_{3}), (\Lambda_{4})]$ and by putting $g\mapsto g/h$ and $f\mapsto fh$ , we get the proof of $(\Lambda_{7})$ and $(\Lambda_{8})$ .

Corollary 2.4. In Theorem 2.2 $[(\Lambda_{7}), (\Lambda_{8})]$ , assuming $\mathbb{T} = \mathbb{R}$ , the following results obtains:

$\begin{equation} (i)\qquad \int_{m}^{n}r^{\sigma}(\wp)g(\wp)d_{\alpha} \wp\leq\int_{m}^{m+\lambda_1}r^{\sigma}(\wp)\zeta(\wp)d_{\alpha} \wp+\int_{n-\lambda_2}^{n}r^{\sigma}(\wp)\zeta(\wp)d_{\alpha} \wp. \end{equation}$

(2.23)

$\begin{equation} (ii)\qquad \int_{m}^{n}r^{\sigma}(\wp)g(\wp)d_{\alpha} \wp\geq\int_{k-\lambda_1}^{k+\lambda_2}r^{\sigma}(\wp)\zeta(\wp)d_{\alpha} \wp. \end{equation}$

(2.24)

Corollary 2.5. In Corollary 2.4 $[(i), (ii)]$ , when we put $\alpha = 1$ and $\phi(\wp) = \wp$ then [47,Theorems 16 and 17] gotten.

Corollary 2.6. In (2.23) and (2.24) letting $\mathbb{T} = \mathbb{Z}$ , gets

$\begin{equation*} (i)\qquad \sum\limits_{\wp = m}^{n-1}r(\wp+1)g(\wp)\wp^{\alpha-1}\leq\sum\limits_{\wp = m}^{m+\lambda_1-1}r(\wp+1)h(\wp)+\sum\limits_{\wp = n-\lambda_2}^{n-1}r(\wp+1)h(\wp)\wp^{\alpha-1}. \end{equation*}$

$\begin{equation*} (ii)\qquad \sum\limits_{\wp = m}^{n-1}r(\wp+1)g(\wp)\wp^{\alpha-1}\geq\sum\limits_{\wp = k-\lambda_1}^{k+\lambda_2-1}r(\wp+1)\zeta(\wp)\wp^{\alpha-1}. \end{equation*}$

Theorem 2.3. Using the same conditions in Lemma 2.3. Letting $w:[m, n]\cap{\mathbb{T}}\rightarrow \mathbb{R}$ be integrable with $0\leq g(\wp)\leq w(\wp)$ $\forall \wp\in[m, n]\cap{\mathbb{T}}$ and

$\begin{equation*} (\Lambda_{9})\qquad \int_{m}^{m+\lambda_1}w(\wp)\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{k}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp, \end{equation*}$

$\begin{equation*} \int_{n-\lambda_2}^{n}w(\wp)\zeta(\wp)\Delta_{\alpha} \wp = \int_{k}^{n}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp. \end{equation*}$

If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

$\begin{equation} \int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp = \int_{m}^{m+\lambda_1}\phi(\wp)w(\wp)\zeta(\wp)\Delta_{\alpha} \wp+\int_{n-\lambda_2}^{n}\phi(\wp)w(\wp)\zeta(\wp)\Delta_{\alpha} \wp, \end{equation}$

(2.25)

then

$\begin{equation} \int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\leq\int_{m}^{m+\lambda_1}r^{\sigma}(\wp)w(\wp)\Delta_{\alpha} \wp+\int_{n-\lambda_2}^{n}r^{\sigma}(\wp)w(\wp)\Delta_{\alpha} \wp. \end{equation}$

(2.26)

$\begin{equation*} (\Lambda_{10})\qquad \int_{k-\lambda_1}^{k}w(\wp)\zeta(\wp)\Delta_{\alpha} \wp = \int_{m}^{k}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp, \end{equation*}$

$\begin{equation*} \int_{k}^{k+\lambda_2}w(\wp)\zeta(\wp)\Delta_{\alpha} \wp = \int_{k}^{n}\zeta(\wp)g(\wp)\Delta_{\alpha} \wp. \end{equation*}$

If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

$\begin{equation} \int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp = \int_{k-\lambda_1}^{k+\lambda_2}\phi(\wp)w(\wp)\zeta(\wp)\Delta_{\alpha} \wp, \end{equation}$

(2.27)

$\begin{equation} \int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\geq\int_{k-\lambda_1}^{k+\lambda_2}r^{\sigma}(\wp)w(\wp)\Delta_{\alpha} \wp. \end{equation}$

(2.28)

The inequalities in (2.26) and (2.28) are reversible if $r^{\sigma}/\zeta\in\mathbb{AH}_2^c [a, b]$ and (2.25), (2.27) hold.

Proof. In Theorem 2.1 $[(\Lambda_{3}), (\Lambda_{4})]$ , $\zeta$ changes $wq$ , $g$ changes $g/w$ and $r$ changes $rw$ .

Corollary 2.7. In (2.26) and (2.28). Letting $\mathbb{T} = \mathbb{R}$ , we have

$\begin{equation} (i)\qquad \int_{m}^{n}r^{\sigma}(\wp)g(\wp)d_{\alpha} \wp\leq\int_{m}^{m+\lambda_1}r^{\sigma}(\wp)w(\wp)d_{\alpha} \wp+\int_{n-\lambda_2}^{n}r^{\sigma}(\wp)w(\wp)d_{\alpha} \wp. \end{equation}$

(2.29)

$\begin{equation} (ii)\qquad \int_{m}^{n}r^{\sigma}(\wp)g(\wp)d_{\alpha} \wp\geq\int_{k-\lambda_1}^{k+\lambda_2}r^{\sigma}(\wp)w(\wp)d_{\alpha} \wp. \end{equation}$

(2.30)

Corollary 2.8. In Corollary 2.7 $[(i), (ii)]$ , letting $\alpha = 1$ and $\phi(\wp) = \wp$ we get [47,Theorems 18 and 19].

Corollary 2.9. In (2.29) and (2.30), crossing $\mathbb{T} = \mathbb{Z}$ , gets

$\begin{equation*} (i)\qquad \sum\limits_{\wp = m}^{n-1}r(\wp+1)g(\wp)\wp^{\alpha-1}\leq\sum\limits_{\wp = m}^{m+\lambda_1-1}r(\wp+1)w(\wp)+\sum\limits_{\wp = n-\lambda_2}^{n-1}r(\wp+1)w(\wp)\wp^{\alpha-1}. \end{equation*}$

$\begin{equation*} (ii)\qquad \sum\limits_{\wp = m}^{n-1}r(\wp+1)g(\wp)\wp^{\alpha-1}\geq\sum\limits_{\wp = k-\lambda_1}^{k+\lambda_2-1}r(\wp+1)w(\wp)\wp^{\alpha-1}. \end{equation*}$

Theorem 2.4. Using the same conditions in Lemma 2.1, and Theorem 2.1 $[(\Lambda_{3}), (\Lambda_{4})]$ with $\psi:[m, n]\cap{\mathbb{T}}\rightarrow \mathbb{R}$ be a integrable: $0\leq \psi(\wp)\leq g(\wp)\leq 1-\psi(\wp)$ .

$(\Lambda_{11})$ If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

$\begin{eqnarray} \qquad &&\int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp\\ && = \int_{m}^{m+\lambda_1}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp-\int_{m}^{k}\big|\phi(\wp)-m-\lambda_1\big|\zeta(\wp)\psi(\wp)\Delta_{\alpha} \wp +\int_{n-\lambda_2}^{n}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp\\ &&\quad +\int_{k}^{n}\big|\phi(\wp)-n+\lambda_2\big|\zeta(\wp)\psi(\wp)\Delta_{\alpha} \wp, \end{eqnarray}$

(2.31)

then

$\begin{eqnarray} &&\int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\\ &&\leq\int_{m}^{m+\lambda_1}r^{\sigma}(\wp)\Delta_{\alpha} \wp -\int_{m}^{k}\Big|\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(m+\lambda_1)}{\zeta(m+\lambda_1)}\Big|\zeta(\wp)\psi(\wp)\Delta_{\alpha} \wp+\int_{n-\lambda_2}^{n}r^{\sigma}(\wp)\Delta_{\alpha} \wp\\ &&\quad +\int_{k}^{n}\Big|\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(n-\lambda_2)}{\zeta(n-\lambda_2)}\Big|\zeta(\wp)\psi(\wp)\Delta_{\alpha} \wp. \end{eqnarray}$

(2.32)

$(\Lambda_{12})$ If $r^{\sigma}/\zeta\in\mathbb{AH}_1^k[m, n]$ and

$\begin{eqnarray} &&\int_{m}^{n}\phi(\wp)\zeta(\wp)g(\wp)\Delta_{\alpha} \wp\\ && = \int_{k-\lambda_1}^{k}\phi(\wp)\zeta(\wp)\Delta_{\alpha} \wp-\int_{m}^{k}\big|\phi(\wp)-k+\lambda_1\big|\zeta(\wp)\psi(\wp)\Delta_{\alpha} \wp\\ &&+\int_{k}^{n}\big|\phi(\wp)-k-\lambda_1\big|\zeta(\wp)\psi(\wp)\Delta_{\alpha} \wp, \end{eqnarray}$

(2.33)

then

$\begin{eqnarray} &&\int_{m}^{n}r^{\sigma}(\wp)g(\wp)\Delta_{\alpha} \wp\\ &&\geq\int_{k-\lambda_1}^{k+\lambda_2}r^{\sigma}(\wp)\Delta_{\alpha} \wp+\int_{m}^{k}\bigg|\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(k-\lambda_1)}{\zeta(k-\lambda_1)}\bigg|\zeta(\wp)\psi(\wp)\Delta_{\alpha} \wp\\ &&\quad-\int_{k}^{n}\bigg|\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(k+\lambda_2)}{\zeta(k+\lambda_2)}\bigg|\zeta(\wp)\psi(\wp) \Delta_{\alpha} \wp. \end{eqnarray}$

(2.34)

If $r^{\sigma}/\zeta\in\mathbb{AH}_2^k [m, n]$ and (2.31) and (2.33) satisfied, we get the reverse of (2.32) and (2.34).

Proof. The same steps of Theorem 2.1 $[(\Lambda_{3}), (\Lambda_{4})]$ with Lemma 2.1, $R_{1}/\zeta:[m, k]\cap{\mathbb{T}}\rightarrow \mathbb{R}$ nonincreasing, $R_{1}/\zeta:[k, n]\cap{\mathbb{T}}\rightarrow \mathbb{R}$ nondecreasing.

Corollary 2.10. In Theorem 2.4 $[(\Lambda_{11}), (\Lambda_{12})]$ , letting $\mathbb{T} = \mathbb{R}$ we get:

$\begin{eqnarray} (i)\qquad &&\int_{m}^{n}r^{\sigma}(\wp)g(\wp)d_{\alpha} \wp\\ &&\leq\int_{m}^{m+\lambda_1}r^{\sigma}(\wp)d_{\alpha} \wp -\int_{m}^{k}\Big|\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(m+\lambda_1)}{\zeta(m+\lambda_1)}\Big|\zeta(\wp)\psi(\wp)d_{\alpha} \wp+\int_{n-\lambda_2}^{n}r^{\sigma}(\wp)d_{\alpha} \wp\\ &&\quad +\int_{k}^{n}\Big|\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(n-\lambda_2)}{\zeta(n-\lambda_2)}\Big|\zeta(\wp)\psi(\wp)d_{\alpha} \wp. \end{eqnarray}$

(2.35)

$\begin{eqnarray} (ii)\qquad &&\int_{m}^{n}r^{\sigma}(\wp)g(\wp)d_{\alpha} \wp\\ &&\geq\int_{k-\lambda_1}^{k+\lambda_2}r^{\sigma}(\wp)d_{\alpha} \wp+\int_{m}^{k}\bigg|\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(k-\lambda_1)}{\zeta(k-\lambda_1)}\bigg|\zeta(\wp)\psi(\wp)d_{\alpha} \wp\\ &&\quad-\int_{k}^{n}\bigg|\frac{r^{\sigma}(\wp)}{\zeta(\wp)}-\frac{r^{\sigma}(k+\lambda_2)}{\zeta(k+\lambda_2)}\bigg|\zeta(\wp)\psi(\wp)d_{\alpha} \wp. \end{eqnarray}$

(2.36)

Corollary 2.11. In (2.35) and (2.36), we put $\alpha = 1$ , with $\phi(\wp) = \wp$ we get [47,Theorems 23 and 24].

Corollary 2.12. Our results (2.35) and (2.36), by using $\mathbb{T} = \mathbb{Z}$ gets

$\begin{eqnarray*} (i)\qquad \nonumber&&\sum\limits_{\wp = m}^{n-1}r(\wp+1)g(\wp)\wp^{\alpha-1}\\ \nonumber&&\leq\sum\limits_{\wp = m}^{m+\lambda_1-1}r(\wp+1)\wp^{\alpha-1} -\sum\limits_{\wp = m}^{k-1}\Big|\frac{r(\wp+1)}{\zeta(\wp)}-\frac{r(m+\lambda_1+1)}{\zeta(m+\lambda_1)}\Big|\zeta(\wp)\psi(\wp)\hat{\nabla} \wp\\ &&\quad+\sum\limits_{\wp = n-\lambda_2}^{n-1}r(\wp+1)\wp^{\alpha-1}+\sum\limits_{\wp = k}^{n-1}\Big|\frac{r(\wp+1)}{\zeta(\wp)}-\frac{r(n-\lambda_2+1)}{\zeta(n-\lambda_2)}\Big|\zeta(\wp)\psi(\wp)\wp^{\alpha-1}. \end{eqnarray*}$

$\begin{eqnarray*} (ii)\qquad\nonumber&&\sum\limits_{\wp = m}^{n-1}r(\wp+1)g(\wp)\wp^{\alpha-1}\\ \nonumber&&\geq\sum\limits_{\wp = k-\lambda_1}^{k+\lambda_2-1}r(\wp+1)\wp^{\alpha-1}+\sum\limits_{\wp = m}^{k-1}\bigg|\frac{r(\wp+1)}{\zeta(\wp)}-\frac{r(k-\lambda_1+1)}{\zeta(k-\lambda_1)} \bigg|\zeta(\wp)\psi(\wp)\wp^{\alpha-1}\\ &&\quad-\sum\limits_{\wp = k}^{n-1}\bigg|\frac{r(\wp+1)}{\zeta(\wp)}-\frac{r(k+\lambda_2+1)}{\zeta(k+\lambda_2)}\bigg|h(\wp)\psi(\wp)\wp^{\alpha-1}. \end{eqnarray*}$

3. Conclusions

In this work, we explore new generalizations of the integral Steffensen inequality given in ^[38,39,43] by the utilization of the $\alpha$ -conformable derivatives and integrals, A few of these results are generalised to time scales. We also obtained the discrete and continuous case of our main results, in order to gain some fresh inequalities as specific cases.

Acknowledgments

The authors extend their appreciation to the Research Supporting Project number (RSP-2022/167), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare no conflict of interest.

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