Identification of diagnostic biomarkers of gestational diabetes mellitus based on transcriptome gene expression and alternations of microRNAs

Xuemei XIA; Xuemei HU; Xuemei XIA; Xuemei HU

doi:10.3934/bioeng.2023014

AIMS Bioengineering

2023, Volume 10, Issue 3: 202-217. doi: 10.3934/bioeng.2023014

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Research article

Identification of diagnostic biomarkers of gestational diabetes mellitus based on transcriptome gene expression and alternations of microRNAs

Xuemei XIA ^,,
Xuemei HU

People's Hospital of Lishui City, Zhejiang Province, China

Received: 03 April 2023 Revised: 02 August 2023 Accepted: 06 August 2023 Published: 15 August 2023

Background

Gestational diabetes mellitus (GDM), characterized by glucose intolerance during pregnancy, poses substantial health risks for both mothers and infants due to the interplay of insulin resistance and β-cell dysfunction. Molecular biomarkers, including SNPs, microRNAs (miRNAs), and proteins, have been linked to GDM development during pregnancy. Notably, miRNA-mediated regulation of gene expression holds pivotal roles in metabolic disorders. This study aims to identify diagnostic biomarkers for GDM and establish a diagnostic model.

Methods

Firstly, gene expression data from GDM samples (N = 9) and normal samples (N = 9) were sourced from the Gene Expression Omnibus (GEO) database. Subsequently, the limma package was employed to discern differentially expressed genes (DEGs), with subsequent functional and enrichment analyses executed using the clusterProfiler package. A comprehensive exploration of genes significantly correlated with GDM was undertaken via weighted gene co-expression network analysis (WGCNA). The construction of a protein-protein interaction (PPI) network was facilitated by STRING, while visualization of hub genes was achieved through Cytoscape. Moreover, the miRNA-mRNA network was established using StarBase. Concurrently, immune infiltration significantly correlated with hub genes was identified.

Results

In this study, 209 DEGs between normal and GDM samples were identified, and these genes were associated with collagen containing extracellular matrix heparin binding and axon guidance, etc. Then, 18 modules were identified by WGCNA and the brown module including 212 genes had a significantly negative correlation with GDM (r = −0.66, P = 0.003). Additionally, five low gene expressions (CXCL12, MEF2C, MMP2, SOX17 and THBS2) and two high gene expressions (BMP4 and SFRP5) were identified as GDM related hub genes. Moreover, hub genes regulated by alternations of miRNAs were established and three hub genes (CXCL12, MEF2C and THBS2) were negatively correlated with activated Natural Killer (NK) cells while two hub genes (BMP4 and SFRP5) were positively correlated with activated NK cells.

Conclusions

This study offers novel hub genes that could contribute to the diagnostic approach for GDM, potentially shedding light on the intricate mechanisms underpinning GDM's developmental pathways.

Keywords:

Citation: Xuemei XIA, Xuemei HU. Identification of diagnostic biomarkers of gestational diabetes mellitus based on transcriptome gene expression and alternations of microRNAs[J]. AIMS Bioengineering, 2023, 10(3): 202-217. doi: 10.3934/bioeng.2023014

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Abstract

Background

Methods

Results

Conclusions

This study offers novel hub genes that could contribute to the diagnostic approach for GDM, potentially shedding light on the intricate mechanisms underpinning GDM's developmental pathways.

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.

Acknowledgments

We sincerely appreciate the investigators and authors who have contributed to this field. This research was supported by the Zhejiang Provincial Medical and Health Technology Project (Grant No. 2022498503).

Conflict of interest

The authors declare there are no competing interests.

Author contributions

Xuemei Xia, analysis and collection of the data, drafting the manuscript; Xuemei Hu, conception and design, analysis and explanation of the data, revising the manuscript. All the authors have read and approved the final manuscript.

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