Enhancing land cover classification in remote sensing imagery using an optimal deep learning model

Abdelwahed Motwake; Aisha Hassan Abdalla Hashim; Marwa Obayya; Majdy M. Eltahir; Abdelwahed Motwake; Aisha Hassan Abdalla Hashim; Marwa Obayya; Majdy M. Eltahir

doi:10.3934/math.2024009

AIMS Mathematics

2024, Volume 9, Issue 1: 140-159. doi: 10.3934/math.2024009

Previous Article Next Article

Research article Special Issues

Enhancing land cover classification in remote sensing imagery using an optimal deep learning model

1.
Department of Electrical and Computer Engineering, International Islamic University Malaysia 53100 Kuala Lumpur, Malaysia
2.
Department of Information Systems, College of Business Administration in Hawtat bani Tamim, Prince Sattam bin Abdulaziz University, Saudi Arabia
3.
Department of Biomedical Engineering, College of Engineering, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
4.
Department of Information Systems, College of Science & Art at Mahayil, King Khalid University, Saudi Arabia

Received: 20 August 2023 Revised: 10 November 2023 Accepted: 13 November 2023 Published: 27 November 2023
MSC : 11Y40

The land cover classification process, accomplished through Remote Sensing Imagery (RSI), exploits advanced Machine Learning (ML) approaches to classify different types of land cover within the geographical area, captured by the RS method. The model distinguishes various types of land cover under different classes, such as agricultural fields, water bodies, urban areas, forests, etc. based on the patterns present in these images. The application of Deep Learning (DL)-based land cover classification technique in RSI revolutionizes the accuracy and efficiency of land cover mapping. By leveraging the abilities of Deep Neural Networks (DNNs) namely, Convolutional Neural Networks (CNN) or Recurrent Neural Networks (RNN), the technology can autonomously learn spatial and spectral features inherent to the RSI. The current study presents an Improved Sand Cat Swarm Optimization with Deep Learning-based Land Cover Classification (ISCSODL-LCC) approach on the RSIs. The main objective of the proposed method is to efficiently classify the dissimilar land cover types within the geographical area, pictured by remote sensing models. The ISCSODL-LCC technique utilizes advanced machine learning methods by employing the Squeeze-Excitation ResNet (SE-ResNet) model for feature extraction and the Stacked Gated Recurrent Unit (SGRU) mechanism for land cover classification. Since 'manual hyperparameter tuning' is an erroneous and laborious task, the hyperparameter selection is accomplished with the help of the Reptile Search Algorithm (RSA). The simulation analysis was conducted upon the ISCSODL-LCC model using two benchmark datasets and the results established the superior performance of the proposed model. The simulation values infer better outcomes of the ISCSODL-LCC method over other techniques with the maximum accuracy values such as 97.92% and 99.14% under India Pines and Pavia University datasets, respectively.

Keywords:

Citation: Abdelwahed Motwake, Aisha Hassan Abdalla Hashim, Marwa Obayya, Majdy M. Eltahir. Enhancing land cover classification in remote sensing imagery using an optimal deep learning model[J]. AIMS Mathematics, 2024, 9(1): 140-159. doi: 10.3934/math.2024009

Related Papers:

[1]	Abigail Wiafe, Pasi Fränti . Affective algorithmic composition of music: A systematic review. Applied Computing and Intelligence, 2023, 3(1): 27-43. doi: 10.3934/aci.2023003
[2]	Abrhalei Tela, Abraham Woubie, Ville Hautamäki . Transferring monolingual model to low-resource language: the case of Tigrinya. Applied Computing and Intelligence, 2024, 4(2): 184-194. doi: 10.3934/aci.2024011
[3]	Pasi Fränti, Sami Sieranoja . Clustering accuracy. Applied Computing and Intelligence, 2024, 4(1): 24-44. doi: 10.3934/aci.2024003
[4]	Tinja Pitkämäki, Tapio Pahikkala, Ileana Montoya Perez, Parisa Movahedi, Valtteri Nieminen, Tom Southerington, Juho Vaiste, Mojtaba Jafaritadi, Muhammad Irfan Khan, Elina Kontio, Pertti Ranttila, Juha Pajula, Harri Pölönen, Aysen Degerli, Johan Plomp, Antti Airola . Finnish perspective on using synthetic health data to protect privacy: the PRIVASA project. Applied Computing and Intelligence, 2024, 4(2): 138-163. doi: 10.3934/aci.2024009
[5]	Francis Nweke, Abm Adnan Azmee, Md Abdullah Al Hafiz Khan, Yong Pei, Dominic Thomas, Monica Nandan . A transformer-driven framework for multi-label behavioral health classification in police narratives. Applied Computing and Intelligence, 2024, 4(2): 234-252. doi: 10.3934/aci.2024014
[6]	Hong Cao, Rong Ma, Yanlong Zhai, Jun Shen . LLM-Collab: a framework for enhancing task planning via chain-of-thought and multi-agent collaboration. Applied Computing and Intelligence, 2024, 4(2): 328-348. doi: 10.3934/aci.2024019
[7]	Elizaveta Zimina, Kalervo Järvelin, Jaakko Peltonen, Aarne Ranta, Kostas Stefanidis, Jyrki Nummenmaa . Linguistic summarisation of multiple entities in RDF graphs. Applied Computing and Intelligence, 2024, 4(1): 1-18. doi: 10.3934/aci.2024001
[8]	Yang Wang, Hassan A. Karimi . Exploring large language models for climate forecasting. Applied Computing and Intelligence, 2025, 5(1): 1-13. doi: 10.3934/aci.2025001
[9]	Marko Niemelä, Mikaela von Bonsdorff, Sami Äyrämö, Tommi Kärkkäinen . Classification of dementia from spoken speech using feature selection and the bag of acoustic words model. Applied Computing and Intelligence, 2024, 4(1): 45-65. doi: 10.3934/aci.2024004
[10]	Libero Nigro, Franco Cicirelli . Property assessment of Peterson's mutual exclusion algorithms. Applied Computing and Intelligence, 2024, 4(1): 66-92. doi: 10.3934/aci.2024005

Abstract

1. Introduction

The study of integral inequality is an interesting area for research in mathematical analysis [1,2]. The fundamental integral inequalities can be instrumental in cultivating the subjective properties of convexity. The existence of massive literature surrounding integral inequalities for convex functions [-] depicts the importance of this topic. The most beautiful fact about convex function is that, it has a very elegant representation based on an inequality presented when the functional value of a linear combination of two points in its domain does not exceed the linear combination of the functional values at those two points. Fractional calculus owes its starting point to whether or not the importance of a derivative to an integer order could be generalized to a fractional order which is not an integer. Following this unique conversation between L'Hopital and Leibniz, the concept of fractional calculus grabbed the eye of some extraordinary researchers like Euler, Laplace, Fourier, Lacroix, Abel, Riemann, and Liouville. Over time, fractional operators have been differentiated with their singularity, locality and having general forms with the improvements made in their kernel structures. In this sense, based on the basic concepts of fractional analysis, Riemann-Liouville(R-L) and Caputo operators, various new trends have been successful. Fractional integral inequalities are marvelous tools for building up the qualitative and quantitative properties of preinvex functions. There has been a ceaseless development of intrigue in such a region of research so as to address the issues of different utilizations of these variants. In $\rm 1938$ , Ostrowski inequality established the following useful and interesting integral inequality, (see [12] and [13]). This review assumed a vital part in growing and getting varieties of well-known integral inequalities with the assistance of fractional integral operators. Then again, by characterizing various forms of Riemann-Liouville(R-L) fractional operator somewhat recently, new forms and refinements of integral inequalities involving differentiable functions have been presented. Studies in the field of fractional calculus have carried another point of view and direction in different fields of applied sciences. It has revealed insight into numerous real-life issues with the utilizations of recently characterized fractional operators.

For recent result and their related some generalizations, variants and extensions concerning Ostrowski inequality (see [9,10,14-17]).

The aim of this paper is to establish some integral inequalities for functions whose derivatives in absolute value are preinvex. Now we recall some notions in invexity analysis which will be used through the paper (see [20,21,24,26,28]) and references therein.

Let $g:\mathcal K \rightarrow \Re$ and $\eta : \mathcal K \times \mathcal K \rightarrow \Re$ , where $\mathcal K$ is a nonempty set in $\Re^n,$ be continuous functions.

Definition [] A function $g:\mathcal K\subseteq \Re = \left( -\infty , \infty \right) \rightarrow \Re$ is said to be convex, if we have

$\begin{equation*} g\left( v\mathfrak{c}+\left( 1-v\right) \mathfrak{e}\right) \leq vg\left( \mathfrak{c}\right) +\left( 1-v\right) g\left( \mathfrak{e}\right). \end{equation*}$

for all $\mathfrak{c}, \mathfrak{e}\in \mathcal K\;$ and $v\in \left[0, 1\right].$

Definition [] The set $\mathcal K \subset \Re^n$ is said to be invex with respect to $\eta(., .)$ , if for every $\mathfrak{c}, \mathfrak{e}\in \mathcal K$ and $v\in[0, 1]$

$\begin{equation*} \mathfrak{c}+v\eta(\mathfrak{e}, \mathfrak{c})\in \mathcal K. \end{equation*}$

The above set $\mathcal K$ is also called $\eta$ -connected set.

It is obvious that every convex set is invex with respect to $\eta(\mathfrak{e}, \mathfrak{c}) = \mathfrak{e}-\mathfrak{c}$ but there exist invex sets which are not convex [20].

Definition The function $g$ on the invex set $\mathcal K$ is said to be preinvex with respect to $\eta$ if

$\begin{equation*} g\left( \mathfrak{c}+v\eta \left( \mathfrak{e}, \mathfrak{c}\right) \right) \ \leq\ \left( 1-v\right) \ g\left( \mathfrak{c}\right) \ +\ v\ g\left( \mathfrak{e}\right) \ , \ \ \ \ \ \ \ \forall \mathfrak{c}\ , \ \mathfrak{e}\in \mathcal K\ , \ \ v\in \left[ 0, 1\right]. \end{equation*}$

The function $-g$ is said to be preconcave if and only if $g$ is preinvex .

The important note that every convex function is a preinvex function but the converse is not true []. For example $g(v) = -|v|, \ \ \forall \ v\in \Re$ , is not convex function but it is preinvex function with respect to

$\begin{equation*} \eta \left( \mathfrak{e}, \mathfrak{c}\right) = \left\{ \begin{array}{c} \mathfrak{e}-\mathfrak{c}\ \ \ \ \ \ \ \ if\ \ \mathfrak{c} \mathfrak{e}\geq 0, \\ \mathfrak{c}-\mathfrak{e}\ \ \ \ \ \ \ \ if\ \ \mathfrak{c} \mathfrak{e} < 0. \end{array} \right. \end{equation*}$

We also want the following hypothesis regarding the function $\eta$ which is due to Mohan et al. []. Condition- $\mathcal C$ : Let $\mathcal K \subset \Re^n$ be an open invex subset with respect to $\eta : \mathcal K \times \mathcal K \rightarrow \Re$ . For any $\mathfrak{c}, \mathfrak{e} \in \mathcal K$ and $v \in [0, 1]$

$\begin{eqnarray} \eta \left( \mathfrak{e}, \mathfrak{e}+v\ \eta \left( \mathfrak{c}, \mathfrak{e}\right) \right) = \ \ -v\;\eta \left( \mathfrak{c}, \mathfrak{e}\right), \\ \eta \left( \mathfrak{c}, \mathfrak{e}+v\ \eta \left( \mathfrak{c}, \mathfrak{e}\right) \right) = \ \ \left( 1-v\right) \;\eta \left( \mathfrak{c}, \mathfrak{e}\right). \end{eqnarray}$

(1.1)

For any $\mathfrak{c}, \mathfrak{e}\in \mathcal K$ and $v_{1}, v_{2} \in [0, 1]$ from condition $\mathcal C$ , we have

$\begin{eqnarray*} \eta \left( \mathfrak{e}+v_{2}\ \eta \left( \mathfrak{c}, \mathfrak{e}\right) \ , \ \mathfrak{e}+v_{1}\ \eta \left( \mathfrak{c}, \mathfrak{e}\right) \right) = \ \ \left( v_{2}-v_{1}\right) \eta \left( \mathfrak{c}, \mathfrak{e}\right). \end{eqnarray*}$

If $g$ is a preinvex function on $[\mathfrak{c}, \mathfrak{c}+\eta(\mathfrak{e}, \mathfrak{c})]$ and the mapping $\eta$ satisfies condition $\mathcal C$ , then for every $v \in [0, 1]$ , from Eq (1.1), it yields that

$\begin{eqnarray} \label{1.2} \left\vert g\left( \mathfrak{c}+v\eta \left( \mathfrak{e}, \mathfrak{c}\right) \right) \right\vert = \ \left\vert g\left( \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right) \right) +\left( 1-v\right) \eta \left( \mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right) \right) \right\vert \ \ \ \ \ \ \ \ \ \ \ \ \\ \leq \ v\ \left\vert g\left( \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right) \right) \right\vert +\left( 1-v\right) \left\vert g\left( \mathfrak{c}\right) \right\vert, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{eqnarray}$

and

$\begin{eqnarray*} \label{1.3} \left\vert g\left( \mathfrak{c}+\left( 1-v\right) \eta \left( \mathfrak{e}, \mathfrak{c}\right) \right) \right\vert = \ \left\vert g\left( \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right) \right) +v\eta \left( \mathfrak{c}, \mathfrak{c}+\eta \left(\mathfrak{e}, \mathfrak{c}\right) \right) \right\vert \ \ \ \ \ \\ \leq \ \left( 1-v\right) \ \left\vert g\left( \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right) \right) \right\vert +v\left\vert g\left( \mathfrak{c}\right) \right\vert .\notag \end{eqnarray*}$

There are many vector functions that satisfy the condition $\mathcal C$ in [], which trivial case $\eta \left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e}$ . For example suppose $\mathcal K = \Re\backslash \{0\}$ and

$\begin{eqnarray*} \eta \left( \mathfrak{e}, \mathfrak{c}\right) = \left\{ \begin{array}{c} \mathfrak{e}-\mathfrak{c}\ \ \ \ \ \ \ \ if\ \ \mathfrak{c} > 0, \mathfrak{e} > 0 \\ \mathfrak{e}-\mathfrak{c}\ \ \ \ \ \ \ \ if\ \ \mathfrak{c} < 0, \mathfrak{e} < 0 \\ -\mathfrak{e}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ otherwise \end{array} \right. \end{eqnarray*}$

The set $\mathcal K$ is invex set and $\eta$ satisfies the condition $\mathcal C$ .

Noor et al. [23], proved the following Hermite-Hadamard type inequalities.

Theorem 1.1. Let $g:\mathcal K = [\mathfrak{c}, \mathfrak{c}+\eta(\mathfrak{e}, \mathfrak{c})]\to (0, \infty)$ be a preinvex function on the interval of real numbers $\mathcal K^0$ with $\eta(\mathfrak{e}, \mathfrak{c})>0$ , then the following inequalities hold:

$\begin{eqnarray*} g\left( \frac{2\mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right) }{2}\right) \leq \frac{1}{ \eta \left( \mathfrak{e}, \mathfrak{c}\right) }\int_{\mathfrak{c}}^{\mathfrak{c}+\; \eta \left(\mathfrak{e}, \mathfrak{c}\right) }g\left( x\right) \ dx\leq \frac{g\left( \mathfrak{c}\right) \ +\ g\left( \mathfrak{e}\right) }{2}. \end{eqnarray*}$

Then Riemann-Liouville(R-L) fractional integrals of order $\varepsilon >0$ with $\mathfrak{c} \geq 0$ are defined as follows:

$\begin{equation*} J_{\mathfrak{c}^{+}}^{\varepsilon }g\left( \mathfrak{z}\right) = \ \frac{1}{\Gamma \left( \varepsilon \right) }\int_{\mathfrak{c}}^{\mathfrak{z}}\left( \mathfrak{z}-v\right) ^{\varepsilon -1}\ g \left( v\right) \ dv\ , \ \ \ \ \mathfrak{z} > \mathfrak{c}, \ \end{equation*}$

and

$\begin{equation*} J_{\mathfrak{e}^{-}}^{\varepsilon }g\left( \mathfrak{z}\right) = \ \frac{1}{\Gamma \left( \varepsilon \right) }\int_{\mathfrak{z}}^{\mathfrak{e}}\left( v-\mathfrak{z}\right) ^{\varepsilon -1}\ g\left( v\right) \ dv\ , \ \ \ \ \mathfrak{z} < \mathfrak{e}. \end{equation*}$

In [30], Sarikaya et al. also described the inequality in fractional integral version. In this study, considering the above mentioned theoretical framework, firstly, an integral identity which is candidate to produce Ostrowski type inequalities has been proved. With the help of such identity like Hölder, Power mean, Young's inequalities, Hölder-İşcan, Improved power means inequality and convexity, a new type of inequality, Ostrowski type inequalities, has been obtained.

2. Main results

In this section, we give Ostrowski inequalities for Riemann-Liouville(R-L) fractional integrals operator are obtained for a differentiable functions on $(\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right))$ . For this purpose, we give a new identity that involve Riemann-Liouville(R-L) fractional integrals operator whose second derivatives are preinvex functions.

Lemma 2.1. Suppose that a mapping $g:[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)]\rightarrow \Re$ istwice differentiable with $\mathfrak{c}< \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)$ . If $g^{{\prime \prime }}\in \mathcal L_{1}[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)],$ then for all $\mathfrak{z}\in\lbrack \mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)]$ and $\varepsilon >0,$ the following equality

$\begin{align} & \frac{\eta^{\varepsilon +1}\left( \mathfrak{z}, \mathfrak{c}\right) -\eta^{\varepsilon +1}\left(\mathfrak{e}, \mathfrak{z}\right)}{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g^{{\prime }}\left( \mathfrak{z}\right) -\ \frac{\eta^{\varepsilon }\left( \mathfrak{z}, \mathfrak{c} \right) +\eta^{\varepsilon }\left( \mathfrak{e}, \mathfrak{z}\right) }{\eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\left( \mathfrak{z}\right) \\ & +\frac{\Gamma \left( \varepsilon +1\right) }{ \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\{J_{\mathfrak{[\mathfrak{c}+\eta \left( \mathfrak{z}, \mathfrak{c}\right)]}^{-}}^{\varepsilon }g\left( \mathfrak{c} \right) +J_{[\mathfrak{\mathfrak{e}+\eta \left( \mathfrak{z}, \mathfrak{e}\right)]}^{+}}^{\varepsilon }g\left( \mathfrak{e}\right) g\} \\ & = \frac{\eta^{\varepsilon +2}\left( \mathfrak{z}, \mathfrak{c}\right)}{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }\ \int_{0}^{1}v^{\varepsilon +1}g^{{ \prime \prime }}\left( \mathfrak{c}+v\eta \left( \mathfrak{z}, \mathfrak{c}\right)\right) dv\ \\ & +\frac{\eta^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right)}{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }\ \int_{0}^{1}v^{\varepsilon +1}g^{{ \prime \prime }}\left( \mathfrak{e}+v\eta \left( \mathfrak{z}, \mathfrak{e}\right) \right)dv, \end{align}$

(2.1)

satisfies for $v\in [0, 1]$ .

Proof. Let us assume that

$\begin{align} & \frac{\eta^{\varepsilon +2}\left( \mathfrak{z}, \mathfrak{c}\right)}{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }\ \int_{0}^{1}v^{\varepsilon +1}g^{{ \prime \prime }}\left( \mathfrak{c}+v\eta \left( \mathfrak{z}, \mathfrak{c}\right)\right) dv\ \\ & +\frac{\eta^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right)}{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }\ \int_{0}^{1}v^{\varepsilon +1}g^{{ \prime \prime }}\left( \mathfrak{e}+v\eta \left( \mathfrak{z}, \mathfrak{e}\right) \right)dv, \\& I = \frac{\eta^{\varepsilon +2}\left( \mathfrak{z}, \mathfrak{c}\right)}{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }\ I_{1}+ \frac{\eta^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right)}{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }\ I_{2} , \end{align}$

(2.2)

where

$\begin{align*} & I_{1} = \int_{0}^{1}v^{\varepsilon +1}g^{{ \prime \prime }}\left( \mathfrak{c}+v\eta \left( \mathfrak{z}, \mathfrak{c}\right)\right) dv\ \\& = \frac{v^{\varepsilon +1}g^{{\prime }}\left( \mathfrak{c}+v\eta \left( \mathfrak{z}, \mathfrak{c}\right)\right) } {\eta\left( \mathfrak{z}, \mathfrak{c}\right)} g|_{0}^{1}-\int_{0}^{1}\frac{\left( \varepsilon +1\right) v^{\varepsilon }\ g^{{\prime }}\left( \mathfrak{c}+v\eta \left( \mathfrak{z}, \mathfrak{c}\right)\right) }{\eta(\mathfrak{z}, \mathfrak{c})}dv \\& = \; \frac{g^{{\prime }}\left( \mathfrak{z}\right) }{\eta\left( \mathfrak{z}, \mathfrak{c}\right)}-\frac{\varepsilon +1}{\eta\left( \mathfrak{z}, \mathfrak{c}\right)} \int_{0}^{1}v^{\varepsilon }\ g^{{\prime }}\left( \mathfrak{c}+v\eta \left( \mathfrak{z}, \mathfrak{c}\right)\right) dv \\& = \; \frac{g^{{\prime }}\left( \mathfrak{z}\right) }{\eta \left( \mathfrak{z}, \mathfrak{c}\right)}-\frac{\varepsilon +1}{ \eta ^{2}\left( \mathfrak{z}, \mathfrak{c}\right)}g\left( \mathfrak{z}\right) +\frac{\varepsilon \left( \varepsilon +1\right) }{\eta ^{2}\left( \mathfrak{z}, \mathfrak{c}\right)}\int_{0}^{1}v^{\varepsilon -1}\ g\left( \mathfrak{c}+v\eta \left( \mathfrak{z}, \mathfrak{c}\right)\right) dv \\& = \; \frac{g^{{\prime }}\left( \mathfrak{z}\right) }{\eta \left( \mathfrak{z}, \mathfrak{c}\right)}-\frac{\varepsilon +1}{\eta ^{2}\left( \mathfrak{z}, \mathfrak{c}\right)}g\left( \mathfrak{z}\right) \ +\frac{\Gamma \left( \varepsilon +2\right) }{ \eta ^{\varepsilon +2}\left( \mathfrak{z}, \mathfrak{c}\right) }\ J_{\mathfrak{[\mathfrak{c}+\eta \left( \mathfrak{z}, \mathfrak{c}\right)]}^{-}}^{\varepsilon }g\left( \mathfrak{c}\right), \end{align*}$

and similarly

$\begin{align*} & I_{2} = \ \int_{0}^{1}v^{\varepsilon +1}g^{{\prime \prime }}\left( \mathfrak{e}+v\eta \left( \mathfrak{z}, \mathfrak{e}\right) \right) dv \\& = \frac{v^{\varepsilon +1}g^{{\prime }}\left( \mathfrak{e}+v\eta \left( \mathfrak{z}, \mathfrak{e}\right) \right) } {\eta\left( \mathfrak{z}, \mathfrak{e}\right)} g|_{0}^{1}-\int_{0}^{1}\frac{\left( \varepsilon +1\right) v^{\varepsilon }\ g^{{\prime }}\left( \mathfrak{e}+v\eta \left( \mathfrak{z}, \mathfrak{e}\right) \right) }{\eta(\mathfrak{z}, \mathfrak{e})}dv \\& = \; \frac{g^{{\prime }}\left( \mathfrak{z}\right) }{\eta\left( \mathfrak{z}, \mathfrak{e}\right)}-\frac{\varepsilon +1}{\eta\left( \mathfrak{z}, \mathfrak{c}\right)} \int_{0}^{1}v^{\varepsilon }\ g^{{\prime }}\left( \mathfrak{c}+v\eta \left( \mathfrak{z}, \mathfrak{c}\right)\right) dv \\& = \; -\frac{g^{{\prime }}\left( \mathfrak{z}\right) }{\eta \left( \mathfrak{e}, \mathfrak{z}\right)}-\frac{\varepsilon +1}{ \eta ^{2}\left( \mathfrak{z}, \mathfrak{e}\right)}g\left( \mathfrak{z}\right) +\frac{\varepsilon \left( \varepsilon +1\right) }{\eta ^{2}\left( \mathfrak{z}, \mathfrak{e}\right)}\int_{0}^{1}v^{\varepsilon -1}\ g\left( \mathfrak{e}+v\eta \left( \mathfrak{z}, \mathfrak{e}\right)\right) dv \\& = \; -\frac{g^{{\prime }}\left( \mathfrak{z}\right) }{\eta \left( \mathfrak{e}, \mathfrak{z}\right)}-\frac{\varepsilon +1}{\eta ^{2}\left( \mathfrak{e}, \mathfrak{z}\right)}g\left( \mathfrak{z}\right) \ +\frac{\Gamma \left( \varepsilon +2\right) }{ \eta ^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right) }\ J_{\mathfrak{[\mathfrak{e}+\eta \left( \mathfrak{z}, \mathfrak{e}\right)]}^{+}}^{\varepsilon }g\left( \mathfrak{e}\right), \end{align*}$

Combining $I_{1}$ and $I_{2}$ with (2.2), we get (2.3).

Remark 2.1. If we set $\varepsilon = 1$ and $\eta\left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e}$ in Lemma 2.1, we get (Lemma 1 in [11]).

Theorem 2.1. Assume that all the assumptions as defined in Lemma 2.1 and $|g^{{\prime \prime }}|$ is preinvex function on $[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)],$ then for all $\varepsilon >0,$ the following inequality

$\begin{align} & g|\frac{\eta^{\varepsilon +1}\left( \mathfrak{z}, \mathfrak{c}\right) -\eta^{\varepsilon +1}\left(\mathfrak{e}, \mathfrak{z}\right)}{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g^{{\prime }}\left( \mathfrak{z}\right) -\ \frac{\eta^{\varepsilon }\left( \mathfrak{z}, \mathfrak{c} \right) +\eta^{\varepsilon }\left( \mathfrak{e}, \mathfrak{z}\right) }{\eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\left( \mathfrak{z}\right) \\ & +\frac{\Gamma \left( \varepsilon +1\right) }{ \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\{J_{\mathfrak{[\mathfrak{c}+\eta \left( \mathfrak{z}, \mathfrak{c}\right)]}^{-}}^{\varepsilon }g\left( \mathfrak{c} \right) +J_{[\mathfrak{\mathfrak{e}+\eta \left( \mathfrak{z}, \mathfrak{e}\right)]}^{+}}^{\varepsilon }g\left( \mathfrak{e}\right) g\}g| \\ & \leq \frac{\eta^{\varepsilon +2}\left( \mathfrak{z}, \mathfrak{c}\right) }{\left( \varepsilon +1\right)\left( \varepsilon +3\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\{\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert \; +\left\vert g^{{\prime \prime }}\left( \mathfrak{c}\right) \right\vert \frac{1}{\varepsilon +2 }g\} \\ & +\ \frac{\eta^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right) }{\left( \varepsilon +1\right)\left( \varepsilon +3\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\{\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert \; +\left\vert g^{{\prime \prime }}\left( \mathfrak{e}\right) \right\vert \frac{1}{ \varepsilon +2 }g\}. \end{align}$

(2.3)

satisfies for $v\in \lbrack 0, 1].$

Proof. From Lemma 2.1 and since $|g^{{\prime \prime}}|$ is preinvex function on $[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)],$ we obtain

$\begin{align*} & g|\frac{\eta^{\varepsilon +1}\left( \mathfrak{z}, \mathfrak{c}\right) -\eta^{\varepsilon +1}\left(\mathfrak{e}, \mathfrak{z}\right)}{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g^{{\prime }}\left( \mathfrak{z}\right) -\ \frac{\eta^{\varepsilon }\left( \mathfrak{z}, \mathfrak{c} \right) +\eta^{\varepsilon }\left( \mathfrak{e}, \mathfrak{z}\right) }{\eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\left( \mathfrak{z}\right) \\ & +\frac{\Gamma \left( \varepsilon +1\right) }{ \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\{J_{\mathfrak{[\mathfrak{c}+\eta \left( \mathfrak{z}, \mathfrak{c}\right)]}^{-}}^{\varepsilon }g\left( \mathfrak{c} \right) +J_{[\mathfrak{\mathfrak{e}+\eta \left( \mathfrak{z}, \mathfrak{e}\right)]}^{+}}^{\varepsilon }g\left( \mathfrak{e}\right) g\}g| \\& \notag \leq \ \frac{\eta^{\varepsilon +2}\left( \mathfrak{z}, \mathfrak{c}\right)}{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }\ \int_{0}^{1}v^{\varepsilon +1}|g^{{ \prime \prime }}\left( \mathfrak{c}+v\eta \left( \mathfrak{z}, \mathfrak{c}\right)\right)| dv\ \notag \\ & +\frac{\eta^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right)}{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }\ \int_{0}^{1}v^{\varepsilon +1}|g^{{ \prime \prime }}\left( \mathfrak{e}+v\eta \left( \mathfrak{z}, \mathfrak{e}\right) \right)|dv \\& \notag \leq \ \frac{\eta^{\varepsilon +2}\left( \mathfrak{z}, \mathfrak{c}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }\ \ \int_{0}^{1}v^{\varepsilon +1}g\{v\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert +\left( 1-v\right) \left\vert g^{{\prime \prime }}\left( \mathfrak{c}\right) \right\vert g\}dv \\& \notag +\; \frac{ \eta^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) } \int_{0}^{1}v^{\varepsilon +1}g\{v\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert +\left( 1-v\right) \left\vert g^{{\prime \prime }}\left( \mathfrak{e}\right) \right\vert g\}dv \\& \notag \leq \frac{\eta^{\varepsilon +2}\left( \mathfrak{z}, \mathfrak{c}\right) }{\left( \varepsilon +1\right)\left( \varepsilon +3\right)\eta \left( \mathfrak{e}, \mathfrak{c}\right) }g\{\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert \; +\left\vert g^{{\prime \prime }}\left( \mathfrak{c}\right) \right\vert \frac{1}{\varepsilon +2 }g\} \notag \\ & +\ \frac{\eta^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right) }{\left( \varepsilon +1\right)\left( \varepsilon +3\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\{\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert \; +\left\vert g^{{\prime \prime }}\left( \mathfrak{e}\right) \right\vert \frac{1}{\varepsilon +2 }g\}. \end{align*}$

This completes the proof.

Remark 2.2. If we set $\varepsilon = 1$ and $\eta\left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e},$ then from Theorem 2.1, we get (Theorem 4 in []) with $s = 1$ .

Corollary 2.1. By using Theorem 2.1 with $|g^{{\prime \prime }}|\leq \mathcal M,$ we get the following inequality

Remark 2.3. If we set $\varepsilon = 1$ and $\eta\left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e},$ then from Corollary 2.1, we recapture (Theorem 2.1, [32]).

Corollary 2.2. If we set $\eta\left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e}$ and $\mathfrak{z} = \frac{\mathfrak{c}+\mathfrak{e}}{2},$ in Corollary 2.1, we get the mid-point inequality

$\begin{eqnarray*} &&\left\vert \frac{\Gamma \left( \varepsilon +1\right) }{ \left( \mathfrak{e}-\mathfrak{c}\right) }g\{J_{\left( \frac{\mathfrak{c}+\mathfrak{e}}{2} \right) ^{-}}^{\varepsilon }g\left( \mathfrak{c}\right) +J_{\left( \frac{\mathfrak{c}+ \mathfrak{e}}{2}\right) ^{+}}^{\varepsilon }g\left( \mathfrak{e}\right) g\}-\left( \frac{ \mathfrak{e}-\mathfrak{c}}{2}\right) ^{\varepsilon -1}g\left( \mathfrak{z}\right) \right\vert \\ &\leq &\; \mathcal M\; \frac{\left( \mathfrak{e}-\mathfrak{c}\right) ^{\varepsilon +1}}{2^{\varepsilon +1}}\; g(\frac{1}{\left( \varepsilon +1\right) \left( \varepsilon +2\right) }g). \end{eqnarray*}$

Theorem 2.2. Assume that all the assumptions as defined in Lemma 2.1 and $|g^{{\prime \prime }}|^q,$ $q>1$ is preinvex function on $[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)],$ then for all $\varepsilon >0,$ the following inequality

$\begin{align} & g|\frac{\eta^{\varepsilon +1}\left( \mathfrak{z}, \mathfrak{c}\right) -\eta^{\varepsilon +1}\left(\mathfrak{e}, \mathfrak{z}\right)}{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g^{{\prime }}\left( \mathfrak{z}\right) -\ \frac{\eta^{\varepsilon }\left( \mathfrak{z}, \mathfrak{c} \right) +\eta^{\varepsilon }\left( \mathfrak{e}, \mathfrak{z}\right) }{\eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\left( \mathfrak{z}\right) \\ & +\frac{\Gamma \left( \varepsilon +1\right) }{ \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\{J_{\mathfrak{[\mathfrak{c}+\eta \left( \mathfrak{z}, \mathfrak{c}\right)]}^{-}}^{\varepsilon }g\left( \mathfrak{c} \right) +J_{[\mathfrak{\mathfrak{e}+\eta \left( \mathfrak{z}, \mathfrak{e}\right)]}^{+}}^{\varepsilon }g\left( \mathfrak{e}\right) g\}g| \\ & \leq \ \left( \frac{1}{\left( \varepsilon +1\right) p+1}\right) ^{\frac{1}{p}}\; \\ & \times g[\frac{\eta^{\varepsilon +2}\left( \mathfrak{z}, \mathfrak{c}\right) }{\left( \varepsilon +1\right) \eta \left( \mathfrak{e}, \mathfrak{c}\right) }\left( \frac{\left\vert g^{ {\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left\vert g^{{ \prime \prime }}\left( \mathfrak{c}\right) \right\vert ^{q}}{2}\right) ^{\frac{1}{ q}}+\frac{\eta^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }\left( \frac{\left\vert g^{{ \prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left\vert g^{{ \prime \prime }}\left( \mathfrak{e}\right) \right\vert ^{q}}{2}\right) ^{\frac{1 }{q}}g], \end{align}$

(2.4)

satisfies for $v\in \lbrack 0, 1],$ where $q^{-1}+p^{-1} = 1.$

Proof. Suppose that $p>1.$ From Lemma 2.1, by using the well-known Hölder integral inequality and the preinvexity of $|g^{{\prime \prime}}|^q$ , we obtain

(2.5)

$\begin{align} & \leq \ \frac{\eta^{\varepsilon +2}\left( \mathfrak{z}, \mathfrak{c}\right)}{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }\ \int_{0}^{1}v^{\varepsilon +1}|g^{{ \prime \prime }}\left( \mathfrak{c}+v\eta \left( \mathfrak{z}, \mathfrak{c}\right)\right)| dv\ \notag \\ & +\frac{\eta^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right)}{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }\ \int_{0}^{1}v^{\varepsilon +1}|g^{{ \prime \prime }}\left( \mathfrak{e}+v\eta \left( \mathfrak{z}, \mathfrak{e}\right) \right)|dv \end{align}$

(2.6)

$\begin{align} & \leq \ \frac{\eta^{\varepsilon +2}\left( \mathfrak{z}, \mathfrak{c}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }\ \ \left( \int_{0}^{1}v^{\left( \varepsilon +1\right) p}dv\right) ^{\frac{1}{p}}\left( \int_{0}^{1}\left\vert g^{{\prime \prime }}\left( \mathfrak{c}+v\eta \left( \mathfrak{z}, \mathfrak{c}\right)\right) \right\vert ^{q}dv\right) ^{\frac{1}{q}} \\& +\; \frac{\eta^{\varepsilon +2} \left( \mathfrak{e}, \mathfrak{z}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }\left( \int_{0}^{1}v^{\left( \varepsilon +1\right) p}dv\right) ^{\frac{1}{p}}\left( \int_{0}^{1}\left\vert g^{{\prime \prime }}\left( \mathfrak{e}+v\eta \left( \mathfrak{z}, \mathfrak{e}\right) \right) \right\vert ^{q}dv\right) ^{\frac{1}{q}} . \end{align}$

(2.7)

Since $|g^{{\prime \prime}}|^q$ is preinvexity on $[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)],$ we obtain

$\begin{align} & \int_{0}^{1}\left\vert g^{{\prime \prime }}\left( \mathfrak{c}+v\eta \left( \mathfrak{z}, \mathfrak{c}\right)\right) \right\vert ^{q}dv\leq \int_{0}^{1}g\{v\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left( 1-v\right) \left\vert g^{{\prime \prime }}\left( \mathfrak{c}\right) \right\vert ^{q}g\}dv \\& = \ \frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left\vert g^{{\prime \prime }}\left( \mathfrak{c}\right) \right\vert ^{q}}{2}, \end{align}$

(2.8)

and

$\begin{align} & \int_{0}^{1}\left\vert g^{{\prime \prime }}\left( \mathfrak{e}+v\eta \left( \mathfrak{z}, \mathfrak{e}\right) \right) \right\vert ^{q}dv\leq \int_{0}^{1}g\{v\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left( 1-v\right) \left\vert g^{{\prime \prime }}\left( \mathfrak{e}\right) \right\vert ^{q}g\}dv \\& = \ \frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left\vert g^{{\prime \prime }}\left( \mathfrak{e}\right) \right\vert ^{q}}{2}. \end{align}$

(2.9)

By using (2.8) and (2.9) with (2.7), we obtain

This completes the proof.

Remark 2.4. If we set $\varepsilon = 1$ and $\eta\left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e},$ then from Theorem 2.2, we get ( Theorem 5, []) with $s = 1$ .

Corollary 2.3. Using Theorem 2.2 with $|g^{{\prime \prime }}|\leq \mathcal M,$ we get

Corollary 2.4. If in Corollary 2.3, we set $\eta\left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e}$ and $\mathfrak{z} = \frac{\mathfrak{c}+\mathfrak{e}}{2},$ then we get the mid-point inequality

Theorem 2.3. Assume that all the assumptions as defined in Lemma 2.1 and $|g^{{\prime \prime }}|^q,$ $q\geq1$ is preinvex function on $[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)],$ then for all $\varepsilon >0,$ the following inequality

$\begin{align} & g|\frac{\eta^{\varepsilon +1}\left( \mathfrak{z}, \mathfrak{c}\right) -\eta^{\varepsilon +1}\left(\mathfrak{e}, \mathfrak{z}\right)}{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g^{{\prime }}\left( \mathfrak{z}\right) -\ \frac{\eta^{\varepsilon }\left( \mathfrak{z}, \mathfrak{c} \right) +\eta^{\varepsilon }\left( \mathfrak{e}, \mathfrak{z}\right) }{\eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\left( \mathfrak{z}\right) \\ & +\frac{\Gamma \left( \varepsilon +1\right) }{ \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\{J_{\mathfrak{[\mathfrak{c}+\eta \left( \mathfrak{z}, \mathfrak{c}\right)]}^{-}}^{\varepsilon }g\left( \mathfrak{c} \right) +J_{[\mathfrak{\mathfrak{e}+\eta \left( \mathfrak{z}, \mathfrak{e}\right)]}^{+}}^{\varepsilon }g\left( \mathfrak{e}\right) g\}g| \\ & \leq \ \left( \frac{1}{\varepsilon +2}\right) ^{1-\frac{1}{q}}\; g[\frac{ \eta^{\varepsilon +2}\left( \mathfrak{z}, \mathfrak{c}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }\left( \frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}}{\varepsilon +3}+\frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{c}\right) \right\vert ^{q}}{\left( \varepsilon +2\right) \left( \varepsilon +3\right) }\right) ^{\frac{1}{q}} \\ & +\frac{\eta^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }\left( \frac{\left\vert g^{{ \prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}}{\varepsilon +3}+\frac{ \left\vert g^{{\prime \prime }}\left( \mathfrak{e}\right) \right\vert ^{q}}{ \left( \varepsilon +2\right) \left( \varepsilon +3\right) }\right) ^{\frac{1}{q}}g] , \end{align}$

(2.10)

satisfies for $v\in \lbrack 0, 1].$

Proof. Suppose that $q\geq1.$ From Lemma 2.1, by using the power-mean integral inequality and preinvexity of $|g^{{\prime \prime }}|^q$ , we obtain

$\begin{align} & \leq \ \frac{\eta^{\varepsilon +2}\left( \mathfrak{z}, \mathfrak{c}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }\ \ \left( \int_{0}^{1}v^{\varepsilon +1}dv\right) ^{1-\frac{1}{q}}\left( \int_{0}^{1}\left\vert g^{{\prime \prime }}\left( \mathfrak{c}+v\eta \left( \mathfrak{z}, \mathfrak{c}\right)\right) \right\vert ^{q}dv\right) ^{\frac{1}{q}} \\& +\; \frac{\eta^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }\left( \int_{0}^{1}v^{\varepsilon +1}dv\right) ^{1-\frac{1}{q}}\left( \int_{0}^{1}\left\vert g^{{\prime \prime }}\left( \mathfrak{e}+v\eta \left( \mathfrak{z}, \mathfrak{e}\right) \right) \right\vert ^{q}dv\right) ^{\frac{1}{q}} . \end{align}$

(2.11)

Since $|g^{{\prime \prime}}|^q$ is preinvexity on $[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)],$ we obtain

$\begin{align} & \int_{0}^{1}v^{\varepsilon +1}\left\vert g^{{\prime \prime }}\left( \mathfrak{c}+v\eta \left( \mathfrak{z}, \mathfrak{c}\right)\right) \right\vert ^{q}dv\leq \int_{0}^{1}v^{\varepsilon +1}g\{v\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left( 1-v\right) \left\vert g^{{\prime \prime }}\left( \mathfrak{c}\right) \right\vert ^{q}g\}dv \\& = \ \frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}}{ \varepsilon +3}+\frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{c}\right) \right\vert ^{q}}{\left( \varepsilon +2\right) \left( \varepsilon +3\right) } \end{align}$

(2.12)

and

$\begin{align} & \int_{0}^{1}v^{\varepsilon +1}\left\vert g^{{\prime \prime }}\left( \mathfrak{e}+v\eta \left( \mathfrak{z}, \mathfrak{e}\right) \right) \right\vert ^{q}dv\leq \int_{0}^{1}v^{\varepsilon +1}g\{v\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left( 1-v\right) \left\vert g^{{\prime \prime }}\left( \mathfrak{e}\right) \right\vert ^{q}g\}dv \\& = \ \frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}}{ \varepsilon +3}+\frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{e}\right) \right\vert ^{q}}{\left( \varepsilon +2\right) \left( \varepsilon +3\right) }. \end{align}$

(2.13)

By using (2.12) and (2.13) with (2.11), we obtain

This completes the proof.

Remark 2.5. If we set $\varepsilon = 1$ and $\eta\left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e},$ then from Theorem 2.3, we get (Theorem 6, [])with $s = 1$ .

Corollary 2.5. Under the same assumptions in Theorem 2.3 with $|g^{{\prime \prime }}|\leq \mathcal M,$ we get the following inequality

Corollary 2.6. If in Corollary 2.5, we set $\eta\left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e}$ and $\mathfrak{z} = \frac{\mathfrak{c}+\mathfrak{e}}{2},$ then we get the mid-point inequality

Theorem 2.4. Assume that all the assumptions as defined in Lemma 2.1 and $|g^{{\prime \prime }}|^q,$ $q>1$ is preinvex function on $[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)],$ then for all $\varepsilon >0,$ the following inequality

$\begin{align} & g|\frac{\eta^{\varepsilon +1}\left( \mathfrak{z}, \mathfrak{c}\right) -\eta^{\varepsilon +1}\left(\mathfrak{e}, \mathfrak{z}\right)}{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g^{{\prime }}\left( \mathfrak{z}\right) -\ \frac{\eta^{\varepsilon }\left( \mathfrak{z}, \mathfrak{c} \right) +\eta^{\varepsilon }\left( \mathfrak{e}, \mathfrak{z}\right) }{\eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\left( \mathfrak{z}\right) \\ & +\frac{\Gamma \left( \varepsilon +1\right) }{ \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\{J_{\mathfrak{[\mathfrak{c}+\eta \left( \mathfrak{z}, \mathfrak{c}\right)]}^{-}}^{\varepsilon }g\left( \mathfrak{c} \right) +J_{[\mathfrak{\mathfrak{e}+\eta \left( \mathfrak{z}, \mathfrak{e}\right)]}^{+}}^{\varepsilon }g\left( \mathfrak{e}\right) g\}g| \\ & \leq \ \frac{\eta^{\varepsilon +2}\left( \mathfrak{z}, \mathfrak{c}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\{\frac{1}{\left( \left( \varepsilon +1\right) p+1\right) p}+\frac{\left\vert g^{{\prime \prime } }\left( \mathfrak{z}\right) \right\vert ^{q}+\left\vert g^{{\prime \prime } }\left( \mathfrak{c}\right) \right\vert ^{q}}{2q}g\} \\ & +\; \frac{\eta^{\varepsilon +2} \left( \mathfrak{e}, \mathfrak{z}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\{\frac{1}{\left( \left( \varepsilon +1\right) p+1\right) p}+\frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left\vert g^{{\prime \prime }}\left( \mathfrak{e}\right) \right\vert ^{q}}{2q}g\}, \end{align}$

(2.14)

satisfies for $v\in \lbrack 0, 1].$

Proof. From Lemma 2.1, we obtain

By using the Young's inequality as

$\begin{align*} & xy \ < \ \frac{1}{p}x^p + \frac{1}{q}y^q. \end{align*}$

$\begin{align*} & g|\frac{\eta^{\varepsilon +1}\left( \mathfrak{z}, \mathfrak{c}\right) -\eta^{\varepsilon +1}\left(\mathfrak{e}, \mathfrak{z}\right)}{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g^{{\prime }}\left( \mathfrak{z}\right) -\ \frac{\eta^{\varepsilon }\left( \mathfrak{z}, \mathfrak{c} \right) +\eta^{\varepsilon }\left( \mathfrak{e}, \mathfrak{z}\right) }{\eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\left( \mathfrak{z}\right) \\ & +\frac{\Gamma \left( \varepsilon +1\right) }{ \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\{J_{\mathfrak{[\mathfrak{c}+\eta \left( \mathfrak{z}, \mathfrak{c}\right)]}^{-}}^{\varepsilon }g\left( \mathfrak{c} \right) +J_{[\mathfrak{\mathfrak{e}+\eta \left( \mathfrak{z}, \mathfrak{e}\right)]}^{+}}^{\varepsilon }g\left( \mathfrak{e}\right) g\}g| \\& \notag \leq \ \frac{\eta^{\varepsilon +2}\left( \mathfrak{z}, \mathfrak{c}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\{\frac{1}{p}\int_{0}^{1}v^{\left( \varepsilon +1\right) p}dv+\frac{1}{q}\int_{0}^{1}\left\vert g^{{\prime \prime }}\left( \mathfrak{c}+v\eta \left( \mathfrak{z}, \mathfrak{c}\right)\right) \right\vert ^{q}dvg\} \\& \notag +\ \; \frac{\eta^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }\; \; g\{\frac{1}{p}\int_{0}^{1}v^{\left( \varepsilon +1\right) p}dv+\frac{ 1}{q}\int_{0}^{1}\left\vert g^{{\prime \prime }}\left( \mathfrak{e}+v\eta \left( \mathfrak{z}, \mathfrak{e}\right)\right) \right\vert ^{q}dvg\} \\& \notag \leq \ \frac{\eta ^{\varepsilon +2}\left( \mathfrak{z}, \mathfrak{c}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\{\frac{1}{p}\int_{0}^{1}v^{\left( \varepsilon +1\right) p}dv+\frac{1}{q}\int_{0}^{1}g\{v\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left( 1-v\right) \left\vert g^{{\prime \prime }}\left( \mathfrak{c}\right) \right\vert ^{q}g\}g\} \\& \notag +\ \frac{\eta^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }\ g\{\frac{1}{p}\int_{0}^{1}v^{\left( \varepsilon +1\right) p}dv+\frac{ 1}{q}\int_{0}^{1}g\{v\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left( 1-v\right) \left\vert g^{{\prime \prime }}\left( \mathfrak{e}\right) \right\vert ^{q}g\}g\} \\& \notag \leq \ \frac{\eta^{\varepsilon +2}\left( \mathfrak{z}, \mathfrak{c}\right) }{\left( \alpha +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\{ \frac{1}{\left( \left( \varepsilon +1\right) p+1\right) p}+\frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left\vert g^{{\prime \prime }}\left( \mathfrak{c}\right) \right\vert ^{q}}{2q}g\} \\& \notag +\; \frac{\eta^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\{\frac{1}{\left( \left( \varepsilon +1\right) p+1\right) p}+\frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left\vert g^{{\prime \prime }}\left( \mathfrak{e}\right) \right\vert ^{q}}{2q}g\}. \end{align*}$

This completes the proof.

Corollary 2.7. If we set $\eta\left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e}$ and $\varepsilon = 1$ in Theorem 2.4, we get

$\begin{align*} & \left\vert \frac{1\ }{\mathfrak{e}-\mathfrak{c}}\int_{\mathfrak{c}}^{\mathfrak{e}}g\left( u\right) du-g\left( \mathfrak{z}\right) +\left( \mathfrak{z}-\frac{\mathfrak{c}+\mathfrak{e} }{2}\right) g^{{\prime }}\left( \mathfrak{z}\right) \right\vert \\ & \leq \frac{\left( \mathfrak{z}-\mathfrak{c}\right) ^{3}}{2\left( \mathfrak{e}-\mathfrak{c} \right) }g[\frac{1}{\left( 2p+1\right) p}+\frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left\vert g^{{\prime \prime }}\left( \mathfrak{c}\right) \right\vert ^{q}}{2q}g]+\frac{\left( \mathfrak{e}-\mathfrak{z}\right) ^{3}}{2\left( \mathfrak{e}-\mathfrak{c}\right) }g[ \frac{1}{\left( 2p+1\right) p}+\frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left\vert g^{{\prime \prime }}\left( \mathfrak{e}\right) \right\vert ^{q}}{2q}g]. \end{align*}$

Corollary 2.8. If in Theorem 2.4, we set $\eta\left( \mathfrak{c}, \mathfrak{e}\right) = \mathfrak{c}-\mathfrak{e}$ and $\mathfrak{z} = \frac{\mathfrak{c}+\mathfrak{e}}{2},$ then we get the mid-point inequality

$\begin{align*} & \left\vert \frac{\Gamma \left( \varepsilon +1\right) }{ \left( \mathfrak{e}-\mathfrak{c}\right) }g\{J_{\left( \frac{\mathfrak{c}+\mathfrak{e}}{2} \right) ^{-}}^{\varepsilon }g\left( \mathfrak{c}\right) +J_{\left( \frac{\mathfrak{c}+ \mathfrak{e}}{2}\right) ^{+}}^{\varepsilon }g\left( \mathfrak{e}\right) g\}-\left( \frac{\mathfrak{e}-\mathfrak{c}}{2}\right) ^{\varepsilon -1}g\left( \mathfrak{z}\right) \right\vert \\ & \leq \frac{\left( \mathfrak{e}-\mathfrak{c}\right) ^{\varepsilon +1}}{2^{\varepsilon +2}\left( \varepsilon +1\right) }g\{\frac{2}{\left( \left( \varepsilon +1\right) p+1\right) p}+\frac{\left\vert g^{{\prime \prime }}\left( \frac{\mathfrak{c}+ \mathfrak{e}}{2}\right) \right\vert ^{q}+\left\vert g^{{\prime \prime }}\left( \mathfrak{c}\right) \right\vert ^{q}}{2q}+\frac{\left\vert g^{{\prime \prime } }\left( \frac{\mathfrak{c}+\mathfrak{e}}{2}\right) \right\vert ^{q}+\left\vert g^{{ \prime \prime }}\left( \mathfrak{e}\right) \right\vert ^{q}}{2q}g\}. \end{align*}$

Theorem 2.5. Assume that all the assumptions as defined in Lemma 2.1 and $|g^{{\prime \prime }}|^q,$ $q>1$ is preinvex function on $[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)],$ then for all $\varepsilon >0,$ the following inequality

(2.15)

$\begin{align*} & \leq \ \frac{\eta^{\varepsilon +2}\left( \mathfrak{z}, \mathfrak{c}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g[\left( \frac{1}{\left( \varepsilon p+p+1\right) \left( \varepsilon p+p+2\right) }\right) ^{\frac{1}{p}}\left( \frac{1}{6}\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\frac{1}{3}\left\vert g^{{\prime \prime }}\left( \mathfrak{c}\right) \right\vert ^{q}\right) ^{\frac{1}{q}} \notag \\ & +\; \left( \frac{1}{\left( \varepsilon +1\right) p+2}\right) ^{\frac{1}{p}}\left( \frac{1}{3}\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\frac{1}{6}\left\vert g^{{\prime \prime }}\left( \mathfrak{c}\right) \right\vert ^{q}\right) ^{\frac{1}{q}}g] \notag \\ & +\ \frac{\eta^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g[\; \left( \frac{1}{\left( \varepsilon p+p+1\right) \left( \varepsilon p+p+2\right) }\right) ^{\frac{1}{p}}\left( \frac{1}{6}\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\frac{1}{3}\left\vert g^{{\prime \prime }}\left( \mathfrak{e}\right) \right\vert ^{q}\right) ^{\frac{1}{q}} \notag \\ & +\; \left( \frac{1}{\left( \varepsilon +1\right) p+2}\right) ^{\frac{1}{p}}\left( \frac{1}{3}\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\frac{1}{6}\left\vert g^{{\prime \prime }}\left( \mathfrak{e}\right) \right\vert ^{q}\right) ^{\frac{1}{q}}g], \notag \end{align*}$

satisfies for $v\in \lbrack 0, 1],$ where $q^{-1}+p^{-1} = 1.$

Proof. From Lemma 2.1, by using the Hölder-İşcan integral inequality (see in []) and the preinvexity of $|g^{{\prime \prime}}|^q$ , we obtain

$\begin{align*} & \leq \ \frac{\eta^{\varepsilon +2}\left( \mathfrak{z}, \mathfrak{c}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g[\left( \int_{0}^{1}\left( 1-v\right) v^{\left( \varepsilon +1\right) p}dv\right) ^{\frac{1}{p}}\left( \int_{0}^{1}\left( 1-v\right) \left\vert g^{{\prime \prime }}\left( \mathfrak{c}+v\eta \left( \mathfrak{z}, \mathfrak{c}\right)\right) \right\vert ^{q}dv\right) ^{\frac{1}{q}}\\& +\left( \int_{0}^{1}v^{\left( \varepsilon +1\right) p+1}dv\right) ^{\frac{1}{p} }\left( \int_{0}^{1}v\left\vert g^{{\prime \prime }}\left( \mathfrak{c}+v\eta \left( \mathfrak{z}, \mathfrak{c}\right)\right) \right\vert ^{q}dv\right) ^{\frac{1}{q}}g] \\& +\ \frac{\eta^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g[\left( \int_{0}^{1}\left( 1-v\right) v^{\left( \varepsilon +1\right) p}dv\right) ^{\frac{1}{p}}\left( \int_{0}^{1}\left( 1-v\right) \left\vert g^{{\prime \prime }}\left( \mathfrak{e}+v\eta \left( \mathfrak{z}, \mathfrak{e}\right) \right) \right\vert ^{q}dv\right) ^{\frac{1}{q}} \end{align*}$

$\begin{align*} & +\left( \int_{0}^{1}v^{\left( \varepsilon +1\right) p+1}dv\right) ^{\frac{1}{p} }\left( \int_{0}^{1}v\left\vert g^{{\prime \prime }}\left( \mathfrak{e}+v\eta \left( \mathfrak{z}, \mathfrak{e}\right) \right) \right\vert ^{q}dv\right) ^{\frac{1}{q}}g] \\& \notag \leq \ \frac{\eta^{\alpha +2}\left( \mathfrak{z}, \mathfrak{c}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g[\left( \int_{0}^{1}\left( 1-v\right) v^{\left( \varepsilon +1\right) p}dv\right) ^{\frac{1}{p}}\left( \int_{0}^{1}\left( 1-v\right) \{v\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left( 1-v\right) \left\vert g^{{\prime \prime }}\left( \mathfrak{c}\right) \right\vert ^{q}\}dv\right) ^{\frac{1}{q}} \end{align*}$

$\begin{align*} & +\left( \int_{0}^{1}v^{\left( \varepsilon +1\right) p+1}dv\right) ^{\frac{1}{p} }\left( \int_{0}^{1}v\{v\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left( 1-v\right) \left\vert g^{{\prime \prime }}\left( \mathfrak{c}\right) \right\vert ^{q}\}dv\right) ^{\frac{1}{q}}g] \\& \notag +\ \frac{\eta^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g[\left( \int_{0}^{1}\left( 1-v\right) v^{\left( \varepsilon +1\right) p}dv\right) ^{\frac{1}{p}}\left( \int_{0}^{1}\left( 1-v\right) \{v\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left( 1-v\right) \left\vert g^{{\prime \prime }}\left( \mathfrak{e}\right) \right\vert ^{q}\}dv\right) ^{\frac{1}{q}} \\& \notag +\left( \int_{0}^{1}v^{\left( \varepsilon +1\right) p+1}dv\right) ^{\frac{1}{p} }\left( \int_{0}^{1}v\{v\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left( 1-v\right) \left\vert g^{{\prime \prime }}\left( \mathfrak{e}\right) \right\vert ^{q}\}dv\right) ^{\frac{1}{q}}g]. \end{align*}$

After simplification, we get (2.15). This completes the proof.

Corollary 2.9. Using the same assumptions in Theorem 2.5 with $|g^{{\prime \prime }}|\leq \mathcal M,$ we get

Theorem 2.6. Assume that all the assumptions as defined in Lemma 2.1 and $|g^{{\prime \prime }}|^q,$ $q\geq1$ is preinvex function on $[\mathfrak{c}, \mathfrak{c}+\eta \left( \mathfrak{e}, \mathfrak{c}\right)],$ then for all $\varepsilon >0,$ the following inequality

$\begin{align} & g|\frac{\eta^{\varepsilon +1}\left( \mathfrak{z}, \mathfrak{c}\right) -\eta^{\varepsilon +1}\left(\mathfrak{e}, \mathfrak{z}\right)}{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g^{{\prime }}\left( \mathfrak{z}\right) -\ \frac{\eta^{\varepsilon }\left( \mathfrak{z}, \mathfrak{c} \right) +\eta^{\varepsilon }\left( \mathfrak{e}, \mathfrak{z}\right) }{\eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\left( \mathfrak{z}\right) \\ & +\frac{\Gamma \left( \varepsilon +1\right) }{ \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g\{J_{\mathfrak{[\mathfrak{c}+\eta \left( \mathfrak{z}, \mathfrak{c}\right)]}^{-}}^{\varepsilon }g\left( \mathfrak{c} \right) +J_{[\mathfrak{\mathfrak{e}+\eta \left( \mathfrak{z}, \mathfrak{e}\right)]}^{+}}^{\varepsilon }g\left( \mathfrak{e}\right) g\}g| \\ & \leq \ \frac{\eta^{\varepsilon +2}\left( \mathfrak{z}, \mathfrak{c}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g[\ \left( \frac{1}{\left( \varepsilon +2\right) \left( \varepsilon +3\right) }\right) ^{1-\frac{1}{q}}\left( \frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q} }{\left( \varepsilon +3\right) \left( \varepsilon +4\right) }+\frac{2\left\vert g^{{ \prime \prime }}\left( \mathfrak{c}\right) \right\vert ^{q}}{\left( \varepsilon +2\right) \left( \varepsilon +3\right) \left( \varepsilon +4\right) }\right) ^{\frac{1 }{q}} \end{align}$

(2.16)

$\begin{align} & +\left( \frac{1}{\varepsilon +3}\right) ^{1-\frac{1}{q}}\left( \frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}}{\varepsilon +4}+ \frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{c}\right) \right\vert ^{q}}{ \left( \varepsilon +3\right) \left( \varepsilon +4\right) }\right) ^{\frac{1}{q}}g] \\ & +\frac{\eta^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g[\ \left( \frac{1}{\left( \varepsilon +2\right) \left( \varepsilon +3\right) }\right) ^{1-\frac{1}{q}}\left( \frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q} }{\left( \varepsilon +3\right) \left( \varepsilon +4\right) }+\frac{2\left\vert g^{{ \prime \prime }}\left( \mathfrak{e}\right) \right\vert ^{q}}{\left( \varepsilon +2\right) \left( \varepsilon +3\right) \left( \varepsilon +4\right) }\right) ^{\frac{1 }{q}} \\ & +\left( \frac{1}{\varepsilon +3}\right) ^{1-\frac{1}{q}}\left( \frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}}{\varepsilon +4}+ \frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{e}\right) \right\vert ^{q} }{\left( \varepsilon +3\right) \left( \varepsilon +4\right) }\right) ^{\frac{1}{q}} g], \end{align}$

satisfies for $v\in \lbrack 0, 1],$ where $q^{-1}+p^{-1} = 1.$

Proof. From Lemma 2.1, improved power-mean integral inequality(see in []) and the preinvexity of $|g^{{\prime \prime}}|^q$ , we obtain

$\begin{align*} & +\frac{\eta^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g[\left( \int_{0}^{1}\left( 1-v\right) v^{\varepsilon +1}dv\right) ^{1- \frac{1}{q}}\left( \int_{0}^{1}\left( 1-v\right) v^{\varepsilon +1}\left\vert g^{{\prime \prime }}\left( \mathfrak{e}+v\eta \left( \mathfrak{z}, \mathfrak{e}\right) \right) \right\vert ^{q}dv\right) ^{\frac{1}{q}} \\& \notag +\ \left( \int_{0}^{1}v^{\varepsilon +2}dv\right) ^{1-\frac{1}{q}}\left( \int_{0}^{1}v^{\varepsilon +2}\left\vert g^{{\prime \prime }}\left( \mathfrak{e}+v\eta \left( \mathfrak{z}, \mathfrak{e}\right) \right) \right\vert ^{q}dv\right) ^{\frac{1}{q}}g] \\& \notag \leq \ \frac{\eta^{\varepsilon +2}\left( \mathfrak{z}, \mathfrak{c}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g[\left( \int_{0}^{1}\left( 1-v\right) v^{\varepsilon +1}dv\right) ^{1-\frac{1}{q}} \\& \times \left( \int_{0}^{1}\left( 1-v\right) v^{\varepsilon +1}\{v\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left( 1-v\right) \left\vert g^{{\prime \prime }}\left( \mathfrak{c}\right) \right\vert ^{q}\}dv\right) ^{\frac{1}{q}} \\& \notag +\ \left( \int_{0}^{1}v^{\varepsilon +2}dv\right) ^{1-\frac{1}{q}}\left( \int_{0}^{1}v^{\varepsilon +2}\{v\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left( 1-v\right) \left\vert g^{{\prime \prime }}\left( \mathfrak{c}\right) \right\vert ^{q}\}dv\right) ^{\frac{1}{q}}g] \\& \notag +\frac{\eta^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g[\left( \int_{0}^{1}\left( 1-v\right) v^{\varepsilon +1}dv\right) ^{1- \frac{1}{q}}\left( \int_{0}^{1}\left( 1-v\right) v^{\varepsilon +1}\{v\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left( 1-v\right) \left\vert g^{{\prime \prime }}\left( \mathfrak{e}\right) \right\vert ^{q}\}dv\right) ^{\frac{1}{q}} \\& \notag +\ \left( \int_{0}^{1}v^{\varepsilon +2}dv\right) ^{1-\frac{1}{q}}\left( \int_{0}^{1}v^{\varepsilon +2}\{v\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}+\left( 1-v\right) \left\vert g^{{\prime \prime }}\left( \mathfrak{e}\right) \right\vert ^{q}\}dv\right) ^{\frac{1}{q}}g] \end{align*}$

$\begin{align*} & +\left( \frac{1}{ \varepsilon +3}\right) ^{1-\frac{1}{q}}\left( \frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}}{\varepsilon +4}+\frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{c}\right) \right\vert ^{q}}{\left( \varepsilon +3\right) \left( \varepsilon +4\right) }\right) ^{\frac{1}{q}}g] \\& \notag +\frac{\eta^{\varepsilon +2}\left( \mathfrak{e}, \mathfrak{z}\right) }{\left( \varepsilon +1\right) \eta\left( \mathfrak{e}, \mathfrak{c}\right) }g[\ \left( \frac{1}{\left( \varepsilon +2\right) \left( \varepsilon +3\right) }\right) ^{1-\frac{1}{q}}\left( \frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}}{\left( \varepsilon +3\right) \left( \varepsilon +4\right) }+\frac{2\left\vert g^{{\prime \prime }}\left( \mathfrak{e}\right) \right\vert ^{q}}{\left( \varepsilon +2\right) \left( \varepsilon +3\right) \left( \varepsilon +4\right) }\right) ^{\frac{1}{q}} \\& \notag +\left( \frac{1}{\varepsilon +3}\right) ^{1-\frac{1}{q}}\left( \frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{z}\right) \right\vert ^{q}}{\varepsilon +4}+\frac{\left\vert g^{{\prime \prime }}\left( \mathfrak{e}\right) \right\vert ^{q}}{\left( \varepsilon +3\right) \left( \varepsilon +4\right) } \right) ^{\frac{1}{q}}g]. \end{align*}$

This completes the proof.

Corollary 2.10. Using the same assumption of Theorem 2.6 with $|g^{{\prime \prime }}|\leq \mathcal M,$ we get

3. Modified bessel function

We recall the first kind modified Bessel function $\Im_m,$ which has the series representation (see [42], p.77)

$\begin{equation*} \Im_m(\zeta) = \Sigma_{n\geq 0}\frac{\left( \frac{\zeta }{2}\right) ^{m+2n}}{n!\Gamma \left( m+n+1\right) }. \end{equation*}$

where $\zeta \in \Re$ and $m>-1,$ while the second kind modified Bessel function $g_m$ (see [42], p.78) is usually defined as

$\begin{equation*} g_{m}\left( \zeta \right) = \frac{\pi }{2}\ \frac{\Im_{-m}\left( \zeta \right) -\Im_{m}\left( \zeta \right) }{\sin m\pi }. \end{equation*}$

Consider the function $\Omega_m\left( \zeta \right): \Re \rightarrow [1, \infty)$ defined by

$\begin{equation*} \Omega_m\left( \zeta \right) = 2^{m}\Gamma \left( m+1\right) \zeta ^{-m}\hbar_{m}\left( \zeta \right) , \end{equation*}$

where $\Gamma$ is the gamma function.

The first order derivative formula of $\Omega_m\left( \zeta \right)$ is given by [42]:

$\begin{equation} \Omega^{{\prime }}_m\left( \zeta \right) = \frac{\zeta }{2\left( m+1\right) }\Omega_{m+1}\left( \zeta \right) \end{equation}$

(3.1)

and the second derivative can be easily calculated from (3.1) as

$\begin{equation} \Omega^{{\prime \prime }}_m\left( \zeta \right) = \frac{\zeta ^{2}}{4\left( m+1\right) \left( m+2\right) }\ \Omega_{m+2}\left( \zeta \right)\ +\ \ \frac{1}{2\left( m+1\right) }\Omega_{m+1}\left( \zeta \right) . \end{equation}$

(3.2)

and the third derivative can be easily calculated from (3.2) as

$\begin{equation} \Omega _{m}^{{\prime \prime \prime }}\left( \zeta \right) = \frac{\zeta ^{3} }{4\left( m+1\right) \left( m+2\right) \left( m+3\right) }\Omega _{m+3}\left( \zeta \right) +\frac{3\zeta }{4\left( m+1\right) \left( m+2\right) }\Omega _{m+2}\left( \zeta \right) . \end{equation}$

(3.3)

Proposition 3.1. Suppose that $m>-1$ and $0<\mathfrak{c}<\mathfrak{e}.$ Then we get the inequality

$\begin{align*} & g|\frac{\Omega _{m}\left( \mathfrak{e}\right) -\Omega _{m}\left( \mathfrak{c}\right) }{ \mathfrak{e}-\mathfrak{c}}-\frac{\mathfrak{z}}{2\left( m+1\right) }\Omega _{m+1}\left( \mathfrak{z}\right) +\left( \mathfrak{z}- \frac{\mathfrak{c}+\mathfrak{e}}{2}\right) \\&\times g\{\frac{\mathfrak{z}^{2}}{4\left( m+1\right) \left( m+2\right) } \Omega _{m+2}\left( \mathfrak{z}\right) +\frac{1}{2\left( m+1\right) }\Omega _{m+1}\left( \mathfrak{z}\right) g\}g| \\& \leq \frac{\left( \mathfrak{z}-\mathfrak{c}\right) ^{3}}{2\left( \mathfrak{e}-\mathfrak{c}\right) }g[\frac{1}{ \left( 2p+1\right) p}+\frac{1}{2q}g\{\left( \frac{\mathfrak{z}^{3}}{8\left( m+1\right) \left( m+2\right) \left( m+3\right) }\Omega _{m+3}\left( \mathfrak{z}\right) +\frac{3\mathfrak{z}}{ 4\left( m+1\right) \left( m+2\right) }\Omega _{m+2}\left( \mathfrak{z}\right) \right) ^{q} \\& +\left( \frac{\mathfrak{c}^{3}}{8\left( m+1\right) \left( m+2\right) \left( m+3\right) }\Omega _{m+3}\left( \mathfrak{z}\right) +\frac{3\mathfrak{c}}{4\left( m+1\right) \left( m+2\right) }\Omega _{m+2}\left( \mathfrak{c}\right) \right) ^{q}g\}g] \\& +\frac{\left( \mathfrak{e}-\mathfrak{z}\right) ^{3}}{2\left( \mathfrak{e}-\mathfrak{c}\right) }g[\frac{1}{ \left( 2p+1\right) p}+\frac{1}{2q}g\{\left( \frac{\mathfrak{z}^{3}}{8\left( m+1\right) \left( m+2\right) \left( m+3\right) }\Omega _{m+3}\left( \mathfrak{z}\right) +\frac{3\mathfrak{z}}{ 4\left( m+1\right) \left( m+2\right) }\Omega _{m+2}\left( \mathfrak{z}\right) \right) ^{q} \\& +\left( \frac{\mathfrak{e}^{3}}{8\left( m+1\right) \left( m+2\right) \left( m+3\right) }\Omega _{m+3}\left( \mathfrak{z}\right) +\frac{3\mathfrak{e}}{4\left( m+1\right) \left( m+2\right) }\Omega _{m+2}\left( \mathfrak{e}\right) \right) ^{q}g\}g]. \end{align*}$

Proof. The assertion follows immediately from Corollary 2.7 using $g\left(\zeta\right) = \Omega^{\prime} _{m}(\zeta),$ $\zeta>0$ and the identities (3.2) and (3.3).

4. Conclusions

In this paper, we have defined an idea of fractional integral inequalities whose second derivatives are preinvex functions. We also investigated and proved a new lemma for the second derivatives of Riemann-Liouville fractional integral operator. Some new special cases are discovered in the form of corollaries. We hope that the strategies of this paper will motivate the researchers working in functional analysis, information theory and statistical theory. It is quite open to think about Ostrowski variants for generalized integral operators having Atangana-Baleanu operator etc. by applying generalized preinvexity. The results, which we have presented in this article, will potentially motivate researchers to study analogous and more general integral inequalities for various other kinds of fractional integral operators.

Conflict of interest

All authors have no conflict of interest.

References

[1]	T. Kwak, Y. Kim, Semi-supervised land cover classification of remote sensing imagery using CycleGAN and EfficientNet, KSCE J. Civ. Eng., 27 (2023), 1760–1773. https://doi.org/10.1007/s12205-023-2285-0 doi: 10.1007/s12205-023-2285-0
[2]	T. He, S. Wang, Multi-spectral remote sensing land-cover classification based on deep learning methods, J. Supercomput., 77 (2021), 2829–2843. https://doi.org/10.1007/s11227-020-03377-w doi: 10.1007/s11227-020-03377-w
[3]	L. Wang, J. Wang, Z. Liu, J. Zhu, F. Qin, Evaluation of a deep-learning model for multispectral remote sensing of land use and crop classification, Crop. J., 10 (2022), 1435–1451. https://doi.org/10.1016/j.cj.2022.01.009 doi: 10.1016/j.cj.2022.01.009
[4]	A. Tzepkenlis, K. Marthoglou, N. Grammalidis, Efficient deep semantic segmentation for land cover classification using sentinel imagery, Remote Sens., 15 (2023), 2027. https://doi.org/10.3390/rs15082027 doi: 10.3390/rs15082027
[5]	Y. Li, Y. Zhou, Y. Zhang, L. Zhong, J. Wang, J. Chen, DKDFN: Domain knowledge-guided deep collaborative fusion network for multimodal unitemporal remote sensing land cover classification, ISPRS J. Photogramm. Remote Sens., 186 (2022), 170–189. https://doi.org/10.1016/j.isprsjprs.2022.02.013 doi: 10.1016/j.isprsjprs.2022.02.013
[6]	L. Bergamasco, F. Bovolo, L. Bruzzone, A dual-branch deep learning architecture for multisensor and multitemporal remote sensing semantic segmentation, IEEE J. STARS, 16 (2023), 2147–2162. https://doi.org/10.1109/JSTARS.2023.3243396 doi: 10.1109/JSTARS.2023.3243396
[7]	X. Yuan, Z. Chen, N. Chen, J. Gong, Land cover classification based on the PSPNet and superpixel segmentation methods with high spatial resolution multispectral remote sensing imagery, J. Appl. Remote Sens., 15 (2021), 034511. https://doi.org/10.1117/1.JRS.15.034511 doi: 10.1117/1.JRS.15.034511
[8]	J. Yan, J. Liu, L. Wang, D. Liang, Q. Cao, W. Zhang, et al., Land-cover classification with time-series remote sensing images by complete extraction of multiscale timing dependence, IEEE J. STARS, 15 (2022), 1953–1967. https://doi.org/10.1109/JSTARS.2022.3150430
[9]	J. Kim, Y. Song, W. K. Lee, Accuracy analysis of multi-series phenological landcover classification using U-Net-based deep learning model–Focusing on the Seoul, Republic of Korea–, Korean J. Remote Sens., 37 (2021), 409–418. https://doi.org/10.7780/kjrs.2020.37.3.4
[10]	V. Yaloveha, A. Podorozhniak, H. Kuchuk, Convolutional neural network hyperparameter optimization applied to land cover classification, Radioelectron. Comput. Syst., 2022,115–128. https://doi.org/10.32620/reks.2022.1.09 doi: 10.32620/reks.2022.1.09
[11]	A. Temenos, N. Temenos, M. Kaselimi, A. Doulamis, N. Doulamis, Interpretable deep learning framework for land use and land cover classification in remote sensing using SHAP, IEEE Geosci. Remote Sens. Lett., 20 (2023), 8500105. https://doi.org/10.1109/LGRS.2023.3251652 doi: 10.1109/LGRS.2023.3251652
[12]	X. Cheng, X. He, M. Qiao, P. Li, S. Hu, P. Chang, et al., Enhanced contextual representation with deep neural networks for land cover classification based on remote sensing images, Int. J. Appl. Earth Obse., 107 (2022), 102706. https://doi.org/10.1016/j.jag.2022.102706
[13]	B. Ekim, E. Sertel, Deep neural network ensembles for remote sensing land cover and land use classification, Int. J. Digit. Earth, 14 (2021), 1868–1881. https://doi.org/10.1080/17538947.2021.1980125 doi: 10.1080/17538947.2021.1980125
[14]	V. N. Vinaykumar, J. A. Babu, J. Frnda, Optimal guidance whale optimization algorithm and hybrid deep learning networks for land use land cover classification, EURASIP J. Adv. Signal Process., 2023 (2023), 13. https://doi.org/10.1186/s13634-023-00980-w doi: 10.1186/s13634-023-00980-w
[15]	M. Luo, S. Ji, Cross-spatiotemporal land-cover classification from VHR remote sensing images with deep learning based domain adaptation, ISPRS J. Photogramm. Remote Sens., 191 (2022), 105–128. https://doi.org/10.1016/j.isprsjprs.2022.07.011 doi: 10.1016/j.isprsjprs.2022.07.011
[16]	W. Zhou, C. Persello, A. Stein, Building usage classification using a transformer-based multimodal deep learning method, 2023 Joint Urban Remote Sensing Event (JURSE), 2023. https://doi.org/10.1109/JURSE57346.2023.10144168 doi: 10.1109/JURSE57346.2023.10144168
[17]	G. P. Joshi, F. Alenezi, G. Thirumoorthy, A. K. Dutta, J. You, Ensemble of deep learning-based multimodal remote sensing image classification model on unmanned aerial vehicle networks, Mathematics, 9 (2021), 2984. https://doi.org/10.3390/math9222984 doi: 10.3390/math9222984
[18]	R. Li, S. Zheng, C. Duan, L. Wang, C. Zhang, Land cover classification from remote sensing images based on multi-scale fully convolutional network, Geo-Spat. Inf Sci., 25 (2022), 278–294. https://doi.org/10.1080/10095020.2021.2017237 doi: 10.1080/10095020.2021.2017237
[19]	A. Tariq, F. Mumtaz, Modeling spatio-temporal assessment of land use land cover of Lahore and its impact on land surface temperature using multi-spectral remote sensing data, Environ. Sci. Pollut. Res., 30 (2022), 23908–23924. https://doi.org/10.1007/s11356-022-23928-3 doi: 10.1007/s11356-022-23928-3
[20]	A. Tariq, J. Yan, F. Mumtaz, Land change modeler and CA-Markov chain analysis for land use land cover change using satellite data of Peshawar, Pakistan, Phys. Chem. Earth Parts A/B/C, 128 (2022), 103286.
[21]	A. Tariq, F. Mumtaz, M. Majeed, X. Zeng, Spatio-temporal assessment of land use land cover based on trajectories and cellular automata Markov modelling and its impact on land surface temperature of Lahore district Pakistan, Environ. Monit. Assess., 195 (2023), 114. https://doi.org/10.1007/s10661-022-10738-w doi: 10.1007/s10661-022-10738-w
[22]	A. Tariq, H. Shu, CA-Markov chain analysis of seasonal land surface temperature and land use land cover change using optical multi-temporal satellite data of Faisalabad, Pakistan, Remote Sens., 12 (2020), 3402. https://doi.org/10.3390/rs12203402
[23]	T. Chen, H. Qin, X. Li, W. Wan, W. Yan, A Non-Intrusive Load Monitoring Method Based on Feature Fusion and SE-ResNet, Electronics, 12 (2023), 1909. https://doi.org/10.3390/electronics12081909 doi: 10.3390/electronics12081909
[24]	H. Long, Y. He, Y. Xu, C. You, D. Zeng, H. Lu, Optimal allocation research of distribution network with DGs and SCs by improved sand cat swarm optimization algorithm, IAENG Int. J. Comput. Sci., 2023.
[25]	A. Al Hamoud, A. Hoenig, K. Roy, Sentence subjectivity analysis of a political and ideological debate dataset using LSTM and BiLSTM with attention and GRU models, J. King Saud Univ.-Com., 34 (2022), 7974–7987. https://doi.org/10.1016/j.jksuci.2022.07.014 doi: 10.1016/j.jksuci.2022.07.014
[26]	L. Kong, H. Liang, G. Liu, S. Liu, Research on wind turbine fault detection based on the fusion of ASL-CatBoost and TtRSA, Sensors, 23 (2023), 6741. https://doi.org/10.3390/s23156741 doi: 10.3390/s23156741
[27]	S. Rajalakshmi, S. Nalini, A. Alkhayyat, R. Q. Malik, Hyperspectral remote sensing image classification using improved metaheuristic with deep learning, Comput. Syst. Sci. Eng., 46 (2023), 1673–1688. https://doi.org/10.32604/csse.2023.034414 doi: 10.32604/csse.2023.034414

This article has been cited by:

1.	Mahyar Abbasian, Elahe Khatibi, Iman Azimi, David Oniani, Zahra Shakeri Hossein Abad, Alexander Thieme, Ram Sriram, Zhongqi Yang, Yanshan Wang, Bryant Lin, Olivier Gevaert, Li-Jia Li, Ramesh Jain, Amir M. Rahmani, Foundation metrics for evaluating effectiveness of healthcare conversations powered by generative AI, 2024, 7, 2398-6352, 10.1038/s41746-024-01074-z
2.	Muhammad Asif, Monica Palmirani, 2024, Chapter 4, 978-3-031-68210-0, 34, 10.1007/978-3-031-68211-7_4
3.	Andrea Zielinski, Simon Hirzel, Sonja Arnold-Keifer, 2024, Enhancing Digital Libraries with Automated Definition Generation, 9798400710933, 1, 10.1145/3677389.3702536

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)