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     {\huge СИБИРСКИЕ \ ЭЛЕКТРОННЫЕ} \\[2mm]  %Headings in Russian
     {\huge МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ} \\[2mm]
     {\large Siberian Electronic Mathematical Reports} \\[1mm]
     {\LARGE\tt{http://semr.math.nsc.ru}}\\[0.5mm]
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     {\flushleft\it Том ??, стр. ??--?? (20??) \hspace{65mm}{\rm\small УДК 517.218.244,517.518.86}} %Volume, pages in Russian,
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     {\rm\small DOI~10.33048/semi.2019.16.xxx}\hphantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}{\rm\small MSC 26D10,26A51,26A33}
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\begin{document}
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\title[Weighted fractional variants of Hadamard  inequalities ]{New generalized weighted fractional variants of Hermite-Hadamard
	inequalities with applications}
\author{{J. E. N\'apoles, B. Bayraktar, S. I. Butt}}%
\address{Juan E. N\'apoles Vald\'es
\newline\hphantom{iii} UNNE FaCENA  and   UTN-FRRE,
\newline\hphantom{iii} Ave. Libertad 5450 and French 414 Resistencia,
\newline\hphantom{iii} Corrientes 3400 and Chaco 3500, Argentina}%
%\newline\hphantom{iii},
\email{jnapoles@exa.unne.edu.ar} 
%\newline\hphantom{iii} French 414, Resistencia,
%\newline\hphantom{iii} Chaco 3500, Argentina}%
\email{jnapoles@frre.utn.edu.ar}
%%%%%%%%%%%%%%%
\address{ Bahtiyar Bayraktar
	\newline\hphantom{iii} Bursa Uludag University Faculty of Education,	
	\newline\hphantom{iii} Gorukle Campus,
	\newline\hphantom{iii} Bursa, Turkey}%
\email{bbayraktar@uludag.edu.tr}
%%%%%%%%%%%%%%%
\address{ Saad Ihsan Butt
	\newline\hphantom{iii} COMSATS University, 	
	\newline\hphantom{iii}  Lahore Campus,
	\newline\hphantom{iii} Islamabad, Pakistan}%
\email{saadihsanbutt@gmail.com}
%%%%%%%%%%%%%%%


\thanks{\sc N\'apoles J. E., Bayraktar B.,  Butt S. I.,
New generalized weighted fractional variants of Hermite-Hadamard
inequalities with applications}
\thanks{\copyright \ 2023 N\'apoles J. E., Bayraktar B.,  Butt S. I.}
%\thanks{\rm The work is supported by RFFI (grant 03-01-11111)}
\thanks{\it Received  Febriary, ??, 2023, published ??, ??,  20??.}%


\semrtop \vspace{1cm}
\maketitle {\small
\begin{quote}
\noindent{\sc Abstract.} Integral  inequalities play a fundamental \, role in various mathematical fields, which have led to new methods and theoretical developments, both pure and applied. The need for searching for precise inequalities, in which the notion of convexity plays an important role, has a high impact on approximation theory calculus. In this paper, we first obtained a new version of the weighted fractional identity that led us to obtain new variants of weighted Trapezoid, Simpson and Midpoint inequalities. We then presented several refinements of it in the framework of weighted integrals for the modified second type $(h,m)$--convex functions. \medskip

\noindent{\bf Keywords:} Hermite--Hadamard integral inequality, H\"{o}lder's inequality,  $(h,m)-$convex modified functions, weighted fractional integrals operators.
 \end{quote}
}


\section{Introduction}

\label{sec1} Let $\varsigma _{1},\varsigma _{2}\in \mathbb{R}$ with $%
\varsigma _{1}<\varsigma _{2}$. A function $\psi :I=[\varsigma
_{1},\varsigma _{2}]\rightarrow \mathbb{R}$ is said to be convex if 
\begin{equation*}
	\psi \left( \zeta \theta +(1-\zeta )\vartheta \right) \leq \zeta {\psi }{%
		(\theta )}+(1-\zeta )\psi (\vartheta )
\end{equation*}%
holds for all $\zeta \in \lbrack 0,1]$ and $\theta ,\vartheta \in I$. If the
function is concave, then the inequalities will be in reverse order.

In the last few decades, convex functions have become the object of
extensive research by scientists from different countries. Today, this
concept in the literature has a lot of generalizations and extensions. In 
\cite{NRS}, the authors presented a fairly wide range of different classes
 convexity.

Of the theory of inequalities, the most important one is the
Hermite--Hadamard inequality: 
\begin{equation}
	\psi \left( \frac{\varsigma _{1}+\varsigma _{2}}{2}\right) \leq \frac{1}{%
		\varsigma _{2}-\varsigma _{1}}\int_{\varsigma _{1}}^{\varsigma _{2}}\psi
	(x)dx\leq \frac{\psi (\varsigma _{1})+\psi (\varsigma _{2})}{2},
	\label{Hadamard}
\end{equation}

holds for convex functions on an interval.

The ground breaking inequality helps to predict estimations of the mean
values of a convex function and it is pertinent mention that it also
provides a refine estimation of Jensen gap.

Interesting results for integral inequalities by using fractional operators
can be found, for example, \cite{Bayrak0, Bayrak1, BN, BN-Udm, BKN, But0, But1,  Drag,  DA, But2, Sitnik, Wang} and the references therein. 

For example, the authors in \cite{Bayrak0, Bayrak1, BN, BN-Udm} obtained new results in terms of fractional integral operators of the Riemann--Liouville for   $(s,m)$,  $(\alpha,m)$ and $(h,m)-$convex functions. In \cite{BKN}, Bermudo et al. presented some variants of the Hermite--Hadamard
inequality general convex functions in the context of $q-$calculus. 
In \cite{But1}, Butt and Pe\v{c}ari\'{c} obtained some generalized Hermite--Hadamard's integral inequalities for the monotone functions of the form $f/h$. 
%The authors in \cite{But2, But3} obtained new and general integral inequalities for the convex functions by using Atangana--Baleanu integral operators. 
In \cite{But0}, the authors presented an article in which they obtained new Hermite--Hadamard inequalities of the Jensen--Mercer type for a harmonically convex function through fractional integrals.  In \cite{GNG}, Guzm\'{a}n et al. studied the most important properties of the generalized integral operators and obtained different inequalities of the Hermite--Hadamard type for the weighted symmetric convex and $h-$ symmetrized convex functions. 
%In \cite{Kunt}, Kunt et al. proved new fractional inequalities of the trapezoid and midpoint type for differentiable convex functions by using only the left Riemann--Liouville fractional integrals.
%For the $(h,m)-$convex functions, Matloka in the study \cite{Matloka1}, established several Hadamard type inequalities. 
In \cite{Bayrak2}, the  authors have defined new integral operators to obtain new Hermite--Hadamard inequalities for $h-$convex functions.

Dragomir and Agarwal in \cite{DA}, by using the convexity of the first
derivatives of the function, obtained the following estimate for the
trapezoid inequality:

\begin{theorem}
	Let $\varsigma _{1},\varsigma _{2}\in\mathbb{R}$ with $\varsigma
	_{1}<\varsigma _{2}, \psi:I=[\varsigma _{1},\varsigma _{2}]\rightarrow 
	\mathbb{R}$ and $\psi \in C^{1}(\varsigma _{1},\varsigma _{2})$. If \ $|\psi
	^{\prime }|$ is convex on $[\varsigma _{1},\varsigma _{2}]$, then the
	inequality holds: 
	\begin{equation}
		\left\vert \frac{\psi \left( \varsigma _{1}\right) +\psi (\varsigma _{2})}{2}%
		-\frac{1}{\varsigma _{2}-\varsigma _{1}}\int_{\varsigma _{1}}^{\varsigma
			_{2}}\psi (\zeta )d\zeta \right\vert \leq \frac{\varsigma _{2}-\varsigma _{1}%
		}{8}\left( \left\vert \psi ^{\prime }(\varsigma_{1})\right\vert +\left\vert
		\psi ^{\prime }(\varsigma _{2})\right\vert \right).  \label{Dr1}
	\end{equation}
\end{theorem}

%The following definitions are successive extensions of the concept of a convex function and will be the basis for our article \cite{BN}:
The definitions of sequentially expanding the concept of convex functions,
which form the basis of our article, are given below \cite{BN, BN-Udm}.

\begin{definition}
	\label{d:Hmc1} Let $h:[0,1]\rightarrow (0,1]$, $\psi :[0,+\infty
	)\rightarrow \lbrack 0,+\infty ).$ If inequality 
	\begin{equation*}
		\psi \left( \zeta \theta +m(1-\zeta )\vartheta \right) \leq h^{s}(\zeta
		)\psi (\theta )+m(1-h^{s}(\zeta ))\psi (\vartheta )
	\end{equation*}%
	is fulfilled for all $\zeta \in \lbrack 0,1]$ and $\theta ,\vartheta \in
	\lbrack 0,+\infty )$, then the function $\psi $ is called the $(h,m)-$convex
	modified of the first type on $[0,+\infty ).$ Here $m\in \lbrack 0,1]$ and $%
	s\in \lbrack -1,1].$
\end{definition}

\begin{definition}
	\label{d:Hmc2} et $h:[0,1]\rightarrow (0,1]$, $\psi :[0,+\infty )\rightarrow
	\lbrack 0,+\infty ).$ If inequality 
	\begin{equation*}
		\psi \left( \zeta \theta +m(1-\zeta )\vartheta \right) \leq h^{s}(\zeta
		)\psi (\theta )+m(1-h(\zeta ))^{s}\psi (\vartheta )
	\end{equation*}%
	is fulfilled for all $\zeta \in \lbrack 0,1]$ and $\theta ,\vartheta \in
	\lbrack 0,+\infty )$, then the function $\psi $ is called the $(h,m)-$convex
	modified of the second type on $[0,+\infty ).$ Here $m\in \lbrack 0,1]$ and $s\in \lbrack -1,1].$
\end{definition}

We denote by $N_{h,m}^{s,1}[0,+\infty )$ and $N_{h,m}^{s,2}[0,+\infty )$ the
class of all $(h,m)-$convex modified functions of the first and second type
respectively on the $[0,+\infty )$.

\begin{remark}
	From Definitions \ref{d:Hmc1} and \ref{d:Hmc2}, we have the following
	classifications of functions on $[0,+\infty )$.
	
	\begin{itemize}
		\item[(i01)] If $h(\zeta )=\zeta $, then $\psi $ recaptures the class of $m$--convex functions, see \cite{Toa}.
		
		\item[(i02)] If $h(\zeta )=\zeta ^{\alpha }$ with $a\in (0,1]$, then $\psi $
		recaptures the class of $(\alpha ,s,m)$--convex functions, see \cite{XGQ2020}.
		
		\item[(i03)] If $s=1$, then $\psi $ recaptures the class of $(h,m)$--convex
		function.
		
		\item[(i04)] If $h(\zeta )=\zeta ,s\in (0,1]$ and $m=1$, then $\psi $
		recaptures the class of $s$--convex functions, see \cite{HM}.
		
		\item[(i05)] If $h(\zeta )=\zeta ,m=1$ and $s\in \lbrack -1,1]$, then $\psi $
		recaptures the extended class of $s$--convex functions, see \cite{XQ}.
		
		\item[(i06)] If $h(\zeta )=\zeta $, $m=s=1$, then $\psi $ is a convex
		function.
	\end{itemize}
\end{remark}

To begin with, we will give a definition of Riemann--Liouville fractional
integrals, and then, for a better understanding of the topic, we will give a
definition of weighted integral operators(with $0\leq \varsigma _{1}<\zeta
<\varsigma _{2}\leq \infty $ and real number $\kappa \geq 0$), which will be
the basis for our article.

\begin{definition}
	\label{d:rl} Let $\psi \in L_{1}[\varsigma _{1},\varsigma _{2}]$, $\alpha
	\in \mathbb{C}$ and $\Re (\alpha )>0$. Right and left fractional integrals
	defined below%
	\begin{equation*}
		{}^{\alpha }I_{\varsigma _{1}^{{+}}}\psi (x)=\frac{1}{\Gamma (\alpha )}%
		\int_{\varsigma _{1}}^{x}{(x-\zeta )^{{\alpha }-1}}\psi (\zeta )d\zeta
		,\quad x>\varsigma _{1},
	\end{equation*}
	\begin{equation*}
		{}^{\alpha }I_{\varsigma _{2}^{{-}}}\psi (x)=\frac{1}{\Gamma (\alpha )}%
		\int_{x}^{\varsigma _{2}}{(\zeta -x)^{{\alpha }-1}}\psi (\zeta )d\zeta
		,\quad x<\varsigma _{2},
	\end{equation*}%
	are called right and left Riemann--Liouville fractional integrals,
	respectively. \ Here $\Gamma (\alpha )$ is the gamma function.
\end{definition}

\begin{definition}
	\label{d:wi} Let $\psi \in L_{1}(\varsigma _{1},\varsigma _{2})$, $%
	w:[0,\infty )\rightarrow \lbrack 0,\infty )$ \ and $w\in C[0,\infty )$ with
	derivatives piecewise continuous on $[0,\infty )$ up to the second order
	inclusive. Integral operators defined below 
	\begin{equation*}
		J_{\varsigma _{1}^{+}}^{w}\psi (x)=\int_{\varsigma _{1}}^{x}{w^{\prime
			}\left( \frac{x-\zeta }{\frac{\varsigma _{2}-\varsigma _{1}}{1+\kappa }}%
			\right) \psi (\zeta )d\zeta },\quad x>\varsigma _{1},
	\end{equation*}
	\begin{equation*}
		J_{\varsigma _{2}^{-}}^{w}\psi (x)=\int_{x}^{\varsigma _{2}}{w^{\prime
			}\left( \frac{\zeta -x}{\frac{\varsigma _{2}-\varsigma _{1}}{1+\kappa }}%
			\right) \psi (\zeta )d\zeta },\quad x<\varsigma _{2}.
	\end{equation*}%
	called weighted fractional integrals (right and left, respectively).
\end{definition}

\begin{remark}
	If we take $w^{\prime }(\zeta )=\frac{(\varsigma _{2}-\varsigma _{1})\zeta
		^{\alpha -1}}{\Gamma (\alpha )}$ and $\kappa =0$, we get the
	Riemann--Liouville fractional integral, but if we take $w^{\prime }(\zeta
	)\equiv 1$ it turns into a classical Riemann integral.
\end{remark}
In the present article, we presented some new unified variants of weighted
Hadamard inequality  \eqref{Hadamard} by employing generalized convexity for
generalized fractional integral operators given in Definition \ref{d:wi}.
Several special cases are discussed as well.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Main results}

\label{sec2} The results are obtained with the help of the following lemma.

\begin{lemma}
	\label{l:1} Let $\psi :[\varsigma _{1},\varsigma _{2}]\subset \mathbb{R}%
	\rightarrow \mathbb{R}$ with $\psi \in C^{1}(\varsigma _{1},\varsigma _{2})$
	and let $w:[0,\infty )\rightarrow \lbrack 0,\infty ),$ $w\in C[0,\infty )$
	with the first and second order derivatives piecewise continuous on $%
	[0,\infty )$. If $\psi ^{\prime }\in L_{1}(\varsigma _{1},\varsigma _{2})$,
	then equality holds: 
	\begin{align}
		& \left[ \psi (\varsigma _{1})+\psi (\varsigma _{2})\right]w(0) -\left[
		\psi \left( \frac{\kappa \varsigma _{1}+\varsigma _{2}}{1+\kappa }\right)
		+\psi \left( \frac{\varsigma _{1}+\kappa \varsigma _{2}}{1+\kappa }\right) %
		\right]w(1)  \label{e:3} \\
		& \quad\quad\quad\quad\quad +\frac{1+\kappa }{\varsigma _{2}-\varsigma _{1}}\left[ {J_{\left( {%
					\frac{\varsigma _{1}+\kappa \varsigma _{2}}{1+\kappa }}\right) ^{+}}^{w}\psi
			(\varsigma _{2})}+{J_{\left( {\frac{\kappa \varsigma _{1}+\varsigma _{2}}{%
						1+\kappa }}\right) ^{-}}^{w}\psi (\varsigma _{1})}\right]  \notag \\
		& =\frac{\varsigma _{2}-\varsigma _{1}}{1+\kappa }\left[ \int_{0}^{1}{%
			w(\zeta )\psi ^{\prime }\left( \frac{\zeta }{1+\kappa }\varsigma _{1}+\frac{%
				1+\kappa -\zeta }{1+\kappa }\varsigma _{2}\right) }d\zeta \right.  \notag \\
		& \qquad\qquad\quad\quad\quad\quad \left. -\int_{0}^{1}{w(\zeta )\psi ^{\prime }\left( \frac{1+\kappa
				-\zeta }{1+\kappa }\varsigma _{1}+\frac{\zeta }{1+\kappa }\varsigma
			_{2}\right) }d\zeta \right] .  \notag
	\end{align}
\end{lemma}

\begin{proof}
	For simplicity, 
	\begin{align*}
		&\int_{0}^{1}{\ w(\zeta )\psi ^{\prime }\left( \frac{\zeta }{1+\kappa }%
			\varsigma _{1}+\frac{1+\kappa -\zeta }{1+\kappa }\varsigma _{2}\right) }d\zeta \\
		&\qquad\qquad-\int_{0}^{1}{\ w(\zeta )\psi ^{\prime }\left( \frac{1+\kappa -\zeta 
			}{1+\kappa }\varsigma _{1}+\frac{\zeta}{1+\kappa }\varsigma _{2}\right) }%
		d\zeta:={\ I}_{1}-{\ I}_{2}. 
	\end{align*}
	By integrating by parts in $I_{1}$, we have 
	\begin{align*}
		& {I}_{1}=\frac{1+\kappa }{\varsigma _{2}-\varsigma _{1}}\left[ w(0)\psi
		(\varsigma_{2})-w(1)\psi\left(\frac{\varsigma _{1}+\kappa \varsigma _{2}}{%
			1+\kappa }\right)\right] \\
		& \quad\quad\quad+\frac{(1+\kappa )}{(\varsigma _{2}-\varsigma _{1})}%
		\int_{0}^{1}w^{\prime}(\zeta )\psi \left( \frac{\zeta }{1+\kappa }\varsigma
		_{1}+\frac{1+\kappa -\zeta }{1+\kappa }\varsigma _{2}\right) d\zeta \\
		& =\frac{1+\kappa }{\varsigma _{2}-\varsigma _{1}}\left[ w(0)\psi (\varsigma
		_{2})-w(1)\psi\left(\frac{\varsigma _{1}+\kappa \varsigma _{2}}{1+\kappa }%
		\right)\right] \\
		& \quad\quad\qquad+\frac{(1+\kappa )^{2}}{(\varsigma _{2}-\varsigma _{1})^{2}}{%
			J_{\left( {\frac{\varsigma _{1}+\kappa \varsigma _{2}}{1+\kappa }}\right)
				^{+}}^{w}\psi (\varsigma _{2})},
	\end{align*}
	since 
	\begin{equation*}
		\int_{0}^{1}w^{\prime }(\zeta )\psi \left( \frac{\zeta }{1+\kappa }\varsigma
		_{1}+\frac{1+\kappa -\zeta }{1+\kappa }\varsigma _{2}\right) d\zeta =\frac{%
			1+\kappa }{\varsigma _{2}-\varsigma _{1}}\int_{\frac{\varsigma _{1}+\kappa
				\varsigma _{2}}{1+\kappa }}^{\varsigma _{2}}w^{\prime }\left[\frac{\varsigma
			_{2}-z}{\frac{\varsigma _{2}-\varsigma _{1}}{1+\kappa }}\right] \psi (z)dz.
	\end{equation*}
	
	Analogously for $I_{2}$, we have 
	\begin{equation*}
		{I}_{2}=\frac{1+\kappa }{\varsigma _{2}-\varsigma _{1}}\left[ w(1)\psi (%
		\frac{\kappa \varsigma_{1}+\varsigma _{2}}{1+\kappa })-w(0)\psi (\varsigma
		_{1})\right] -{\frac{(1+\kappa )^{2}}{(\varsigma _{2}-\varsigma _{1})^{2}}%
			J_{\left( {\frac{\kappa \varsigma _{1}+\varsigma _{2}}{1+\kappa }}\right)
				^{-}}^{w}\psi (\varsigma _{1})}.
	\end{equation*}
	Substituting in $I_{1}-I_{2}$, grouping appropriately and multiplying by $%
	\frac{\varsigma _{2}-\varsigma _{1}}{1+\kappa}$, we obtain the inequality %
	\eqref{e:3}.
\end{proof}

The following remarks have the mission of illustrating the scope and
strength of the previous result. The choice of special weight functions in
the Lemma \ref{l:1} leads us to many interesting results published in the
literature.

\begin{remark}
	\label{r:1} Special cases of the Lemma \ref{l:1}:	
	\begin{itemize}
		\item[(i01)] If we take $\kappa =0$ in \eqref{e:3}, then we get the
		following new general weighted identity.  
		\begin{align}
			& \left[ w(0)-w(1)\right] \left[ \psi (\varsigma _{2})+\psi (\varsigma _{1})\right] 
			+\frac{1}{\varsigma _{2}-\varsigma _{1}}\left( J_{\varsigma
				_{1}^{+}}^{w}\psi (\varsigma _{2})+J_{\varsigma _{2}^{-}}^{w}\psi (\varsigma
			_{1})\right) \label{e:41}\\
			& =(\varsigma _{2}-\varsigma _{1})\left[ \int_{0}^{1}{w(\zeta )\psi ^{\prime
				}\left( \zeta \varsigma _{1}+(1-\zeta )\varsigma _{2}\right) }d\zeta \right. \notag\\
			&\qquad\qquad\qquad\qquad\left. -\int_{0}^{1}{w(\zeta )\psi ^{\prime }\left( (1-\zeta )\varsigma _{1}+\zeta\varsigma _{2}\right) }d\zeta \right].  \notag
		\end{align}
		
		\item[(i02)] If we take $\kappa =0,w(\zeta )=1-\zeta ^{\alpha }$ in %
		\eqref{e:3}, substituting $x$ for $\varsigma _{1}$ in $I_{1}$ and $x$ for $%
		\varsigma _{2}$ in $I_{2}$, then we will have \cite[Lemma 2]{OAK}.
		
		\item[(i03)] If we take $\kappa =0, w(\zeta )=(1-\zeta )^{\alpha }-\zeta
		^{\alpha }$ and work only with $I_{1}$, then we get \cite[Lemma 2]{SSYB};
		furthermore, if we substitute $\varsigma _{2}$ for $r\in (\varsigma
		_{1},\varsigma _{2}]$, then we will have \cite[Lemma 3]{ODY}. 
		%\item[(iv)] If we take $\kappa =1, w(\zeta )=(1-\zeta )^{\alpha }$, then we obtain \cite[Lemma 1]{MB}.
		\end{itemize}
	\end{remark}
	\begin{remark}
		\label{r:1-1}Other special cases of the Lemma \ref{l:1}:	
	\begin{itemize}
		\item[(i04)] If we take $\kappa =1,w(\zeta )=\zeta $ in \eqref{e:3}, then we
		get 
		\begin{align}
			& \left[ w(0)-w(1)\right] \left[ \psi (\varsigma _{2})+\psi (\varsigma _{1})%
			\right]  \label{e:42} \\
			& \quad\quad\quad +\frac{1}{\varsigma _{2}-\varsigma _{1}}\left( J_{\left( \frac{%
					\varsigma _{1}+\varsigma _{2}}{2}\right) ^{+}}^{w}\psi (\varsigma
			_{2})+J_{\left( \frac{\varsigma _{1}+\varsigma _{2}}{2}\right) ^{-}}^{w}\psi
			(\varsigma _{1})\right)  \notag \\
			& =\frac{\varsigma _{2}-\varsigma _{1}}{4}\left[ \int_{0}^{1}\zeta \psi
			^{\prime }\left( \frac{\zeta }{2}\varsigma _{1}+\frac{2-\zeta }{2}\varsigma
			_{2}\right) d\zeta -\int_{0}^{1}\zeta \psi ^{\prime }\left( \frac{2-\zeta }{2%
			}\varsigma _{1}+\frac{\zeta }{2}\varsigma _{2}\right) d\zeta \right]. 
			\notag
		\end{align}
		
		\item[(i05)] If we take $\kappa =1,w(\zeta )=1-\zeta $, by replacing in %
		\eqref{e:3} $1-\zeta $ by $\zeta $, then we obtain 
		\begin{align}
			& \frac{\psi (\varsigma _{1})+\psi (\varsigma _{2})}{2}-\frac{1}{\varsigma
				_{2}-\varsigma _{1}}\int_{\varsigma _{1}}^{\varsigma _{2}}\psi (\zeta )d\zeta
			\label{e:5} \\
			& =\frac{\varsigma _{2}-\varsigma _{1}}{4}\left[ \int_{0}^{1}{\zeta \psi
				^{\prime }\left( \frac{1-\zeta }{2}\varsigma _{1}+\frac{1+\zeta }{2}%
				\varsigma _{2}\right) }d\zeta \right. \notag \\
			&\qquad\qquad\qquad\qquad\left. -\int_{0}^{1}{\zeta \psi ^{\prime }\left( 
				\frac{1+\zeta }{2}\varsigma _{1}+\frac{1-\zeta }{2}\varsigma _{2}\right) }%
			d\zeta \right] ,  \notag
		\end{align}%
		which is given as in \cite[Lemma 2.1]{ADK}.
	\end{itemize}
\end{remark}

Our first and main result was obtained from Lemma \ref{l:1} in the following
theorem.

\begin{theorem}
	\label{t:1} Let $\psi :\left[ \varsigma _{1},\frac{\varsigma _{2}}{m}\right]
	\rightarrow \mathbb{R}$ be a $L^{1}$ function. If $\left\vert \psi ^{\prime
	}\right\vert \in N_{h,m}^{s,2}\left[ \varsigma _{1},\frac{\varsigma _{2}}{m}%
	\right] $, we have the following inequality: 
	\begin{align}
		& \left\vert \mathbb{D}+{\frac{1+\kappa }{\varsigma _{2}-\varsigma _{1}}}%
		\left[ {J_{\left( {\frac{\varsigma _{1}+\kappa \varsigma _{2}}{1+\kappa }}%
				\right) ^{+}}^{w}\psi (\varsigma _{2})}+{J_{\left( {\frac{\kappa \varsigma
						_{1}+\varsigma _{2}}{1+\kappa }}\right) ^{-}}^{w}\psi (\varsigma _{1})}%
		\right] \right\vert   \label{e:6} \\
		& \leq \frac{\varsigma _{2}-\varsigma _{1}}{1+\kappa }\left[ \left\vert \psi
		^{\prime }(\varsigma _{1})\right\vert \mathbb{B}+m\left\vert \psi ^{\prime
		}\left( \frac{\varsigma _{2}}{m}\right) \right\vert \mathbb{C}\right] , 
		\notag
	\end{align}%
	where 
	\begin{align*}
		\mathbb{D}& =\left[ \psi (\varsigma _{1})+\psi (\varsigma _{2})\right]w(0)
		-\left[ \psi \left( \frac{\kappa \varsigma _{1}+\varsigma _{2}}{1+\kappa 
		}\right) +\psi \left( \frac{\varsigma _{1}+\kappa \varsigma _{2}}{1+\kappa }%
		\right) \right]w(1), \\
		\mathbb{B}& =\int_{0}^{1}\left\vert w(\zeta )\right\vert \left[ h^{s}\left( 
		\frac{\zeta }{1+\kappa }\right) +h^{s}\left( \frac{1+\kappa -\zeta }{%
			1+\kappa }\right) \right] d\zeta ,
	\end{align*}%
	\begin{equation*}
		\mathbb{C}=\int_{0}^{1}\left\vert w(\zeta )\right\vert \left[ \left(
		1-h\left( \frac{1+\kappa -\zeta }{1+\kappa }\right) \right) ^{s}+\left(
		1-h\left( \frac{\zeta }{1+\kappa }\right) \right) ^{s}\right] d\zeta .
	\end{equation*}
\end{theorem}

\begin{proof}
	From Lemma \ref{l:1},  we obtain 
	\begin{align}
		& \left\vert \int_{0}^{1}{w(\zeta )\psi ^{\prime }\left( \frac{\zeta }{%
				1+\kappa }\varsigma _{1}+\frac{1+\kappa -\zeta }{1+\kappa }\varsigma_{2}\right) }d\zeta \right. \label{e:60} \\
		&\left. \qquad\qquad-\int_{0}^{1}{w(\zeta )\psi ^{\prime }\left( \frac{1+\kappa -\zeta }{1+\kappa }\varsigma _{1}+\frac{\zeta }{1+\kappa }\varsigma
			_{2}\right) }d\zeta \right\vert  \notag \\
		& \leq \int_{0}^{1}\left\vert {w(\zeta )}\right\vert {\left\vert \psi
			^{\prime }\left( \frac{\zeta }{1+\kappa }\varsigma _{1}+\frac{1+\kappa
				-\zeta }{1+\kappa }\varsigma _{2}\right) \right\vert }d\zeta  \notag \\
		& \qquad\qquad +\int_{0}^{1}\left\vert {w(\zeta )}\right\vert {\left\vert \psi
			^{\prime }\left( \frac{1+\kappa -\zeta }{1+\kappa }\varsigma _{1}+\frac{	\zeta }{1+\kappa }\varsigma _{2}\right) \right\vert }d\zeta.  \notag
	\end{align}%
	Taking into account modified $(h,m)$--convexity of $\left\vert \psi ^{\prime
	}\right\vert $, we get 
	\begin{align}
		\left\vert \psi ^{\prime }\left( \frac{\zeta }{1+\kappa }\varsigma _{1}+%
		\frac{1+\kappa -\zeta }{1+\kappa }\varsigma _{2}\right) \right\vert \leq&
		\left\vert \psi ^{\prime }(\varsigma _{1})\right\vert h^{s}\left( \frac{%
			\zeta }{1+\kappa }\right) \label{e:61} \\
		&+m\left\vert \psi ^{\prime }(\frac{\varsigma _{2}}{%
			m})\right\vert \left( 1-h\left( \frac{\zeta }{1+\kappa }\right) \right) ^{s} \notag		
	\end{align}
	and 
	\begin{align}
		\left\vert \psi ^{\prime }\left( \frac{1+\kappa -\zeta }{1+\kappa }%
		\varsigma _{1}+\frac{\zeta }{1+\kappa }\varsigma _{2}\right) \right\vert \leq& \left\vert \psi ^{\prime }(\varsigma _{1})\right\vert h^{s}\left( 
		\frac{1+\kappa -\zeta }{1+\kappa }\right)
		\label{e:62} \\
		& +m\left\vert \psi ^{\prime }(\frac{%
			\varsigma _{2}}{m})\right\vert \left( 1-h\left( \frac{1+\kappa -\zeta }{%
			1+\kappa }\right) \right) ^{s}.  \notag
	\end{align}%
	By substituting \eqref{e:61}  and \eqref{e:62} in \eqref{e:60}, we have the
	required result.
\end{proof}

\begin{remark}
	\label{r:11} If we take $\kappa =0$, we get some special cases of the
	Theorem \ref{t:1}:
	
	\begin{itemize}
		\item[(i01)] Theorem 2.1 from \cite{BOP} (case $q=1$), using $w(\zeta)=1-2\zeta$ by working only with $I_{1}$ for $s=1,h(\zeta )=\zeta $ and $m-$convex functions.
		
		\item[(i02)] Theorem 2.2 from \cite{DA}, by using $I_{2}$ with  $w(\zeta )=1-\ 2\zeta $.
		
		\item[(i03)] Theorem 2.4 of \cite{FRZ} for $h(\zeta )=\zeta $ and $m=s=1$
		with $I_{1}$ and $w(\zeta )=(1-\zeta )^{\alpha }-\zeta ^{\alpha }.$
		
		\item[(i04)] Theorem 2.2 of \cite{Kirma}, for an appropriate $w(\zeta )$
		function.
		
		\item[(i05)] Theorem 2.3 of \cite{MBI}, with $I_{2}$ \ and $w(\zeta )=1-\
		2\zeta $,
		
		\item[(i06)] Theorem 7 from \cite{OAK} (see Remark \ref{r:1}(ii)).
		
		\item[(i07)] Theorem 3.1 from \cite{QZX}, where $I_{2}$ was used and the
		interval $[0,1]$ was divided by using $w_{1}=\lambda -\zeta $ for $[0,\frac{1%
		}{2}]$ and $w_{2}=\mu -\zeta $ for $[\frac{1}{2},1]$, where $\lambda $ and $%
		\mu $ real numbers such that $0\leq \lambda \leq \frac{1}{2}\leq \mu \leq 1$.
		
		\item[(i08)] Theorem 3 of \cite{SSYB} by taking $w(\zeta )=(1-\zeta
		)^{\alpha }-\zeta ^{\alpha }$ and $I_{2}.$
		
		%\item[(i09)]The first part of Theorem 2.1 of \cite{Tunc1}, statement for fractional integrals of the Riemann-Liouville type, by using only $I_{2}$ and $w(\zeta )=(1-\zeta )^{\alpha }-\zeta ^{\alpha }$, in function class $h$-convex
		
		\item[(i09)] Theorem 5 of \cite{YOK} \ by working with $w(\zeta )=(1-\zeta)^{\alpha }-\zeta ^{\alpha }$ and by using only $I_{2}$.
	\end{itemize}
\end{remark}

\begin{corollary}
	\label{Corol1} If we choose $w(\zeta )=1-2\zeta ,h(\zeta )=\zeta ,m=s=1$ and 
	$\kappa =1$ in Theorem \ref{t:1}, then we get the Simpson type inequality: 
	\begin{align}
		&\left\vert \frac{1}{2}\left[ \frac{\psi (\varsigma _{1})+\psi (\varsigma
			_{2})}{2}+\psi \left( \frac{\varsigma _{1}+\varsigma _{2}}{2}\right) \right] 
		\mathbb{-}{\frac{1}{\varsigma _{2}-\varsigma _{1}}}\int_{\varsigma
			_{1}}^{\varsigma _{2}}{\psi (\zeta )d\zeta }\right\vert  \label{111} \\
		&\leq \frac{\varsigma _{2}-\varsigma _{1}}{16}\left( \left\vert \psi
		^{\prime }(\varsigma _{1})\right\vert +\left\vert \psi ^{\prime }(\varsigma
		_{2})\right\vert \right) .  \notag
	\end{align}
\end{corollary}

\begin{proof}
	From \eqref{e:6}, we get 
	\begin{align*}
		\mathbb{D} &=\left[\psi (\varsigma _{1})+\psi (\varsigma _{2})\right]w(0)%
		-\left[\psi\left(\frac{\kappa \varsigma _{1}+\varsigma _{2}}{1+\kappa}%
		\right)+\psi\left(\frac{\varsigma_{1}+\kappa \varsigma _{2}}{1+\kappa }%
		\right)\right]w(1) \\
		&=\left[\psi (\varsigma _{1})+\psi (\varsigma _{2})\right] +\ 2\psi\left(%
		\frac{\varsigma_{1}+\varsigma _{2}}{2}\right),
	\end{align*}
	since $w^{\prime }(\zeta )=-2$, 
	\begin{align*}
		& \left\vert \mathbb{D}+{\frac{2}{\varsigma _{2}-\varsigma _{1}}\left[{%
				J_{\left( \frac{\varsigma _{1}+\varsigma _{2}}{2}\right) ^{+}}^{w}\psi
				(\varsigma _{2})}+{J_{\left( \frac{\varsigma _{1}+\varsigma _{2}}{2}\right)
					^{-}}^{w}\psi (\varsigma _{1})}\right]}\right\vert \\
		& =\left\vert \psi (\varsigma _{1})+\psi (\varsigma _{2})+\ 2\psi\left(\frac{%
			\varsigma _{1}+\varsigma_{2}}{2}\right)\mathbb{-}{\frac{4}{\varsigma
				_{2}-\varsigma _{1}}}\int_{\varsigma _{1}}^{\varsigma _{2}}{\psi (\zeta
			)d\zeta }\right\vert,
	\end{align*}
	\begin{align*}
		\mathbb{B} &=\int_{0}^{1}\left\vert 1-2\zeta \right\vert \left[ h\left(\frac{%
			\zeta}{2}\right)+h\left(\frac{2-\zeta }{2}\right)\right] d\zeta
		=\int_{0}^{1}\left\vert 1- 2\zeta\right\vert \left[ \frac{\zeta }{2}+\frac{%
			2-\zeta }{2}\right] d\zeta =\frac{1}{2}, \\
		\mathbb{C} &=\int_{0}^{1}\left\vert 1-2\zeta \right\vert \left[
		\left(1-h\left( \frac{2-\zeta }{2}\right) \right) +\left( 1-h\left( \frac{%
			\zeta }{2}\right) \right) \right] d\zeta \\
		&=\int_{0}^{1}\left\vert 1-2\zeta \right\vert \left[ 1-\frac{2-\zeta }{2}+1-%
		\frac{\zeta }{2}\right] d\zeta =\int_{0}^{1}\left\vert 1-\ 2\zeta
		\right\vert d\zeta =\frac{1}{2},
	\end{align*}
	and 
	\begin{align*}
		\frac{\varsigma _{2}-\varsigma _{1}}{1+\kappa }\left\{ \left\vert \psi
		^{\prime }(\varsigma_{1})\right\vert \mathbb{B}+m\left\vert \psi ^{\prime
		}\left(\frac{\varsigma _{2}}{m}\right)\right\vert \mathbb{C}\right\} &=\frac{%
			\varsigma _{2}-\varsigma _{1}}{2}\left[ \frac{\left\vert \psi ^{\prime
			}(\varsigma _{1})\right\vert }{2}\mathbb{+}\frac{\left\vert \psi ^{\prime
			}(\varsigma _{2})\right\vert }{2}\right] \\
		&=\frac{\varsigma _{2}-\varsigma _{1}}{4}\left( \left\vert \psi ^{\prime
		}(\varsigma_{1})\right\vert +\left\vert \psi ^{\prime }(\varsigma
		_{2})\right\vert \right).
	\end{align*}
	Thus, we have  
	\begin{align*}
		&\left\vert \psi (\varsigma _{1})+\psi (\varsigma _{2})+\ 2\psi\left(\frac{%
			\varsigma _{1}+\varsigma _{2}}{2}\right)\mathbb{-}{\frac{4}{\varsigma
				_{2}-\varsigma _{1}}}\int_{\varsigma _{1}}^{\varsigma _{2}}{\psi (\zeta
			)d\zeta }\right\vert \\
		&\qquad\qquad\qquad\leq \frac{\varsigma _{2}-\varsigma _{1}}{4}\left(
		\left\vert\psi ^{\prime }(\varsigma _{1})\right\vert +\left\vert \psi
		^{\prime }(\varsigma _{2})\right\vert \right).
	\end{align*}
	By dividing both parts of the last inequality by $4$, we obtain \eqref{111}.
	The proof is completed.
\end{proof}

\begin{corollary}
	\label{Corol2} If we choose $w(\zeta )=1-\ 2\zeta, h(\zeta)=\zeta, m=s=1$
	and $\kappa =0$ in Theorem \ref{t:1}, then we get Trapezoidal inequality %
	\eqref{Dr1}.
\end{corollary}

\begin{proof}
	From \eqref{e:6}, we get 
	\begin{align*}
		\mathbb{D} &=\left[\psi (\varsigma _{1})+\psi (\varsigma _{2})\right]w(0)%
		-\left[\psi\left(\frac{\kappa \varsigma _{1}+\varsigma _{2}}{1+\kappa }%
		\right)+\psi\left(\frac{\varsigma _{1}+\kappa \varsigma _{2}}{1+\kappa }%
		\right)\right]w(1) \\
		&=\left[\psi (\varsigma _{1})+\psi (\varsigma _{2})\right]w(0) -\left[
		\psi (\varsigma_{2})+\psi (\varsigma _{1})\right] \\
		&=2\left[ \psi (\varsigma _{1})+\psi (\varsigma _{2})\right]w(1),
	\end{align*}
	since $w^{\prime }(\zeta )=-2$,  
	\begin{align*}
		& \left\vert \mathbb{D}+{\frac{1}{\varsigma _{2}-\varsigma _{1}}\left( {%
				J_{\varsigma_{1}^{+}}^{w}\psi (\varsigma _{2})}+{J_{\varsigma
					_{2}^{-}}^{w}\psi (\varsigma _{1})}\right) }\right\vert \\
		& =\left\vert 2\left[ \psi (\varsigma _{1})+\psi (\varsigma _{2})\right]-{%
			\frac{4}{\varsigma _{2}-\varsigma _{1}}}\int_{\varsigma _{1}}^{\varsigma
			_{2}}{\psi (\zeta )d\zeta }\right\vert ,
	\end{align*}
	\begin{align*}
		\mathbb{B} &=\int_{0}^{1}\left\vert 1-2\zeta \right\vert \left[ h(\frac{\zeta%
		}{2})+h(\frac{2-\zeta }{2})\right] d\zeta =\int_{0}^{1}\left\vert 1-2\zeta
		\right\vert \left[ \frac{\zeta }{2}+\frac{2-\zeta }{2}\right] d\zeta =\frac{1%
		}{2}, \\
		\mathbb{C} &=\int_{0}^{1}\left\vert 1-2\zeta \right\vert \left[
		\left(1-h\left( \frac{2-\zeta }{2}\right) \right) +\left( 1-h\left( \frac{%
			\zeta }{2}\right) \right) \right] d\zeta \\
		&=\int_{0}^{1}\left\vert 1-2\zeta \right\vert \left[ 1-\frac{2-\zeta }{2}+1-%
		\frac{\zeta }{2}\right] d\zeta =\int_{0}^{1}\left\vert 1-2\zeta \right\vert
		d\zeta =\frac{1}{2},
	\end{align*}
	and 
	\begin{align*}
		\frac{\varsigma _{2}-\varsigma _{1}}{1+\kappa }\left\{ \left\vert \psi
		^{\prime }(\varsigma_{1})\right\vert \mathbb{B}+m\left\vert \psi ^{\prime
		}\left(\frac{\varsigma _{2}}{m}\right)\right\vert \mathbb{C}\right\}
		&=\left( \varsigma _{2}-\varsigma _{1}\right) \left[ \frac{\left\vert \psi
			^{\prime }(\varsigma _{1})\right\vert }{2}\mathbb{+}\frac{\left\vert \psi
			^{\prime }(\varsigma _{2})\right\vert }{2}\right] \\
		&=\frac{\varsigma _{2}-\varsigma _{1}}{2}\left( \left\vert \psi ^{\prime
		}(\varsigma_{1})\right\vert +\left\vert \psi ^{\prime }(\varsigma
		_{2})\right\vert \right).
	\end{align*}
	Thus, we have    
	\begin{equation*}
		\left\vert 2\left[\psi (\varsigma _{1})+\psi (\varsigma _{2})\right]-{\frac{4%
			}{\varsigma _{2}-\varsigma _{1}}}\int_{\varsigma _{1}}^{\varsigma _{2}}{\psi
			(\zeta )d\zeta }\right\vert \leq \frac{\varsigma _{2}-\varsigma _{1}}{2}%
		\left( \left\vert \psi ^{\prime}(\varsigma _{1})\right\vert +\left\vert \psi
		^{\prime }(\varsigma _{2})\right\vert\right) .
	\end{equation*}
	By dividing both parts of the last inequality by $4$, we obtain \eqref{Dr1}.
	The proof is completed.
\end{proof}
By imposing additional assumptions on $\left\vert \psi ^{\prime}\right\vert,$
we can obtain new refinements of known results.

\begin{theorem}
	\label{t:7}  If $\psi :\left[ \varsigma _{1}, \frac{\varsigma _{2}}{m}\right]
	\rightarrow \mathbb{R}$ is a $L^{1}$ function and if $\left\vert
	\psi ^{\prime }\right\vert ^{q}\in N_{h,m}^{s,2}\left[ \varsigma _{1},\frac{%
		\varsigma _{2}}{m}\right] $, then  for $q\geq 1$ the following inequality 
	\begin{align}
		& \left\vert \mathbb{D}+\frac{1+\kappa }{\varsigma _{2}-\varsigma _{1}}\left[
		{J_{\left( \frac{\varsigma _{1}+\kappa \varsigma _{2}}{1+\kappa }\right)
				^{+}}^{w}\psi (\varsigma _{2})}+{J_{\left( \frac{\kappa \varsigma
					_{1}+\varsigma _{2}}{1+\kappa }\right) ^{-}}^{w}\psi (\varsigma _{1})}\right]
		\right\vert   \label{e:7} \\
		& \leq \frac{\varsigma _{2}-\varsigma _{1}}{1+\kappa }B_{q}\left\{ \left(
		\left\vert \psi ^{\prime }\left( \varsigma _{1}\right) \right\vert
		^{q}C_{11}+m\left\vert \psi ^{\prime }\left( \frac{\varsigma _{2}}{m}\right)
		\right\vert ^{q}C_{12}\right) ^{\frac{1}{q}}\right.   \notag \\
		& \qquad\qquad\qquad\qquad \left. +\left( \left\vert \psi ^{\prime }\left( \varsigma
		_{1}\right) \right\vert ^{q}C_{21}+m\left\vert \psi ^{\prime }\left( \frac{%
			\varsigma _{2}}{m}\right) \right\vert ^{q}C_{22}\right) ^{\frac{1}{q}%
		}\right\}   \notag
	\end{align}%
	holds, with $\mathbb{D}$ as before, 	
	\begin{align*}
			B_{q}&={\left( \int_{0}^{1}\left\vert {w(\zeta )}\right\vert ^{p}d\zeta
			\right) }^{\frac{1}{p}}={\left( \int_{0}^{1}\left\vert {w(\zeta )}%
			\right\vert ^{\frac{q}{q-1}}d\zeta \right) }^{1-\frac{1}{q}},\\
		C_{11}& =\int_{0}^{1}h^{s}\left( \frac{\zeta }{1+\kappa }\right) d\zeta
		,\quad C_{12}=\int_{0}^{1}\left( 1-h\left( \frac{\zeta }{1+\kappa }\right)
		\right) ^{s}d\zeta , \\
		C_{21}& =\int_{0}^{1}h^{s}\left( \frac{1+\kappa -\zeta }{1+\kappa }\right)
		d\zeta \quad \text{and}\quad C_{22}=\int_{0}^{1}\left( 1-h\left( \frac{%
			1+\kappa -\zeta }{1+\kappa }\right) \right) ^{s}d\zeta .
	\end{align*}
\end{theorem}

\begin{proof}
	By proceeding as in the previous Theorem, from Lemma \ref{l:1}, we have 
	\begin{align}
		&\left\vert \int_{0}^{1}{w(\zeta )\psi ^{\prime }\left( \frac{\zeta }{%
				1+\kappa }\varsigma _{1}+\frac{1+\kappa -\zeta }{1+\kappa }\varsigma_{2}\right) }d\zeta \right. \label{e:70} \\
		&\qquad\qquad\left.-\int_{0}^{1}{\ w(\zeta )\psi ^{\prime }\left( \frac{1+\kappa -\zeta }{1+\kappa }\varsigma _{1}+\frac{\zeta}{1+\kappa }\varsigma_{2}\right) }d\zeta \right\vert  \notag \\
		& \qquad\leq \int_{0}^{1}{\ }\left\vert {w(\zeta )}\right\vert {\left\vert
			\psi^{\prime }\left( \frac{\zeta }{1+\kappa }\varsigma _{1}+\frac{1+\kappa-\zeta }{1+\kappa }\varsigma _{2}\right) \right\vert }d\zeta  \notag \\
		& \quad\qquad\qquad+\int_{0}^{1}\left\vert {w(\zeta )}\right\vert {\left\vert\psi
			^{\prime }\left( \frac{1+\kappa -\zeta }{1+\kappa }\varsigma _{1}+\frac{\zeta }{1+\kappa }\varsigma _{2}\right) \right\vert }d\zeta.   \notag
	\end{align}
	By using the H\"{o}lder's inequality in each of the integrals of \eqref{e:70}
	leads us to 
	\begin{align}
		& \int_{0}^{1}{\ }\left\vert {w(\zeta )}\right\vert {\left\vert \psi
			^{\prime}\left( \frac{\zeta }{1+\kappa }\varsigma _{1}+\frac{1+\kappa -\zeta 
			}{1+\kappa }\varsigma _{2}\right) \right\vert }d\zeta  \label{e:h1}\\
		& \leq \left( \int_{0}^{1}\left\vert w(\zeta )\right\vert ^{p}d\zeta \right)
		^{\frac{1}{p}}\left( \int_{0}^{1}\left\vert \psi ^{\prime }\left( \frac{%
			\zeta }{1+\kappa }\varsigma _{1}+\frac{1+\kappa -\zeta }{1+\kappa }\varsigma
		_{2}\right) \right\vert^{q}d\zeta \right) ^{\frac{1}{q}}\notag  
	\end{align}
	and 
	\begin{align}
		& \int_{0}^{1}\left\vert {w(\zeta )}\right\vert {\left\vert \psi
			^{\prime}\left( \frac{1+\kappa -\zeta }{1+\kappa }\varsigma _{1}+\frac{\zeta 
			}{1+\kappa }\varsigma _{2}\right) \right\vert }d\zeta    \label{e:h2} \\
		& \leq \left( \int_{0}^{1}\left\vert w(\zeta )\right\vert ^{p}d\zeta \right)
		^{\frac{1}{p}}\left( \int_{0}^{1}\left\vert \psi ^{\prime }\left( \frac{%
			1+\kappa -\zeta }{1+\kappa }\varsigma _{1}+\frac{\zeta }{1+\kappa }\varsigma
		_{2}\right) \right\vert^{q}d\zeta \right) ^{\frac{1}{q}}\notag
	\end{align}
	with $\frac{1}{p}+\frac{1}{q}=1$. By considering the $(h,m)$-convexity of
	the second type of $\left\vert \psi ^{\prime }\right\vert ^{q}$, we obtain 
	\begin{align}
		& \int_{0}^{1}\left\vert \psi ^{\prime }\left( \frac{\zeta }{1+\kappa }%
		\varsigma _{1}+ \frac{1+\kappa -\zeta }{1+\kappa }\varsigma _{2}\right)
		\right\vert ^{q}d\zeta\leq \left\vert \psi ^{\prime }\left( \varsigma _{1}\right) \right\vert
		^{q}\int_{0}^{1}h^{s}\left( \frac{\zeta }{1+\kappa }\right) d\zeta  \label{e:73} \\
		&\qquad\qquad\qquad\qquad\qquad\qquad +m\left\vert \psi ^{\prime }\left( \frac{\varsigma _{2}}{m}\right)
		\right\vert ^{q}\int_{0}^{1}\left( 1-h\left( \frac{\zeta }{1+\kappa }\right)
		\right) ^{s}d\zeta  \notag
	\end{align}%
	and 
	\begin{align}
		& \int_{0}^{1}\left\vert \psi ^{\prime }\left( \frac{1+\kappa -\zeta }{%
			1+\kappa } \varsigma _{1}+\frac{\zeta }{1+\kappa }\varsigma _{2}\right)
		\right\vert ^{q}d\zeta \leq \left\vert \psi ^{\prime }\left( \varsigma _{1}\right) \right\vert
		^{q}\int_{0}^{1}h^{s}\left( \frac{1+\kappa -\zeta }{1+\kappa }\right) d\zeta  \label{e:74} \\
		&\qquad\qquad\qquad\qquad\qquad\qquad+m\left\vert \psi ^{\prime }\left( \frac{\varsigma _{2}}{m}\right)
		\right\vert ^{q}\int_{0}^{1}\left( 1-h\left( \frac{1+\kappa -\zeta }{%
			1+\kappa }\right) \right) ^{s}d\zeta.  \notag
	\end{align}
	By denoting and required and substituting \eqref {e:73}, \eqref {e:74} in %
	\eqref{e:h1} and \eqref {e:h2}, we obtain the desired result (\ref{e:7}).
\end{proof}

\begin{remark}
	If we take $\kappa =0$, we get some special cases of the Theorem \ref{t:7}:
	
	\begin{itemize}
	
		%	Consider $\kappa =0$ and $\left\vert \psi ^{\prime }\right\vert ^{q}$, $q>1$,
		%	convex; from above result, we obtain Theorem 1 (second part) of \cite{MB}.
		
		%	\item[(2)]By taking $\kappa =1$, we can get the following special cases:
		
		\item[(i.01)] Theorem 11 of \cite{TB2} by using $I_{1}$,  with $w(\zeta )=(1-\zeta)^{\alpha }-\zeta ^{\alpha }$.
		
		\item[(i.02)] Theorem 3.2 of \cite{FRA} by using $I_{1}$,  with $w(\zeta )=(1-\zeta	)^{\alpha }-\zeta ^{\alpha }$.
		
		\item[(i.03)] Theorem 6 of \cite{YOK} by using $I_{1}$ with $w(\zeta )=(1-\zeta)^{\alpha }-\zeta ^{\alpha }$.
		
		\item[(i.04)] Theorem 2.1 (second part) and Theorem 2.2 of \cite{BOP}
		by using $I_{1}$ with $w(\zeta )=1-2\zeta $ and $m-$convex functions.
		
		\item[(i.05)] Theorem 2.3 \cite{DA} by using  $I_{1}$ with  $w(\zeta )=1-2\zeta$.
		
		%\item[(i.06)]Theorem 6 of \cite{TB1},
		
		\item[(i.06)] Theorem 8 of \cite{OAK}, for $s-$convex (see Remarks \ref{r:1}%
		(ii)).
		
		%\item[(i.07)]  the second part of Theorem 1 of \cite{OKA},
		
		\item[(i.07)] Theorem 1 of \cite{PP} by using $I_{1}$ with  $		w(\zeta )=1-2\zeta $.
		
		%\item[(i.09)] Theorem	2.11 of \cite{MBI}
		
		\item[(i.08)] Theorem 2.1 of \cite{Tunc1} (see Remarks \ref{r:11}(i09)).
		
		\item[(i.09)] Theorem 2 of \cite{AD} with $w(\zeta )=p(\zeta )$.
	\end{itemize}
\end{remark}

%%%%%%%%%++++++++++++++++++++++

\begin{remark}
	\label{R001}For $\kappa=1$ we have a new result for $(h,m)-$convex functions:
	
	\begin{corollary}
		If $\phi :\left[ \varsigma _{1},\frac{\varsigma _{2}}{m}\right] \rightarrow   
		\mathbb{R}$ is a $L^{1}$ function and if $\left\vert \psi
		^{\prime }\right\vert ^{q}\in N_{h,m}^{s,2}\left[ \varsigma _{1},\frac{%
			\varsigma _{2}}{m}\right] $, then for $q\geq 1\,$ the following inequality
		
		\begin{align}
			& \left\vert \mathbb{D}+\frac{2}{\varsigma _{2}-\varsigma _{1}}\left( {%
				J_{\varsigma _{1}+}^{w}\phi (\frac{\varsigma _{1}+\varsigma _{2}}{2})}+{%
				J_{\varsigma _{2}-}^{w}\phi (\frac{ \varsigma _{1}+\varsigma _{2}}{2})}%
			\right) \right\vert  \label{e:7-1} \\
			& \leq \frac{\varsigma _{2}-\varsigma _{1}}{2}\cdot B_{q}\cdot \left\{
			\left( \left\vert \phi ^{\prime }\left( \varsigma _{1}\right) \right\vert
			^{q}C_{11}+m\left\vert \phi ^{\prime }\left( \frac{\varsigma _{2}}{m}\right)
			\right\vert ^{q}C_{12}\right) ^{\frac{1}{q}}\right.  \notag \\
			& \qquad\qquad\qquad\qquad\left. +\left( \left\vert \phi ^{\prime }\left( \varsigma _{1}\right)
			\right\vert ^{q}C_{21}+m\left\vert \phi ^{\prime }\left( \frac{\varsigma _{2}%
			}{m} \right) \right\vert ^{q}C_{22}\right) ^{\frac{1}{q}}\right\}  \notag
		\end{align}
		
		holds, with $\mathbb{D}$ as before, $B_{q}={\left( \int_{0}^{1}\left\vert {\
				w(t)}\right\vert ^{p}\,dt\right) }^{\frac{1}{p}}={\left(
			\int_{0}^{1}\left\vert {w(t)}\right\vert ^{\frac{q}{q-1}}\,dt\right) }^{1- 
			\frac{1}{q}},$ 
		\begin{align*}
			C_{11} &=\int_{0}^{1}h^{s}\left( \frac{\zeta}{2}\right) d\zeta,\text{ \ \ \ \ }
			C_{12}=\int_{0}^{1}\left( 1-h\left( \frac{\zeta}{2}\right) \right) ^{s}d\zeta, \\
			C_{21} &=\int_{0}^{1}h^{s}\left( \frac{2-\zeta}{2}\right)
			d\zeta, C_{22}=\int_{0}^{1}\left( 1-h\left( \frac{2-\zeta}{2}\right) \right) ^{s}d\zeta.
		\end{align*}
	\end{corollary}
\end{remark}

\begin{remark}
	\label{R002}With $\kappa=1$ and the change of variables $t \rightarrow (t+1)$, we obtain Theorem 6 from \cite{TB1} for convex functions (i.e. $m=s=1$
	and $h(t)=t$), with $w_{1}(t)=\frac{t b+(1-t)a}{4}$ in $I_{1}$ and $%
	w_{2}(t)=- \frac{ta+(1-t)b}{4}$ for $I_{2}$.
\end{remark}

\begin{remark}
	Under the above assumptions also we have the Theorem 11 of \cite{TB2}.
\end{remark}

%%%%%%%%%++++++++++++++++++++++

\begin{theorem}
	\label{t:8} If $\psi :\left[ \varsigma _{1},\frac{\varsigma _{2}}{m}\right]
	\rightarrow \mathbb{R}$ is a $L^{1}$ function and if $\left\vert \psi
	^{\prime }\right\vert ^{q}\in N_{h,m}^{s,2} \left[ \varsigma _{1},\frac{%
		\varsigma _{2}}{m}\right] $, then for $q>1 $ we have 
	\begin{align}
		& \left\vert \mathbb{D}+\frac{1+\kappa }{\varsigma _{2}-\varsigma _{1}}%
		\left( {J_{\left( \frac{\varsigma _{1}+\kappa \varsigma _{2}}{1+\kappa }%
				\right) ^{+}}^{w}\psi (\varsigma _{2})}+{J_{\left( \frac{\kappa \varsigma
					_{1}+\varsigma _{2}}{1+\kappa }\right) ^{-}}^{w}\psi (\varsigma _{1})}%
		\right) \right\vert   \label{e:8} \\
		& \qquad\leq \frac{\varsigma _{2}-\varsigma _{1}}{1+\kappa }D\left\{ \left(
		\left\vert \psi ^{\prime }\left( \varsigma _{1}\right) \right\vert
		^{q}D_{11}+m\left\vert \psi ^{\prime }\left( \frac{\varsigma _{2}}{m}\right)
		\right\vert ^{q}D_{12}\right) ^{\frac{1}{q}}\right.   \notag \\
		& \qquad\qquad\qquad \left. +\left( \left\vert \psi ^{\prime }\left( \varsigma
		_{1}\right) \right\vert ^{q}D_{21}+m\left\vert \psi ^{\prime }\left( \frac{%
			\varsigma _{2}}{m}\right) \right\vert ^{q}D_{22}\right) ^{\frac{1}{q}%
		}\right\}   \notag
	\end{align}%
	with $\mathbb{D}$ as before, $D={\left( \int_{0}^{1}\left\vert {{w}(\zeta )}%
		\right\vert d\zeta \right) }^{1-\frac{1}{q}}$, 
	\begin{align*}
		D_{11}& =\int_{0}^{1}\left\vert w(\zeta )\right\vert h^{s}\left( \frac{\zeta 
		}{1+\kappa }\right) d\zeta ,\qquad D_{12}=\int_{0}^{1}\left\vert w(\zeta
		)\right\vert \left( 1-h\left( \frac{\zeta }{1+\kappa }\right) \right)
		^{s}d\zeta, \\
		D_{21}& =\int_{0}^{1}\left\vert w(\zeta )\right\vert h^{s}\left( \frac{%
			1+\kappa -\zeta }{1+\kappa }\right) d\zeta, \quad 
		D_{22}=\int_{0}^{1}\left\vert w(\zeta )\right\vert \left( 1-h\left( \frac{%
			1+\kappa -\zeta }{1+\kappa }\right) \right) ^{s}d\zeta .
	\end{align*}
\end{theorem}

\begin{proof}
	As before, taking into account Lemma \ref{l:1}, we have 
	\begin{align}
		& \left\vert \int_{0}^{1}\left\vert {w(\zeta )}\right\vert {\psi ^{\prime
			}\left( \frac{\zeta }{1+\kappa }\varsigma _{1}+\frac{1+\kappa -\zeta }{1+\kappa }\varsigma _{2}\right) }d\zeta \right.  \label{e:80} \\
		&\qquad\qquad\left. -\int_{0}^{1}\left\vert {w(\zeta )}\right\vert {\psi ^{\prime }\left( \frac{1+\kappa -\zeta }{1+\kappa }\varsigma _{1}+\frac{\zeta }{1+\kappa }\varsigma _{2}\right) }d\zeta	\right\vert  \notag \\
		& \qquad	 \leq \int_{0}^{1}{\left\vert {w(\zeta )}\right\vert \left\vert \psi	^{\prime }\left( \frac{\zeta }{1+\kappa }\varsigma _{1}+\frac{1+\kappa
				-\zeta }{1+\kappa }\varsigma _{2}\right) \right\vert }d\zeta  \notag \\
		& \qquad\qquad +\int_{0}^{1}\left\vert {w(\zeta )}\right\vert {\left\vert \psi^{\prime }\left( \frac{1+\kappa -\zeta }{1+\kappa }\varsigma _{1}+\frac{%
				\zeta }{1+\kappa }\varsigma _{2}\right) \right\vert }d\zeta \notag
	\end{align}
	and by using well-known power mean inequality, we have 
	\begin{align}
		& \int_{0}^{1}{\left\vert {w(\zeta )}\right\vert \left\vert \psi ^{\prime
			}\left( \frac{\zeta }{1+\kappa }\varsigma _{1}+\frac{1+\kappa -\zeta }{%
				1+\kappa }\varsigma _{2}\right) \right\vert }d\zeta  \label{e:81} \\
		& \leq \left( \int_{0}^{1}\left\vert {w(\zeta )}\right\vert d\zeta \right)
		^{1-\frac{1}{q}}\left( \int_{0}^{1}\left\vert w(\zeta )\right\vert
		\left\vert \psi ^{\prime }\left( \frac{\zeta }{1+\kappa }\varsigma _{1}+%
		\frac{1+\kappa -\zeta }{1+\kappa }\varsigma _{2}\right) \right\vert
		^{q}d\zeta \right) ^{\frac{1}{q}}  \notag
	\end{align}%
	and 
	\begin{align}
		& \int_{0}^{1}{\ }\left\vert {w(\zeta )}\right\vert {\left\vert \psi
			^{\prime }\left( \frac{1+\kappa -\zeta }{1+\kappa }\varsigma _{1}+\frac{%
				\zeta }{1+\kappa }\varsigma _{2}\right) \right\vert }d\zeta  \label{e:82} \\
		& \leq \left( \int_{0}^{1}\left\vert w(\zeta )\right\vert d\zeta \right) ^{1-%
			\frac{1}{q}}\left( \int_{0}^{1}\left\vert w(\zeta )\right\vert \left\vert
		\psi ^{\prime }\left( \frac{1+\kappa -\zeta }{1+\kappa }\varsigma _{1}+\frac{%
			\zeta }{1+\kappa }\varsigma _{2}\right) \right\vert ^{q}d\zeta \right) ^{%
			\frac{1}{q}}.  \notag
	\end{align}%
	By using the modified $(h,m)$-convexity of $\left\vert \psi ^{\prime
	}\right\vert ^{q}$, we get 
	\begin{align}
		& \int_{0}^{1}\left\vert {w(\zeta )}\right\vert {\left\vert \psi ^{\prime
			}\left( \frac{\zeta }{1+\kappa }\varsigma _{1}+\frac{1+\kappa -\zeta }{%
				1+\kappa }\varsigma _{2}\right) \right\vert ^{q}}d\zeta  \label{e:83} \\
		& \leq \int_{0}^{1}\left\vert {w(\zeta )}\right\vert {\left[ h^{s}\left( 
			\frac{\zeta }{1+\kappa }\right) \left\vert \psi ^{\prime }(\varsigma
			_{1})\right\vert ^{q}+m\left( 1-h\left( \frac{\zeta }{1+\kappa }\right)
			\right) ^{s}\left\vert \psi ^{\prime }(\frac{\varsigma _{2}}{m})\right\vert
			^{q}\right] d\zeta }  \notag \\
		& =\left\vert \psi ^{\prime }(\varsigma _{1})\right\vert
		^{q}\int_{0}^{1}w(\zeta )h^{s}\left( \frac{\zeta }{1+\kappa }\right) d\zeta \notag \\
		&\qquad\qquad+m\left\vert \psi ^{\prime }(\frac{\varsigma _{2}}{m})\right\vert
		^{q}\int_{0}^{1}\left\vert {w(\zeta )}\right\vert \left( 1-h\left( \frac{%
			\zeta }{1+\kappa }\right) \right) ^{s}d\zeta.  \notag
	\end{align}
	
	Similarly, 
	\begin{align}
		& \int_{0}^{1}{\left\vert {w(\zeta )}\right\vert \left\vert \psi ^{\prime
			}\left( \frac{1+\kappa -\zeta }{1+\kappa }\varsigma _{1}+\frac{\zeta }{%
				1+\kappa }\varsigma _{2}\right) \right\vert ^{q}}d\zeta  \label{e:84} \\
		& \leq \left\vert \psi ^{\prime }(\varsigma _{1})\right\vert
		^{q}\int_{0}^{1}\left\vert {w(\zeta )}\right\vert h^{s}\left( \frac{1+\kappa
			-\zeta }{1+\kappa }\right) d\zeta \notag \\
		&\qquad\qquad +m\left\vert \psi ^{\prime }(\frac{%
			\varsigma _{2}}{m} )\right\vert ^{q}\int_{0}^{1}\left\vert {w(\zeta )}%
		\right\vert \left( 1-h\left( \frac{1+\kappa -\zeta }{1+\kappa }\right)
		\right) ^{s}d\zeta.  \notag
	\end{align}
	
	If we put \eqref {e:83} and \eqref {e:84} in \eqref {e:81} and in %
	\eqref{e:82}, it allows us to obtain the inequality \eqref {e:8}. The proof
	is completed.
\end{proof}

\begin{remark}
	Results known in the literature, which can be obtained from Theorem \ref{t:8}%
	, by taking $\kappa =0$, we can get the following special cases:
	
	\begin{itemize}
		\item[(i.01)] For the $s$-convex functions we can obtain the Theorem 12 of 
		\cite{TB2} by using $I_{1}$ with $w(\zeta )=(1-\zeta)^{\alpha }-\zeta ^{\alpha }$.
		
		\item[(i.02)] Theorem 3.2 of \cite{QZX} (see Remarks \ref{r:11}(i09)).
		
		%(2.03) Theorem 2.13 of \cite{MBI},
		
		\item[(i.03)] Theorem 5 of \cite{AD}, by using $I_{1}$ for $w(t)=p(t)$.
		
		\item[(i.04)] Theorem 9 of \cite{OAK} (see Remarks \ref{r:1}(ii)) and $%
		\vert\psi\vert^{q}$ is $s-$convex.
		
		%(2.06)Theorem 7 of \cite{TB1},
		
		%(2.07)Theorem 2 of 	\cite{OKA},
		
		\item[(i.05)] Theorem 2.3 of \cite{BOP}  by using $I_{1}$ with $	w(\zeta )=1-2\zeta $.
		
		\item[(i.06)] Theorem 7 of \cite{YOK} by using $I_{1}$ with $w(\zeta
		)=(1-\zeta)^{\alpha }-\zeta ^{\alpha }$ and $\vert\psi\vert^{q}$ is $s-$convex.
		
		\item[(i.10)] Theorem 3.6 of \cite{FRA} by using $I_{1}$  with $w(\zeta
		)=(1-\zeta)^{\frac{\alpha}{k} }-\zeta ^{\frac{\alpha}{k}}$ and $\vert\psi\vert^{q}$ is $(h-m)-$convex.
	\end{itemize}
\end{remark}

\begin{remark}
	With $\kappa=1$ and using the new variables $\zeta \rightarrow t+1$, we
	obtain Theorem 7 from \cite{TB1} for convex functions, $m=s=1$ and $h(t)=t$,
	with $w(t)=\frac{(tb+(1-t)a}{4}$ for $I_{1}$ and $w(t)=-\frac{(ta+(1-t)b}{4}$
	for $I_{2}$.
\end{remark}

\begin{remark}
	Under the assumptions of above remark, also we have the Theorem 12 of \cite%
	{TB2}.
\end{remark}

\begin{theorem}
	\label{t:0} Let $\psi :\left[ \varsigma _{1},\varsigma _{2}\right]
	\rightarrow \mathbb{R}$ is a $L^{1}$ function. If $\ \left\vert \psi
	^{\prime }\right\vert $ on the interval $\left[ \varsigma _{1},\varsigma _{2}%
	\right] $\ satisfies the condition of the Lagrange Theorem, then the
	following inequality holds:%
	\begin{align}
		& \left\vert \mathbb{D}+{\frac{1+\kappa }{\varsigma _{2}-\varsigma _{1}}%
			\left( {J_{\left( \frac{\varsigma _{1}+\kappa \varsigma _{2}}{1+\kappa }%
					\right) ^{+}}^{w}\psi (\varsigma _{2})}+{J_{\left( \frac{\kappa \varsigma
						_{1}+\varsigma _{2}}{1+\kappa }\right) ^{-}}^{w}\psi (\varsigma _{1})}%
			\right) }\right\vert  \label{Lag0} \\
		& \leq \frac{\left( \varsigma _{2}-\varsigma _{1}\right) \left\vert \psi
			^{\prime \prime }(\zeta )\right\vert }{1+\kappa }\int_{0}^{1}\left\vert {%
			w(\zeta )}\right\vert \left\vert 1-\frac{2\zeta }{1+\kappa }\right\vert
		d\zeta ,  \notag
	\end{align}%
	where $\mathbb{D}$ was defined earlier, 
	\begin{equation*}
		\zeta \in 
		\begin{cases}
			\left( c,d\right) , & \text{for }\zeta \leq \frac{1+\kappa }{2}; \\ 
			\left( d,c\right) , & \text{for }\zeta >\frac{1+\kappa }{2},%
		\end{cases}%
		c=\frac{\zeta }{1+\kappa }\varsigma _{1}+\frac{1+\kappa -\zeta }{1+\kappa }%
		\varsigma _{2},\quad d=\frac{1+\kappa -\zeta }{1+\kappa }%
		\varsigma _{1}+\frac{\zeta }{1+\kappa }\varsigma _{2}.
	\end{equation*}
\end{theorem}

\begin{proof}
	From Lemma \ref{l:1}  and modulus properties, we can write 
	\begin{align}
		& \left\vert \mathbb{D}+{\frac{1+\kappa }{\varsigma _{2}-\varsigma _{1}}%
			\left( {J_{\left( \frac{\varsigma _{1}+\kappa \varsigma _{2}}{1+\kappa }%
					\right) ^{+}}^{w}\psi (\varsigma _{2})}+{J_{\left( \frac{\kappa \varsigma
						_{1}+\varsigma _{2}}{1+\kappa }\right) ^{-}}^{w}\psi (\varsigma _{1})}%
			\right) }\right\vert  \label{Lag1} \\
		& \leq \int_{0}^{1}\left\vert {w(\zeta )}\right\vert {\left\vert \psi
			^{\prime }\left( \frac{\zeta }{1+\kappa }\varsigma _{1}+\frac{1+\kappa
				-\zeta }{1+\kappa }\varsigma _{2}\right) -\psi ^{\prime }\left( \frac{%
				1+\kappa -\zeta }{1+\kappa }\varsigma _{1}+\frac{\zeta }{1+\kappa }\varsigma
			_{2}\right) \right\vert }d\zeta .  \notag
	\end{align}%
	By using the Lagrange Theorem condition, for the right side of the
	inequality \eqref{Lag1}, we get 
	\begin{align}
		& {\left\vert \psi ^{\prime }\left( \frac{\zeta }{1+\kappa }\varsigma _{1}+%
			\frac{1+\kappa -\zeta }{1+\kappa }\varsigma _{2}\right) -\psi ^{\prime
			}\left( \frac{1+\kappa -\zeta }{1+\kappa }\varsigma _{1}+\frac{\zeta }{%
				1+\kappa }\varsigma _{2}\right) \right\vert }  \label{Lag2} \\
		& \leq \left\vert \psi ^{\prime \prime }(\zeta )\right\vert \left\vert \frac{%
			\zeta }{1+\kappa }\varsigma _{1}+\frac{1+\kappa -\zeta }{1+\kappa }\varsigma
		_{2}-\frac{1+\kappa -\zeta }{1+\kappa }\varsigma _{1}-\frac{\zeta }{1+\kappa 
		}\varsigma _{2}\right\vert  \notag \\
		& =\left\vert \psi ^{\prime \prime }(\zeta )\right\vert \left\vert \left( 1-%
		\frac{\ 2\zeta }{1+\kappa }\right) \left( \varsigma _{2}-\varsigma
		_{1}\right) \right\vert .  \notag
	\end{align}%
	By taking into account \eqref{Lag1} and \eqref{Lag2}, we have \eqref{Lag0}.
\end{proof}

\begin{corollary}
	If we choose $w(\zeta )=1-\ 2\zeta $ and $\kappa =0$ in Theorem \ref{t:0},
	then we get 
	\begin{equation}  \label{Lag4}
		\left\vert \frac{\psi (\varsigma _{2})+\psi (\varsigma _{1})}{2}\mathbb{-}{%
			\frac{1}{\varsigma_{2}-\varsigma _{1}}}\int_{\varsigma _{1}}^{\varsigma _{2}}%
		{\psi (\zeta )d\zeta }\right\vert \leq \frac{\varsigma _{2}-\varsigma _{1}}{%
			12}\left\vert \left\vert \psi ^{\prime \prime}\right\vert \right\vert
		_{\infty },
	\end{equation}%
	where $\left\vert \left\vert \psi ^{\prime \prime }\right\vert
	\right\vert_{\infty }=\underset{\zeta \in \left[ \varsigma _{1},\varsigma
		_{2}\right] }{\max }\left( \left\vert {\psi }^{\prime \prime }{(\zeta )}%
	\right\vert \right).$
\end{corollary}

\begin{proof}
	From \eqref{Lag0}, we get 
	\begin{align*}
		\mathbb{D} &= w(0)\left( \psi (\varsigma _{1})+\psi (\varsigma
		_{2})\right)-w(1)\left( \psi (\frac{\kappa \varsigma _{1}+\varsigma _{2}}{%
			1+\kappa })+\psi (\frac{\varsigma_{1}+\kappa \varsigma _{2}}{1+\kappa }%
		)\right) \\
		&=\left( w(0)-w(1)\right) \left( \psi (\varsigma _{2})+\psi (\varsigma
		_{1})\right)=2\left( \psi (\varsigma _{2})+\psi (\varsigma _{1})\right),
	\end{align*}%
	since $w^{\prime }(\zeta )=-2$, 
	\begin{align*}
		& \left\vert \mathbb{D}+{\frac{2}{\varsigma _{2}-\varsigma _{1}}\left( {%
				J_{\varsigma_{1}^{+}}^{w}\psi (\varsigma _{2})}+{J_{\varsigma
					_{2}^{-}}^{w}\psi (\varsigma _{1})}\right) }\right\vert \\
		& =\left\vert 2\left[ \psi (\varsigma _{2})+\psi (\varsigma _{1})\right]-{%
			\frac{4}{\varsigma _{2}-\varsigma _{1}}}\int_{\varsigma _{1}}^{\varsigma
			_{2}}{\psi (\zeta )d\zeta }\right\vert
	\end{align*}
	and 
	\begin{equation*}
		\int_{0}^{1}\left\vert {w(\zeta )}\right\vert \left\vert 1-\frac{2\zeta }{%
			1+\kappa }\right\vert d\zeta =\int_{0}^{1}\left\vert 1-2\zeta \right\vert \left\vert 1-2\zeta \right\vert d\zeta =\int_{0}^{1}\left( 1-2\zeta\right)^{2}d\zeta =\frac{1}{3}.
	\end{equation*}
	Thus, we have   
	\begin{equation*}
		\left\vert 2\left[ \psi (\varsigma _{2})+\psi (\varsigma _{1})\right] 
		\mathbb{-}{\frac{4}{\varsigma _{2}-\varsigma _{1}}}\int_{\varsigma
			_{1}}^{\varsigma _{2}}{\psi (\zeta )d\zeta }\right\vert \leq \frac{\left(
			\varsigma _{2}-\varsigma _{1}\right) \left\vert \left\vert\psi ^{\prime
				\prime }\right\vert \right\vert _{\infty }}{3}.
	\end{equation*}%
	By dividing both parts of the last inequality by $4$, we obtain \eqref{Lag4}.
\end{proof}

\begin{remark}
	For $\varsigma _{2}-\varsigma _{1}>1$, it is easy to verify that the upper
	estimate \eqref{Lag4} was better than the estimate obtained in \cite[	Proposition 2]{Sar0} and confirm in a number of subsequent studies, for
	example, see \cite[Corollary 3.1]{Bayrak0} and \cite[Corollary 2]{Bayrak1},
	for more details.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Some application to special means}

\label{sec3} Let us present some applications in which the obtained results
can be used. Known mean values of the real numbers $\alpha $ and $\beta $;
see Pearce and J. Pe\v{c}ari\'{c} \cite{PP}:

\begin{itemize}
	\item[(i)] Arithmetic mean: $A(\alpha ,\beta )=\frac{\alpha +\beta }{2};$
	
	\item[(ii)] Quadratic mean: $Q(\alpha ,\beta )=\sqrt{\alpha ^{2}+\beta^{2}};$
	
	\item[(iii)] Geometric mean: $G(\alpha ,\beta )=\sqrt{\alpha \beta },$ where 
	$\alpha \beta \geq 0;$
	
	\item[(iv)] Harmonic mean: $H(\alpha ,\beta )=\frac{2\alpha \beta }{\alpha
		+\beta },$ where $\alpha +\beta \neq 0;$
	
	\item[(v)] Logarithmic mean: $L(\alpha ,\beta )=\frac{\beta -\alpha }{\ln
		\beta-\ln \alpha },$ where $\alpha,\beta >0$ and $\alpha \neq \beta$.
\end{itemize}

Now, by using some results, we give special means of positive real numbers
to some applications.

\begin{example}
	In Corollary \ref{Corol1}, if we take $\psi(x)=\sqrt{1+x^{2}}$ defined on
	the interval $(0,\varsigma _{2})$ as a function, then we get 
	\begin{align*}
		&\left\vert A\left\{ A\left[ 1,Q\left( \varsigma _{2},1\right) \right]
		,Q\left(\frac{\varsigma _{2}}{2},1\right) \right\} -A\left[ Q\left(
		\varsigma _{2},1\right), \ln \left( 2A\left( \varsigma _{2},Q\left(
		\varsigma _{2},1\right) \right) \right) ^{\frac{1}{\varsigma _{2}}}\right]
		\right\vert\\
		&\qquad \leq \frac{\varsigma _{2}^{2}}{16}Q^{-1}\left( \varsigma
		_{2},1\right) 
	\end{align*}
	and from \eqref{Lag4}, we have 
	\begin{equation*}
		\left\vert A(1,Q\left( \varsigma _{2},1\right) )-A\left[ Q\left( \varsigma
		_{2},1\right),\ln \left( 2A\left( \varsigma _{2},Q\left( \varsigma
		_{2},1\right) \right) \right) ^{\frac{1}{\varsigma _{2}}}\right] \right\vert
		\leq \frac{\varsigma _{2}}{12}.
	\end{equation*}
\end{example}

\begin{example}
	In Corollary \ref{Corol1}, if we take $\psi(x)=\frac{1}{x^{2}}$ defined on
	the interval $(\varsigma _{1},\varsigma _{2}),$ where $\varsigma
	_{1},\varsigma _{2}\in \mathbb{R}^{+}$ as a function, then we get 
	\begin{equation*}
		\left\vert A\left[ H^{-1}\left( \varsigma _{1}^{2},\varsigma
		_{2}^{2}\right), A^{-2}\left( \varsigma _{1},\varsigma _{2}\right) \right]
		-G^{-2}\left( \varsigma_{1}^{2},\varsigma _{2}^{2}\right) \right\vert \leq 
		\frac{\varsigma _{2}-\varsigma _{1}}{3}H^{-1}\left(\varsigma _{1}^{\frac{3}{2%
		}},\varsigma _{2}^{\frac{3}{2}}\right)
	\end{equation*}
	and from \eqref{Lag4}, we have 
	\begin{equation*}
		\left\vert H^{-1}\left( \varsigma _{1}^{2},\varsigma _{2}^{2}\right)
		-G^{-2}\left( \varsigma_{1}^{2},\varsigma _{2}^{2}\right) \right\vert \leq 
		\frac{\varsigma _{2}-\varsigma _{1}}{2\varsigma_{1}^{4}}.
	\end{equation*}
\end{example}

\begin{example}
	In Corollary \ref{Corol1}, if we take $\psi(x)=\frac{1}{x}$ defined on the
	interval $(\varsigma _{1},\varsigma _{2}),$ where $\varsigma _{1},\varsigma
	_{2}\in \mathbb{R}^{+}$ as a function, then we get 
	\begin{equation*}
		\left\vert A\left[ H^{-1}\left( \varsigma _{1},\varsigma _{2}\right)
		,A^{-1}\left( \varsigma_{1},\varsigma _{2}\right) \right] -L^{-1}\left(
		\varsigma _{1},\varsigma _{2}\right) \right\vert \leq \frac{\varsigma
			_{2}-\varsigma _{1}}{16}H^{-1}\left( \varsigma
		_{1}^{2},\varsigma_{2}^{2}\right)
	\end{equation*}
	and from \eqref{Lag4}, we have 
	\begin{equation*}
		\left\vert H^{-1}\left( \varsigma _{1},\varsigma _{2}\right) -L^{-1}\left(
		\varsigma _{1},\varsigma_{2}\right) \right\vert \leq \frac{\varsigma
			_{2}-\varsigma _{1}}{6\varsigma _{1}^{3}}.
	\end{equation*}
\end{example}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Conclusions}

\label{sec4} In this article, we considered new integral inequalities of the
Hadamard type for the weighted integrals. Throughout the article, we have
shown that many known results are our particular cases.
The generality of the results lies in the fact that they are valid for
various classes of convex functions defined in some closed subset of a
non-negative numerical semi--axis, for example, $s-$convex in the second
sense, $h-$convex, $m-$convex, and for a number of other composite classes
of convex functions. These results can easily be extended to the case of $(h,m)-$convex functions of the first type.
\bigskip
\begin{thebibliography}{1}




%\bibitem{1} D.G. Northcott, {\it Ideal Theory}, Cambridge University Press, Cambridge, 1953.

%\bibitem{2} I.I. Ivanov, {\it Some properties of commutative algebras}, Siberian Electronic Mathematical Reports, {\bf 1} (2004), 142--143.

%\bibitem{3} A.V. Tetenov, M. Samuel, D.A. Vauilin, {\it On dendrites generated by polyhedral systems and their ramification points}, Trudy Inst. Mat. Mekh. UrO RAN, \textbf{23}:4 (2017), 281--291.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{ADK} M. Alomari, M. Darus, U. S. K\i rmac\i , {\it Refinements of
	Hadamard-type inequalities for quasi--convex functions with applications to
	trapezoidal formula and to special means}, Computers and Mathematics with
Applications, {\bf 59} (2010) 225--232. %nujen

\bibitem{AD} M. Alomari, M. Darus, {\it Some Ostrowski type inequalities for
convex functions with applications}, RGMIA Res. Rep. Coll.,  {\bf 13}:2 (2010),
Article 3, http://ajmaa.org/RGMIA/v13n2.php   %nujen

\bibitem{BOP} M. K. Bakula, M. E. \"{O}demir, J. Pe\v{c}ari\'{c},
{\it Inequalities for $m-$Convex and $(\alpha ,m)-$Convex Functions}, JIPAM, {\bf 9}:4 (2008), art. 96. %nujen

\bibitem{Bayrak0} B. Bayraktar, {\it Some integral inequalities of
Hermite-Hadamard type for differentiable $(s,m)-$convex functions via
fractional integrals}, TWMS J. App. Eng. Math. {\bf 10}:3 (2020), 625--637. %nujen

\bibitem{Bayrak1} B. Bayraktar, V. Kudaev, {\it Some new integral inequalities for $(s,m)-$convex and $(\alpha ,m)-$convex functions}, Bulletin of the Karaganda University-Mathematics, {\bf 94}:2 (2019), 15--25. Doi:10.31489/2019M2/15-25. %????

\bibitem{BN-Udm} B. Bayraktar, J. E. N\'{a}poles, {\it New Generalized Integral	Inequalities Via $(h,m)-$Convex Modified Functions}, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, {\bf 60} (2022), 1--13, DOI: 10.35634/2226-3594-2022-60-NN  %nujen

\bibitem{BN} B. Bayraktar, J. E. N\'{a}poles, {\it Hermite--Hadamard Weighted Integral Inequalities For $(h,m)-$Convex Modified Functions}, Fractional Differential Calculus, {\bf 12}:2 (2022), 235--248, doi:10.7153/fdc-2022-12-15  %nenujen




\bibitem{BKN} S. Bermudo, P. K\'orus, J. E. N\'apoles, {\it On $q-$
Hermite-Hadamard inequalities for general convex functions}, Acta Math.
Hungar. {\bf 162} (2020), 364--374  https://doi.org/10.1007/s10474-020-01025-6  

\bibitem{But1} S.I. Butt, J. Pe\v{c}ari\'{c},{ \it Generalized Hermite-Hadamard's inequality}, Proc. A. Razmadze Math. Inst, {\bf 163} 2013,  9--27. 

\bibitem{But2} S.I. Butt, S. Yousaf, A.O. Akdemir, M.A. Dokuyucu, {\it New Hadamard-type integral inequalities via a general form of fractional integral operators}, Chaos, Solitons \& Fractals, {\bf 148} (2013),  111025  

\bibitem{But0} S. I. Butt, S. Yousaf, A. Asghar, K. A. Khan, H. R. Moradi,
{\it New Fractional Hermite-Hadamard-Mercer Inequalities for Harmonically Convex
Function,} Journal of Function Spaces, {\bf 2021} (2021), Article ID 5868326,
https://doi.org/10.1155/2021/5868326  

\bibitem{DA} S. S. Dragomir, R.P. Agarwal, {\it Two inequalities for
differentiable mappings and applications to special means of real numbers
and trapezoidal formula}, Appl. Math. Lett., {\bf 11}:5 (1998), 91--95. 

\bibitem{Drag} S. S. Dragomir, Charles E.M. Pearce, {\it Selected Topics on Hermite-Hadamard Inequalities and Applications}, RGMIA, (2002).

\bibitem{FRA} G. Farid, A. U. Rehman, Q. U. Ain, {\it $k-$fractional integral
inequalities of Hadamard type for $(h-m)-$convex functions}, Computational
Methods for Differential Equations,{\bf 8}:1 ( 2020), 119--140,
DOI:10.22034/cmde.2019.9462

\bibitem{FRZ} G. Farid, A. Rehman, M. Zahra, {\it On Hadamard inequalities for $%
k- $-fractional integrals}, Nonlinear Functional Analysis and Applications,
{\bf 21}:3 (2016), 463--478, http://nfaa.kyungnam.ac.kr/journal-nfaa  

\bibitem{GNG} P. M. Guzm\'an, J. E. N\'apoles V., Y. Gasimov, {\it Integral
inequalities within the framework of generalized fractional integrals},
Fractional Differential Calculus, {\bf 11}:1 (2021), 69--84,
doi:10.7153/fdc-2021-11-05

\bibitem{HM} H. Hudzik, L. Maligranda, {\it Some remarks on $s-$convex functions},
Aequationes Math. {\bf 48}:1 (1994), 100--111.

\bibitem{KiKi} E. Kikianty,  {\it Hermite-Hadamard Inequality in the Geometry of Banach Spaces}.  Thesis is presented in fulfilment of the requirements for the degree of Doctor of Philosophy, School of Engineering and Science Faculty of Health, Engineering and Science Victoria University, (2010).

\bibitem{Kirma} U. S. K\i rmac\i , {\it Inequalities for differentiable mappings
and applications to special means of real numbers and to midpoint formula},
Applied Mathematics and Computation, {\bf 147} (2004), 137--146

%\bibitem{Kunt} M. Kunt, D. Karapinar, S. Turhan, I. Iscan, { \it The left Riemann-Liouville fractional Hermite--Hadamard type inequalities for convex functions},  Mtematica Slovaca,  {\bf 69}:4 (2019),  773--784, DOI10.1515/ms-2017-0261

%\bibitem{Matloka1} M. Matloka, { \it On some integral inequalities for $(h,m)-$convex functions}, Mathematical Economics, {\bf 9}:16 (2013), 55--70.

%\bibitem{MB}
%P. O. Mohammed, I. Brevik, A New Version of the Hermite-Hadamard Inequality for Riemann-Liouville Fractional Integrals,
%Symmetry 2019, 12, 610; doi:10.3390/sym12040610

\bibitem{MBI} M. Muddassar, M. I. Bhatti, W. Irshad, {\it Generalisation of
integral inequalities of Hermite-Hadamard type through convexity}, Bull.
Aust. Math. Soc. {\bf 88}:2 (2013), 320--330.

\bibitem{Bayrak2} J. E. N\'{a}poles, B. Bayraktar, {\it On The Generalized Inequalities of the Hermite-Hadamard type}, FILOMAT, {\bf 35}:14 (2021), 4917--4924, https://doi.org/10.2298/FIL2114917N 

\bibitem{NRS} J. E. N\'apoles Vald\'es, F. Rabossi, A. D. {\it Samaniego, Convex Functions: Ariadne's Thread Or Charlotte's Spiderweb?}, Advanced Mathematical
Models \& Applications, {\bf 5}:2 (2020), 176--191.

\bibitem{OAK} M. E. \"{O}zdemir, M.Avc\i , H. Kavurmaci, {\it Hermite-Hadamard
type inequalities for $s-$convex and $s-$ concave functions via fractional
integrals}, arxiv:1202,0380v1, 2 feb 2012.

\bibitem{ODY} M. E. \"{O}demir, S. S. Dragomir, \c{C}. Y\i ld\i z, {\it The
Hadamard's inequality for convex function via fractional integrals}, Acta
Math. Sci.,  {\bf 33}:5 (2013), 1293--1299, doi: 10.1016/S0252-9602(13)60081-8

%%&&&

%\bibitem{OKA} M. E. \"{O}zdemir, H. Kavurmac\i , M. Avc\i, {\it Ostrowski Type Inequalities For Convex Functions, Tamkang Journal Of Mathematics}, {\bf 45}:4 (2014), 335-340, doi:10.5556/j.tkjm.45.2014.1143

\bibitem{PP} C. E. M. Pearce, J. Pe\v{c}ari\'{c},  {\it Inequalities for
differentiable mappings with application to special means and quadrature
formulae}, Appl. Math. Lett., {\bf 13}:2 (2000), 51--55.

\bibitem{QZX} F. Qi, T. Y. Zhang, B. Y. Xi, {\it Hermite-Hadamard type integral
inequalities for functions whose first derivatives are of convexity},
Ukrainian Mathematical Journal,  {\bf 67}:4 (2015), 555--567,
DOI:10.1007/s11253-015-1103-3 %%&&&&

\bibitem{Sar0} M.Z. Sar\i kaya, N. Aktan,  {\it On the generalization of
some integral inequalities and their applications}. Math. Comput. Modelling,
{\bf 54} (2011),  2175--2182.

\bibitem{SSYB} M. Z. Sar\i kaya, E. Set, H. Yald\i z, N. Ba\c{s}ak,
{\it Hermite-Hadamard's inequalities for fractional integrals and related
fractional inequalities}, Math. Comput. Modelling, {\bf 57}:9-10 (2013),
2403--2407.

%\bibitem{But3} E. Set, S.I. Butt, A.O. Akdemir, A. Karaoglan, T. Abdeljawad, {\it New integral inequalities for differentiable convex functions via Atangana-Baleanu fractional integral operators}, Chaos, Solitons \& Fractals {\bf 143} (2021), 110554, DOI: 10.1016/j.chaos.2020.110554 %%(((())))

\bibitem{Sitnik} H. M. Srivastava, S. Mehrez, S. M. Sitnik, {\it Hermite--Hadamard-Type Integral Inequalities for Convex Functions and Their Applications}”, Mathematics (MDPI),  Mathematics, {\bf 10}:17, (2022) 3127, https://doi.org/10.3390/math10173127 

\bibitem{Toa} G. Toader, {\it Some generalizations of the convexity}, Proceedings
of the Colloquium on Approximation and Optimization, University Cluj-Napoca,
(1985), 329--338.

\bibitem{Tunc1} M. Tun\c{c}, {\it On New Inequalities For $h-$Convex Functions
Via Riemann-Liouville Fractional Integration}, FILOMAT, {\bf 27}:4 (2013), 559--565, https://doi.org/10.2298/FIL1304559T

\bibitem{TB1} M. Tun\c{c}, S. Balge\c{c}ti, {\it Some inequalities for
differentiable convex functions with applications}, 
http://arxiv.org/pdf/1406.7217.pdf, submitted.

\bibitem{TB2} M. Tun\c{c}, S. Balge\c{c}ti,{ \it Integral Inequalities for
Mappings Whose Derivatives Are $s-$Convex in the Second Sense and
Applications to Special Means for Positive Real Numbers}, Turkish Journal of
Analysis and Number Theory,  {\bf 4}:2 (2016), 48-53,
DOI:10.12691/tjant-4-2-5

\bibitem{Wang} J. R. Wang,  M. Fe\v{c}kan,  {\it Fractional Hermite--Hadamard Inequalities. Series: Fractional Calculus in Applied Sciences and Engineering}, 5  Publisher: De Gruyter,  2018

\bibitem{XQ} B. Y. Xi, F. Qi, {\it Inequalities of Hermite-Hadamard type for
extended $s-$convex functions and applications to means}, J. Nonlinear
Convex. Anal. {\bf 16}:5 (2015), 873--890.

\bibitem{XGQ2020} B. Y. Xi, D.D. Gao, F. Qi, {\it Integral inequalities of
	Hermite-Hadamard type for $(\alpha,s)-$convex and $(\alpha,s,m)-$convex
	functions}, Italian Journal Of Pure And Applied Mathematics, 
{\bf 44} (2020),  499--510

\bibitem{YOK} C. Y\i ld\i z, M. E. \"{O}zdemir, H. Kavurmac\i, {\it Fractional
	Integral Inequalities via $s-$Convex Functions}, Turkish Journal of Analysis
and Number Theory, {\bf 5}:1 (2017),  18--22, doi: 10.12691/tjant-5-1-4.


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