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\begin{document}
\title[Positivity of Sums and Integrals fo`r $n$-convex functions]{Positivity of Sums and Integrals for $n$-convex functions Via The Fink Identity and New Green Functions}
%\author{A. Agli\'{c} Aljinovi\'{c}}
%\address{University of Zagreb, Faculty of Electrical Engineering and Computing,
%Department of applied mathematics, Unska 3, 10 000 Zagreb, Croatia}
%\email{andrea.aglic@fer.hr}
\author{Asif R. Khan$^{1}$}
\address{1-Department of Mathematics, Faculty of Science, University of Karachi, University Road, Karachi-75270, Pakistan}
\email{asifrk@uok.edu.pk}
\author{Josip Pe\v{c}ari\'{c}$^{2}$}
\address{2-University of Zagreb, Faculty of textile technology, Prilaz baruna
Filipovi\'{c}a 28A, 10000 Zagreb, Croatia}
%\address{3 -Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan}
%\email{pecaric@hazu.hr}
%\author{Marjan Praljak$^{3}$}
%\address{4-University Of Zagreb, Faculty Of Food Technology and Biotechnology, Zagreb, Croatia }
%\email{mpraljak@pbf.hr}
%\author{Sanja Varo\v{s}anec$^{4}$}
%\address{5 -Department of Mathematics, University Of Zagreb, Zagreb, Croatia }
%\email{varosans@math.hr}
\date{February 5, 2018}
\subjclass[2010]{26A51, 26D15, 26D20}
\keywords{$n$-convex functions, Fink identity, green function, \v{C}eby\v{s}ev functional}


%\thanks{\textbf{Acknowledgement:} The research of the second and third author has been fully supported by Croatian Science Foundation under the project 5435}


\begin{abstract}
We consider positivity of sum $\sum_{i=1}^np_if(x_i)$ involving convex functions of higher order. Analogous for integral $\int_a^bp(x)f(g(x))dx$ is also given. Representation of a function $f$ via the Fink identity and the Green function leads us to identities for which we obtain conditions for positivity of the mentioned sum and integral. We obtain bounds for integral remainders which occur in those identities as well as corresponding mean value theorems.

\end{abstract}
\maketitle

\section{Introduction}
In \cite{old green} we proved various results related to general linear inequalities via Fink identity with and without Green function (see also \cite{book}). Recently, in \cite{butt} the authors have introduced new Green type functions. Our main objective of present article is to further extend results of \cite{old green} using new definitions stated in \cite{butt}.

To recall the definitions of generalized convex function and related concepts and results we refer to interested readers the following references
\cite{Asif-popoviciu}, \cite{Asif-majorization} and \cite{redbook}.

In the sequel we use the notation $AC[a,b]$ for class of absolutely continuous functions defined on a real interval  $[a,b]$ and  by $(\xi-s)^k_+$, $k\in \mathbb{N}_0$, we will mean the following
$$(\xi-s)^k_+=\left\{ \begin{array}{ll}
				(\xi-s)^k, & \mbox{ if } \xi\geq s \\
				0,   & \mbox{ if } \xi< s.
\end{array}
\right.
$$
Now we recall the Fink identity to prove many interesting results. The following theorem is proved by A. M. Fink in \cite{Fink}.
\begin{proposition}\label{FinkThm}
Let $a,b \in \mathbb {R}$, $f:\left[a,b\right]\rightarrow \mathbb {R}$, $n\geq 1$ and $f^{\left(n-1\right)}$ is absolutely continuous on $\left[a,b\right]$. Then
\begin{eqnarray}\label{Fink}
&&f\left(x\right)=\frac{n}{b-a}\int_{a}^{b}f\left(t\right)dt  \notag
-\sum_{k=1}^{n-1} \frac{n-k}{k!} \left( \frac{f^{\left(k-1\right)}\left(a\right)\left(x-a\right)^{k}-f^{\left(k-1\right)}\left(b\right)\left(x-b\right)^{k}}{b-a}\right) \notag \\
&&+ \frac{1}{\left( n-3\right) !\left(b-a\right)}
\int_{a}^{b}\left(x-t\right)^{n-1}P^{\left[a,b\right]}\left(t,x\right)f^{\left(n\right)}\left(t\right)dt,
\end{eqnarray}
%\begin{eqnarray}\label{Fink}
%f(x)=&&\frac{n}{b-a}\int_{a}^{b}f\left(t\right)dt-\sum_{k=1}^{n-1}\left( \frac{n-k}{k!}%
%\right) \left( \frac{f^{\left(k-1\right)}\left(a\right)\left(x-a\right)^{k}-f^{\left(k-1\right)}\left(b\right)\left(x-b\right)^{k}}{b-a}\right) \notag \\
%&&+ \frac{1}{\left( n-3\right) !\left(b-a\right)}
%\int_{a}^{b}\left(x-t\right)^{n-1}P^{\left[a,b\right]}\left(t,x\right)f^{\left(n\right)}\left(t\right)dt,
%\end{eqnarray}
where
\begin{equation}\label{Fink[a,b]}
P^{\left[a,b\right]}\left(t,x\right)=\left\{
\begin{array}{ll}
t-a, & a\leq {t}\leq {x}\leq {b}, \\
t-b, & a\leq {x}<{t}\leq {b}.%
\end{array}%
\right.
\end{equation}
\end{proposition}

Pe\v cari\' c in \cite{Jessen-III} proved the following result (see also \cite[p.262]{redbook}):
\begin{proposition}\label{prop6}
The inequality
\begin{equation} \label{discopineq}
\sum_{i=1}^m p_if(x_i)\ge0
\end{equation}
holds for all convex functions $f$ if and only if the $m-$tuples $\mathbf{x}=(x_1,\ldots,x_m),\,\mathbf{p}=(p_1,\ldots,p_m)\in \mathbb{R}^m$ satisfy
\begin{equation} \label{cond1}
\sum_{i=1}^m p_i=0 \quad \hbox{ and } \quad \sum_{i=1}^mp_i|x_i-x_k|\ge 0 \hbox{ for } k\in\{1,\ldots,m\}.
\end{equation}
\end{proposition}

Since $
\sum_{i=1}^mp_i|x_i-x_k| = 2 \sum_{i=1}^m p_i (x_i - x_k)_+ - \sum_{i=1}^mp_i (x_i-x_k),
$
where $y_+ = \max (y,0)$, it is easy to see that condition (\ref{cond1}) is equivalent to
\begin{equation} \label{cond2}
\sum_{i=1}^m p_i=0, \quad \sum_{i=1}^m p_i x_i = 0 \quad \hbox{ and } \quad \sum_{i=1}^mp_i (x_i-x_k)_+\ge 0 \hbox{ for } k\in\{1,\ldots,m-1\}.
\end{equation}

The following result is due to Popoviciu \cite{Pop-40a, Pop-40b} (see \cite{ redbook, Pop-44} also).

\begin{proposition}\label{prop7}
Let $n\geq 2$. Inequality $(\ref{discopineq})$ holds for all $n$-convex functions $f:[a,b]\to \mathbb{R}$ if and only if the $m-$tuples $\mathbf{x} \in [a,b]^m$, $\mathbf{p}\in \mathbb{R}^m$ satisfy
\begin{gather}
%\begin{split}
\sum_{i=1}^m p_i x_i^k=0, \quad \hbox{ for all } k\in\{0,1,\ldots,n-1\} \label{cond4} \\
\sum_{i=1}^mp_i (x_i-t )_+^{n-1}\ge 0, \quad \hbox{ for every } t\in [a,b]. \label{cond4b}
%\end{split}
\end{gather}
\end{proposition}

\begin{proposition}\label{prop4}
Let $n\geq 2$, $p:[\alpha, \beta] \to \mathbb{R}$ and $g:[\alpha, \beta] \to [a,b]$. % be such that the linear operator $\bar{A} (f) = \int_{\alpha}^{\beta} p(x) f(g(x)) \, dx$ is continuous.
Then, the inequality
\begin{equation} \label{intopineq}
\int_{\alpha}^{\beta} p(x) f(g(x)) \, dx\ge 0
\end{equation}
holds for all $n$-convex functions $f:[a,b] \to \mathbb{R}$ if and only if
\begin{gather} \label{cond5}
\begin{split}
\int_{\alpha}^{\beta} p(x) g(x)^k \, dx = 0, \quad \hbox{ for all } k\in \{0,1,\ldots,n-1\} \\
\int_{\alpha}^{\beta} p(x) \left( g(x) - t \right)_+^{n-1} \, dx \geq 0, \quad \hbox{ for every } t\in [a,b].
\end{split}
\end{gather}
\end{proposition}
After this introductory section, we continue with section 2 where identities for $\sum_{i=1}^np_if(x_i)$ and $\int_a^bp(x)f(g(x))dx$ are given using the Fink identity and new Green functions. Also we consider inequalities for $n$-convex functions which are based on these identities. Section 3 is devoted to estimations of functions $A_k$ by using \v Ceby\v{s}ev, G\"{r}uss and Ostrowski type inequalities and the H\"{o}lder inequality. In the last section we give mean value theorems for functionals $A_k$, $k\in \{1,2\}$.
\section{Popoviciu type identities and inequalities via the Fink identity and New Green functions} \label{Sect2G}
In this section we obtain some discrete and integral identities and the corresponding linear inequalities using new Green functions and applying the Fink identity.
As a special choice  of Abel-Gontscharoff polynomial for `two-point right focal' interpolating polynomial for $n=2$ could be stated as (see \cite{abel}):
\begin{equation}\label{Glem-eqi1}
f(\xi)=f(a)+(\xi-a)f{'}(b)+\int\limits_{a}^{b}{G_{1}(\xi,t)f{''}(t)}dt,
\end{equation}
where $G_{1}(s, t) $ is Green's function for `two-point right focal problem' defined as
\begin{equation}\label{AbelG-eq1}
G_1(s,t)=\left\{
          \begin{array}{ll}
            \displaystyle a-t, &  a\le t \le s,\\
           \displaystyle a-s,  & s\le t \le b.
           \end{array}
         \right.
\end{equation}
%\end{remark}
Motivated by Abel-Gontscharoff identity \eqref{Glem-eqi1} and related Green's function $(\ref{AbelG-eq1})$, we recall some new types of Green functions $G_l:[a,b]\times[a,b]\rightarrow \mathbb{R},$ $(l=2,3,4,)$ defined as in \cite{butt}:
\begin{equation}\label{AbelG-eq2}
G_2(s,t) = \left\{
          \begin{array}{ll}
            \displaystyle s-b, &  a\le t \le s,\\
           \displaystyle t-b ,  & s\le t \le b.
           \end{array}
         \right.
\end{equation}
\begin{equation}\label{AbelG-eq3}
G_3(s,t) = \left\{
          \begin{array}{ll}
            \displaystyle s-a, &  a\le t \le s,\\
           \displaystyle t-a,  & s\le t \le b.
           \end{array}
         \right.
\end{equation}
\begin{equation}\label{AbelG-eq4}
G_4(s,t) = \left\{
          \begin{array}{ll}
            \displaystyle b-t, &  a\le t \le s,\\
           \displaystyle b-s,  & s\le t \le b.
           \end{array}
         \right.
\end{equation}
In \cite{butt}, it is also shown that  %The graphical representations of $G_k,\,\,\,k=1,2,3,4,$ is depicted in the following Figure \ref{fig01}
%\begin{figure}[h]
  % Requires \usepackage{graphicx}
 % \includegraphics[width=10cm]{fig01}\\
%  \caption{Graph of Green functions for fix $w$.}\label{fig01}
%\end{figure}
all four Green functions are symmetric and continuous. Moreover, all functions are convex
with respect to both variables $s$ and $t$. From these functions we can obtain new identities, given in following lemma:
\begin{lemma}\label{Glem}
Let $f:[a,b]\to \mathbb{R}$ be twice differentiable function and $G_l, \,\,\, (l=1,2,3,4)$ are defined in \eqref{Glem-eqi1}, \eqref{Glem-eqi2}, \eqref{Glem-eqi3} and \eqref{Glem-eqi4}. Then the following identities holds:
\begin{eqnarray}\label{Glem-eqi2}
&&f(\xi)=f(b)+(b-\xi)f{'}(a)+\int\limits_{a}^{b}{G_2(\xi,t)f{''}(t)}dt,\\
\label{Glem-eqi3}
&&f(\xi)=f(b)-(b-a)f{'}(b)+(\xi-a)f^{'}(a)+\int\limits_{a}^{b}{G_3(\xi,t)f{''}(t)}dt,
\\ \label{Glem-eqi4}
&&f(\xi)=f(a)+(b-a)f{'}(a)-(b-\xi)f^{'}(b)+\int\limits_{a}^{b}{G_4(\xi,t)f{''}(t)}dt.
\end{eqnarray}
\end{lemma}
We can easily obtain these identities by using integration by parts by using respective Green function.
Now we state here main results related to the Fink identity and the Green function.
\begin{theorem}\label{FMTiu}
Fix $l\in\{1,2,3,4\}$. Let $f:\left[a,b\right]\rightarrow \mathbb {R}$ be such that for $n\geq 3$, $f^{\left(n-1\right)}$ is absolutely continuous.
Let $x_i,y_i\in \left[a,b\right]$, $p_i \in \mathbb R$ for $i\in\{1,\ldots,m\}$ be such that $\sum_{i=1}^mp_i=0$ and $\sum_{i=1}^mp_ix_i=0$ and let $P^{\left[a,b\right]}\left(t,x\right)$ be the same as defined in $\left( \ref{Fink[a,b]}\right)$. If $G_l$ are the Green functions as defined in $(\ref{AbelG-eq1})-(\ref{AbelG-eq4})$, then we have
\begin{eqnarray}\label{FinkMainIdentityG}
&&\sum_{i=1}^{m}p_{i}f\left( x_{i}\right) =\sum_{k=0}^{n-3}\left( \frac{n-k-2}{k!\left( b-a\right) }\right)
\int_{a}^{b}\left( \sum_{i=1}^{m}p_{i}G_l\left( x_{i},s\right)
\right)    \notag \\
&&\times\left( f^{\left( k+1\right) }\left( b\right) \left( s-b\right)
^{k}-f^{\left( k+1\right) }\left( a\right) \left( s-a\right) ^{k}\right) ds+%
\frac{1}{\left( n-3\right) !\left( b-a\right) }   \notag \\
&&\times\int_{a}^{b}f^{\left( n\right) }\left( t\right) \left( \int_{a}^{b}
\sum_{i=1}^{m}p_{i}G_l\left( x_{i},s\right)\left( s-t\right) ^{n-3}P^{\left[ a,b\right] }\left( t,s\right)
ds\right) dt.
\end{eqnarray}%
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     1st Main Result Fink Identity Proof   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
First consider four identities $(\ref{Glem-eqi1}), (\ref{Glem-eqi2}),(\ref{Glem-eqi3})$ and $(\ref{Glem-eqi4})$, and putting $x=\xi_i$ in all these identities, multiplying each with $p_i$, and then summing  over each identity for $i\in\{1, \ldots ,m\}$ and using conditions that
$\sum_{i=1}^{m}p_i =0$, $\sum_{i=1}^m p_i\xi_i =0$ we get by fixing $l\in\{1,2,3,4\}$
\begin{equation}\label{mainres}
\sum_{i=1}^{m}p_{i}f (\xi_{i})=\int_a^b \left( \sum_{i=1}^{m}p_i G_l(\xi_i,t) \right) f''(t) dt.
\end{equation}
Differentiating Fink identity twice we easily get
\begin{eqnarray}\label{FinkD}
f''\left( x\right)  &=&\sum_{k=0}^{n-3}\frac{n-k-2}{%
k!} \frac{f^{\left( k+1\right) }\left( b\right) \left(
x-b\right) ^{k}-f^{\left( k+1\right) }\left( a\right) \left( x-a\right) ^{k}%
}{b-a}   \notag \\
&&+\frac{1}{\left( n-3\right) !\left( b-a\right) }\int_{a}^{b}\left(
x-t\right) ^{n-3}P^{\left[ a,b\right] }\left( t,x\right) f^{\left( n\right)
}\left( t\right) dt,
\end{eqnarray}
and by using $\left(\ref{FinkD}\right)$ in $\left(\ref{mainres}\right)$, we have
\begin{eqnarray*}
&&\sum_{i=1}^{m}p_{i}f\left( x_{i}\right) =\int_{a}^{b}\left( \sum_{i=1}^{m}p_{i}G_l\left( x_{i},s\right)
\right)   \notag \\
&&\times\sum_{k=0}^{n-3} \frac{n-k-2}{k!} \frac{f^{\left(
k+1\right) }\left( b\right) \left( s-b\right) ^{k}-f^{\left( k+1\right)
}\left( a\right) \left( s-a\right) ^{k}}{b-a} ds  \notag \\
&&+\frac{1}{\left( n-3\right) !\left( b-a\right) }\int_{a}^{b} \sum_{i=1}^{m}p_{i}G_l\left( x_{i},s\right)
\left(\int_{a}^{b}\left(
s-t\right) ^{n-3}P^{\left[ a,b\right] }\left( t,s\right) f^{\left( n\right)
}\left( t\right) dt\right)ds.
\end{eqnarray*}%
Now by interchanging the integral and summation in the second term and by applying Fubini's theorem in the last term, we have $\left(\ref{FinkMainIdentityG}\right)$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     Integral Version: 1st Main Result Fink Identity     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The following theorem is the integral version of Theorem \ref{FMTiu}.
\begin{theorem}\label{FMTintiu}
Fix $l\in \{1,2,3,4\}$.  Let $f:\left[a,b\right]\rightarrow \mathbb {R}$ be such that for $n\geq 3$, $f^{\left(n-1\right)}$ is absolutely continuous on $\left[a,b\right]$ and let $p: \left[\alpha,\beta\right]\rightarrow \mathbb{R}$ and $g:\left[\alpha,\beta\right]\rightarrow \left[a,b\right]$ be integrable functions such that $\int_\alpha^\beta p(x)dx=0$ and $\int_\alpha^\beta p(x)g(x)dx=0$. Let $P^{\left[a,b\right]}\left(t,x\right)$ be the same as defined in $\left( \ref{Fink[a,b]}\right)$.
If $G_l$ are the Green functions as defined in $(\ref{AbelG-eq1})-(\ref{AbelG-eq4})$, then we have
\begin{eqnarray}
&&\int_{\alpha}^{\beta}p\left( x\right) f\left( g \left( x\right) \right)dx=\sum_{k=0}^{n-3} \frac{n-k-2}{k!\left( b-a\right) }\int_{a}^{b}\left( \int_{\alpha}^{\beta}p\left( x\right) G_l\left( g \left(
x\right) ,s\right) dx\right)\notag\\ &&\left( f^{\left( k+1\right) }\left( b\right) \left( s-b\right)
^{k}-f^{\left( k+1\right) }\left( a\right) \left( s-a\right) ^{k}\right) ds+%
\frac{1}{\left( n-3\right) !\left( b-a\right) } \notag \\
&&\times\int_{a}^{b}f^{\left( n\right) }\left( t\right) \left( \int_{a}^{b}
\left(\int_{\alpha}^{\beta}p\left( x\right) G_l\left( g \left( x\right) ,s\right) dx\right)
\left( s-t\right) ^{n-3}P^{\left[ a,b\right] }\left( t,s\right)
ds\right) dt.\label{FinkMainIdentity1int}
\end{eqnarray}
\end{theorem}
\begin{proof}
Since the proof is similar to that of the previous theorem, we omit the details.
\end{proof}
Here we introduce some notations which will be used in rest of the paper:
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
   \Omega_1^{[a,b]}(m,\mathbf{x},\mathbf{p},t)&=&\int_{a}^{b} \sum_{i=1}^{m}p_{i}G_l\left( x_{i},s\right) \left( s-t\right)
^{n-3}P^{\left[ a,b\right] }\left( t,s\right) ds, \label{omega3} \\
   \Omega_2^{[a,b]}([\alpha, \beta],g,p,t)&=&\int_{a}^{b}
\int_{\alpha}^{\beta}p\left( x\right) G_l\left( g \left( x\right) ,s\right) dx
\left( s-t\right) ^{n-3}P^{\left[ a,b\right] }\left( t,s\right)
ds.\nonumber\\
\label{omega4}
\end{eqnarray}
\begin{eqnarray}\nonumber
   A_1^{[a,b]}(m,\mathbf{x},\mathbf{p},f)&=&\sum_{i=1}^{m}p_{i}f\left( x_{i}\right)- \sum_{k=0}^{n-3}\left( \frac{n-k-2}{k!\left( b-a\right) }\right)
\int_{a}^{b} \sum_{i=1}^{m}p_{i}G_l\left( x_{i},s\right)
   \notag \\
&\times& \left( f^{\left( k+1\right) }\left( b\right) \left( s-b\right)
^{k}-f^{\left( k+1\right) }\left( a\right) \left( s-a\right) ^{k}\right) ds \label{A3}\\
A_2^{[a,b]}([\alpha, \beta],g,p,f)&=&\int_{\alpha}^{\beta}p\left( x\right) f\left( g\left( x\right) \right)
dx \nonumber\\&-& \sum_{k=0}^{n-3}\left( \frac{n-k-2}{k!\left( b-a\right) }\right)
\int_{a}^{b}\left( \int_{\alpha}^{\beta}p\left( x\right) G_l\left( g \left(
x\right) ,s\right) dx\right)  \notag \\
&\times& \left( f^{\left( k+1\right) }\left( b\right) \left( s-b\right)
^{k}-f^{\left( k+1\right) }\left( a\right) \left( s-a\right) ^{k}\right) ds.\label{A4}
\end{eqnarray}
The following theorem is our second main result of this section:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    Fink:  2nd Main Result n-convexity     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{F9}
Let all the assumptions of Theorem \ref{FMTiu} be satisfied and let for $n\geq3$, the inequality
\begin{equation}\label{F2.1G}
 \Omega_1^{[a,b]}(m,\mathbf{x},\mathbf{p},t)\geq 0
\end{equation}
holds. If $f$ is $n$-convex, then we have
\begin{eqnarray}\label{F2.2G}
&&A_1^{[a,b]}(m,\mathbf{x},\mathbf{p},f)\ge0.
\end{eqnarray}%
If opposite inequality holds in $\left(\ref{F2.1G}\right)$, then $\left(\ref{F2.2G}\right)$ holds in the reverse direction.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Fink:  2nd Main Result n-convexity Proof    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Since $f^{\left(n-1\right)}$ is absolutely continuous on $\left[a,b\right]$, $f^{\left(n\right)}$ exists almost everywhere. As $f$ is $n$-convex, applying definition, we have, $f^{\left(n\right)}\left(x\right)\geq 0$ for all $x\in \left[a,b\right]$. Now by using $f^{\left(n\right)}\geq 0$ and $\left(\ref{F2.1G}\right)$ in $\left(\ref{FinkMainIdentityG}\right)$, we have $\left(\ref{F2.2G}\right)$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Integral Version:  Fink:  2nd Main Result n-convexity     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{corollary}\label{Cor1G}
Let all the assumptions of Theorem $\ref{FMTiu}$ be satisfied. In addition we let
$$\sum_{i=1}^mp_i(x_i-x_k)_+\ge0\quad\text{for}\quad k\in\{1,\ldots,m\}.$$
Let $n$ be even and $n>3$. If the function $f:\left[a,b\right]\rightarrow \mathbb {R}$ is $n$-convex, then inequality $(\ref{F2.2G})$ is satisfied, $i.~e.$
\begin{eqnarray}  \label{F2.3G}
&&\sum_{i=1}^{m}p_{i}f\left( x_{i}\right) \geq   \notag  \sum_{k=0}^{n-3}\frac{n-k-2}{k!\left( b-a\right) }
\int_{a}^{b}\sum_{i=1}^{m}p_{i}G_l\left( x_{i},s\right)   \notag \\
&&\times \left( f^{\left( k+1\right) }\left( b\right) \left( s-b\right)
^{k}-f^{\left( k+1\right) }\left( a\right) \left( s-a\right) ^{k}\right) ds.
\end{eqnarray}
Further if $f^{\left(k+1\right)}\left(a\right)\leq 0$ and $(-1)^kf^{\left(k+1\right)}\left(b\right)\geq 0$ for $k\in\{0,1,\ldots,n-3\}$ then $\sum_{i=1}^mp_if(x_i)\ge0$.
    %and we have inequality
%\begin{equation}\label{fconvex}
%\sum_{i=1}^{m}p_{i}f\left(x_{i}\right)\geq \sum_{i=1}^{m}p_{i}f\left(y_{i}\right).
%\end{equation}
\end{corollary}
%%\begin{corollary}\label{Cor1}
%%%Let all the assumptions of Theorem \ref{FMT} be satisfied and let ${\bf x}=\left(x_1,\ldots,x_m\right)$, ${\bf y}=\left(y_1,\ldots,y_m\right)$ be two decreasing real $m$-tuples such that $\left(\ref{mc1}\right)$ and $\left(\ref{mc2}\right)$ holds.\\
%%Let all the assumptions of Theorem \ref{FMT} be satisfied and
%%let the function $f:\left[a,b\right]\rightarrow \mathbb{R}$ be $n$-convex
%%for even $n$, where $n>3$. Let $\mathbf{x}=\left(x_1,\ldots,x_m\right)$ and $%
%%\mathbf{y}=\left(y_1,\ldots,y_m\right)$ be two decreasing real $m$-tuples
%%such that $\left(\ref{mc1}\right)$ and $\left(\ref{mc2}\right)$ holds.
%%\begin{itemize}
%%\item[(i)] %Let $n$ be even and $n>3$. If the function $f:\left[a,b\right]\rightarrow \mathbb {R}$ is $n$-convex, then $\left(\ref{F2.2}\right)$ holds.
%%If $\vartheta \left( x\right) :={\left( x-t\right) ^{n-1}}k^{%
%%\left[ a,b\right] }\left( t,x\right) $, where $x,t \in \left[a,b\right]$, then $\left( \ref{F2.2}\right) $
%%holds.
%%\item[(ii)] Let the inequality $\left(\ref{F2.2}\right)$ be satisfied. If $\tilde{\zeta}%
%%\left(x\right):=\left(x-a\right)^k$ and $\gamma\left(x\right):=\left(x-b\right)^k$, where $x \in \left[a,b\right]$ and $k=1,\ldots,m$ and if for even $k$; $f^{\left(k-1\right)}\left(a\right)$ $\leq 0$ and $f^{\left(k-1\right)}\left(b\right)\geq 0$ and for odd $k$; $f^{\left(k-1\right)}\left(a\right)\leq 0$ and $f^{\left(k-1\right)}\left(b\right)\leq 0$, then the right hand side of $\left( \ref{F2.2}\right)$ is non-negative and we have inequality
%%\begin{equation}\label{fconvex}
%%\sum_{i=1}^{m}p_{i}f\left(x_{i}\right)\geq \sum_{i=1}^{m}p_{i}f\left(y_{i}\right).
%%\end{equation}
%%\end{itemize}
%%\end{corollary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   Fink: 1st Corollary,  majorization theorem for the two decreasing $m$-tuples Proof   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof} We fix $l\in \{1,2,3,4\}$ and $n>3$.
As $\mathbf{x}$ and $\mathbf{p}$ are real $m$-tuples such that they satisfy the assumption $(\ref{cond2})$, by using the convex function $x\mapsto G_l\left( x,s\right)$  in $\left(\ref{discopineq}\right)$, we obtain
\begin{equation}\label{GCE}
\sum_{i=1}^{m}p_{i}G_l\left( x_{i},s\right)
\geq 0.
\end{equation}
For $a \leq s \leq t$, it is easy to see that
\begin{equation}\label{G1}
\int_{a}^{t}\sum_{i=1}^{m}p_{i}G_l\left(
x_{i},s\right) \left(
s-t\right) ^{n-3}P^{\left[ a,b\right] }\left( t,s\right) ds \geq 0
\end{equation}
holds for even $n$. Now as $f$ is $n$-convex for even $n$, by applying Theorem \ref{F9}, we get $\left( \ref{F2.3G} \right)$.

If $a \leq s \leq b$ and $k\in\{0,\ldots,n-3\}$, then from assumptions $f^{\left(k+1\right)}\left(a\right)\leq 0$ and $(-1)^kf^{\left(k+1\right)}\left(b\right)\geq 0$ we have that
\begin{equation}
f^{(k+1)}(b)\left(s-b\right)^{k}-f^{(k+1)}(a)\left(s-a\right)^{k}\geq 0,  \label{C1G}
\end{equation}
So, from inequalities $\left( \ref{F2.3G}\right)$,  $\left( \ref{GCE}\right)$ and $\left( \ref{C1G}\right)$ the non-negativity of the right hand side of $\left( \ref{F2.3G}\right)$ is immediate.
\end{proof}

An integral version of our second main result states that:
\begin{theorem}\label{F10}
Let all the assumptions of Theorem $\ref{FMTintiu}$ be satisfied and let for $n\geq 3$, the inequality
\begin{equation} \label{omega4.1}
\Omega_2^{[a,b]}([\alpha, \beta],g,p,t)\geq 0
\end{equation}%
holds. If $f$ is $n$-convex, then we have
\begin{eqnarray}\label{F2.2intG}
&&A_2^{[a,b]}([\alpha, \beta],g,p,f)\ge0.
\end{eqnarray}%
If opposite inequality holds in $\left(\ref{omega4}\right)$, then $\left(\ref{F2.2intG}\right)$ holds in the reverse direction.
\end{theorem}
\begin{proof}
The idea of the proof is the same as that of the proof of Theorem \ref{F9}. By using $f^{\left(n\right)}\geq 0$ and $\left(\ref{omega4}\right)$ in $\left(\ref{FinkMainIdentity1int}\right)$, we have $\left(\ref{F2.2intG}\right)$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   Fink: 1st Corollary,  majorization theorem for the two decreasing $m$-tuples   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Integral Version: Fink: 1st Corollary,  majorization theorem for the two decreasing $m$-tuples   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{corollary}
Let all the assumptions of Theorem $\ref{FMTintiu}$ be satisfied. In addition we let
$$\int_{\alpha}^{\beta} p(x) \left( g(x) - t \right)_+^{n-1} \, dx \geq 0, \quad \hbox{ for every } t\in [a,b].$$
Let $n$ be even and $n>3$. If the function $f:\left[a,b\right]\rightarrow \mathbb {R}$ is $n$-convex, then we have
\begin{eqnarray}\label{F2.2Gint}
&&\int_{\alpha}^{\beta}p\left( x\right) f\left( g \left( x\right) \right)
dx\geq
\sum_{k=0}^{n-3} \frac{n-k-2}{k!\left( b-a\right)}
\int_{a}^{b}\left( \int_{\alpha}^{\beta}p\left( x\right) G_l\left( g
\left( x\right) ,s\right) dx\right)   \notag \\
&&\times \left( f^{\left( k+1\right) }\left( b\right) \left( s-b\right)
^{k}-f^{\left( k+1\right) }\left( a\right) \left( s-a\right) ^{k}\right) ds.
\end{eqnarray}%
Further if $f^{\left(k+1\right)}\left(a\right)\leq 0$ and $(-1)^kf^{\left(k+1\right)}\left(b\right)\geq 0$ for $k\in\{0,1\ldots,n-3\}$, then the right hand side of $\left( \ref{F2.2Gint}\right)$ is non-negative.
   % and we have inequality
%\begin{eqnarray*}\label{ineqfirst}
%\int_{\alpha}^{\beta}p\left( z\right) f\left( \varphi \left( z\right) \right)dz
%\geq\int_{\alpha}^{\beta}p\left( z\right) f\left( \psi \left( z\right) \right)dz.
%\end{eqnarray*}
\end{corollary}
\begin{proof}
The proof is analogous to the proof of Corollary $\ref{Cor1G}$ but instead of Theorem $\ref{F9}$, we apply Theorem $\ref{F10}$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   Fink: 2nd Corollary,  majorization theorem for the two majorized $m$-tuples   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%For the two $m$-tuples ${\bf x}$ and ${\bf y}$ such that ${\bf x}\succ {\bf y}$, the following corollary presents a refinement of the majorization-type inequality.
%\begin{corollary}
%Let all the assumptions of Theorem \ref{FMT} be satisfied and let ${\bf x}=\left(x_1,\ldots,x_m\right)$ and ${\bf y}=\left(y_1,\ldots,y_m\right)$ be two real $m$-tuples such that ${\bf x}\succ {\bf y}$.
%\begin{itemize}
%\item[$\left(i\right)$] Let $n$ be even and $n>3$. If the function $f:\left[a,b\right]\rightarrow \mathbb {R}$ is $n$-convex, then we have
%\begin{eqnarray}\label{Cor2}
%&&\sum_{i=1}^{m}f\left( x_{i}\right) -\sum_{i=1}^{m}f\left(
%y_{i}\right)\geq \notag \\
%&& \sum_{k=0}^{n-3}\left( \frac{n-k-2}{k!\left( b-a\right) }\right)
%\int_{a}^{b}\left( \sum_{i=1}^{m}G_l\left( x_{i},s\right)
%-\sum_{i=1}^{m}G_l\left( y_{i},s\right) \right)   \notag \\
%&& \times \left( f^{\left( k+1\right) }\left( b\right) \left( s-b\right)
%^{k}-f^{\left( k+1\right) }\left( a\right) \left( s-a\right) ^{k}\right) ds.
%\end{eqnarray}
%\item[$\left(ii\right)$] Let the inequality $\left(\ref{Cor2}\right)$ be satisfied. If for even $k$; $f^{\left(k+1\right)}\left(a\right)\leq 0$ and $f^{\left(k+1\right)}\left(b\right)\geq 0$ and for odd $k$; $f^{\left(k+1\right)}\left(a\right)\leq 0$ and $f^{\left(k+1\right)}\left(b\right)\leq 0$, then the right hand side of $\left( \ref{Cor2}\right)$ is non-negative.
%    % and we have inequality
%%\begin{equation*}\label{fconvex1}
%%\sum_{i=1}^{m}f\left(x_{i}\right)\geq \sum_{i=1}^{m}f\left(y_{i}\right).
%%\end{equation*}
%\end{itemize}
%\end{corollary}
%\begin{corollary}
%%Let $f:[a,b]\rightarrow \mathbb {R}$ be such that for $n\geq 1$, $f^{(n-1)}$ is absolutely continuous. Let $x_i,y_i\in [a,b]$, $p_i \in \mathbb R$ $\left(i=1,\ldots,m\right)$
%Let all the assumptions of Theorem \ref{FMT} be satisfied and let the function $f:\left[a,b\right]\rightarrow \mathbb {R}$ be $n$-convex for even $n$, where $n>3$. Let ${\bf x}=\left(x_1,\ldots,x_m\right)$ and ${\bf y}=\left(y_1,\ldots,y_m\right)$ be two real $m$-tuples such that ${\bf x}\succ {\bf y}$.
%\begin{itemize}
%\item[(i)] If $\vartheta \left( x\right) :={\left( x-t\right) ^{n-1}}P^{\left[ a,b\right] }\left( t,x\right) $, where $x,t \in \left[a,b\right]$ , then we have
%\begin{eqnarray}
%&&\sum_{i=1}^{m}f\left(x_{i}\right)-\sum_{i=1}^{m}f\left(y_{i}\right) \geq \label{Cor2}\\
%&&\sum_{k=1}^{n-1}\left( \frac{n-k}{\text{ }k!\left( b-a\right) }\right)
%\left(
%\begin{array}{c}
%f^{\left(k-1\right)}\left(a\right)\left(
%\sum_{i=1}^{m}\left(y_{i}-a\right)^{k}-\sum_{i=1}^{m}\left(x_{i}-a\right)^{k}\right)  \\
%-f^{\left(k-1\right)}\left(b\right)\left(
%\sum_{i=1}^{m}\left(y_{i}-b\right)^{k}-\sum_{i=1}^{m}\left(x_{i}-b\right)^{k}\right)
%\end{array}
%\right).\notag
%\end{eqnarray}
%\item[(ii)] Let the inequality $\left(\ref{Cor2}\right)$ be satisfied.  If $\tilde{\zeta}%
%\left(x\right):=\left(x-a\right)^k$ and $\gamma\left(x\right):=\left(x-b\right)^k$, where $x \in \left[a,b\right]$ and $k=1,\ldots,m$ and if for even $k$; $f^{\left(k-1\right)}\left(a\right)\leq 0$ and $f^{\left(k-1\right)}\left(b\right)\geq 0$ and for odd $k$; $f^{\left(k-1\right)}\left(a\right)\leq 0$ and $f^{\left(k-1\right)}\left(b\right)\leq 0$, then the right hand side of $\left( \ref{Cor2}\right)$ is non-negative and we have inequality
%\begin{equation*}\label{fconvex1}
%\sum_{i=1}^{m}f\left(x_{i}\right)\geq \sum_{i=1}^{m}f\left(y_{i}\right).
%\end{equation*}
%\end{itemize}
%\end{corollary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   Fink: 2nd Corollary,  majorization theorem for the two majorized $m$-tuples Proof  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{proof}
%\begin{itemize}
%\item[(i)]
%As ${\bf x}=\left(x_1,\ldots,x_m\right)$ and ${\bf y}=\left(y_1,\ldots,y_m\right)$ be two real $m$-tuples such that ${\bf x}\succ {\bf y}$ and as $G \left(x,s\right)$
% is convex, by applying Theorem \ref{Majorization} for the convex function $G \left(x,s\right)$, we have
%\begin{equation*}\label{Cor2.1}
%\sum_{i=1}^{m}G_l\left( x_{i},s\right)
%-\sum_{i=1}^{m}G_l\left( y_{i},s\right)\geq 0.
%\end{equation*}
%Now following the same steps as given in the proof of Corollary $\ref{Cor1}$ $\left(i\right)$, we obtain $\left( \ref{F2.1}\right)$ for each $p_i =1$ $\left(i=1,\ldots,m\right)$. Now as $f$ is $n$-convex for even $n$, where $n>3$, we apply Theorem \ref{F2} for each $p_i =1$ $\left(i=1,\ldots,m\right)$ and $\left(\ref{Cor2}\right)$ is immediate.
%\item[(ii)]
%It is similar to the proof of Corollary $\ref{Cor1}$ $\left(ii\right)$.
%\end{itemize}
%\end{proof}

%
\section{Related inequalities for $n$-convex functions at a point} \label{Sect3}

In this section we will give related results for the class of $n$-convex functions at a point introduced in \cite{PPW}.

\begin{definition} \label{defKI}
Let $I$ be an interval in $\mathbb{R}$, $c$ a point in the interior of $I$ and $n\in \mathbb{N}$. A function $ f: I\to \mathbb{R} $ is said to be $n$-convex at point $c$ if there exists a constant $K$ such that the function
\begin{equation*}
F(x) = f(x) - \frac{K}{(n-3)!} x^{n-1}
\end{equation*}
is $(n-1)$-concave on $I\cap (-\infty,c]$ and $(n-1)$-convex on $I\cap [c,\infty)$. A function $f$ is said to be $n$-concave at point $c$ if the function $-f$ is $n$-convex at point $c$.
\end{definition}

Let $e_i$ denote the monomials $e_i (x) = x^i$, $i\in \mathbb{N}_0$.
First we state main results for  discrete case.
\begin{theorem}
\label{disc1atc}
Let $c\in (a,b)$, $\mathbf{x} \in [a,c]^m$, $\mathbf{y} \in [c,b]^l$, $\mathbf{p} \in \mathbb{R}^m$, $\mathbf{q} \in \mathbb{R}^l$ and $f:[a,b]\to \mathbb{R}$ be a function such that $f^{(n-1)}$ is absolutely continuous.
Let $\Omega_1^{[\cdot,\cdot]}(\cdot,\cdot,\cdot,t)$ and $A_1^{[\cdot,\cdot]}(\cdot,\cdot,\cdot,f)$be defined as in $(\ref{omega3})$ and $(\ref{A3})$ and satisfy the following conditions:
\begin{equation}\label{4.1}
\Omega_1^{[a,c]}(m,\mathbf{x},\mathbf{p},t)\ge0 \quad \text{for every}\quad t \in [a,c],
\end{equation}
\begin{equation}\label{4.2}
\Omega_1^{[c,b]}(l,\mathbf{y},\mathbf{q},t)\ge0 \quad \text{for every}\quad t \in [c,b],
\end{equation}
and
\begin{equation}\label{4.3}
A_1^{[a,c]}(m,\mathbf{x},\mathbf{p},e_n)= A_1^{[c,b]}(l,\mathbf{y},\mathbf{q},e_n).
\end{equation}
If $f$ is $(n+1)$-convex at point $c$, then
\begin{equation}\label{4.4}
A_1^{[a,c]}(m,\mathbf{x},\mathbf{p},f)\le A_1^{[c,b]}(l,\mathbf{y},\mathbf{q},f).
\end{equation}
If inequalities in $(\ref{4.1})$ and $(\ref{4.2})$ are reversed, then $(\ref{4.4})$ holds with the reverse sign of inequality.
\end{theorem}
\begin{proof}
Let $F = f - \frac{K}{n!} e_{n}$ be as in Definition \ref{defKI}, {\it i.~e.}, the function $F$ is $n$-concave on $[a,c]$ and $n$-convex on $[c,b]$. Applying Theorem \ref{F9} to $F$ on the interval $[a,c]$ and on the interval $[c,b]$ we have
\begin{equation*}
A_1^{[a,c]}(m,\mathbf{x},\mathbf{p},F)\le 0 \le A_1^{[c,b]}(l,\mathbf{y},\mathbf{q},F).
\end{equation*}
Using definition of $F$ we obtain
\begin{equation*}
A_1^{[a,c]}(m,\mathbf{x},\mathbf{p},f)-\frac{K}{n!}A_1^{[a,c]}(m,\mathbf{x},\mathbf{p},e_n)\le A_1^{[c,b]}(l,\mathbf{y},\mathbf{q},f)-\frac{K}{n!}A_1^{[c,b]}(l,\mathbf{y},\mathbf{q},e_n).
\end{equation*}
Since equality (\ref{4.3}) is valid we get
\begin{equation*}
A_1^{[a,c]}(m,\mathbf{x},\mathbf{p},f)\le A_1^{[c,b]}(l,\mathbf{y},\mathbf{q},f).
\end{equation*}
\end{proof}

\begin{remark}
A closer look at the proof of Theorem \ref{disc1atc} gives us that a similar result hold if instead equality (\ref{4.3}) we consider a positivity  of the difference
$$K\left( A_k^{[c,b]}(l,\mathbf{y},\mathbf{q},e_n) - A_k^{[a,c]}(m,\mathbf{x},\mathbf{p},e_n) \right) \geq 0$$
\end{remark}


\begin{corollary} \label{cor1atc}
Let $j_1, j_2,n\in \mathbb{N}$, $2\leq j_1,j_2 \leq n$ and let $f:[a,b]\to \mathbb{R}$ be $(n+1)$-convex at point $c$. Let $m$-tuples $\mathbf{x} \in [a,c]^m$ and $\mathbf{p} \in \mathbb{R}^m$ satisfy $(\ref{cond4})$ and $(\ref{cond4b})$ with $n$ replaced by $j_1$, let $l$-tuples $\mathbf{y} \in [c,b]^l$ and $\mathbf{q} \in \mathbb{R}^l$ satisfy
\begin{gather*}
\sum_{i=1}^l q_i y_i^k=0, \quad \hbox{ for all } k=0,1,\ldots,j_2 -1 \\
\sum_{i=1}^l q_i (y_i-t )_+^{j_2-1}\ge 0, \quad \hbox{ for every } t\in [y_{(1)}, y_{(l-n+1)}]
\end{gather*}
and let $(\ref{4.3})$ holds. If $n-j_1$ and $n-j_2$ are even, then $(\ref{4.4})$ holds.
\end{corollary}
\begin{remark}
For idea of the proof  see \cite[pp. 171-172]{book}..
\end{remark}
Integral analogous of previous theorem may be stated as:
\begin{theorem}\label{thm1con}
Let $c\in (a,b)$ and let $g:[\alpha, \beta]\to [a,c]$, $p:[\alpha, \beta]\to \mathbb{R}$, $h:[\gamma, \delta]\to [c,b]$, $q:[\gamma, \delta]\to \mathbb{R}$ be integrable functions. Let $f:I\to \mathbb{R}$, $[a,b]\subset  I$ be a function such that $f^{(n-1)}$ is absolutely continuous.
Let $\Omega_2^{[\cdot,\cdot]}(\cdot,\cdot,\cdot,t)$ and $A_2^{[\cdot,\cdot]}(\cdot,\cdot,\cdot,f)$ be defined as in $(\ref{omega4})$ and $(\ref{A4})$ satisfy the following conditions:
\begin{equation}\label{4.5}
\Omega_2^{[a,c]}([\alpha, \beta],g,p,t)\ge0 \quad \text{for every}\quad t \in [a,c],
\end{equation}
\begin{equation}\label{4.6}
\Omega_2^{[c,b]}([\gamma, \delta],h,q,t)\ge0 \quad \text{for every}\quad t \in [c,b],
\end{equation}
and
\begin{equation}\label{4.7}
A_2^{[a,c]}([\alpha, \beta],g,p,e_n)= A_2^{[c,b]}([\gamma, \delta],h,q,e_n).
\end{equation}
If $f$ is $(n+1)$-convex at point $c$ (for $k=3$, $n\ge3$), then
\begin{equation}\label{4.8}
A_2^{[a,c]}([\alpha, \beta],g,p,f)\le A_2^{[c,b]}([\gamma, \delta],h,q,f).
\end{equation}
If inequalities in $(\ref{4.5})$ and $(\ref{4.6})$ are reversed, then $(\ref{4.8})$ holds with the reverse sign of inequality.
\end{theorem}
\begin{corollary}\label{cor1con}
Let $j_1, j_2,n\in \mathbb{N}$, $2\leq j_1,j_2 \leq n$ and let $f:[a,b]\to \mathbb{R}$ be $(n+1)$-convex at point $c$. Let integrable functions $g:[\alpha, \beta]\to [a,c]$, $p:[\alpha, \beta]\to \mathbb{R}$ satisfy $(\ref{cond5})$ with $n$ replaced by $j_1$, let $h:[\gamma, \delta]\to [c,b]$, $q:[\gamma, \delta]\to \mathbb{R}$ satisfy
\begin{gather*}
\begin{split}
\int_{\gamma}^{\delta} q(x) h(x)^k \, dx = 0, \quad \hbox{ for all } k\in \{0,1,\ldots,j_2-1\} \\
\int_{\gamma}^{\delta} q(x) \left( h(x) - t \right)_+^{j_2-1} \, dx \geq 0, \quad \hbox{ for every } t\in [c,b].
\end{split}
\end{gather*}
and let $(\ref{4.7})$ holds. If $n-j_1$ and $n-j_2$ are even, then $(\ref{4.4})$ holds.
\end{corollary}

\section{Bounds for $A_k^{[\cdot,\cdot]}(\cdot,\cdot,\cdot,f)$ and $R_n^k$} \label{Sect4}
\section{Bounds for identities related to generalized linear inequalities}
Let $f,h:[a,b]\rightarrow\mathbb{R}$ be two Lebesgue integrable functions. We
consider the \v{C}eby\v{s}ev functional
\begin{equation}
T(f,h)=\frac{1}{b-a}\int_{a}^{b}f(x)h(x)dx-\left(  \frac{1}{b-a}\int_{a}%
^{b}f(x)dx\right)  \left(  \frac{1}{b-a}\int_{a}^{b}h(x)dx\right)  . \label{T}%
\end{equation}
The following results can be found in \cite{Dragomir}:

\begin{proposition}
\label{prop4.11} Let $f:[a,b]\rightarrow\mathbb{R}$ be a Lebesgue integrable
function and let $h:[a,b]\rightarrow\mathbb{R}$ be an absolutely continuous
function with $(\cdot-a)(b-\cdot)[h^{\prime}]^{2}\in L[a,b]$. Then we have the
inequality
\begin{equation}
|T(f,h)|\leq\frac{1}{\sqrt{2}}\left(  \frac{1}{b-a}|T(f,f)|\int_{a}%
^{b}(x-a)(b-x)[h^{\prime}(x)]^{2}dx\right)  ^{1/2}. \label{T1}%
\end{equation}
The constant $\frac{1}{\sqrt{2}}$ in $(\ref{T1})$ is the best possible.
\end{proposition}

\begin{proposition}
\label{prop5} Let $h : [a, b] \to\mathbb{R}$ be a monotonic nondecreasing
function and let $f : [a, b] \to\mathbb{R}$ be an absolutely continuous
function such that $f^{\prime}\in L_{\infty}[a, b]$. Then we have the
inequality
\begin{equation}
\label{T2}|T(f,h)|\le\frac{1}{2(b-a)}\|f^{\prime}\|_{\infty}\int_{a}%
^{b}(x-a)(b-x)dh(x).
\end{equation}
The constant $\frac{1}{2}$ in $(\ref{T2})$ is the best possible.
\end{proposition}
We use the well-known H\"olders inequality and bound for the \v{C}eby\v{s}ev functional $T(f,h)$.
This bound is given in the following proposition in which the pre-Gr\"uss inequality is given \cite{Matic-Pecaric}.

\begin{proposition}
\label{prop4.1} Let $f,h:[a,b]\rightarrow\mathbb{R}$ be Lebesgue integrable
functions such that $fh:[a,b]\in L(a,b)$. If
$$\gamma \le h(x)\le \Gamma \quad \text{for} \quad x\in [a,b],$$
then
\begin{equation}
|T(f,h)|\leq\frac{1}{2}(\Gamma-\gamma)\sqrt{T(f,f)}, \label{pre-G}%
\end{equation}
\end{proposition}

Now by using aforementioned results, we are going to obtain generalizations of
the result proved in the previous section.

\begin{remark}
For the sake of brevity, in present and next sections at some places we will use the notations $A_k(f)=A^{[\cdot,\cdot]}_k(\cdot,\cdot,\cdot,f)$ and $\Omega_k(t)=\Omega^{[\cdot,\cdot]}_k(\cdot,\cdot,\cdot,t)$  for $k\in\{1,2\}$ as defined in Theorems $\ref{F9}$ and $\ref{F10}$.
\end{remark}

Now, we are ready to state main results of this section:
\begin{theorem}
Let $f:[a,b]\rightarrow\mathbb{R}$ be such that $f^{(n)}$ is an absolutely continuous function for
$n\in\mathbb{N}$ with $(.-a)(b-.)[f^{(n+1)}]^{2}\in L[a,b]$. Then it holds for $k\in\{1,2\}$
\begin{eqnarray}\nonumber
A_k(f)=\frac{\left[  f^{(n-1)}(b)-f^{(n-1)}(a)\right]  }{(n-3)!(b-a)}\int
_{a}^{b}\Omega_k(s)ds+R_{n}^{k}(f;a,b),\label{disineq2}
\end{eqnarray}
where the remainder $R_{n}^{k}(f;a,b)$ satisfies the estimation
\begin{equation}\label{remainderdis1}
|R_{n}^{k}(f;a,b)|\leq\frac{1}{(n-3)!}\left(  \frac{(b-a)}{2}\left\vert
T(\Omega_k,\Omega_k)\int_{a}^{b}(s-a)(b-s)[f^{(n+1)}(s)]^{2}ds\right\vert \right)
^{1/2}.
\end{equation}
\end{theorem}
%\begin{theorem}
%Let $f:[a,b]\rightarrow\mathbb{R}$ be such that $f\in C^{n}[a,b]$ for
%$n\in\mathbb{N}$ and $f^{(n)}$ be an absolutely continuous functions with $(.-a)(b-.)[f^{(n+1)}]^{2}\in L[a,b]$ and $x_{i}\in\lbrack a,b]$ and $p_{i}\in\mathbb{R}$ $(i\in\{1,\ldots,m\})$ such that $\sum_{i=0}^{m}p_i=0$ and let the
%functions $T_{n}$, $T$ and $\delta$ be defined in $(\ref{Tn}),\;(\ref{T})$ and
%$(\ref{delta})$ respectively. Then it holds
%\begin{eqnarray}\nonumber
%  \sum_{i=1}^{m}p_{i}f\left(  x_{i}\right) &=&\frac{1}{b-a}\left[  \sum_{k=0}^{n-2}\frac{1}{k!\left(  k+2\right)
%}f^{\left(  k+1\right)  }\left(  a\right)\sum_{i=1}^{m}p_{i}
%\left(  x_{i}-a\right)  ^{k+2}  \right.  \\  &-&\left.\sum_{k=0}^{n-2}\frac{1}{k!\left(  k+2\right)
%}f^{\left(  k+1\right)  }\left(  b\right)\sum_{i=1}^{m}p_{i}  \left(
%x_{i}-b\right)  ^{k+2}  \right]\nonumber\\
%&+&\frac{\left[  f^{(n-1)}(b)-f^{(n-1)}(a)\right]  }{(n-3)!(b-a)}\int
%_{a}^{b}\delta(t)dt+R_{n}^{1}(f;a,b),\label{ineq1}
%\end{eqnarray}
%where the remainder $R_{n}^{1}(f;a,b)$ satisfies the estimation
%\begin{equation}
%|R_{n}^{1}(f;a,b)|\leq\frac{1}{(n-3)!}\left(  \frac{b-a}{2}\left\vert
%T(\delta,\delta)\int_{a}^{b}(t-a)(b-t)[f^{(n+1)}(t)]^{2}dt\right\vert \right)
%^{1/2}. \label{remainder}%
%\end{equation}
%
%\end{theorem}

\begin{proof}
Fix $k\in\{1,2\}$. If we apply Proposition \ref{prop4.1} for $f\rightarrow\Omega_k$ and $h\rightarrow
f^{(n)}$, then we obtain
\begin{multline*}
\left\vert \frac{1}{b-a}\int_{a}^{b}\Omega_k(t)f^{(n)}(t)dt-\left(  \frac
{1}{b-a}\int_{a}^{b}\Omega_k(t)dt\right)  \left(  \frac{1}{b-a}\int_{a}%
^{b}f^{(n)}(t)dt\right)  \right\vert \\
  \leq\frac{1}{\sqrt{2}}\left(  \frac{1}{b-a}|T(\Omega_k,\Omega_k)|\int_{a}%
^{b}(t-a)(b-t)[f^{(n+1)}(t)]^{2}dt\right)  ^{1/2}.
\end{multline*}
Therefore we have
\begin{eqnarray*}
\frac{1}{(n-3)!}\int_{a}^{b}\Omega_k(t)f^{(n)}(t)dt=\frac{\left[
f^{(n-1)}(b)-f^{(n-1)}(a)\right]  }{(n-3)!(b-a)}\int_{a}^{b}\Omega_k(t)dt
+R_{n}^{k}(f;a,b).
\end{eqnarray*}
where $R_{n}^{k}(f;a,b)$ satisfies inequality (\ref{remainderdis1}). Now from
identities $(\ref{FinkMainIdentityG})$ and $(\ref{FinkMainIdentity1int})$ for $k\in\{1,2\}$ respectively, we obtain $(\ref{disineq2})$.
\end{proof}
By using Proposition \ref{prop5} we obtain the following Gr\"uss type inequality.

%\begin{theorem}
%Let $f:[a,b]\rightarrow\mathbb{R}$ be such that $f\in C^{n}[a,b]$ for
%$n\in\mathbb{N}$ with $f^{(n)}\geq0$ on $[a,b]$ and $p_{i}\in\mathbb{R}$ $(i\in\{1,\ldots,m\})$ such that $\sum_{i=0}^{m}p_i=0$. Also let the functions $T$
%and $\delta$ be defined in $(\ref{T})$ and $(\ref{delta})$ respectively. Then
%we have the representation $(\ref{ineq1})$ and the remainder $R_{n}%
%^{1}(f;a,b)$ satisfies the following condition
%\begin{equation}
%|R_{n}^{1}(f;a,b)|\leq\frac{1}{(n-3)!}\Vert\delta^{\prime}\Vert_{\infty
%}\left[  \frac{f^{(n-1)}(b)+f^{(n-1)}(a)}{2}-\frac{f^{(n-2)}(b)-f^{(n-2)}%
%(a)}{b-a}\right]  . \label{remainder2}%
%\end{equation}
%
%\end{theorem}
\begin{theorem}
Let $f:[a,b]\rightarrow\mathbb{R}$ be such that $f^{(n)}$ is an absolutely continuous function for
$n\in\mathbb{N}$ with $(.-a)(b-.)[f^{(n+1)}]^{2}\in L[a,b]$ with $f^{(n+1)}\geq0$ on $[a,b]$. Then
we have the representation $(\ref{disineq2})$ and the remainder $R_{n}%
^{k}(f;a,b)$ satisfies the following condition for $k\in\{1,2\}$
\begin{multline}\label{disremainder2}
|R_{n}^{k}(f;a,b)|\leq \frac{1}{(n-3)!}\Vert\Omega_k^{\prime}\Vert_{\infty
}\left\{  \frac{b-a}{2}\left[f^{(n-1)}(b)+f^{(n-1)}(a)\right]\right.\\ \left.-\left[f^{(n-2)}(b)-f^{(n-2)}
(a)\right]\right\}  .
\end{multline}
\end{theorem}
\begin{proof}
Fix $k\in\{1,2\}$. If we apply Proposition \ref{prop5} for $f\rightarrow \Omega_k$ and $h\rightarrow
f^{(n)}$, then we obtain
\begin{multline*}
\left\vert \frac{1}{b-a}\int_{a}^{b}\Omega_k(t)f^{(n)}(t)dt-\left(  \frac
{1}{b-a}\int_{a}^{b}\Omega_k(t)dt\right)  \left(  \frac{1}{b-a}\int_{a}%
^{b}f^{(n)}(t)dt\right)  \right\vert \\
\leq\frac{1}{2(b-a)}\Vert\Omega_k^{\prime}\Vert_{\infty}\int_{a}%
^{b}(t-a)(b-t)f^{(n+1)}(t)dt.
\end{multline*}
Since
\begin{multline}\label{disgruss}
\int_{a}^{b}(t-a)(b-t)f^{(n+1)}(t)dt=\int_{a}^{b}(2t-a-b)f^{(n)}%
(t)dt\\
 =(b-a)\left[  f^{(n-1)}(b)+f^{(n-1)}(a)\right]  -2\left[  f^{(n-2)}%
(b)-f^{(n-2)}(a)\right].
\end{multline}
Therefore, by using the identities $(\ref{FinkMainIdentityG})$ and $(\ref{FinkMainIdentity1int})$ for $k\in\{1,2\}$ respectively and $(\ref{disgruss})$ we deduce
$(\ref{disremainder2})$.
\end{proof}
%Hence we have another similar result:
%\begin{theorem}
%Let $f:[a,b]\rightarrow\mathbb{R}$ be such that $f^{(n)}$ is an absolutely continuous function for
%$n\in\mathbb{N}$ with $(.-a)(b-.)[f^{(n+1)}]^{2}\in L[a,b]$ with $f^{(n+1)}\geq0$ on $[a,b]$ and let $x_i$ and $p_i$, $i\in \{1,\ldots, m\}$ satisfy condition of $(\ref{cond1b})$. Also let the functions $T$ and $\Omega_2$ be defined in $(\ref{T})$ and $(\ref{F2})$ respectively. Then
%we have the representation $(\ref{disineq3})$ and the remainder $R_{n}%
%^{2}(f;a,b)$ satisfies the following condition
%\begin{multline}
%|R_{n}^{2}(f;a,b)|\leq\frac{1}{(n-3)!} \Vert{\Omega_2}^{\prime}\Vert_{\infty
%}\left\{  \frac{b-a}{2}\left[f^{(n-1)}(b)+f^{(n-1)}(a)\right]\right.\\ \left.-\left[f^{(n-2)}(b)-f^{(n-2)}
%(a)\right]\right\}.
%\end{multline}
%\end{theorem}

\begin{theorem} For $k=1$ we assume that $\mathbf{x}$ and $\mathbf{p}$ satisfy the assumptions of Theorem $\ref{FMTiu}$ and for $k=2$ we assume that $x$ and $p$ satisfy the assumptions of Theorem $\ref{FMTintiu}$.
\item[$(i)$] Let $k\in \{1,2\}$. Let $f:I\rightarrow\mathbb{R}$, $[a,b]\subseteq I$, be such that $f^{(n)}$ is an absolutely continuous function and
$$\gamma \le f^{(n)}(x)\le \Gamma \quad \text{for} \quad x\in [a,b].$$
Then
\begin{eqnarray}
  A_k(f)=\frac{\left[  f^{(n-1)}(b)-f^{(n-1)}(a)\right]  }{(n-3)!(b-a)}\int
_{a}^{b}\Omega_k(t)dt+R_{n}^{k}(f;a,b),\label{ineq1}
\end{eqnarray}
where the remainder $R_{n}^{k}(f;a,b)$ satisfies the estimation
\begin{equation}
|R_{n}^{k}(f;a,b)|\leq\frac{b-a}{2(n-3)!}(\Gamma-\gamma)\sqrt{T(\Omega_k,\Omega_k)}.
\label{remainder}%
\end{equation}
 \end{theorem}

\begin{proof}
Fix $k\in \{1,2\}$. Using definition of $A_k$ and result from the second section we have
\begin{eqnarray*}
  A_k(f)&=&\frac{1}{(n-3)!}\int_{a}^{b}f^{(n)}(t)\Omega_k(t)dt\\
  &=&\frac{1}{(n-3)!(b-a)}\int_{a}^{b}f^{(n)}(t)dt\int_a^b\Omega_k(t)dt+R_{n}^{k}(f;a,b)\\
  &=&\frac{\left[  f^{(n-1)}(b)-f^{(n-1)}(a)\right]  }{(n-3)!(b-a)}\int_{a}^{b}\Omega_k(t)dt+R_{n}^{k}(f;a,b),
\end{eqnarray*}
where
\begin{eqnarray*}
R_{n}^{k}(f;a,b)=\frac{1}{(n-3)!}\left(\int_{a}^{b}f^{(n)}(t)\Omega_k(t)dt- \frac{1}{b-a}\int_{a}^{b}f^{(n)}(s)ds\int_{a}^{b}\Omega_k(t)dt\right).
\end{eqnarray*}
If we apply Proposition \ref{prop4.1} for $f\rightarrow\Omega_k$ and $h\rightarrow
f^{(n)}$, then we obtain
$$|R_{n}^{k}(f;a,b)|=|T(\Omega_k,f^{(n)})|\leq\frac{b-a}{2(n-3)!}(\Gamma-\gamma)\sqrt{T(\Omega_k,\Omega_k)}.$$
%The proof for $k\in\{3,4\}$ is done in a similar manner.
\end{proof}
%Using the same method as we used in the previous theorem and other type of bounds for the \v Ceby\v sev functional we are able to give another estimation for a remainder. The following theorem gives us some Ostrowski-type inequalities.
%As usual the symbol $L_{p}\left[  a,b\right]  $ $\left(  1\leq p<\infty\right)  $
%denotes the space of $p$-power integrable functions on the interval $\left[
%a,b\right]  $ equipped with the norm%
%\[
%\left\Vert f\right\Vert _{p}=\left(  \int_{a}^{b}\left\vert f\left(  t\right)
%\right\vert ^{p}dt\right)  ^{\frac{1}{p}}<\infty%
%\]
%and $L_{\infty}\left[  a,b\right] $ denotes the space of essentially bounded
%functions on $\left[  a,b\right]  $ with the norm%
%\[
%\left\Vert f\right\Vert _{\infty}=\text{ess}\sup_{t\in\left[  a,b\right]
%}\left\vert f\left(  t\right)  \right\vert .
%\]
%
 \begin{theorem}
%Let all the assumptions of Theorem $\ref{01}$ hold. Furthermore,
\begin{itemize}
\item[$(i)$] Let $k\in \{1,2\}$. Let $(q,r)$ be  a pair of conjugate exponents, that is, $1\leq q,r\leq\infty$, $\frac{1}{q}+\frac{1}{r}=1$. Let $f^{(n)}\in L_{q}\left[  a,b\right]  $ for some
$n\in\mathbb{N}$, $n>1$. Further, for $k=1$ we assume that $\mathbf{x}$ and $\mathbf{p}$ satisfy the assumptions of Theorem $\ref{FMTiu}$ and for $k=2$ we assume that $x$ and $p$ satisfy the assumptions of Theorem $\ref{FMTintiu}$.
 Then we have
\begin{eqnarray}
\left|A_k(f) \right|
  \leq\frac{1}{(n-3)!}\| f^{(n)}\|_q \|\Omega_k \|_{r}.\label{sharp}
\end{eqnarray}
The constant on the right hand side of $(\ref{sharp})$ is sharp for
$1<q\leq\infty$ and the best possible for $q=1$.
\end{itemize}\end{theorem}

\begin{proof}
Fix $k\in\{1,2\}$. From definition of $A_k$ and results from the second section, applying the H\"older inequality we get
$$|A_k(f)|=\left|\frac{1}{\left( n-3\right) ! }
\int_{a}^{b}f^{\left(n\right)}(t)\Omega_k(t)dt\right|\le \|f^{(n)}\|_q\|\lambda_k\|_r$$
where we denoted
$\frac{1}{\left( n-3\right) ! }\Omega_k$  by $\lambda_k$.

The sharpness of the constant $\left(  \int_{a}^{b}\left\vert
\lambda_k(t)\right\vert ^{r}ds\right)  ^{1/r}$ can be proved by considering the following function $f$ for
which the equality in $(\ref{sharp})$ is obtained.

For $1<q<\infty$ we take $f$ to be such that $f^{(n)}(s)=sgn\lambda_k(t)\cdot|\lambda_k(t)|^{1/(q-1)}$,

while for $q=\infty$, we define $f$ such that
$
f^{(n)}(t)=sgn\lambda_k(t).
$
The fact that (\ref{sharp}) is the best possible for $q=1$, can be proved as in \cite[Thm 12]{APP}.
\end{proof}

\section{Mean Value Results}

In this section we consider mean value theorems involving $A_k$.
 Throughout the section we use this agreement  that if $k \in \{1,2\}$, then $n\ge3$. Further $k=1$ we assume that $\mathbf{x}$ and $\mathbf{p}$ satisfy the assumptions of Theorem \ref{FMTiu} and for $k=2$ we assume that $\mathbf{x}$ and $\mathbf{p}$ satisfy the assumptions of Theorem \ref{FMTintiu}.

\begin{theorem}\label{th3}
Let $k\in \{1,2\}$ and let us consider $A_k$ as a functional on $C^{n}[a,b]$. If corresponding conditions from set $\{(\ref{omega3}),(\ref{omega4})\}$ related to the fixed $k$, hold, then there exists $\xi_{k} \,\,\in\,[a,b]$ such that
\begin{equation}\label{3-8}
A_k(f)=f^{(n)}(\xi_{k})  A_k(f_{0}),
\end{equation}
where $f_0(x)=\frac{x^n}{n!}$.
\end{theorem}

\begin{proof}
Let us define functions
\[
F_1(x)=M f_{0}(x)-f(x)
\]
and
\[
F_2(x)=f(x)-L f_{0}(x)
\]
where $L$ and $M$ are minimum and maximum of the image of $[a,b]$, i.e.,
\[
F^{(n)}([a,b])=[L,M]
\]
Then $F_1$ and $F_2$ are $n-$convex. Hence $A_k (F_1)\geq0$ and $A_k (F_2)\geq0$ and
\[
L  A_k(f_{0})\le A_k (f)\leq M  A_k(f_{0}).
\]
If  $A_k(f_{0})=0$, then the statement obviously holds.

If $A_k(f_{0})\not=0$, then $\frac{A_k(f)}{A_k(f_0)}\in[L,M]=f^{(n)}([a,b])$, so  there exist $\xi_k\in [a,b]$ such that $\frac{A_k(f)}{A_k(f_0)}=f^{(n)}(\xi_k)$.
\end{proof}
Applying Theorem \ref{th3} on function $\omega= A_k(h)f- A_k(f)h$, we get the following result.

\begin{theorem}\label{th2.9}
Let $k\in \{1,2\}$ and let us consider $A_k$ as a functional on $C^{n}[a,b]$. If corresponding conditions from set $\{(\ref{omega3}),(\ref{omega4})\}$ related to the fixed $k$, hold, then there exists $\xi_{k} \,\,\in\,[a,b]$ such that
\[
\frac{ A_k(f)}{ A_k(h)}=\frac{f^{(n)}(\xi_{k})}{h^{(n)}(\xi
_{k})}%
\]
assuming that both the denominators are non-zero.
\end{theorem}
%\begin{proof}
%Fix $k\in\{1,2\}$. Let $h\in C^{n}[a,b]$ be defined as
%\[
%.
%\]
%Using Theorem \ref{th3} there exists $\xi_{k}$ such that
%\[
%0= A_k(\omega)=\omega^{(n)}(\xi_{k}) A_k(f_{0})
%\]
%or
%\[
%[ A_k(h)f^{(n)}(\xi_{k})- A_k(f)h^{(n)}(\xi_{k})]\Lambda
%_{k}(f_{0})=0
%\]
%which gives us the required result.
%\end{proof}
\begin{remark}
If the inverse of $\frac{f^{(n)}}{h^{(n)}}$ exists, then from the above mean
value theorems we can give generalized means
\begin{equation}\label{inverse-mean}
\xi_{k}=\left(  \frac{f^{(n)}}{h^{(n)}}\right)
^{-1}\left(  \frac{  A_k(f)}{  A_k(h)}\right).
\end{equation}
\end{remark}
\begin{remark}
Using the same method as in \cite{APP}, we can construct new families of exponentially convex functions and Cauchy type means.
\end{remark}

%\subsection{Logarithmically Convex Functions}
%A number of important inequalities arise from the logarithmic convexity of
%some functions as one can see in \cite{MO-new}.
%
%Now, we recall some definitions. The following definition is originally given
%by Jensen in 1906 \cite{Jensen-1906}. Here $I$ is an interval in $\mathbb{R}$.
%
%\begin{definition}
%A function $f:I\to(0,\infty)$ is called $\log-$convex in $J-$sense if
%the inequality
%\[
%f^{2}\left(  {\frac{x_{1}+x_{2}}{2}} \right)  \leq f\left(  x_{1}\right)
%f\left(  x_{2}\right)
%\]
%holds for each $x_{1},x_{2} \in I$.
%\end{definition}
%
%\begin{definition}
%\cite[p.~7]{redbook} A function $f:I\to(0,\infty)$ is called \emph{$\log
%-$convex} if the inequality
%\[
%f(\lambda x_{1}+(1-\lambda)x_{2})\leq{[f(x_{1})]}^{\lambda} {[f(x_{2}%
%)]}^{(1-\lambda)}
%\]
%holds for each $x_{1}, x_{2} \in I$ and $\lambda\in[0,1]$.
%\end{definition}
%
%\begin{remark}\label{log-remark}
%A function $\log$-convex in the $J-$sense is $\log$-convex if it is continuous
%as well.
%\end{remark}
%
\bigskip

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\end{document}