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\begin{document}
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\begin{center}{\footnotesize Khayyam J. Math. 1 (2015), no. 1, 45--61}\\\end{center}
\noindent\parbox{2.85cm}{\includegraphics*[keepaspectratio=true,scale=0.24]{KJM.jpg}}
\vspace{0.5cm}

\title[Generalizations of Steffensen's inequality]{Generalizations of Steffensen's inequality by Abel-Gontscharoff polynomial}

\author[J. Pe\v{c}ari\'{c}, A. Peru\v{s}i\'{c}, K. Smoljak]{Josip Pe\v{c}ari\'{c}$^1$, Anamarija Peru\v{s}i\'{c}$^2$ and Ksenija Smoljak$^{3*}$}

\address{$^1$ Faculty of Textile Technology, University of Zagreb,
Prilaz baruna Filipovi\'{c}a~28a, \hbox{10000 Zagreb,} Croatia}
\email{pecaric@element.hr}

\address{$^2$ Faculty of Civil Engineering, University of Rijeka,
Radmile Matej\v ci\' c 3, \hbox{51000 Rijeka,} Croatia}
\email{anamarija.perusic@gradri.hr}


\address{$^3$ Faculty of Textile Technology, University of Zagreb,
Prilaz baruna Filipovi\'{c}a~28a, \hbox{10000 Zagreb,} Croatia}
\email{ksmoljak@ttf.hr}


\dedicatory{{\rm Communicated by A.R. Mirmostafaee}}
\subjclass[2010]{Primary 26D15; Secondary 26D20.}

\keywords{Steffensen's inequality, Abel-Gontscharoff polynomial, Ostrowski type inequality, 
$n-$exponential convexity.}

\date{Received: 28 June 2014;  Accepted: 1 September  2014.
\newline \indent $^{*}$ Corresponding author}
\begin{abstract}
In this paper generalizations of Steffensen's inequality using Abel-Gontscharoff interpolating polynomial are obtained.
Moreover, in a special case generalizations by Abel-Gontscharoff polynomial reduce to known weaker conditions for Steffensen's inequality.
Furthermore, Ostrowski type inequalities related to obtained generalizations are given.
\end{abstract} \maketitle



%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%
Let $-\infty< a<  b < \infty$ and let $ a\leq a_{1} <  a_{2}<...<  a_{n} \leq b $ be the given points.
For $f\in C^{n} [ a, b ]$  \emph{Abel-Gontscharoff interpolating polynomial $P_{AG}$} of degree $(n-1)$ satisfying 
Abel-Gontscharoff conditions
\begin{equation*}
 P_{AG}^{(i)}( a_{i+1}) = f^{(i)}( a_{i+1}), \quad  0\leq i \leq n-1
\end{equation*}
exists uniquely (\cite{D}, \cite{G}).\\
This conditions in particular include two-point right focal conditions
\begin{align*}
P_{ AG2}^{(i)}( a_1)  &  = f^{(i)}( a_1), \,\,\, 0 \leq i \leq \alpha,\\
P_{ AG2}^{(i)}( a_2 )  &  = f^{(i)}( a_2 ), \,\,\, \alpha +1 \leq i \leq n-1, \,\, a\leq a_1<a_2\leq b.
\end{align*}

First, we give representations of Abel-Gontscharoff interpolating polynomial. For details and proofs see \cite{AW}.
\begin{theorem}
\label{tm:ag}
 Abel-Gontscharoff interpolating polynomial $P_{AG}$ of the function $f$ can be expressed as
\begin{equation*}
%\label{P_ag}
 P_{AG}(t)=\sum_{i=0}^{n-1} T_i(t)f^{(i)}(a_{i+1}),
\end{equation*}
where $T_0(t)=1$ and $T_i$, $1\leq i\leq n-1$ is the unique polynomial of degree $i$ satisfying
\begin{align*}
 T_i^{(k)}(a_{k+1})&=0, \, \, 0\leq k\leq i-1\\
T_i^{(i)}(a_{i+1}) &= 1
\end{align*}
and it can be written as
\begin{align}
 T_i(t)&= \frac{1}{1! 2!\cdots i!}
\begin{vmatrix}
 1 & a_1 & a_1^2 &\dots & a_1^{i-1} & a_1^i \\
0 & 1 & 2a_2 &\dots & (i-1)a_2^{i-2} & ia_2^{i-1} \\
\vdots & \vdots & \vdots & \dots & \vdots & \vdots \\
0 & 0 & 0 &\dots & (i-1)! & i!a_i \\
1 & t & t^2 &\dots & t^{i-1} & t^i \\
\end{vmatrix}\nonumber \\
&=\int_{a_1}^t\int_{a_2}^{t_1}\cdots \int_{a_i}^{t_{i-1}} dt_i dt_{i-1} \cdots dt_1, \,\, (t_0=t). \label{T_ag}
\end{align}
\end{theorem}

In particular, we have
\begin{equation*}
 \begin{split}
  T_0(t)&=1 \\
T_1(t)&=t-a_1 \\
T_2(t)&= \frac{1}{2} \left[ t^2-2a_2t+a_1(2a_2-a_1)\right].
 \end{split}
\end{equation*}

\begin{corollary}
\label{cor:ag}
 The two-point right focal interpolating polynomial $P_{AG2}$ of the function $f$ can be written as
\begin{equation*}
\begin{split}
 P_{AG2}(t)=\sum_{i=0}^\alpha &\frac{(t-a_1)^i}{i!} f^{(i)}(a_1) \\
&+\sum_{j=0}^{n-\alpha -2} \left[ \sum_{i=0}^j \frac{(t-a_1)^{\alpha +1+i} (a_1-a_2)^{j-i}}{(\alpha +1+i)! (j-i)!}\right]
f^{(\alpha +1+j)}(a_2).
\end{split}
\end{equation*}
\end{corollary}


The associated error $e_{AG}(t)=f(t)-P_{AG}(t)$ can be represented in terms of the Green's function $g_{AG}(t,s)$ of the boundary value problem
$$z^{(n)} =0,\, z^{(i)}( a_{i+1})=0,\,\, 0 \leq i \leq n-1 $$
and appears as (see \cite{AW}):
\begin{equation}
\label{green_ag}
g_{AG}(t,s)=
\begin{cases}
\sum\limits_{i=0}^{k-1} \frac{T_i(t)}{(n-i-1)!} (a_{i+1}-s)^{n-i-1}, & a_k\leq s\leq t;  \\
-\sum\limits_{i=k}^{n-1} \frac{T_i(t)}{(n-i-1)!} ( a_{i+1}-s)^{n-i-1}, & t\leq s\leq a_{k+1} \\
& k=0, 1,\dots ,n \, \, (a_0=a, a_{n+1}=b)
\end{cases}
\end{equation}

Corresponding to the two-point right focal conditions Green's function $g_{AG2}(t,s)$ of the boundary value problem
$$z^{(n)} =0,\, z^{(i)}( a_{1})=0,\,\, 0 \leq i \leq \alpha, \,\, z^{(i)}(a_2)=0, \alpha +1\leq i\leq n-1 $$
is given by (see \cite{AW}):
\begin{equation}
\label{green_ag2}
g_{AG2}(t,s)=  \frac{1}{(n-1)!}
\begin{cases}
\sum\limits_{i=0}^{\alpha} \binom{n-1}{i} (t-a_1)^i (a_{1}-s)^{n-i-1}, & a\leq s\leq t;  \\
-\sum\limits_{i=\alpha +1}^{n-1} \binom{n-1}{i} (t-a_1)^i ( a_{1}-s)^{n-i-1}, & t\leq s\leq b.
\end{cases}
\end{equation}

Further, for $a_1\leq s$, $t\leq a_2$ the following inequalities hold
\begin{equation*}
 (-1)^{n-\alpha -1} \frac{\partial^i g_{AG2}(t,s)}{\partial t^i}\geq 0, \quad 0\leq i\leq \alpha
\end{equation*}
\begin{equation*}
 (-1)^{n-i} \frac{\partial^i g_{AG2}(t,s)}{\partial t^i}\geq 0, \quad \alpha +1\leq i\leq n-1.
\end{equation*}

\begin{theorem}
\label{thm:ag}
Let $f \in C^{n}[a,b]$, and let $P_{AG}$ be its Abel-Gontscharoff interpolating polynomial. Then
\begin{align}
f(t)  &  = P_{AG}(t)+ e_{AG}(t)\nonumber\\
&= \sum_{i=0}^{n-1} T_i(t)f^{(i)}(a_{i+1})+ \int\limits_{ a}^{ b } g_{AG}(t,s) f^{(n)}(s) ds
\label{f_ag}
\end{align}
where $T_i$ is defined by \eqref{T_ag} and $g_{AG}(t,s)$ is defined by \eqref{green_ag}.
\end{theorem}

\begin{theorem}
\label{thm:ag2}
Let $f \in C^{n}[a,b]$, and let $P_{AG2}$ be its two-point right focal Abel-Gontscharoff interpolating polynomial. Then
\begin{align}
f(t)  &  = P_{AG2}(t)+ e_{AG2}(t)\nonumber\\
&=\sum_{i=0}^\alpha \frac{(t-a_1)^i}{i!} f^{(i)}(a_1)
+\sum_{j=0}^{n-\alpha -2} \left[ \sum_{i=0}^j \frac{(t-a_1)^{\alpha +1+i} (a_1-a_2)^{j-i}}{(\alpha +1+i)! (j-i)!}\right]
f^{(\alpha +1+j)}(a_2) \nonumber\\
&\quad \quad + \int\limits_{ a}^{ b } g_{AG2}(t,s) f^{(n)}(s) ds
\label{f_ag2}
\end{align}
where $g_{AG2}(t,s)$ is defined by \eqref{green_ag2}.
\end{theorem}

Finally, we recall the well-known Steffensen inequality which reads, \cite{STEFF}:

\begin{theorem}
\label{Steff}
Suppose that $f$ is decreasing  and  $g$ is integrable on $[a,b]$ with
 $0\leq g\leq 1$ and $\lambda=\int_a^bg(t)dt.$ Then we have
\begin{equation}
\label{stef1}
 \int_{b-\lambda}^{b} f(t)dt \leq\int_a^b f(t)g(t)dt\leq\int_a^{a+\lambda} f(t)dt.
\end{equation}
The inequalities are reversed for $f$ increasing.
\end{theorem}

Since its appearance in 1918 Steffensen's inequality has been the subject of investigation by many mathematicians.
Various papers have been devoted to generalizations and refinements of Steffensen's inequality and its connection
to other important inequalities. In the following theorem we recall weaker conditions on the function $g$ obtained by
Milovanovi\' c and Pe\v cari\' c in \cite{MIL1}.

\begin{theorem}\label{thm-mil1}
Let $f$ and $g$ be integrable functions on $[a,b]$ such that $f$ is decreasing and let $\lambda=\int_a^b g(t)dt$.
\begin{itemize}
\item[a)] If
\begin{equation}
\int_a^x g(t)dt \leq x-a \ \ \mbox{and} \ \ \int_x^b g(t)dt \geq 0 \ \ \ \mbox{for every $x\in [a,b]$,}
\label{mil1-1}
\end{equation}
then the second inequality in (\ref{stef1}) holds.
\item[b)] If
\begin{equation}
\int_x^b g(t)dt \leq b-x \ \ \mbox{and} \ \ \int_a^x g(t)dt \geq 0 \ \ \ \mbox{for every $x\in [a,b]$,}
\label{mil1-2}
\end{equation}
then the first inequality  in (\ref{stef1}) holds.
\end{itemize}
\end{theorem}

Steffensen's inequality is important not only in the theory of inequalities but also in many applications such as statistics,
functional equations, special functions, time scales etc. Some of these applications can be found in
\cite{AN1}, \cite{Cerone2}, \cite{CCM}, \cite{GO1}, \cite{GO2}, \cite{GO3}, and \cite{OY}.

The aim of this paper is to obtain new generalizations of
Steffensen's inequality using Abel-Gontscharoff interpolating
polynomial. Our new generalizations involve $n-$ convex function $f$
instead of restricting it to be a decreasing function as in
Steffensen's inequality. As a special case of these generalizations
(for $n=1$) we obtain weaker conditions for Steffensen's inequality
given in Theorem~\ref{thm-mil1}. These new generalizations in the
end enable us to construct linear functionals whose action on
particularly chosen families of functions give us exponentially
convex functions. However, there is lack of examples of this
functions since there are no operative criteria to recognize this
type of functions, so our constructed examples are valuable addition
to the theory of that functions. We also get additional results
about Ostrowski type inequalities.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Difference of integrals on two intervals}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec2}
If $[a,b]\cap [c,d]\neq \emptyset$ we have four possible cases for two intervals $[a,b]$ and $[c,d]$.
We observe cases $[c,d]\subset [a,b]$ and $[a,b]\cap [c,d]=[c,b]$ since other two cases are obtained by changing
$a\leftrightarrow c$ and $b\leftrightarrow d$.

In this paper by $T_{w,n}^{[a,b]}$ %and  $T_{u,n}^{[c,d]}$ 
we denote
\begin{equation*}
T_{w,n}^{[a,b]}=\sum_{i=0}^{n-1}  f^{(i)}(a_{i+1})\int\limits_{a}^{b} w(t) T_{i}(t)dt
\end{equation*}
where $T_i$ is defined by \eqref{T_ag}.

\begin{theorem}
\label{thm:gen_ag}
Let $f:\left[ a,b\right] \cup \left[ c,d\right] \rightarrow \R$ be of class $C^{n}$ on $[a,b]\cup [c,d]$
for some $n\geq1$. Let $w:\left[ a,b\right] \rightarrow \R$ and $u:\left[ c,d\right] \rightarrow \R $. Then if
$\left[ a,b\right] \cap \left[ c,d\right] \neq \emptyset $ we have
\begin{equation}
\begin{split}
\label{diff_ag}
\int_{a}^{b} w\left( t\right) f\left( t\right) dt -\int_{c}^{d}u\left(t\right) f\left( t\right) dt
-T_{w,n}^{[a,b]} +T_{u,n}^{[c,d]} =\int_a^{\max \left\{ b,d\right\} } K_{n}\left( s\right) f^{\left( n\right) }\left( s\right) ds,
\end{split}
\end{equation}
where
in case $\left[ c,d\right] \subseteq \left[ a,b\right], $
\begin{equation}
 \label{K1_ag}
K_{n}\left( s\right) =\left\{
\begin{array}{cc}
 \int_{a}^{b} w(t) g_{AG}\left(t,s\right) dt, & s\in \left[ a,c\right] , \\
&  \\
\int_{a}^{b} w(t) g_{AG}\left(t,s\right) dt-\int_{c}^{d} u(t) g_{AG}\left(t,s\right) dt, & s\in \left\langle c,d\right] , \\
&  \\
\int_{a}^{b} w(t) g_{AG}\left(t,s\right) dt , & s\in \left\langle d,b\right],
\end{array}
\right.
\end{equation}
and in case $\left[ a,b\right] \cap \left[ c,d\right] =\left[ c,b\right], $
\begin{equation}
\label{K2_ag}
K_{n}\left( s\right) =\left\{
\begin{array}{cc}
\int_{a}^{b} w(t) g_{AG}\left(t,s\right) dt& s\in \left[ a,c\right]  , \\
&  \\
\int_{a}^{b} w(t) g_{AG}\left(t,s\right) dt-\int_{c}^{d} u(t) g_{AG}\left(t,s\right) dt, & s\in \left\langle c,b\right] , \\
&  \\
-\int_{c}^{d} u(t) g_{AG}\left(t,s\right) dt , & s\in \left\langle b,d\right].
\end{array}
\right.
\end{equation}
\end{theorem}
\begin{proof}
 Multiplying identity \eqref{f_ag} by $w(t)$, then integrating from $a$ to $b$ and using Fubini's theorem we obtain
\begin{align}
\int\limits_a^b w(t)f(t)dt  &= \sum_{i=0}^{n-1} f^{(i)}(a_{i+1}) \int\limits_a^b w(t) T_i(t)dt+ \int\limits_{ a}^{ b } f^{(n)}(s)
\left( \int\limits_a^b w(t) g_{AG}(t,s) dt \right) ds.
\label{id1_ag}
\end{align}
Furthermore, multiplying identity \eqref{f_ag} by $u(t)$, then integrating from $c$ to $d$ and using Fubini's theorem we obtain similar
identity to identity \eqref{id1_ag}. Now subtracting these two identities we obtain \eqref{diff_ag}.
\end{proof}

\begin{remark}
 \rm
\label{rm:ag}
Using two-point right focal Abel-Gontscharoff polynomial, i.e. using \eqref{f_ag2},
inequality \eqref{diff_ag} becomes
\begin{equation*}
\begin{split}
\label{diff_ag2}
\int_{a}^{b} w\left( t\right) f\left( t\right) dt -\int_{c}^{d}u\left(t\right) f\left( t\right) dt
-Q_{w,n}^{[a,b]} +Q_{u,n}^{[c,d]} =\int_a^{\max \left\{ b,d\right\} } K_{n}\left( s\right) f^{\left( n\right) }\left( s\right) ds,
\end{split}
\end{equation*}
where $g_{AG}(t,s)$ is replaced by $g_{AG2}(t,s)$ in definition of $K_n(s)$ and by $Q_{w,n}^{[a,b]}$ we denote
\begin{equation*}
\begin{split}
Q_{w,n}^{[a,b]}&=\sum_{i=0}^{\alpha}  \frac{f^{(i)}(a_{1})}{i!}\int\limits_{a}^{b} w(t) (t-a_1)^idt \\
&+\sum_{j=0}^{n-\alpha -2} f^{(\alpha +1+j)}(a_2) \left[ \sum_{i=0}^j \frac{(a_1-a_2)^{j-i}}{(\alpha +1+j)!(j-i)!} \int\limits_a^b
w(t) (t-a_1)^{\alpha +1+i} dt \right].
\end{split}
\end{equation*}
\end{remark}

\begin{theorem}
\label{thm:opceniti_ag}
Let $f:\left[ a,b\right] \cup \left[ c,d\right] \rightarrow \R$ be $n-$convex on $[a,b]\cup [c,d]$ and
let $w:\left[ a,b\right] \rightarrow \R$ and $u:\left[ c,d\right] \rightarrow \R$.
Then if $\left[ a,b\right] \cap \left[ c,d\right] \neq \emptyset $ %$x\in \left[ a,b\right] \cap [c,d]$
and
\begin{equation}\label{Kn_1_ag}
 K_n(s)\geq 0,
\end{equation}
 we have
\begin{equation}
\label{opci_ag}
\int_{a}^{b} w\left( t\right) f\left( t\right) dt -T_{w,n}^{\left[ a,b\right] } \geq
\int_{c}^{d} u(t)f\left( t\right) dt - T_{u,n}^{\left[c,d\right]}
\end{equation}
where in case $\left[ c,d\right] \subseteq \left[ a,b\right], $ $K_n(s)$ is defined by \eqref{K1_ag} and
in case $\left[ a,b\right] \cap \left[ c,d\right] =\left[ c,b\right], $ $K_n(s)$ is defined by \eqref{K2_ag}.
\end{theorem}
\begin{proof}
Since $f$ is $n$-convex, without loss of generality we can assume that $f$ is $n-$times differentiable and $f^{(n)}\geq 0$
see \cite[p.\ 16 and p.\ 293]{PPT}.
Now we can apply Theorem~\ref{thm:gen_ag} to obtain \eqref{opci_ag}.
\end{proof}

\begin{remark}
 \rm
As in Remark~\ref{rm:ag}, using two-point right focal Abel-Gontscharoff polynomial, inequality \eqref{opci_ag} becomes
\begin{equation*}
\label{opci_ag2}
\int_{a}^{b} w\left( t\right) f\left( t\right) dt -Q_{w,n}^{\left[ a,b\right] } \geq
\int_{c}^{d} u(t)f\left( t\right) dt - Q_{u,n}^{\left[c,d\right]}.
\end{equation*}
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Generalization of Steffensen's inequality by Abel-Gontscharoff polynomial}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For a special choice of weights and intervals in results from previous section we obtain generalizations of Steffensen's inequality.
\begin{theorem}
\label{thm:gen_ag_steff_a}
Let $f:\left[ a,b\right] \cup \left[ a,a+\lambda\right] \rightarrow \R$ be $n-$convex on $[a,b]\cup [a,a+\lambda]$ for some $n\geq 1$
 and let $w:\left[ a,b\right] \rightarrow \R$.
Then if
\begin{equation}\label{Kn2_ag}
K_n(s)\geq 0,
\end{equation}
 we have
\begin{equation}
\label{steff_ag}
\int_{a}^{b} w\left( t\right) f\left( t\right) dt -T_{w,n}^{\left[ a,b\right]} \geq
\int_{a}^{a+\lambda} f\left( t\right) dt - T_{1,n}^{\left[a,a+\lambda\right]}
\end{equation}
where in case $a\leq a+\lambda \leq b, $
\begin{equation}
\label{K1_a_ag}
K_{n}\left( s\right) =\left\{
\begin{array}{cc}
\int_{a}^{b} w(t) g_{AG}\left(t,s\right) dt - \int_{a}^{a+\lambda} g_{AG}\left(t,s\right) dt, & s\in [a,a+\lambda] , \\
&  \\
\int_{a}^{b} w(t) g_{AG}\left(t,s\right)dt , & s\in \left\langle a+\lambda,b\right],
\end{array}
\right.
\end{equation}
and in case $a\leq b\leq a+\lambda, $
\begin{equation}
\label{K2_a_ag}
K_{n}\left( s\right) =\left\{
\begin{array}{cc}
 \int_{a}^{b} w(t) g_{AG}\left(t,s\right)dt - \int_{a}^{a+\lambda} g_{AG}\left(t,s\right)dt, & s\in [a,b] , \\
&  \\
-\int_{a}^{a+\lambda} g_{AG}\left(t,s\right)dt , & s\in \left\langle b,a+\lambda\right].
\end{array}
\right.
\end{equation}
\end{theorem}
\begin{proof}
 We take $c=a$, $d=a+\lambda$ and $u(t)=1$ in Theorem~\ref{thm:opceniti_ag}.
\end{proof}

\begin{remark}
\rm
 For $n=1$ and $\lambda\leq b-a$, $K_1(s)$ becomes
\begin{equation*}
K_{1}\left( s\right) =\left\{
\begin{array}{cc}
-\int_{a}^{s} w(t) dt + s-a, & s\in [a,a+\lambda] , \\ &  \\
\int_{s}^{b} w(t) dt , & s\in \left\langle a+\lambda,b\right].
\end{array}
\right.
\end{equation*}
So, if
\begin{equation}
\label{w1}
\int_a^s w(t)dt \leq s-a \quad\text{ for } a\leq s\leq a+\lambda
\end{equation}
 and
\begin{equation}
\label{w2}
\int_s^b w(t)dt\geq 0 \quad\text{ for } a+\lambda <s\leq b
\end{equation}
 and $f$
is increasing, from Theorem~\ref{thm:gen_ag_steff_a} we have
$$ \int_a^b w(t)f(t)dt -f(a+\lambda) \int_a^b w(t)dt \geq \int_a^{a+\lambda} f(t)dt -\lambda f(a+\lambda). $$
Furthermore, for $\lambda =\int_a^b w(t)dt$ we obtain the right-hand side of Steffensen's inequality for an increasing function $f$.
In \cite{MIL1} Milovanovi\' c and Pe\v cari\' c showed that conditions (\ref{w1}) and (\ref{w2}) are equivalent to condition
(\ref{mil1-1}), so for $n=1$ Theorem~\ref{thm:gen_ag_steff_a} reduces to Theorem~\ref{thm-mil1} a).
\end{remark}

\begin{theorem}
\label{thm:gen_ag_steff_b}
Let $f:\left[ a,b\right] \cup \left[ b-\lambda,b\right] \rightarrow \R$ be $n-$convex on $[a,b]\cup [b-\lambda,b]$ for some $n\geq 1$
 and let $w:\left[ a,b\right] \rightarrow \R$.
Then if
\begin{equation}\label{Kn3_ag}
K_n(s)\geq 0,
\end{equation}
 we have
\begin{equation}
\label{steff_ag2}
\int_{b-\lambda}^{b} f\left( t\right) dt - T_{1,n}^{\left[b-\lambda,b\right]} \geq
\int_{a}^{b} w\left( t\right) f\left( t\right) dt -T_{w,n}^{\left[ a,b\right]}
\end{equation}
where in case $a\leq b-\lambda \leq b, $
\begin{equation}
 \label{K1_b_ag}
K_{n}\left( s\right) =\left\{
\begin{array}{cc}
-\int_{a}^{b} w(t) g_{AG}\left(t,s\right)dt, & s\in [a,b-\lambda] , \\
&  \\
\int_{b-\lambda}^{b}  g_{AG}\left(t,s\right)dt -\int_{a}^{b} w(t) g_{AG}\left(t,s\right)dt , & s\in \left\langle b-\lambda,b\right],
\end{array}
\right.
\end{equation}
and in case $b-\lambda\leq a\leq  b, $
\begin{equation}
\label{K2_b_ag}
K_{n}\left( s\right) =\left\{
\begin{array}{cc}
\int_{b-\lambda}^{b} g_{AG}\left(t,s\right)dt, & s\in [b-\lambda,a] , \\
&  \\
\int_{b-\lambda}^{b} g_{AG}\left(t,s\right)dt -\int_{a}^{b} w(t) g_{AG}\left(t,s\right)dt, & s\in \left\langle a,b\right].
\end{array}
\right.
\end{equation}
\end{theorem}
\begin{proof}
First we change $a\leftrightarrow c$, $b\leftrightarrow d$ and $w\leftrightarrow u$ in Theorem~\ref{thm:opceniti_ag} and consider
cases $[a,b]\subseteq [c,d]$ and $[a,b]\cap [c,d]=[c,b]$. Then we take $c=b-\lambda$, $d=b$ and $u(t)=1$ to finish the proof.
\end{proof}

\begin{remark}
\rm
 For $n=1$ and $\lambda\leq b-a$, $K_1(s)$ becomes
\begin{equation*}
K_{1}\left( s\right) =\left\{
\begin{array}{cc}
\int_{a}^{s} w(t) dt, & s\in [a,b-\lambda] , \\ &  \\
b-s-\int_{s}^{b} w(t) dt , & s\in \left\langle b-\lambda,b\right].
\end{array}
\right.
\end{equation*}
So, if
\begin{equation}
 \label{w3}
\int_a^s w(t)dt \geq 0 \quad \text{ for } a\leq s\leq b-\lambda
\end{equation}
and
\begin{equation}
 \label{w4}
\int_s^b w(t)dt\leq b-s \quad\text{ for } b-\lambda <s\leq b
\end{equation}
 and $f$ is increasing
from Theorem~\ref{thm:gen_ag_steff_b} we have
$$ \int_{b-\lambda}^b f(t)dt -\lambda f(b-\lambda) \geq \int_a^{b} w(t)f(t)dt -f(b-\lambda)\int_a^b w(t)dt. $$
Furthermore, for $\lambda =\int_a^b w(t)dt$ we obtain the left-hand side of Steffensen's inequality for an increasing function $f$.
Similar as in \cite{MIL1} we can show that conditions (\ref{w3}) and (\ref{w4}) are equivalent to condition (\ref{mil1-2}).
Hence, for $n=1$ Theorem~\ref{thm:gen_ag_steff_b} reduces to Theorem~\ref{thm-mil1} b).
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Estimation of the difference}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec4}

In this section we give Ostrowski type inequalities related to results from previous sections.

\begin{theorem}
\label{tm:ag_Lp}
Suppose that all assumptions of Theorem~\ref{thm:gen_ag} hold. Assume $\left( p,q\right) $ is a
pair of conjugate exponents, that is $1\leq p,q\leq \infty $, $1/p+1/q=1$. Let $\left\vert f^{\left( n\right) }\right\vert
^{p}:\left[ a,b\right]\cup\left[ c,d\right] \rightarrow \R$ be an R-integrable function for some $n\geq1$. Then we have
\begin{equation}
\begin{split}
\label{bound_Lp_ag}
&\left\vert \int_{a}^{b}w(t) f\left( t\right) dt-\int_{c}^{d}u\left( t\right) f\left( t\right) dt-T_{w,n}^{\left[ a,b
\right] } +T_{u,n}^{\left[ c,d\right] } \right\vert \\
&\leq \left\Vert f^{\left( n\right) }\right\Vert _{p}
 \left( \int_{a}^{\max \left\{ b,d\right\} }\left\vert K_{n}\left( s\right) \right\vert ^{q}ds\right) ^{\frac{1}{q}}.
\end{split}
\end{equation}
The constant $\left( \int_{a}^{\max \left\{ b,d\right\}} \left\vert K_{n}\left( s\right) \right\vert ^{q}ds\right)^{1/q}$
in the inequality (\ref{bound_Lp_ag}) is sharp for $1<p\leq \infty $ and the best possible for $p=1$.
\end{theorem}
\begin{proof}
 Using inequality (\ref{diff_ag}) and applying H\"{o}lder's inequality we obtain
\begin{equation*}
\begin{split}
&\left\vert \int_{a}^{b}w(t) f\left( t\right) dt-\int_{c}^{d}u\left( t\right) f\left( t\right) dt-T_{w,n}^{\left[ a,b
\right] } +T_{u,n}^{\left[ c,d\right] } \right\vert \\
&= \left\vert \int_a^{\max\{ b,d\}} K_n(s) f^{(n)}(s) ds \right\vert
\leq \left\Vert f^{\left( n\right) }\right\Vert _{p}
 \left( \int_{a}^{\max \left\{ b,d\right\} }\left\vert K_{n}\left( s\right) \right\vert ^{q}ds\right) ^{\frac{1}{q}}.
\end{split}
\end{equation*}
For the proof of the sharpness of the constant
$\left( \int_{a}^{\max \left\{ b,d\right\} }\left\vert K_{n}\left( s\right) \right\vert ^{q}ds\right) ^{\frac{1}{q}}$
we will find a function $f$ for which the equality in (\ref{bound_Lp_ag}) is obtained.\\
For $1<p<\infty$ take $f$ to be such that
$$ f^{(n)}(s)=\sgn K_n(s) \left\vert K_n(s)\right\vert^\frac{1}{p-1}. $$
For $p=\infty$ take $f^{(n)}(s)=\sgn K_n(s)$.\\
For $p=1$ we will prove that
\begin{equation}
\label{sharp_pom}
\left\vert \int_{a}^{\max \left\{ b,d\right\} } K_{n}\left( s\right) f^{(n)}(s) ds\right\vert
\leq \max_{s\in [a,\max\{ b,d\}]} \left\vert K_n(s)\right\vert \left( \int_a^{\max \{b,d\}} \left\vert f^{(n)}(s)\right\vert
ds \right)
\end{equation}
is the best possible inequality. Suppose that $\left\vert K_n(s)\right\vert$ attains its maximum at $s_0\in [a,\max\{ b,d\}].$
First we assume that $K_n(s_0)>0$. For $\varepsilon$ small enough we define $f_\varepsilon(s)$ by
\begin{equation*}
 f_\varepsilon(s)=
\begin{cases}
0,& a\leq s \leq s_0, \\
\frac{1}{\varepsilon \, n!}(s-s_0)^{n}, &s_0\leq s\leq s_0+\varepsilon, \\
\frac{1}{n!}(s-s_0)^{n-1}, & s_0+\varepsilon \leq s\leq \max\{ b,d\}.
\end{cases}
\end{equation*}
Then for $\varepsilon$ small enough
$$ \left\vert \int_a^{\max\{ b,d\}} K_n(s) f^{(n)}(s)ds \right\vert =\left\vert \int_{s_0}^{s_0+\varepsilon}
K_n(s)\frac{1}{\varepsilon} ds\right\vert =\frac{1}{\varepsilon} \int_{s_0}^{s_0+\varepsilon} K_n(s) ds. $$
Now from inequality (\ref{sharp_pom}) we have
$$ \frac{1}{\varepsilon} \int_{s_0}^{s_0+\varepsilon} K_n(s)ds \leq K_n(s_0) \int_{s_0}^{s_0+\varepsilon} \frac{1}{\varepsilon}
ds =K_n(s_0). $$
Since,
$$ \lim_{\varepsilon \rightarrow 0} \frac{1}{\varepsilon} \int_{s_0}^{s_0+\varepsilon} K_n(s)ds =K_n(s_0) $$
the statement follows. In case $K_n(s_0)<0$ we define
\begin{equation*}
 f_\varepsilon(s)=
\begin{cases}
\frac{1}{n!}(s-s_0-\varepsilon)^{n-1},, & a\leq s\leq s_0, \\
-\frac{1}{\varepsilon \, n!}(s-s_0-\varepsilon)^{n}, &s_0\leq s\leq s_0+\varepsilon, \\
0,& s_0+\varepsilon\leq s \leq \max\{ b,d\}, \\
\end{cases}
\end{equation*}
and the rest of the proof is the same as above.
\end{proof}

\begin{theorem}
\label{tm:ag_Lp_a}
Suppose that all assumptions of Theorem~\ref{thm:gen_ag} for $c=a$ and $d=a+\lambda$ hold.
 Assume $\left( p,q\right)$ is a pair of conjugate exponents,
that is $1\leq p,q\leq \infty $, $1/p+1/q=1$. Let $\left\vert f^{\left( n\right) }\right\vert ^{p}:\left[ a,b\right]\cup
\left[ a,a+\lambda\right] \rightarrow \R$ be an R-integrable function for some $n\geq1$. Let $K_n(s)$ be
defined by (\ref{K1_a_ag}) in case $a\leq a+\lambda \leq b $ and  by (\ref{K2_a_ag}) in case $a \leq b\leq a+\lambda.$ Then we have
\begin{equation}
\begin{split}
\label{bound_Lp_a_ag}
&\left\vert \int_{a}^{b}w(t) f\left( t\right) dt-\int_{a}^{a+\lambda} f\left( t\right) dt-T_{w,n}^{\left[ a,b
\right] } +T_{1,n}^{\left[ a,a+\lambda\right] } \right\vert \\
&\leq \left\Vert f^{\left( n\right) }\right\Vert _{p}
 \left( \int_{a}^{\max \left\{ b,a+\lambda\right\} }\left\vert K_{n}\left( s\right) \right\vert ^{q}ds\right) ^{\frac{1}{q}}.
\end{split}
\end{equation}
The constant $\left( \int_{a}^{\max \left\{ b,a+\lambda\right\}}
\left\vert K_{n}\left( s\right) \right\vert ^{q}ds\right)^{1/q}$ in the inequality (\ref{bound_Lp_a_ag}) is sharp for
$1<p\leq \infty $ and the best possible for $p=1$.
\end{theorem}
\begin{proof}
 We take $c=a$, $d=a+\lambda$ and $u(t)=1$ in Theorem~\ref{tm:ag_Lp}.
\end{proof}

\begin{theorem}
\label{tm:ag_Lp_b}
Suppose that all assumptions of Theorem~\ref{thm:gen_ag} for $c=b-\lambda$ and $d=b$ hold.
Assume $\left( p,q\right)$ is a pair of conjugate exponents,
that is $1\leq p,q\leq \infty $, $1/p+1/q=1$. Let $\left\vert f^{\left( n\right) }\right\vert
^{p}:\left[ a,b\right]\cup\left[ b-\lambda,b\right] \rightarrow \R$ be an R-integrable function for some $n\geq1$. Let $K_n(s)$ be
defined by (\ref{K1_b_ag}) in case $a\leq b-\lambda \leq b $ and by (\ref{K2_b_ag}) in case $b-\lambda\leq a\leq  b$. Then we have
\begin{equation}
\begin{split}
\label{bound_Lp_b_ag}
&\left\vert \int_{b-\lambda}^{b} f\left( t\right) dt-\int_{a}^{b} w(t) f\left( t\right) dt-T_{1,n}^{\left[ b-\lambda,b
\right] } +T_{w,n}^{\left[ a,b\right] } \right\vert \\
&\leq \left\Vert f^{\left( n\right) }\right\Vert _{p}
 \left( \int_{\min \left\{a, b-\lambda\right\} }^{b}\left\vert K_{n}\left( s\right) \right\vert ^{q}ds\right) ^{\frac{1}{q}}.
\end{split}
\end{equation}
The constant $\left( \int_{\min \left\{ a,b-\lambda\right\}}^{b}
\left\vert K_{n}\left( s\right) \right\vert ^{q}ds\right)^{1/q}$ in the inequality (\ref{bound_Lp_b_ag}) is sharp for
$1<p\leq \infty $ and the best possible for $p=1$.
\end{theorem}
\begin{proof}
First we change $a\leftrightarrow c$, $b\leftrightarrow d$ and $w\leftrightarrow u$ in Theorem~\ref{thm:gen_ag} and then
we take $c=b-\lambda$, $d=b$ and $u(t)=1$. The rest of the proof is similar to the proof of Theorem~\ref{tm:ag_Lp}.
\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$n-$ exponetial convexity and exponential convexity}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We begin this section by giving some definitions  and notions which are used frequently in the results.
For more details see e.g. \cite{B}, \cite{JULI} and \cite{jure}.

\begin{definition}
A function $\psi:I\rightarrow\mathbb{R}$ is $n$-exponentially convex
in the Jensen sense on $I$ if
$$\sum\limits_{i,j=1}^n\xi_i\xi_j\,\psi\left(\frac{x_i+x_j}{2}\right)\geq 0,$$
hold for all choices $\xi_1,\ldots,\xi_n\in\mathbb{R}$ and all
choices $x_1,\ldots,x_n\in\ I$. A function
$\psi:I\rightarrow\mathbb{R}$ is $n$-exponentially convex if it is
$n$-exponentially convex in the Jensen sense and continuous on $I$.
\end{definition}

\begin{remark}
\rm
It is clear from the definition that $1$-exponentially convex
function in the Jensen sense is in fact a nonnegative function.
Also, $n$-exponentially convex function in the Jensen sense is
$k$-exponentially convex in the Jensen sense for every
$k\in\mathbb{N},~k\leq n$.
\end{remark}

\begin{definition}
A function $\psi:I\rightarrow\mathbb{R}$ is exponentially convex in
the Jensen sense on $I$ if it is $n$-exponentially convex in the
Jensen sense for all $n\in\mathbb{N}$.

A function $\psi:I\rightarrow\mathbb{R}$ is exponentially convex if
it is exponentially convex in the Jensen sense and continuous.
\end{definition}

\begin{remark}
\rm
\label{log}
It is known that $\psi:I\rightarrow\mathbb{R}$ is
log-convex in the Jensen sense if and only if
$$\alpha^{2}\psi(x)+2\alpha\beta\psi\left(\frac{x+y}{2}\right)+\beta^{2}\psi(y)\geq 0,$$
holds for every $\alpha,\beta\in\mathbb{R}$ and $x,y\in I$. It
follows that a positive function is log-convex in the Jensen sense if and
only if it is $2$-exponentially convex in the Jensen sense.

A positive function is log-convex if and only if it is $2$-exponentially
convex.
\end{remark}
\begin{proposition} \label{nejednakostkonveksne} If $f$ is a convex function on $I$ and if $x_1\leq y_1,~x_2\leq y_2, ~x_1\neq x_2,~y_1\neq y_2$, then the following inequality is valid
$$\frac{f(x_2)-f(x_1)}{x_2-x_1}\leq \frac{f(y_2)-f(y_1)}{y_2-y_1}.$$
If the function $f$ is concave, the inequality is reversed.
\end{proposition}


\begin{definition}
Let $f$ be a real-valued function defined on the segment $[a,b]$.
The  \emph{$n-$th order divided difference} of the function $f$ at
distinct points $x_0, . . . , x_n \in [a,b]$, is defined recursively
(see \cite{atkinson}, \cite{PPT}) by
$$f[x_i]=f(x_i),~~(i=0,\ldots,n)$$
and
$$f[x_0,\ldots,x_n]=\frac{f
[x_1,\ldots,x_n]-f[x_0,\ldots,x_{n-1}]}{x_n-x_0}.$$
\end{definition}

The value
$f[x_0,\ldots,x_n]$ is independent of the order of the points
$x_0,\ldots,x_n$.
Previous definition may be extended to include the case in which some or all of the points coincide. Assuming that $f^{(j-1)}(x)$
exists, we define
\begin{equation}\label{hermit}
f[\underbrace{x,\ldots,x}_{j-times}]=\frac{f^{(j-1)}(x)}{(j-1)!}.
\end{equation}

\bigskip

Motivated by inequalities
(\ref{opci_ag}),(\ref{steff_ag}) and (\ref{steff_ag2}), under
assumptions of Theorems \ref{thm:opceniti_ag},
\ref{thm:gen_ag_steff_a} and \ref{thm:gen_ag_steff_b} we
define following linear functionals:

\begin{equation}
\label{L1_a}
L_1(f)=\int_{a}^{b} w\left( t\right) f\left( t\right) dt-\int_{c}^{d} u(t)f\left( t\right) dt
-T_{w,n}^{\left[ a,b\right] } + T_{u,n}^{\left[c,d\right]}
\end{equation}

\begin{equation}
    \label{L2_b}
L_2(f)=
\int_{a}^{b} w\left( t\right) f\left( t\right) dt -\int_{a}^{a+\lambda} f\left( t\right) dt-T_{w,n}^{\left[ a,b\right]}
+ T_{1,n}^{\left[a,a+\lambda\right]}
\end{equation}


\begin{equation}
\label{L3_c}
L_3(f)=
\int_{b-\lambda}^{b} f\left( t\right) dt-\int_{a}^{b} w\left( t\right) f\left( t\right) dt - T_{1,n}^{\left[b-\lambda,b\right]}
 +T_{w,n}^{\left[ a,b\right]}
\end{equation}

Also, we define $I_1=[a,b]\cup[c,d], \, I_2=[a,b]\cup[a,a+\lambda]
\, ~ \text{and}~  \, I_3=[a,b]\cup[b-\lambda,b]$.


\begin{remark}
\rm
Under the assumptions of Theorems \ref{thm:opceniti_ag},
\ref{thm:gen_ag_steff_a} and \ref{thm:gen_ag_steff_b}
respectively it holds $L_i(f)\geq 0, \, i=1,2,3$ for all $n-$
convex functions $f$.
\end{remark}

Now we will show how to generate means for our list of linear functionals.


\begin{theorem}
\label{thm:lagrange_a} 
Let $f:I_i\rightarrow\mathbb{R}$ $(i=1,2,3)$ be such that $f\in C^{n}(I_i)$. If inequalities in
(\ref{Kn_1_ag}) $(i=1)$, (\ref{Kn2_ag}) $(i=2)$ and (\ref{Kn3_ag}) $(i=3)$
hold, then there exist $\xi_i\in I_i$ such that
\begin{equation}\label{lagrange}
    L_i(f)=f^{(n)}(\xi_i)L_i(\varphi), \quad i=1,2,3
\end{equation}
where $\varphi(x)=\frac{x^{n}}{n!}$.
\end{theorem}

\begin{proof}
Let us denote $m=\min f^{(n)}(x)$ and $M=\max f^{(n)}(x)$. For a given
function $f\in C^{n}(I_i)$ we define functions
$F_1,F_2:I_i\rightarrow\mathbb{R}$ with
 $$F_{1}(x)=\frac{M x^{n}}{n!}-f(x) \quad \text{and} \quad
F_{2}(x)=f(x)-\frac{m x^{n}}{n!}.$$ Now
$F_1^{(n)}(x)=M-f^{(n)}(x)\geq0, x\in I_i$,  so we conclude
$L_i(F_1)\geq0$ and then $L_i(f)\leq M\cdot L_i(\varphi)$.
Similarly, from $F_{2}^{(n)}(x)=f^{(n)}(x)- m\geq0$ we conclude
$m\cdot L_i(\varphi)\leq L_i(f)$. \\If $L_i(\varphi)=0$,
(\ref{lagrange}) holds for all $\xi_i\in I_i$. Otherwise,
$m\leq\frac{L_i(f)}{L_i(\varphi)}\leq M$. Since $f^{(n)}(x)$ is
continuous on $I_i$ there exist $\xi_i\in I_i$ such that
(\ref{lagrange}) holds and the proof is complete.
\end{proof}

\begin{theorem}
\label{thm:cauchy_a} Let $f,g:I_i\rightarrow\mathbb{R}$
$(i=1,2,3)$ be such that $f,g\in C^{n}(I_i)$ and $g^{(n)}(x)\neq 0$ for every $x\in I_i$. If inequalities in
(\ref{Kn_1_ag}) $(i=1)$, (\ref{Kn2_ag}) $(i=2)$ and (\ref{Kn3_ag}) $(i=3)$
hold, then there exist $\xi_i\in I_i$ such that
\begin{equation}\label{cauchy}
\frac{L_{i}(f)} {L_{i}(g)}=\frac{f^{(n)}(\xi_i)}{g^{(n)}(\xi_i)},
\quad i=1,2,3.
\end{equation}
\end{theorem}

\begin{proof}
 We define  functions
$\phi_i(x)=f(x)L_i(g)-g(x)L_i(f), ~i=1,2,3.$ According to Theorem
\ref{thm:lagrange_a} there exists $\xi_i\in I_i$ such that
$L_i(\phi_i)=\phi_i^{(n)}(\xi_i) L_i(\varphi).$ Since $L_i(\phi_i)=0$
it follows $f^{(n)}(\xi_i)L_i(g)-g^{(n)}(\xi_i)L_i(f)=0$ and
(\ref{cauchy}) is proved.
\end{proof}

We will use previously  defined functionals to construct
exponentially convex functions, a special type of convex functions that are
invented by S. N. Bernstein over eighty years ago in \cite{B}.
An elegant method of constructing $n-$ exponentially convex and
exponentially convex functions is given in \cite{JULI}. We use
this method to prove the $n-$exponential convexity for above defined
functionals. In the sequel the notion $\log$ denotes the natural
logarithm function.

\begin{theorem}
\label{thm_neksp2}
 Let $\Omega =\{ f_p: p\in J \}$, where J is an interval in $\mathbb{R}$, be a family of functions defined on an interval $I_i,~ i=1,2,3$ in $\mathbb{R}$ such that
the function $p \mapsto f_p [x_0,\ldots,x_{m}]$ is $n-$exponentially
convex in the Jensen sense on $J$ for every $(m+1)$ mutually
different points $x_0,\ldots,x_{m}\in I_i, ~i=1,2,3$. Let  $L_i$,
$i=1,2,3$ be linear functionals defined by
$(\ref{L1_a})-(\ref{L3_c})$. Then $p\mapsto L_i(f_p)$ is
$n-$exponentially convex
function in the Jensen sense on $J$. \\
If the function $p\mapsto L_i(f_p)$ is continuous on $J$, then it is
$n-$exponentially convex on $J$.
\end{theorem}


\begin{proof}
For $\xi_{j}\in\mathbb{R},~j=1,\ldots,n$ and $p_{j}\in
J,~j=1,\ldots,n$, we define the function
$$g(x)=\sum_{j,k=1}^{n}\xi_{j}\xi_{k}f_{\frac{p_{j}+p_{k}}{2}}(x).$$
Using the assumption that the function $p\mapsto
f_{p}[x_{0},\ldots,x_{m}]$ is $n$-exponentially convex in the Jensen
sense, we have
$$g[x_{0},\ldots,x_{m}]=\sum_{j,k=1}^{n}\xi_{j}\xi_{k}f_{\frac{p_{j}+p_{k}}{2}}[x_{0},\ldots,x_{m}]\geq 0,$$
which in turn implies that $g$ is a $m$-convex function on $J$, so $L_i(g)\geq 0,~i=1,2,3$. Hence
$$\sum_{j,k=1}^{n}\xi_{j}\xi_{k}L_{i}\left(f_{\frac{p_{j}+p_{k}}{2}}\right)\geq 0.$$
We conclude that the function $p\mapsto L_{i}(f_{p})$ is
$n$-exponentially convex on $J$ in the Jensen sense.

If the function $p\mapsto L_{i}(f_{p})$ is also continuous on $J$,
then $p\mapsto L_{i}(f_{p})$ is $n$-exponentially convex by
definition.
\end{proof}


The following corollaries are immediate consequences of the above theorem:

\begin{corollary}\label{cornovo}
Let $\Omega=\{f_{p}:p\in J\}$, where $J$ is an interval in
$\mathbb{R}$, be a family of functions defined on an interval $I_i,
~i=1,2,3$ in $\mathbb{R}$, such that the function $p\mapsto
f_{p}[x_{0},\ldots,x_{m}]$ is exponentially convex in the Jensen
sense on $J$ for every $(m+1)$ mutually different points
$x_{0},\ldots,x_{m}\in I_i$. Let $L_{i},~i=1,2,3$, be linear
functionals defined by (\ref{L1_a})-(\ref{L3_c}). Then $p\mapsto
L_{i}(f_{p})$ is an exponentially convex function in the Jensen
sense on $J$. If the function $p\mapsto L_{i}(f_{p})$ is continuous
on J, then it is exponentially convex on $J$.
\end{corollary}

\begin{corollary}\label{cornovo2}
Let $\Omega=\{f_{p}:p\in J\}$, where $J$ is an interval in
$\mathbb{R}$, be a family of functions defined on an interval $I_i,
~i=1,2,3$ in $\mathbb{R}$, such that the function $p\mapsto
f_{p}[x_{0},\ldots,x_{m}]$ is $2$-exponentially convex in the Jensen
sense on $J$ for every $(m+1)$ mutually different points
$x_{0},\ldots,x_{m}\in I_i$. Let $L_{i},~i=1,2,3$ be linear
functionals defined by (\ref{L1_a})-(\ref{L3_c}). Then the
following statements hold:
\begin{enumerate}
\item[(i)] If the function $p\mapsto L_{i}(f_{p})$ is continuous on $J$, then it is $2$-exponentially convex function on $J$. If $p\mapsto L_{i}(f_{p})$ is additionally strictly positive, then it is also log-convex on $J$.
Furthermore, the following inequality holds true:
\begin{equation*}
    [L_i(f_s)]^{t-r} \leq \left[L_i(f_r)\right]^{t-s}\left[L_i(f_t)\right]^{s-r}
\end{equation*}
  for every choice $r,s,t\in J$, such that $r<s<t$.
\item[(ii)]If the function $p\mapsto L_{i}(f_{p})$ is strictly positive and differentiable on J, then for every $p,q,u,v\in J$, such that $p\leq u$ and $q\leq v$, we have
\begin{equation}\label{minejednakost}
\mu_{p,q}(L_{i},\Omega)\leq \mu_{u,v}(L_{i},\Omega),
\end{equation}
where
\begin{equation}\label{misredina}
\mu_{p,q}(L_{i},\Omega)=\left\{\begin{array}{ll}\left(\frac{L_{i}(f_{p})}{L_{i}(f_{q})}\right)^{\frac{1}{p-q}},& p\neq q,\\
\exp\left(\frac{\frac{d}{dp}L_{i}(f_{p})}{L_{i}(f_{p})}\right),&
p=q,\end{array}\right.
\end{equation}
for $f_{p},f_{q}\in \Omega$.
\end{enumerate}
\end{corollary}

\begin{proof}
\begin{enumerate}
\item[(i)] This is an immediate consequence of Theorem \ref{thm_neksp2} and Remark \ref{log}.
\item[(ii)] Since $p\mapsto L_{i}(f_{p})$ is positive and continuous, by $\mathrm{(i)}$ we have that $p\mapsto L_{i}(f_{p})$ is log-convex on $J$, that is, the function $p\mapsto \log{L_{i}(f_{p})}$ is convex on $J$.
Applying Proposition \ref{nejednakostkonveksne} we get
\begin{equation}\label{convexprop}
\frac{\log L_{i}(f_{p})-\log L_{i}(f_{q})}{p-q}\leq \frac{\log
L_{i}(f_{u})-\log L_{i}(f_{v})}{u-v},
\end{equation}
for $p\leq u,q\leq v,p\neq q,u\neq v$. Hence, we conclude that
$$
\mu_{p,q}(L_{i},\Omega)\leq \mu_{u,v}(L_{i},\Omega).
$$
Cases $p=q$ and $u=v$ follow from (\ref{convexprop}) as limit
cases.
\end{enumerate}
\end{proof}

\begin{remark}
\rm
\label{remnovo}
Note that the results from the above theorem and corollaries still hold
when two of the points $x_{0},\ldots,x_{m}\in I_i, ~i=1,2,3 $
coincide, say $x_{1}=x_{0}$, for a family of differentiable
functions $f_{p}$ such that the function $p\mapsto
f_{p}[x_{0},\ldots,x_{m}]$ is $n$-exponentially convex in the Jensen
sense (exponentially convex in the Jensen sense, log-convex in the
Jensen sense), and furthermore, they still hold when all $m+1$
points coincide for a family of $m$ differentiable functions with
the same property. The proofs use \eqref{hermit} and suitable
characterization of convexity.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Applications to Stolarsky type means}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section, we present several families of functions which fullfill the conditions of Theorem \ref{thm_neksp2},
Corollary \ref{cornovo}, Corollary \ref{cornovo2} and Remark \ref{remnovo}. This enables us to construct a large families of
functions which are exponentially convex. For a discussion related to this problem see \cite{ehm}.

\begin{example}\label{example1}
Consider a family of functions
$$\Omega_{1}=\{f_{p}:\mathbb{R}\rightarrow\mathbb{R}: p\in\mathbb{R}\}$$
defined by
$$f_{p}(x)=\left\{\begin{array}{ll}\frac{e^{px}}{p^{n}}, & p\neq 0,\\
\frac{x^{n}}{n!}, & p=0.\end{array}\right.$$ Here,
$\frac{d^{n}f_{p}}{dx^{n}}(x)=e^{px}>0$ which shows that $f_{p}$ is
$n$-convex on $\mathbb{R}$ for every $p\in \mathbb{R}$ and $p\mapsto
\frac{d^{n}f_{p}}{dx^{n}}(x)$ is exponentially convex by definition.
Using analogous arguing as in the proof of Theorem \ref{thm_neksp2}
we also have that $p\mapsto f_{p}[x_{0},\ldots,x_{m}]$ is
exponentially convex (and so exponentially convex in the Jensen
sense). Using Corollary \ref{cornovo} we conclude that $p\mapsto
L_{i}(f_{p}), i=1,2,3$, are exponentially convex in the Jensen
sense. It is easy to verify that this mapping is continuous
(although mapping $p\mapsto f_{p}$ is not continuous for $p=0$), so
it is exponentially convex. For this family of functions,
$\mu_{p,q}(L_{i},\Omega_{1}),~i=1,2,3$, from \eqref{misredina},
becomes
$$\mu_{p,q}(L_{i},\Omega_{1})=\left\{\begin{array}{ll}
\left(\frac{L_{i}(f_{p})}{L_{i}(f_{q})}\right)^{\frac{1}{p-q}}, & p\neq q, \\
\exp\left(\frac{L_{i}(id\cdot f_{p})}{L_{i}(f_{p})}-\frac{n}{p}\right), & p=q\neq 0, \\
\exp\left(\frac{1}{n+1}\frac{L_{i}(id\cdot
f_{0})}{L_{i}(f_{0})}\right), & p=q=0,\end{array}\right.$$ where
$id$ is the identity function. Also, by Corollary \ref{cornovo2} it
is monotonic function in parameters $p$ and $q$. \\We observe here
that
$\left(\frac{\frac{d^{n}f_{p}}{dx^{n}}}{\frac{d^{n}f_{q}}{dx^{n}}}\right)^{\frac{1}{p-q}}(\log
x)=x$ so using Theorem \ref{thm:cauchy_a} it follows that:
$$M_{p,q}(L_{i},\Omega_{1})=\log \mu_{p,q}(L_{i},\Omega_{1}), ~~~i=1,2,3$$
satisfies
$$\min\{a,c,b-\lambda\}\leq M_{p,q}(L_{i},\Omega_{1})\leq  \max\{b,d,a+\lambda\}, ~~~i=1,2,3.$$
So, $M_{p,q}(L_{i},\Omega_{1})$ is a monotonic mean.
\end{example}

\begin{example}
\label{ex2} Consider a family of functions
$$\Omega_{2}=\{g_{p}:(0,\infty)\rightarrow\mathbb{R}: p\in\mathbb{R}\}$$
defined by
$$g_{p}(x)=\left\{\begin{array}{ll}\frac{x^{p}}{p(p-1)\cdots(p-n+1)}, & p\notin \{0,1,\ldots,n-1\},\\
\frac{x^{j}\log{x}}{(-1)^{n-1-j}j!(n-1-j)!}, &
p=j\in\{0,1,\ldots,n-1\}.\end{array}\right.$$

Here, $\frac{d^{n}g_{p}}{dx^{n}}(x)=x^{p-n}>0$ which shows that
$g_{p}$ is $n-$convex for $x>0$ and $p\mapsto
\frac{d^{n}g_{p}}{dx^{n}}(x)$ is exponentially convex by
definition. Arguing as in Example \ref{example1} we get that the
mappings $p\mapsto L_{i}(g_{p}), i=1,2,3$ are exponentially convex.
 For this family of functions $\mu_{p,q}(L_{i},\Omega_{2}),~i=1,2,3$, from \eqref{misredina},
is now equal to
{\small $$\mu_{p,q}(L_{i},\Omega_{2})=\left\{\begin{array}{ll}
\left(\frac{L_{i}(g_{p})}{L_{i}(g_{q})}\right)^{\frac{1}{p-q}}, & p\neq q, \\
\exp\left((-1)^{n-1}(n-1)!\frac{L_{i}(g_{0}g_{p})}{L_{i}(g_{p})}+\sum\limits_{k=0}^{n-1}\frac{1}{k-p}\right), & p=q\notin \{0,1,\ldots,n-1\}, \\
\exp\left((-1)^{n-1}(n-1)!\frac{L_{i}(g_{0}g_{p})}{2L_{i}(g_{p})}+\sum\limits_{\substack{k=0\\k\neq
p}}^{n-1}\frac{1}{k-p}\right), & p=q\in
\{0,1,\ldots,n-1\}.\end{array}\right.$$}
 Again, using Theorem \ref{thm:cauchy_a} we conclude that
\begin{equation*}
\min
\{a,b-\lambda,c\}\leq\left(\frac{L_{i}(g_{p})}{L_{i}(g_{q})}\right)^{\frac{1}{p-q}}\leq
\max \{a+\lambda,b,d\},~~~i=1,2,3.
\end{equation*}
So, $\mu_{p,q}(L_{i},\Omega_{2}),i=1,2,3$ is a mean.
\end{example}

\begin{example}
\label{ex3} Consider a family of functions
$$\Omega_{3}=\{\phi_{p}:(0,\infty)\rightarrow\mathbb{R}  :p\in(0,\infty)\}$$
defined by
$$\phi_{p}(x)=\left\{\begin{array}{ll}\frac{p^{-x}}{(-\log p)^{n}}, & p\neq 1\\
\frac{x^{n}}{n!}, & p=1.\end{array}\right.$$ Since
$\frac{d^{n}\phi_{p}}{dx^{n}}(x)=p^{-x}$ is the Laplace transform
of a non-negative function (see \cite{WID}) it is exponentially
convex. Obviously $\phi_{p}$ are $n$-convex functions for every
$p>0$. For this family of functions,
$\mu_{p,q}(L_{i},\Omega_{3}),i=1,2,3$ from \eqref{misredina} is
equal to
$$\mu_{p,q}(L_{i},\Omega_{3})=\left\{\begin{array}{ll}
\left(\frac{L_{i}(\phi_{p})}{L_{i}(\phi_{q})}\right)^{\frac{1}{p-q}}, & p\neq q, \\
\exp\left(-\frac{L_{i}(id\cdot \phi_{p})}{p\ L_{i}(\phi_{p})}-\frac{n}{p\log{p}}\right), & p=q\neq 1, \\
\exp\left(-\frac{1}{n+1}\frac{L_{i}(id\cdot
\phi_{1})}{L_{i}(\phi_{1})}\right), &
p=q=1,\end{array}\right.$$where $id$ is the identity function. This
is a monotone function in parameters $p$ and $q$ by
\eqref{minejednakost}. Using Theorem \ref{thm:cauchy_a} it follows
that
$$M_{p,q}(L_{i},\Omega_{3})=-L(p,q)\log \mu_{p,q}(L_{i},\Omega_{3}), ~~~i=1,2,3$$
satisfies $$\min \{a,b-\lambda,c\}\leq M_{p,q}(L_{i},\Omega
_{3})\leq \max \{a+\lambda,b,d\}.$$ So $M_{p,q}(L_{i},\Omega _{3})$
is a monotonic mean.
 $L(p,q)$ is a logarithmic mean defined by
\begin{equation*}
L(p,q)=%
\begin{cases}
\frac{p-q}{\log p-\log q}, & p\neq q \\
p, & p=q.%
\end{cases}%
\end{equation*}
\end{example}


\begin{example}
Consider a family of functions
$$\Omega_{4}=\{\psi_{p}:(0,\infty)\rightarrow\mathbb{R}:p\in(0,\infty)\}$$
defined by
$$\psi_{p}(x)=\frac{e^{-x\sqrt{p}}}{(-\sqrt{p})^{n}}.$$
Since $\frac{d^{n}\psi_{p}}{dx^{n}}(x)=e^{-x\sqrt{p}}$ is the
Laplace transform of a non-negative function (see \cite{WID}) it is
exponentially convex. Obviously $\psi_{p}$ are $n$-convex functions
for every $p>0$. For this family of functions,
$\mu_{p,q}(L_{i},\Omega_{4}),i=1,2,3$  from \eqref{misredina} is
equal to
$$\mu_{p,q}(L_{i},\Omega_{4})=\left\{\begin{array}{ll}
\left(\frac{L_{i}(\psi_{p})}{L_{i}(\psi_{q})}\right)^{\frac{1}{p-q}}, & p\neq q, \\
\exp\left(-\frac{L_{i}(id\cdot
\psi_{p})}{2\sqrt{p}L_{i}(\psi_{p})}-\frac{n}{2p}\right), & p=q,
\end{array}\right.$$
where $id$ is the identity function.
 This is monotone function in
parameters $p$ and $q$ by \eqref{minejednakost}. Using Theorem
\ref{thm:cauchy_a} it follows that
$$M_{p,q}(L_{i},\Omega_{4})=-(\sqrt{p}+\sqrt{q})\log \mu_{p,q}(L_{i},\Omega_{4}), ~~~i=1,2,3$$
satisfies $\min \{a,b-\lambda,c\}\leq M_{p,q}(L_{i},\Omega _{4})\leq  \max \{a+\lambda,b,d\}$, so $%
M_{p,q}(L_{i},\Omega _{4})$ is a monotonic mean.
\end{example}


{\bf Acknowledgement.} The research of Josip Pe\v cari\' c and Ksenija Smoljak has been fully supported by Croatian Science Foundation under 
the project 5435 and the research of Anamarija Peru\v si\' c has been fully supported by 
University of Rijeka under the project 13.05.1.1.02.

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