\documentclass[11pt,twoside]{article}
\usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, color}
\usepackage[bookmarksnumbered, colorlinks]{hyperref} \usepackage{float}
\usepackage{lipsum}
\usepackage{afterpage}
\usepackage[labelfont=bf]{caption}
\usepackage[nottoc,notlof,notlot]{tocbibind} 
%\renewcommand\bibname{References}
\def\bibname{\Large \bf  References}
\usepackage{lipsum}
\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhf{}
\renewcommand{\headrulewidth}{0pt}
\fancyhead[LE,RO]{\thepage}
\thispagestyle{empty}
%\afterpage{\lhead{new value}}

\fancyhead[CE]{M. Shah Hosseini, H. R. Moradi and B. Moosavi} 
\fancyhead[CO]{More about function order of positive operators}



%\topmargin=-1.6cm
\textheight 17.5cm%
\textwidth  12cm %
\topmargin   8mm  %
\oddsidemargin   20mm   %
\evensidemargin   20mm   %
\footskip=24pt     %

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{xca}[theorem]{Exercise}
%\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\renewenvironment{proof}{{\bfseries \noindent Proof.}}{~~~~$\square$}
\makeatletter
\def\th@newremark{\th@remark\thm@headfont{\bfseries}}
\makeatletter



  
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% If you want to insert other packages. Insert them here
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\long\def\symbolfootnote[#1]#2{\begingroup%
%\def\thefootnote{\fnsymbol{footnote}}\footnote[#1]{#2}\endgroup}



 \def \thesection{\arabic{section}}
 

\begin{document}
%\baselineskip 9mm
%\setcounter{page}{}
\thispagestyle{plain}
{\noindent Journal of Mathematical Extension \\
Vol. XX, No. XX, (2014), pp-pp (Will be inserted by layout editor)}\\
ISSN: 1735-8299\\
URL: http://www.ijmex.com\\
‎\vspace*{9mm}
‎
\begin{center}

{\Large \bf 
More About Function Order of Positive Operators\\}



\let\thefootnote\relax\footnote{\scriptsize Received: XXXX; Accepted: XXXX (Will be inserted by editor)}

{\bf Mohsen Shah Hosseini$^*$}\vspace*{-2mm}\\
\vspace{2mm} {\small  Department of Mathematics, Shahr-e-Qods Branch, Islamic Azad University, Tehran, Iran.} \vspace{2mm}

{\bf Hamid Reza Moradi}\vspace*{-2mm}\\
\vspace{2mm} {\small  Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran.} \vspace{2mm}

{\bf  Baharak Moosavi\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{-2mm}\\
\vspace{2mm} {\small   Department of Mathematics, Safadasht Branch, Islamic Azad University, Tehran, Iran.} \vspace{2mm}

\end{center}

\vspace{4mm}


{\footnotesize
\begin{quotation}
{\noindent \bf Abstract.} Let $A$, $B$ be two positive operators on a Hilbert space $\mathcal{H}$ and $p>1$. It is well-known that $B\le A$ does not imply ${{B}^{p}}\le {{A}^{p}}$ in general. In this paper, we prove 
\[B\le A~\text{ implies }~{{B}^{p}}\le {{A}^{p}}+p\zeta {{\mathbf{1}}_{\mathcal{H}}}\] 
and
\[B\le A~\text{ implies }~{{A}^{p}}\le {{B}^{p}}+p\delta {{\mathbf{1}}_{\mathcal{H}}}\] 
where $\zeta $ and $\delta $ are two positive constant and ${{\mathbf{1}}_{\mathcal{H}}}$ is the identity operator. 
\end{quotation}
\begin{quotation}
\noindent{\bf AMS Subject Classification:} Primary 47A63, Secondary 26A51, 26D15, 26B25, 39B62.

\noindent{\bf Keywords and Phrases:} Operator order, Jensen's inequality, convex functions, self-adjoint operators, positive operators.
\end{quotation}}

\section{Introduction}
Let $\mathcal{B}\left( \mathcal{H} \right)$ be the $C^*$--algebra of all bounded linear operators on a Hilbert space $\mathcal{H}$.  As customary, we reserve $m$, $M$ for scalars and ${{\mathbf{1}}_{\mathcal{H}}}$ for the identity operator on $\mathcal{H}$. A self-adjoint operator $A$ is said to be positive (written $A\ge0$) if $\left\langle Ax,x \right\rangle \ge 0$ holds for all $x\in \mathcal{H}$  also an operator $A$ is said to be strictly positive (denoted by $A>0$) if $A$ is positive and invertible. If $A$ and $B$ are self-adjoint, we
write $B\ge A$ in case $B-A\ge0$. The Gelfand map $f\left( t \right)\mapsto f\left( A \right)$ is an isometrical $*$--isomorphism between the ${{C}^{*}}$--algebra $C\left( \sigma \left( A \right) \right)$ of continuous functions on the spectrum $\sigma \left( A \right)$ of a selfadjoint operator $A$ and the ${{C}^{*}}$--algebra generated by $A$ and the identity operator ${{\mathbf{1}}_{\mathcal{H}}}$. If $f,g\in C\left( \sigma \left( A \right) \right)$, then $f\left( t \right)\ge g\left( t \right)$ ($t\in \sigma \left( A \right)$) implies that $f\left( A \right)\ge g\left( A \right)$.

For $A,B\in \mathcal{B}\left( \mathcal{H} \right)$, $A\oplus B$ is the operator defined on $\mathcal{B}\left( \mathcal{H}\oplus \mathcal{H} \right)$ by $\left( \begin{matrix}
A & 0  \\
0 & B  \\
\end{matrix} \right)$. 
A linear map $\Phi:\mathcal{B}\left( \mathcal{H} \right)\to \mathcal{B}\left( \mathcal{K} \right)$ is positive if $\Phi \left( A \right)\ge 0$ whenever $A\ge 0$. It's said to be unital if $\Phi \left( {{\mathbf{1}}_{\mathcal{H}}} \right)={{\mathbf{1}}_{\mathcal{K}}}$. A continuous function $f$ defined on the interval $J$
is called an operator convex function if $f\left( \left( 1-v \right)A+vB \right)\le \left( 1-v \right)f\left( A \right)+vf\left( B \right)$ for every $0<v<1$ and for every pair of bounded self-adjoint operators $A$ and $B$ whose spectra are both in $J$.



Hansen et al. \cite{1} showed if $f:J\to \mathbb{R}$ is an operator convex function, ${{A}_{1}},\ldots ,{{A}_{n}}\in \mathcal{B}\left( \mathcal{H} \right)$ are self-adjoint operators with the spectra in $J$, and  ${{\Phi }_{1}},\ldots ,{{\Phi }_{n}}:\mathcal{B}\left( \mathcal{H} \right)\to \mathcal{B}\left( \mathcal{K} \right)$ are positive linear mappings such that $\sum\nolimits_{i=1}^{n}{{{\Phi }_{i}}\left( {{\mathbf{1}}_{\mathcal{H}}} \right)}={{\mathbf{1}}_{\mathcal{K}}}$, then 
\begin{equation}\label{7}
f\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)} \right)\le \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f\left( {{A}_{i}} \right) \right)}.
\end{equation}
Though in the case of convex function the inequality \eqref{7} does not hold in general, we have the following estimate \cite[Lemma 2.1]{5}:
\begin{equation}\label{19}
f\left( \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)}x,x \right\rangle  \right)\le \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f\left( {{A}_{i}} \right) \right)}x,x \right\rangle
\end{equation}
for any unit vector $x\in \mathcal{K}$. For recent results treating the Jensen operator inequality, we refer the reader to \cite{10, 6, 11}.






The \lq\lq L\"owner-Heinz inequality'' asserts that  $0\le A\le B$ ensures ${{A}^{p}}\le {{B}^{p}}$ for any $p\in \left[ 0,1 \right]$. As is well-known, the L\"owner-Heinz inequality does not always hold for $p>1$. Related to this problem, Furuta \cite{n3} proved:
\begin{theorem}\label{2}
	Let $A,B\in \mathbb{B}\left( \mathcal{H} \right)$ be two positive operators such that $\sigma \left( A \right)\subseteq \left[ m,M \right]$  for some scalars $0<m<M$. If $B\le A$, then 
	\[{{B}^{p}}\le {{K}}\left( m,M,p \right){{A}^{p}}\quad\text{ for }p\ge 1,\] 
	where $K\left( m,M,p \right)$ is the generalized Kantorovich constant:
	\begin{equation}\label{9}
	K\left( m,M,p \right)=\frac{(m{{M}^{p}}-M{{m}^{p}})}{\left( p-1 \right)\left( M-m \right)}{{\left( \frac{p-1}{p}\frac{{{M}^{p}}-{{m}^{p}}}{m{{M}^{p}}-M{{m}^{p}}} \right)}^{p}}\quad\text{ for }p\in \mathbb{R}.
	\end{equation}
\end{theorem}

Theorem \ref{2}, elegantly extended in \cite[Theorem 2.1]{n1} in the following way:
\begin{theorem}\label{th1.2}
	Let $A,B\in \mathbb{B}\left( \mathcal{H} \right)$ be two positive operators satisfying $\sigma \left( A \right),\sigma \left( B \right)\subseteq \left[ m,M \right]$ for some scalars $0<m<M$ and let $f:\left[ m,M \right]\to \mathbb{R}$ be an increasing convex function. If $B\le A$, then for a given $\alpha >0$,
	\begin{equation}\label{5}
	f\left( B \right)\le \alpha f\left( A \right)+\beta {{\mathbf{1}}_{\mathcal{H}}},	
	\end{equation}
	holds for
	\begin{equation}\label{12}
	\beta =\underset{m\le t\le M}{\mathop{\max }}\,\left\{ {{a}_{f}}t+{{b}_{f}}-\alpha f\left( t \right) \right\},
	\end{equation}
	where
	\[{{a}_{f}}\equiv \frac{f\left( M \right)-f\left( m \right)}{M-m}\quad\text{ and }\quad{{b}_{f}}\equiv \frac{Mf\left( m \right)-mf\left( M \right)}{M-m}.\]
\end{theorem}
Of course, the case $f\left( t \right)={{t}^{p}}\left( p\ge 1 \right)$ in Theorem \ref{th1.2}, reduces to Theorem \ref{2} (see \cite[Remark 3.3]{n1}). 
The following converse of Theorem \ref{th1.2} is proven in \cite[Theorem 2.1]{n2}:
\begin{theorem}\label{c}
	Let $A,B\in \mathbb{B}\left( \mathcal{H} \right)$ be two positive operators satisfying $\sigma \left( A \right),\sigma \left( B \right)\subseteq \left[ m,M \right]$ for some scalars $0<m<M$ and let $f:\left[ m,M \right]\to \mathbb{R}$ be a decreasing convex function. If $B\le A$, then for a given $\alpha >0$,
	\begin{equation}\label{11}
	f\left( A \right)\le \alpha g\left( B \right)+\beta {{\mathbf{1}}_{\mathcal{H}}},
	\end{equation}
	holds with $\beta$ as \eqref{12}.
\end{theorem}
We here cite \cite{n5} and \cite{n4} as pertinent references to inequalities of types \eqref{5} and \eqref{11}.

In the current paper extensions of Jensen-type inequalities for the continuous function of self-adjoint operators on complex Hilbert spaces are given, and their applications to the order preserving power inequalities are also presented. We emphasize that our method in this paper is entirely different from that appeared in \cite{n1} and \cite{n2}.

\section{Main Results}
Let $A\in \mathcal{B}\left( \mathcal{H} \right)$ be a self-adjoint operator with $\sigma \left( A \right)\subseteq \left[ m,M \right]$, and let $f\left( t \right)$ be a convex function on $\left[ m,M \right]$, then from \cite{12},  we have for any unit vector $x\in \mathcal{H}$,
\begin{equation*}
f\left( \left\langle Ax,x \right\rangle  \right)\le \left\langle f\left( A \right)x,x \right\rangle.
\end{equation*}
Replace $A$ with $\Phi \left( A \right)$, where $\Phi :\mathcal{B}\left( \mathcal{H} \right)\to \mathcal{B}\left( \mathcal{K} \right)$ is a unital positive linear map, we get
\begin{equation}\label{17}
f\left( \left\langle \Phi \left( A \right)x,x \right\rangle  \right)\le \left\langle f\left( \Phi \left( A \right) \right)x,x \right\rangle
\end{equation}
for any unit vector $x\in \mathcal{K}$.
Assume that ${{A}_{1}},\ldots ,{{A}_{n}}$ are self-adjoint operators on $\mathcal{H}$ with spectra in $J$  and ${{\Phi }_{1}},\ldots ,{{\Phi }_{n}}:\mathcal{B}\left( \mathcal{H} \right)\to \mathcal{B}\left( \mathcal{K} \right)$ are positive linear maps with $\sum\nolimits_{i=1}^{n}{{{\Phi }_{i}}\left( {{\mathbf{1}}_{\mathcal{H}}} \right)}={{\mathbf{1}}_{\mathcal{K}}}$.  Now apply inequality \eqref{17} to the self-adjoint operator $A$ on the Hilbert space $\mathcal{H}\oplus \cdots \oplus \mathcal{H}$ defined by $A={{A}_{1}}\oplus \cdots \oplus {{A}_{n}}$  and the positive linear map $\Phi $ defined on $\mathcal{B}\left( \mathcal{H}\oplus \cdots \oplus \mathcal{H} \right)$  by $\Phi \left( A \right)={{\Phi }_{1}}\left( {{A}_{1}} \right)\oplus \cdots \oplus {{\Phi }_{n}}\left( {{A}_{n}} \right)$. Thus,
\begin{equation}\label{18}
f\left( \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)}x,x \right\rangle  \right)\le \left\langle f\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)} \right)x,x \right\rangle.
\end{equation}

More generalization is discussed as follows:
\begin{lemma}\label{4}
	Let $f:J\to \mathbb{R}$ be a convex and differentiable function on $\overset{o}{\mathop{J}}\,$ (the interior of $J$) whose derivative $f'$ is continuous on $\overset{o}{\mathop{J}}\,$, let ${{A}_{i}},{{B}_{i}}\in \mathcal{B}\left( \mathcal{H} \right)$ self-adjoint operators with the spectra in $\left[ m,M \right]\subset \overset{o}{\mathop{J}}\,$ for $\left( i=1,\ldots ,n \right)$, and let ${{\Phi }_{1}},\ldots ,{{\Phi }_{n}}:\mathcal{B}\left( \mathcal{H} \right)\to \mathcal{B}\left( \mathcal{K} \right)$ be positive linear mappings such that $\sum\nolimits_{i=1}^{n}{{{\Phi }_{i}}\left( {{\mathbf{1}}_{\mathcal{H}}} \right)}={{\mathbf{1}}_{\mathcal{K}}}$. Then for any unit vector $x\in \mathcal{K}$,
	\[\begin{aligned}
	& f'\left( \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}x,x \right\rangle  \right)\left( \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)}x,x \right\rangle -\left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}x,x \right\rangle  \right) \\ 
	& \le \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f\left( {{A}_{i}} \right) \right)}x,x \right\rangle -f\left( \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}x,x \right\rangle  \right) \\ 
	& \le \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f'\left( {{A}_{i}} \right){{A}_{i}} \right)}x,x \right\rangle -\left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}x,x \right\rangle \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f'\left( {{A}_{i}} \right) \right)}x,x \right\rangle .  
	\end{aligned}\]
\end{lemma}
\begin{proof}
	Since $f$ is convex and differentiable on $\overset{o}{\mathop{J}}\,$, then we have for any $t,s\in \left[ m,M \right]$,
	\[f'\left( s \right)\left( t-s \right)\le f\left( t \right)-f\left( s \right)\le f'\left( t \right)\left( t-s \right).\] 
	It is equivalent to
	\begin{equation}\label{6}
	f'\left( s \right)t-f'\left( s \right)s\le f\left( t \right)-f\left( s \right)\le f'\left( t \right)t-f'\left( t \right)s.
	\end{equation}
	If we fix $s\in \left[ m,M \right]$ and apply the continuous functional calculus for ${{A}_{i}}$ $\left( i=1,\ldots ,n \right)$, we get
	\[f'\left( s \right){{A}_{i}}-f'\left( s \right)s{{\mathbf{1}}_{\mathcal{H}}}\le f\left( {{A}_{i}} \right)-f\left( s \right){{\mathbf{1}}_{\mathcal{H}}}\le f'\left( {{A}_{i}} \right){{A}_{i}}-sf'\left( {{A}_{i}} \right).\] 
	Applying the positive linear mappings ${{\Phi }_{i}}$ and summing on i from 1 to $n$, this implies
	\[\begin{aligned}
	f'\left( s \right)\sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)}-f'\left( s \right)s{{\mathbf{1}}_{\mathcal{K}}}&\le \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f\left( {{A}_{i}} \right) \right)}-f\left( s \right){{\mathbf{1}}_{\mathcal{K}}} \\ 
	& \le \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f'\left( {{A}_{i}} \right){{A}_{i}} \right)}-s\sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f'\left( {{A}_{i}} \right) \right)}.  
	\end{aligned}\]
	Therefore, for any unit vector $x\in \mathcal{K}$, we have  
	\[\begin{aligned}
	&f'\left( s \right)\left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)}x,x \right\rangle -f'\left( s \right)s\\
	&\le \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f\left( {{A}_{i}} \right) \right)}x,x \right\rangle -f\left( s \right) \\ 
	& \le \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f'\left( {{A}_{i}} \right){{A}_{i}} \right)}x,x \right\rangle -s\left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f'\left( {{A}_{i}} \right) \right)}x,x \right\rangle .  
	\end{aligned}\]
	Since $\sum\nolimits_{i=1}^{n}{{{\Phi }_{i}}\left( {{\mathbf{1}}_{\mathcal{H}}} \right)}={{\mathbf{1}}_{\mathcal{K}}}$ and $\sigma \left( {{B}_{i}} \right)\subseteq \left[ m,M \right]$, then $\sigma \left( \sum\nolimits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)} \right)\subseteq \left[ m,M \right]$. Thus,  by substituting $s=\left\langle \sum\nolimits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}x,x \right\rangle $, we deduce the desired result.
\end{proof}

We now have all the tools needed to write the proof of the first theorem.
\begin{theorem}\label{1}
	Let all the assumptions of Lemma \ref{4} hold with the additional condition that $f'$ is non-negative. If $\sum\nolimits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}\le \sum\nolimits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)}$, Then
	\begin{equation}\label{8}
	\sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f\left( {{A}_{i}} \right) \right)}\le f\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)} \right)+\delta {{\mathbf{1}}_{\mathcal{K}}}
	\end{equation}
	where
	\[\delta =\underset{\left\| x \right\|=1}{\mathop{\underset{x\in \mathcal{K}}{\mathop{\sup }}\,}}\,\left\{ \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f'\left( {{A}_{i}} \right){{A}_{i}} \right)}x,x \right\rangle -\left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}x,x \right\rangle \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f'\left( {{A}_{i}} \right) \right)}x,x \right\rangle  \right\}.\]
\end{theorem}

\begin{proof}
	It follows from the assumptions that
	\[\begin{aligned}
	0&\le \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f\left( {{A}_{i}} \right) \right)}x,x \right\rangle -f\left( \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}x,x \right\rangle  \right) \\ 
	& \le \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f'\left( {{A}_{i}} \right){{A}_{i}} \right)}x,x \right\rangle -\left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}x,x \right\rangle \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f'\left( {{A}_{i}} \right) \right)}x,x \right\rangle  \\ 
	& \le \underset{\left\| x \right\|=1}{\mathop{\underset{x\in \mathcal{K}}{\mathop{\sup }}\,}}\,\left\{ \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f'\left( {{A}_{i}} \right){{A}_{i}} \right)}x,x \right\rangle -\left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}x,x \right\rangle \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f'\left( {{A}_{i}} \right) \right)}x,x \right\rangle  \right\} \\ 
	& =\delta,  
	\end{aligned}\]
	thanks to Lemma \ref{4}. Therefore, 
	\[\left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f\left( {{A}_{i}} \right) \right)}x,x \right\rangle \le f\left( \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}x,x \right\rangle  \right)+\delta \]
	for any unit vector $x\in \mathcal{K}$. Now we can write, 
	\[\begin{aligned}
	\left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f\left( {{A}_{i}} \right) \right)}x,x \right\rangle &\le f\left( \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}x,x \right\rangle  \right)+\delta  \\ 
	& \le \left\langle f\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)} \right)x,x \right\rangle +\delta  \quad \text{(by \eqref{18})}\\ 
	& =\left\langle f\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)} \right)x,x \right\rangle +\delta \left\langle x,x \right\rangle \quad \text{(since $\left\| x \right\|=1$)} \\ 
	& =\left\langle f\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)} \right)x,x \right\rangle +\left\langle \delta \mathbf{1}_\mathcal{K} x,x \right\rangle  \\ 
	& =\left\langle f\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)} \right)+\delta {{\mathbf{1}}_{\mathcal{K}}}x,x \right\rangle   
	\end{aligned}\]
	for any unit vector $x\in \mathcal{K}$.
	By replacing $x$ by $\frac{y}{\left\| y \right\|}$ where $y$ is any vector in $\mathcal{K}$, we deduce the desired inequality.
\end{proof}

\begin{remark}
	Let all the assumptions of Lemma \ref{4} hold with the additional condition that $f'$ is negative. If $\sum\nolimits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)}\le \sum\nolimits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}$, then  inequality \eqref{8} holds.
\end{remark}

\begin{corollary}
	Let $f:J\to \mathbb{R}$ be a convex and differentiable function on $\overset{o}{\mathop{J}}\,$ whose derivative $f'$ is continuous on $\overset{o}{\mathop{J}}\,$, and let $A,B\in \mathbb{B}\left( \mathcal{H} \right)$ be two self-adjoint operators with the spectra in $\left[ m,M \right]\subset \overset{o}{\mathop{J}}\,$. If $B \le A$, then
	\[f\left( A \right)\le f\left( B \right)+\delta {{\mathbf{1}}_{\mathcal{H}}}\] 
	where
	\[\delta =\underset{\left\| x \right\|=1}{\mathop{\underset{x\in \mathcal{H}}{\mathop{\sup }}\,}}\,\left\{ \left\langle f'\left( A \right)Ax,x \right\rangle -\left\langle Bx,x \right\rangle \left\langle f'\left( A \right)x,x \right\rangle  \right\}.\] 
	In particular, for any $p>1$
	\[B\le A~~\Rightarrow~~ {{A}^{p}}\le {{B}^{p}}+p\delta {{\mathbf{1}}_{\mathcal{H}}}\]
	where $A$, $B$ are two positive operators and
	\[\delta =\underset{\left\| x \right\|=1}{\mathop{\underset{x\in \mathcal{H}}{\mathop{\sup }}\,}}\,\left\{ \left\langle {{A}^{p}}x,x \right\rangle -\left\langle Bx,x \right\rangle \left\langle {{A}^{p-1}}x,x \right\rangle  \right\}.\]
\end{corollary}
The case $\sum\nolimits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)}=\sum\nolimits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}$ implies the following reverse of inequality \eqref{7}. Here we do not use the operator convexity assumption.
\begin{corollary}\label{3}
	Let all the assumptions of Lemma \ref{4} hold. Then
	\[\sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f\left( {{A}_{i}} \right) \right)}\le f\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)} \right)+\delta {{\mathbf{1}}_{\mathcal{K}}}\]
	where
	\[\delta =\underset{\left\| x \right\|=1}{\mathop{\underset{x\in \mathcal{K}}{\mathop{\sup }}\,}}\,\left\{ \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f'\left( {{A}_{i}} \right){{A}_{i}} \right)}x,x \right\rangle -\left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)}x,x \right\rangle \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f'\left( {{A}_{i}} \right) \right)}x,x \right\rangle  \right\}.\]
\end{corollary}

A kind of a converse of Theorem \ref{1} can be considered as follows.
\begin{theorem}\label{21}
	Let all the assumptions of Lemma \ref{4} hold with the additional condition that $f$ is increasing. If $\sum\nolimits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}\le \sum\nolimits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)}$, then
	\begin{equation}\label{22}
	f\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)} \right)\le \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f\left( {{B}_{i}} \right) \right)}+\zeta {{\mathbf{1}}_{\mathcal{K}}}
	\end{equation}
	where
	{\small
		\[\zeta =\underset{\left\| x \right\|=1}{\mathop{\underset{x\in \mathcal{K}}{\mathop{\sup }}\,}}\,\left\{ \left\langle f'\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)} \right)\sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)}x,x \right\rangle -\left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)}x,x \right\rangle \left\langle f'\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)} \right)x,x \right\rangle  \right\}.\]
	}
\end{theorem}

\begin{proof}
	Fix $t\in \left[ m,M \right]$. Since $\left[ m,M \right]$ contains the spectra of the ${{A}_{i}}$ for $i=1,\ldots ,n$ and $\sum\nolimits_{i=1}^{n}{{{\Phi }_{i}}\left( {{\mathbf{1}}_{\mathcal{H}}} \right)}={{\mathbf{1}}_{\mathcal{K}}}$, so the spectra of $\sum\nolimits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}$ is also contained in $\left[ m,M \right]$. Then we may replace $s$ in the inequality \eqref{6} by $\sum\nolimits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}$, via a functional calculus to get
	\[f\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)} \right)-f\left( t \right){{\mathbf{1}}_{\mathcal{K}}}\le f'\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)} \right)\sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}-tf'\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)} \right).\]
	This inequality implies, for any $x\in \mathcal{K}$ with $\left\| x \right\|=1$,
	\begin{equation}\label{20}
	\begin{aligned}
	& \left\langle f\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)} \right)x,x \right\rangle -f\left( t \right) \\ 
	& \le \left\langle f'\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)} \right)\sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}x,x \right\rangle -t\left\langle f'\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)} \right)x,x \right\rangle. 
	\end{aligned}
	\end{equation}
	Substituting $t$ with $\left\langle \sum\nolimits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}x,x \right\rangle $  in \eqref{20}. Thus,
	{\small
		\begin{equation*}
		\begin{aligned}
		0&\le \left\langle f\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)} \right)x,x \right\rangle -f\left( \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}x,x \right\rangle  \right) \quad \text{(by \eqref{18})}\\ 
		& \le \left\langle f'\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)} \right)\sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}x,x \right\rangle -\left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}x,x \right\rangle \left\langle f'\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)} \right)x,x \right\rangle  \\ 
		& \le \underset{\left\| x \right\|=1}{\mathop{\underset{x\in \mathcal{K}}{\mathop{\sup }}\,}}\,\left\{ \left\langle f'\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)} \right)\sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}x,x \right\rangle -\left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}x,x \right\rangle \left\langle f'\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)} \right)x,x \right\rangle  \right\} \\ 
		& =\zeta.
		\end{aligned}
		\end{equation*}
	}
	On the other hand,
	\[\begin{aligned}
	& \left\langle f\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)} \right)x,x \right\rangle  \\ 
	& \le f\left( \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}x,x \right\rangle  \right)+\zeta  \\ 
	& \le f\left( \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)}x,x \right\rangle  \right)+\zeta  \quad \text{(since $f$ is increasing and $\sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}\le \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)}$)}\\ 
	& \le \left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f\left( {{A}_{i}} \right) \right)}x,x \right\rangle +\zeta \quad \text{(by \eqref{19})}  
	\end{aligned}\]
	for any unit vector $x\in \mathcal{K}$, and the proof is complete.
\end{proof}

Like Corollary \ref{3}, one can get the following result. This can be considered as an extension of \eqref{7}, where the assumption operator convexity is dropped.
\begin{corollary}
	Let all the assumptions of Lemma \ref{4} hold. Then
	\[f\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)} \right)\le \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( f\left( {{A}_{i}} \right) \right)}+\zeta {{\mathbf{1}}_{\mathcal{K}}}\]
	where
	{\small
		\[\zeta =\underset{\left\| x \right\|=1}{\mathop{\underset{x\in \mathcal{K}}{\mathop{\sup }}\,}}\,\left\{ \left\langle f'\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)} \right)\sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)}x,x \right\rangle -\left\langle \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)}x,x \right\rangle \left\langle f'\left( \sum\limits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)} \right)x,x \right\rangle  \right\}.\]
	}
\end{corollary}


\begin{corollary}
	Let $f:J\to \mathbb{R}$ be a convex and differentiable function on $\overset{o}{\mathop{J}}\,$ whose derivative $f'$ is continuous on $\overset{o}{\mathop{J}}\,$, and let $A,B\in \mathbb{B}\left( \mathcal{H} \right)$ be two self-adjoint operators with the spectra in $\left[ m,M \right]\subset \overset{o}{\mathop{J}}\,$. If $B \le A$, then
	\[f\left( B \right)\le f\left( A \right)+\zeta {{\mathbf{1}}_{\mathcal{H}}}\]
	where
	\[\zeta =\underset{\left\| x \right\|=1}{\mathop{\underset{x\in \mathcal{H}}{\mathop{\sup }}\,}}\,\left\{ \left\langle f'\left( B \right)Bx,x \right\rangle -\left\langle Bx,x \right\rangle \left\langle f'\left( B \right)x,x \right\rangle  \right\}.\]
	In particular, for any $p>1$
	\[B\le A~~\Rightarrow ~~{{B}^{p}}\le {{A}^{p}}+p\zeta {{\mathbf{1}}_{\mathcal{H}}}\]
	where $A$, $B$ are two positive operators and
	\[\zeta =\underset{\left\| x \right\|=1}{\mathop{\underset{x\in \mathcal{H}}{\mathop{\sup }}\,}}\,\left\{ \left\langle {{B}^{p}}x,x \right\rangle -\left\langle Bx,x \right\rangle \left\langle {{B}^{p-1}}x,x \right\rangle  \right\}.\]
\end{corollary}

\begin{remark}
	Let all the assumptions of Lemma \ref{4} hold with the additional condition that $f$ is decreasing. If $\sum\nolimits_{i=1}^{n}{{{\Phi }_{i}}\left( {{A}_{i}} \right)}\le \sum\nolimits_{i=1}^{n}{{{\Phi }_{i}}\left( {{B}_{i}} \right)}$, then  inequality \eqref{22} holds.
\end{remark}


% BibTeX users please use one of
%\bibliographystyle{spbasic}      % basic style, author-year citations
%\bibliographystyle{spmpsci}      % mathematics and physical sciences
%\bibliographystyle{spphys}       % APS-like style for physics
%\bibliography{}   % name your BibTeX data base

% Non-BibTeX users please use
\begin{center}
\begin{thebibliography}{99} % Enter references in alphabetical order and according to the following format.

%
\bibitem{n5}
S. Furuichi, H. R. Moradi and M. Sababheh, New sharp inequalities for operator means, {\it Linear Multilinear Algebra}., (2018): 1-12. https://doi.org/10.1080/03081087.2018.1461189

\bibitem{5}
S. Furuichi, H. R. Moradi and A. Zardadi, Some new Karamata type inequalities and their applications to some entropies, {\it Rep. Math. Phys}., (2019) (accepted). arXiv:1811.07277.  

\bibitem{n3}
T. Furuta, Operator inequalities associated with H\"older--McCarthy and Kantorovich inequalities, {\it J. Inequal. Appl}., {\bf2} (1998), 137--148.

\bibitem{n4}
I. H. G\"um\"u\c s, H. R. Moradi and M. Sababheh, More accurate operator means inequalities, {\it J. Math. Anal. Appl}., {\bf465}(1) (2018), 267--280.

\bibitem{1}
F. Hansen, J. Pe\v cari\'c and I. Peri\'c, Jensen's operator inequality and it's converses, {\it Math. Scand}., {\bf100} (2007), 61--73.


\bibitem{10}
L. Horv\'ath, K. A. Khan and J. Pe\v cari\'c, Cyclic refinements of the different versions of operator Jensen's inequality, {\it Electron. J. Linear Algebra}., {\bf31}(1) (2016), 125--133.


\bibitem{6}
J. Mi\'ci\'c, H. R. Moradi and S. Furuichi, Choi--Davis--Jensen's inequality without convexity, {\it J. Math. Inequal}., {\bf12}(4) (2018), 1075--1085.


\bibitem{11}
J. Mi\'ci\'c and J. Pe\v cari\'c, Some mappings related to Levinson's inequality for Hilbert space operators, {\it Filomat}., {\bf31}  (2017), 1995--2009.




\bibitem{n1}
J. Mi\'ci\'c, J. Pe\v cari\'c and Y. Seo, Function order of positive operators based on the Mond--Pe\v cari\'c method, {\it Linear Algebra Appl}., {\bf360} (2003), 15--34.

\bibitem{12}
B. Mond and J. Pe\v cari\'c, On Jensen's inequality for operator convex functions, {\it Houston J. Math}., {\bf21} (1995), 739--753.

\bibitem{n2}
J. Pe\v cari\'c and J. Mi\'ci\'c, Some functions reversing the order of positive operators, {\it Linear Algebra Appl}., {\bf396} (2005),  175--187.

\end{thebibliography}
\end{center}



{\small

\noindent{\bf Mohsen Shah Hosseini}

\noindent Assistant Professor of Mathematics

\noindent Department of Mathematics



\noindent Shahr-e-Qods Branch, Islamic Azad University, 

\noindent Tehran, Iran.

\noindent E-mail: mohsen\_shahhosseini@yahoo.com}\\

{\small
\noindent{\bf  Hamid Reza Moradi}

\noindent Postdoctoral researcher

\noindent  Department of Mathematics



\noindent Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University, 


\noindent Mashhad, Iran.

\noindent E-mail: hrmoradi@mshdiau.ac.ir }\\


{\small
	\noindent{\bf  Baharak Moosavi}
	
	\noindent Assistant Professor of Mathematics
	
	\noindent  Department of Mathematics
	

	
	\noindent Safadasht Branch, Islamic Azad University, 
	
	
	\noindent Tehran, Iran.
	
	\noindent E-mail: baharak\_moosavie@yahoo.com}\\
\end{document}