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\newtheorem*{thB}{Theorem B}
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\noindent\parbox{2.95cm}{\includegraphics*[keepaspectratio=true,scale=0.125]{AFA.jpg}}
\noindent\parbox{4.85in}{\hspace{0.1mm}\\[1.5cm]\noindent
\qquad Ann. Funct. Anal. 3 (2012), no. 1, 67--85\\
{\footnotesize \qquad \textsc{\textbf{$\mathscr{A}$}nnals of
\textbf{$\mathscr{F}$}unctional
\textbf{$\mathscr{A}$}nalysis}\\
\qquad ISSN: 2008-8752 (electronic)\\
\qquad URL:
\textcolor[rgb]{0.00,0.00,0.99}{www.emis.de/journals/AFA/}
}\\[.5in]}

\title[Extension of the refined Jensen's operator inequality]{Extension of the refined Jensen's operator inequality with condition on spectra}

\author[J. Mi\'{c}i\'{c}, J. Pe\v {c}ari\'{c}, J. Peri\'{c}]{Jadranka Mi\'{c}i\'{c}$^1$$^{*}$, Josip Pe\v {c}ari\'{c}$^2$ and Jurica Peri\'{c}$^3$}

\address{$^{1}$ Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb,
Ivana Lu\v ci\' ca 5, 10000 Zagreb, Croatia.}
\email{\textcolor[rgb]{0.00,0.00,0.84}{jmicic@fsb.hr}}

\address{$^{2}$ Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovi\'{c}a 30, 10000 Zagreb, Croatia.}
\email{\textcolor[rgb]{0.00,0.00,0.84}{pecaric@hazu.hr}}

\address{$^{3}$ Faculty of Science, Department of Mathematics, University of Split, Teslina 12, 21000 Split,
Croatia.} \email{\textcolor[rgb]{0.00,0.00,0.84}{jperic@pmfst.hr}}

\dedicatory{{\rm Communicated by M. S. Moslehian}}

\subjclass[2010]{Primary 47A63; Secondary 47B15.}


\keywords{Jensen's operator inequality, self-adjoint operator,
positive linear mapping, convex function, quasi-arithmetic mean.}

\date{Received: 6 January 2012; Accepted: 13 January 2012.
\newline \indent $^{*}$ Corresponding author}

\begin{abstract}
We give an extension of the refined Jensen's operator inequality for
$n-$tuples of self-adjoint operators, unital $n-$tuples of positive
linear mappings and real valued continuous convex functions with
conditions on the spectra of the operators.  We also study the order
among quasi-arithmetic means under similar conditions.
\end{abstract} \maketitle



\section{Introduction}
We recall some notations and definitions. Let $\mathcal{B}(H)$ be
the $C^*$-algebra of all bounded linear operators on a Hilbert space
$ H$ and $1_H$ stands for the identity operator.  We define bounds
of a self-adjoint operator $A\in \mathcal{B}(H)$ by
\begin{equation*} \label{eq-mM} m_A = \inf _{\| x\|=1}
\langle Ax,x \rangle \quad \mbox{and} \quad M_A = \sup _{\| x\|=1}
\langle Ax,x \rangle
\end{equation*}
for $x \in H$. If ${\mathsf{Sp}}(A)$  denotes the spectrum of $A$,
then ${\mathsf{Sp}}(A)$ is real and ${\mathsf{Sp}}(A) \subseteq
[m_A,M_A]$.

For an operator $A\in \mathcal{B}(H)$ we define operators $|A|$,
$A^+$, $A^-$ by $$|A|=(A^*A)^{1/2}, \qquad A^+=(|A|+A)/2, \qquad
A^-=(|A|-A)/2.$$ Obviously, if $A$ is self-adjoint, then
$|A|=(A^{2})^{1/2}$ and $A^{+}, A^{-}\geq 0$ (called positive and
negative parts of $A=A^{+}-A^{-}$).


\medskip

B. Mond and J. Pe\v cari\' c in \cite{MP1} proved Jensen's operator
inequality
\begin{equation}\label{OP-MP}
f\left(\sum_{i=1}^{n}w_{i}\Phi_{i}(A_i)\right) \leq
\sum_{i=1}^{n}w_{i} \Phi_i \left( f(A_i) \right),\end{equation}  for
operator convex functions $ f $ defined on an interval $ I, $ where
$\Phi_i :\mathcal{B}(H)\to \mathcal{B}(K)$, $i=1,\ldots,n$, are
unital positive linear mappings, $A_1,\dots,A_n $ are self-adjoint
operators with the spectra in $I$ and $w_1,\dots,w_n $ are
non-negative real numbers with $\sum_{i=1}^{n}w_i=1$.

 F. Hansen, J. Pe\v{c}ari\'{c} and I. Peri\'{c}  gave in \cite{HPP2} a generalization of \eqref{OP-MP} for a unital
field of positive linear mappings. The following discrete version of
their inequality holds
  \begin{equation}\label{OP-HPP}
f\left(\sum_{i=1}^{n}\Phi_{i}(A_i)\right) \leq \sum_{i=1}^{n} \Phi_i
\left( f(A_i) \right),\end{equation}  for operator convex functions
$ f $ defined on an interval $ I, $ where $\Phi_i :\mathcal{B}(H)\to
\mathcal{B}(K)$, $i=1,\ldots,n$, is a unital field of positive
linear mappings (i.e.\ $\sum_{i=1}^{n} \Phi_i(1_H)=1_K$),
$A_1,\dots,A_n $ are self-adjoint operators with the spectra in $I$.

Recently, J. Mi\'{c}i\'{c}, Z. Pavi\'{c} and J. Pe\v{c}ari\'{c}
proved in \cite[Theorem 1]{MPP} that \eqref{OP-HPP} stands without
operator convexity of $f:I \to {\mathbb{R}}$ if a condition on
spectra
\begin{equation*}\label{tA-condition}(m_A,M_A) \cap [m_i,M_i]= {\O} \quad
\text{for}~ i=1,\ldots,n
\end{equation*} holds,
where $m_i$ and $M_i$,  $m_i \leq M_i$ are bounds of $A_i$,
$i=1,\ldots,n$; and $m_A$ and $M_A$, $m_A \leq M_A$, are bounds of
$A=\sum_{i=1}^{n}\Phi_{i}(A_i)$ (provided that the interval $I$
contains all $m_i,M_i$).

Next, they considered in \cite[Theorem 2.1]{MPP3} the case when
$(m_A,M_A) \cap [m_i,M_i]= {\O} $ is valid for several $i\in
\{1,\ldots,n\}$, but not for all $i=1,\ldots,n$ and obtain an
extension of \eqref{OP-HPP}  as follows.

\begin{thA}\label{thA}
Let  $(A_1,\ldots,A_n)$ be an $n-$tuple of self-adjoint operators
$A_i \in B(H)$ with the bounds $m_i$ and $M_i$,  $m_i \leq M_i$,
$i=1,\ldots,n$. Let $(\Phi_1,\ldots,\Phi_n)$ be an $n-$tuple  of
po\-si\-tive linear mappings $\Phi_i:B(H) \rightarrow B(K)$,
 such that $\sum_{i=1}^{n_1}\Phi_i
(1_H)=\alpha\, 1_K$, $\sum_{i=n_1+1}^{n}\Phi_i (1_H)=\beta\, 1_K$,
where $1\leq n_1<n$, $\alpha, \beta>0$ and $\alpha+\beta=1$. Let $m=
\min \{m_1,\ldots,m_{n_1} \}$ and $M= \max \{M_1,\ldots,M_{n_1} \}$.
If
$$(m,M) \cap [m_i,M_i]= {\O} \qquad \text{for} \quad
i=n_1+1,\ldots,n,$$ and one of two equalities
$$\frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(A_i)=\sum_{i=1}^{n}\Phi_i(A_i)=
\frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(A_i)$$ is valid,
 then
\begin{equation}\label{jmA-eq}
\frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f(A_i))\leq
\sum_{i=1}^{n}\Phi_i(f(A_i))\leq\frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f(A_i)),
\end{equation}
holds for every continuous convex function $f:I  \rightarrow
{\mathbb{R}}$ provided that the interval $I$ contains all $m_i,M_i$,
$i=1,\ldots,n,$.

If $f:I \rightarrow {\mathbb{R}}$ is concave, then the reverse
inequality is valid in \eqref{jmA-eq}.
\end{thA}

Very recently, J. Mi\'{c}i\'{c}, J. Pe\v{c}ari\'{c} and J. Peri\'{c}
gave in \cite[Theorem 3]{MPPeric1} the following refinement of
\eqref{OP-HPP} with condition on spectra, i.e.\ a refinement of
\cite[Theorem 3]{MPP} (see also \cite[Corollary 5]{MPP}).

\begin{thB}\label{thB}
Let  $(A_1,\ldots,A_n)$ be an $n-$tuple of self-adjoint operators
$A_i \in B(H)$ with the bounds $m_i$ and $M_i$,  $m_i \leq M_i$,
$i=1,\ldots,n$. Let $(\Phi_1,\ldots,\Phi_n)$ be an $n-$tuple  of
po\-si\-tive linear mappings $\Phi_i:B(H) \rightarrow B(K)$,
$i=1,\ldots,n$, such that $\sum_{i=1}^{n}\Phi_i (1_H)=1_K$. Let
$$(m_A,M_A) \cap [m_i,M_i]= {\O} \quad \text{for}~
i=1,\ldots,n,\qquad \text{and} \qquad m<M,$$ where $m_A$ and $M_A$,
$m_A \leq M_A$, are the bounds of the operator
$A=\sum_{i=1}^{n}\Phi_{i}(A_i)$ and
$$m=\max \left\{ M_i \colon M_i \leq m_A, i\in \{1,\ldots,n\} \right\}, \; M= \min \left\{m_i \colon m_i \geq M_A, i\in \{1,\ldots,n\} \right\}.
$$
If $f:I\to {\mathbb{R}}$ is a continuous convex $($resp.\ concave$)$
function provided that the interval $I$ contains all $m_i,M_i$, then
\begin{eqnarray}
\displaystyle & & f\left(\sum_{i=1}^{n}\Phi_{i}(A_i)\right)\leq
\sum_{i=1}^{n}\Phi_i\left( f(A_i)\right) - \delta_f \widetilde{A}
\leq \sum_{i=1}^{n}\Phi_i\left( f(A_i)\right)\label{t1-eq0} \\
\displaystyle &(\text{resp.} \quad &
f\left(\sum_{i=1}^{n}\Phi_{i}(A_i)\right)\geq
\sum_{i=1}^{n}\Phi_i\left( f(A_i)\right) + \delta_f \widetilde{A}
\geq \sum_{i=1}^{n}\Phi_i\left( f(A_i)\right)\;) \nonumber
\end{eqnarray}
holds, where
%constants
%$\delta_f\equiv\delta_f(\bar{m},\bar{M})$  and $\widetilde{A}=\widetilde{A}_A(\bar{m},\bar{M})$ are defined as follows
\begin{equation*}\label{t1-const}
\begin{array}{rcl}
\delta_f \equiv \delta_f(\bar{m},\bar{M}) &=& f(\bar{m})+f(\bar{M})
- 2 f\left( \frac{\bar{m}+\bar{M}}{2}\right)
\\[0.5ex] (\text{resp.} \quad \delta_f \equiv \delta_f(\bar{m},\bar{M}) &=& 2 f\left( \frac{\bar{m}+\bar{M}}{2}\right) - f(\bar{m})-f(\bar{M})\;),
\\[0.5ex]   \widetilde{A}\equiv \widetilde{A}_A(\bar{m},\bar{M}) &=& \frac{1}{2} 1_K - \frac{1}{\bar{M}-\bar{m}} \left|A - \frac{\bar{m}+\bar{M}}{2} 1_K \right|
\end{array}
\end{equation*}
and \quad$\bar{m} \in [m,m_A]$, $\bar{M} \in [M_A,M]$, $\bar{m}<
\bar{M}$, \quad are arbitrary numbers.
\end{thB}

\medskip

There is an extensive literature devoted to Jensen’s inequality
concerning different refinements and extensive results, see, for
example \cite{ajs, drag, kadm}, \cite{mos}--\cite{wml}.

In this paper we study an extension of Jensen's inequality given in
Theorem~B and a refinement of Theorem~A.  As an application of this
result to the quasi-arithmetic mean with a weight, we give an
extension of results given in \cite{MPPeric1} and a refinement of
ones given in \cite{MPP3}.


%\medskip

\section{Main results}\label{sec2}

To obtain our main result we need a result \cite[Lemma 2]{MPPeric1}
given in the following lemma.

\begin{lmC}\label{lC}
Let  $A$ be a self-adjoint operator $A \in B(H)$ with
${\mathsf{Sp}}(A) \subseteq [m,M]$, for some scalars $m < M$. Then
\begin{eqnarray}
f\left(A \right) &\leq& \frac{M 1_H - A}{M-m} f(m) + \frac{A - m 1_H}{M-m} f(M) -  \delta_f \widetilde{A} \label{l1-eq} \\
\text{$($resp.} \quad f\left(A \right) &\geq& \frac{M 1_H - A}{M-m}
f(m) + \frac{A - m 1_H}{M-m} f(M) +  \delta_f \widetilde{A}
\;\text{$)$} \nonumber
\end{eqnarray}
holds for every continuous convex $($resp.\ concave$)$ function
$f:[m,M]  \rightarrow {\mathbb{R}}$, where
\begin{eqnarray*}
&\delta_f = f(m)+f(M) - 2 f\left( \frac{m+M}{2}\right) \quad
(\text{resp.} \;\delta_f = 2 f\left( \frac{m+M}{2}\right)
-f(m)-f(M)),
\\ & \text{and} \quad \widetilde{A} = \frac{1}{2} 1_H - \frac{1}{M-m} \left|A- \frac{m+M}{2} 1_H \right|.
\end{eqnarray*}
\end{lmC}

We shall give the proof for the convenience of the reader.

\medskip

\noindent \textit{Proof of Lemma C.} We prove only the convex case.

In \cite[Theorem~1, p. 717]{MPF} is prove that
\begin{equation} \label{l0-eq} \begin{array}{ll}  &\min \{p_1, p_2 \} \left[ f(x) + f(y) - 2 f \left( \frac{x+y}{2} \right) \right] \\
\leq& p_1 f (x) + p_2 f (y) - f (p_1 x + p_2 y)
\end{array}
\end{equation}
holds for every convex function $f$ on an interval $I$ and  $x, y
\in I$, $p_1, p_2 \in [0, 1]$ such that $p_1 + p_2 = 1$.

 Putting $x = m, y = M$ in \eqref{l0-eq} it follows that
\begin{equation}\label{l1-eq2}
\begin{array}{rcl}
f\left( p_1 m + p_2 M \right)  &\leq&  p_1 f(m) + p_2 f(M) \\ &-&
\min \{ p_1,p_2 \} \left( f(m) + f(M) - 2f\left(\frac{m+M}{2}
\right)  \right)
\end{array}
\end{equation}
 holds for every $p_1, p_2 \in [0,1]$ such that $p_1+p_2=1$ .
  For any $t \in [m,M]$ we can write
\begin{equation*}
f \left( t \right) =f \left(
\frac{M-t}{M-m}m+\frac{t-m}{M-m}M\right).
\end{equation*}
Then by using \eqref{l1-eq2} for $p_1=\frac{M-t}{M-m}$ and
$p_2=\frac{t-m}{M-m}$ we get
\begin{equation}\label{l1-eq3}
\begin{array}{rcl}
  f (t) & \leq & \displaystyle \frac{M-t}{M-m} f (m) + \frac{t-m}{M-m}  f (M) \\
   & - & \displaystyle \left( \frac{1}{2}  - \frac{1}{M-m} \left| t- \frac{m+M}{2} \right| \right) \left( f(m) +f (M) -2f \left( \frac{m+M}{2} \right) \right),
\end{array}
\end{equation}
since
$$ \min \left\{ \frac{M-t}{M-m} , \frac{t-m}{M-m} \right\} = \frac{1}{2}  - \frac{1}{M-m} \left|t- \frac{m+M}{2} \right| . $$
Finally we use the continuous functional calculus for a self-adjoint
operator $A$: $f,g \in \mathcal{C}(I), \mathop{Sp} (A) \subseteq I$
and $f\geq g$ implies $f(A)\geq g(A)$; and $h(t)=|t|$ implies
$h(A)=|A|$. Then by using \eqref{l1-eq3} we obtain the desired
inequality \eqref{l1-eq}. \hfill $\Box$

\medskip

In the following theorem we give an extension of Jensen's inequality
given in Theorem~B and a refinement of Theorem~A.

\begin{theorem}\label{jmt1}
Let  $(A_1,\ldots,A_n)$ be an $n-$tuple of self-adjoint operators
$A_i \in B(H)$ with the bounds $m_i$ and $M_i$,  $m_i \leq M_i$,
$i=1,\ldots,n$. Let $(\Phi_1,\ldots,\Phi_n)$ be an $n-$tuple  of
po\-si\-tive linear mappings $\Phi_i:B(H) \rightarrow B(K)$,
 such that $\sum_{i=1}^{n_1}\Phi_i
(1_H)=\alpha\, 1_K$, $\sum_{i=n_1+1}^{n}\Phi_i (1_H)=\beta\, 1_K$,
where $1\leq n_1<n$, $\alpha, \beta>0$ and $\alpha+\beta=1$. Let
\\$m_L= \min \{m_1,\ldots,m_{n_1} \}$, $M_R= \max \{M_1,\ldots,M_{n_1}\}$ and
$$ \begin{array}{rcl}
      m&=&\left\{  \begin{array}{l} m_L, \qquad\text{\rm if} \;  \left\{ M_i \colon M_i \leq m_L, i\in \{n_1+1,\ldots,n\} \right\}={\O}, \\ \max\left\{ M_i \colon M_i \leq m_L, i\in \{n_1+1,\ldots,n\} \right\}, \quad \text{\rm otherwise}, \end{array} \right.\\
     M&=& \left\{  \begin{array}{l} M_R, \qquad\text{\rm if} \;  \left\{m_i \colon m_i \geq M_R, i\in \{n_1+1,\ldots,n\} \right\}={\O}, \\ \min\left\{m_i \colon m_i \geq M_R, i\in \{n_1+1,\ldots,n\} \right\}, \quad \text{\rm otherwise}.\end{array} \right.
   \end{array}
$$
If
$$ (m_L,M_R) \cap [m_i,M_i]= {\O} \quad \text{for} \quad
i=n_1+1,\ldots,n, \quad  \quad m<M,$$ and one of two equalities
$$\frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(A_i)=\sum_{i=1}^{n}\Phi_i(A_i)=
\frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(A_i)$$ is valid,
 then
\begin{eqnarray}
  \frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f(A_i))  & \leq &  \frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f(A_i)) + \beta \delta_f \widetilde{A} \leq \sum_{i=1}^{n}\Phi_i(f(A_i))  \nonumber \\
   & \leq & \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f(A_i)) - \alpha \delta_f \widetilde{A} \leq \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f(A_i)), \label{jmt1-eq}
\end{eqnarray}
holds for every continuous convex function  $f:I  \rightarrow
{\mathbb{R}}$ provided that the interval $I$ contains all $m_i,M_i$,
$i=1,\ldots,n$, where
\begin{equation}\label{t1-const}
\begin{array}{c}
\displaystyle \delta_f \equiv \delta_f(\bar{m},\bar{M}) =
f(\bar{m})+f(\bar{M}) - 2 f\left( \frac{\bar{m}+\bar{M}}{2}\right)
\\ \displaystyle    \widetilde{A}\equiv \widetilde{A}_{A,\Phi,n_1,\alpha}(\bar{m},\bar{M}) = \frac{1}{2} 1_K - \frac{1}{\alpha (\bar{M}-\bar{m})}  \sum_{i=1}^{n_1} \Phi_i
   \left( \left|A_i- \frac{\bar{m}+\bar{M}}{2} 1_H \right| \right)
\end{array}
\end{equation}
and \; $\bar{m} \in [m,m_L]$, $\bar{M} \in [M_R,M]$, $\bar{m}<
\bar{M}$, \; are arbitrary numbers.

\bigskip

If $f:I \rightarrow {\mathbb{R}}$ is concave, then the reverse
inequality is valid in \eqref{jmt1-eq}. \end{theorem}

\begin{proof}
We prove only the convex case.

Let us denote
$$A=\frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(A_i),\qquad
B=\frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(A_i), \qquad
C=\sum_{i=1}^{n}\Phi_i(A_i).$$ It is easy to verify that $A=B$ or
$B=C$ or $A=C$ implies $A=B=C$.


\vspace{2mm}

Since $f$ is convex on $[\bar{m},\bar{M}]$ and
${\mathsf{Sp}}(A_i)\subseteq [m_i,M_i] \subseteq [\bar{m},\bar{M}]$
for $i=1,\ldots,n_1$, it follows from Lemma~C that
$$
f \left( A_i \right) \leq \frac{\bar{M} 1_H-
A_i}{\bar{M}-\bar{m}}f(\bar{m})+\frac{ A_i-\bar{m}
1_H}{\bar{M}-\bar{m}}f(\bar{M}) - \delta_{f} \widetilde{A}_i, \qquad
i=1,\ldots,n_1$$ holds, where
 $\delta_{f} = f(\bar{m})+f(\bar{M}) - 2 f\left( \frac{\bar{m}+\bar{M}}{2}\right)$ and
$\widetilde{A}_i = \frac{1}{2} 1_H - \frac{1}{\bar{M}-\bar{m}}
\left|A_i- \frac{\bar{m}+\bar{M}}{2} 1_H \right|.$ Applying a
positive linear mapping $\Phi_i$ and summing, we obtain
$$ \begin{array}{rcl}
\sum_{i=1}^{n_1} \Phi_i \left( f (A_i) \right) &\leq&
\frac{ \bar{M} \alpha 1_K- \sum_{i=1}^{n_1} \Phi_i(A_i)}{\bar{M}-\bar{m}}f(\bar{m})+\frac{\sum_{i=1}^{n_1}\Phi_i(A_i)- \bar{m} \alpha 1_K}{\bar{M}-\bar{m}}f(\bar{M}) \\
   &-&  \delta_{f} \left( \frac{\alpha}{2} 1_K - \frac{1}{\bar{M}-\bar{m}} \sum_{i=1}^{n_1} \Phi_i
   \left( \left|A_i- \frac{\bar{m}+\bar{M}}{2} 1_H \right| \right) \right),
   \end{array} $$
 since $\sum_{i=1}^{n_1}  \Phi_i(1_H) =\alpha 1_K$. It follows that
\begin{equation}\label{jmt1-eq4}
\frac{1}{\alpha }\sum_{i=1}^{n_1} \Phi_i \left(f (A_i) \right) \leq
\frac{ \bar{M}  1_K- A}{\bar{M}-\bar{m}}f(\bar{m})+\frac{A- \bar{m}
1_K}{\bar{M}-\bar{m}}f(\bar{M}) - \delta_f \widetilde{A},
\end{equation}
where $\widetilde{A} = \frac{1}{2} 1_K - \frac{1}{\alpha
(\bar{M}-\bar{m})} \sum_{i=1}^{n_1} \Phi_i
   \left( \left|A_i- \frac{\bar{m}+\bar{M}}{2} 1_H \right| \right) $.


In addition, since $f$ is convex on all $[m_i,M_i]$ and
$(\bar{m},\bar{M}) \cap [m_i,M_i]= {\O}$ for $i=n_1+1,\ldots,n$,
then
\begin{equation*}\label{jmt1-eq3}
    f(A_i) \geq \frac{\bar{M}1_H-A_i}{\bar{M}-\bar{m}}f(\bar{m})+\frac{A_i-\bar{m}1_H}{\bar{M}-\bar{m}}f(\bar{M}),  \qquad i=n_1+1,\ldots,n.
\end{equation*}
It follows
\begin{equation}\label{jmt1-eq5}
\frac{1}{\beta }\sum_{i=n_1+1}^{n} \Phi_i \left(f (A_i) \right) -
\delta_f \widetilde{A} \geq \frac{ \bar{M}  1_K-
B}{\bar{M}-\bar{m}}f(\bar{m})+\frac{B- \bar{m}
1_K}{\bar{M}-\bar{m}}f(\bar{M}) - \delta_f \widetilde{A}.
\end{equation}
Combining \eqref{jmt1-eq4} and \eqref{jmt1-eq5} and taking into
account that $A= B$, we obtain
\begin{equation}\label{jmt1-eq6}
\frac{1}{\alpha }\sum_{i=1}^{n_1} \Phi_i \left(f (A_i) \right) \leq
\frac{1}{\beta }\sum_{i=n_1+1}^{n} \Phi_i \left(f (A_i) \right) -
\delta_f \widetilde{A}.
\end{equation}
Next, we obtain
\begin{eqnarray*}
& & \frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f(A_i)) \\
&=&
\sum_{i=1}^{n_1}\Phi_i(f(A_i))+\frac{\beta}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f(A_i)) \quad (\text{by} \; \alpha+ \beta =1) \nonumber\\
&\leq&\sum_{i=1}^{n_1}\Phi_i(f(A_i))+\sum_{i=n_1+1}^{n}\Phi_i(f(A_i)) - \beta  \delta_f \widetilde{A} \qquad\qquad (\text{by} \; \eqref{jmt1-eq6}) \nonumber\\
&\leq& \frac{\alpha}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f(A_i)) - \alpha
\delta_f \widetilde{A} +\sum_{i=n_1+1}^{n}\Phi_i(f(A_i)) - \beta
\delta_f \widetilde{A} \qquad (\text{by} \; \eqref{jmt1-eq6})
\nonumber\\&=& \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f(A_i)) -
\delta_f \widetilde{A} \quad (\text{by} \; \alpha+ \beta =1),
\nonumber
\end{eqnarray*}
which gives the following double inequality
\begin{equation*}\label{jmt1-eq2}
\frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f(A_i)) \leq
\sum_{i=1}^{n}\Phi_i(f(A_i)) - \beta  \delta_f \widetilde{A}  \leq
\frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f(A_i)) - \delta_f
\widetilde{A}.
\end{equation*}
Adding  $\beta \delta_f \widetilde{A}$ in the above inequalities, we
get
\begin{equation}
 \frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f(A_i)) + \beta \delta_f \widetilde{A} \leq \sum_{i=1}^{n}\Phi_i(f(A_i)) \leq \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f(A_i)) - \alpha \delta_f \widetilde{A}. \label{jmt1-eq7}
\end{equation}
%Similarly as above, replacing $\sum_{i=1}^{n}\Phi_i(f(A_i)) - \beta  \delta_f \widetilde{A} $ by $\sum_{i=1}^{n}\Phi_i(f(A_i))$ in \eqref{jmt1-eq2}, we get

Now, we remark that $\delta_f\geq0$ and $\widetilde{A}\geq0$.
(Indeed, since $f$ is convex, then $f\left( (\bar{m}+\bar{M})/2
\right) \leq  (f(\bar{m})+f(\bar{M}))/2$, which implies that
$\delta_f \geq 0$. Also, since $$ {\mathsf{Sp}}(A_i) \subseteq
[\bar{m},\bar{M}] \quad \Rightarrow \quad  \left|A_i-
\frac{\bar{M}+\bar{m}}{2} 1_H \right| \leq \frac{\bar{M}-\bar{m}}{2}
1_H, \qquad \text{for $i=1,\ldots,n_1$}, $$ then $$
\sum_{i=1}^{n_1} \Phi_i
   \left( \left|A_i- \frac{\bar{M}+\bar{m}}{2} 1_H \right| \right) \leq \frac{\bar{M}-\bar{m}}{2} \alpha 1_K,
$$ which gives
$$ 0 \leq \frac{1}{2} 1_K - \frac{1}{\alpha (\bar{M}-\bar{m})} \sum_{i=1}^{n_1} \Phi_i
   \left( \left|A_i- \frac{\bar{M}+\bar{m}}{2} 1_H \right| \right)= \widetilde{A}.\qquad )$$
Consequently, the following inequalities
\begin{eqnarray*}
  & & \frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f(A_i)) \leq \frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f(A_i)) + \beta \delta_f \widetilde{A}, \\
    & & \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f(A_i)) - \alpha \delta_f \widetilde{A} \leq \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f(A_i)),
\end{eqnarray*} hold, which with \eqref{jmt1-eq7} proves the desired series inequalities \eqref{jmt1-eq}.

%Finally, since $\sum_{i=1}^{n}\Phi_i(f(A_i)) + \alpha \delta_f \widetilde{A} \leq \sum_{i=1}^{n}\Phi_i(f(A_i)) + %\delta_f \widetilde{A} $ for all $\alpha \in (0,1)$ and  $\sum_{i=1}^{n}\Phi_i(f(A_i)) + \alpha \delta_f %\widetilde{A} \leq \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f(A_i))$  by \eqref{jmt1-eq7}, it follows that
% $\sum_{i=1}^{n}\Phi_i(f(A_i)) +  \delta_f \widetilde{A} \leq \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f(A_i))$. %Then we obtain RHS of \eqref{jmt1-eq}.
 \end{proof}

\begin{example}\label{ex1}  We observe the matrix case of Theorem~\ref{jmt1} for $f(t)=t^4$, which is the convex function but not operator convex, $n=4$, $n_1=2$ and the bounds of matrices as in Figure~1.
%
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=0.35]{Figure1} \caption{An example a convex function and  the bounds of four operators}
\end{center}
\end{figure}
\\
We show an example such that
\begin{eqnarray}
& \frac{1}{\alpha} \left( \Phi_1(A_1^{4}) + \Phi_2(A_2^{4})  \right) < \frac{1}{\alpha} \left( \Phi_1(A_1^{4}) + \Phi_2(A_2^{4})  \right) + \beta \delta_{f} \widetilde{A} \nonumber \\
 & < \Phi_1(A_1^{4}) + \Phi_2(A_2^{4}) +\Phi_3(A_3^{4}) + \Phi_4(A_4^{4}) \label{ex1-eq} \\
& < \frac{1}{\beta} \left( \Phi_3(A_3^{4}) + \Phi_4(A_4^{4})
\right) - \alpha \delta_{f} \widetilde{A}
 <\frac{1}{\beta} \left( \Phi_3(A_3^{4}) + \Phi_4(A_4^{4})  \right) \nonumber
\end{eqnarray}
holds, where $\delta_{f}=\bar{M}^4 + \bar{m}^4- (\bar{M} +
\bar{m})^{4} \slash 8$ and
$$ \widetilde{A} = \frac{1}{2}I_2 - \frac{1}{\alpha (\bar{M}-\bar{m})} \left( \Phi_1 \left( |A_1 - \frac{\bar{M} + \bar{m}}{2} I_h|\right) + \Phi_2 \left( |A_2 - \frac{\bar{M} + \bar{m}}{2} I_3|\right) \right).
$$
We define mappings $\Phi_i : M_3(\mathbb{C}) \rightarrow
M_2(\mathbb{C})$ as follows: $\Phi_i((a_{jk})_{1\leq j,k\leq 3}) =
\frac{1}{4}(a_{jk})_{1\leq j,k\leq 2}$, $i=1,\ldots,4$.  Then
$\sum_{i=1}^4 \Phi_i(I_3)=I_2$ and $\alpha=\beta= \frac{1}{2}$.
\\
   Let
  \begin{eqnarray*}
    & A_1=2 \begin{pmatrix} 2 & 9/8 & 1 \\
9/8 &  2 & 0 \\
1 & 0 & 3
\end{pmatrix},\qquad
     A_2=3 \begin{pmatrix} 2 & 9/8 & 0 \\
9/8 &  1 & 0 \\
0 & 0 & 2
\end{pmatrix}, \\ & A_3=-3 \begin{pmatrix} 4 & 1/2 & 1 \\
1/2 &  4 & 0 \\
1 & 0 & 2
\end{pmatrix}, \qquad  A_4=12 \begin{pmatrix} 5/3 & 1/2 & 0 \\
1/2 &  3/2 & 0 \\
0 & 0 & 3
\end{pmatrix}.
  \end{eqnarray*}
Then $m_1=1.28607$, $M_1=7.70771$, $m_2=0.53777$, $M_2=5.46221$,
$m_3=-14.15050$, $M_3=-4.71071$, $m_4=12.91724$, $M_4=36.$, so
$m_L=m_2$, $M_R=M_1$, $m=M_3$ and $M=m_4$ (rounded to five decimal
places). Also,
$$\frac{1}{\alpha} \left( \Phi_1(A_1) + \Phi_2(A_2)  \right) =\frac{1}{\beta} \left( \Phi_3(A_3) + \Phi_4(A_4)  \right) = \begin{pmatrix} 4 & 9/4  \\
9/4 &  3
\end{pmatrix},$$
and
\begin{eqnarray*}
  A_{f} \equiv \frac{1}{\alpha} \left( \Phi_1(A_1^{4}) + \Phi_2(A_2^{4})  \right) &=& \begin{pmatrix} 989.00391 & 663.46875  \\
663.46875 &  526.12891
\end{pmatrix}, \\
C_{f} \equiv \Phi_1(A_1^{4}) + \Phi_2(A_2^{4}) +\Phi_3(A_3^{4}) + \Phi_4(A_4^{4}) &=& \begin{pmatrix} 68093.14258 & 48477.98437  \\
48477.98437 &  51335.39258
\end{pmatrix},
\\
B_{f} \equiv \frac{1}{\beta} \left( \Phi_3(A_3^{4}) + \Phi_4(A_4^{4})  \right) &=& \begin{pmatrix} 135197.28125 & 96292.5  \\
96292.5 &  102144.65625
\end{pmatrix}.
\end{eqnarray*}
Then
\begin{equation}\label{ex1-eq1}
A_{f} < C_{f} < B_{f}
\end{equation}
holds (which is consistent with \eqref{jmA-eq}).

We will choose three pairs of numbers $(\bar{m},\bar{M})$,
$\bar{m}\in [-4.71071,0.53777]$, $\bar{M}\in [7.70771,12.91724]$ as
follows:
\\[1ex]
\textbf{i)} \; $\bar{m} = m_L=0.53777$, $\bar{M}=M_R=7.70771$, then \\[1ex] $ \widetilde{\Delta}_1=\beta \delta_{f} \widetilde{A}= 0.5 \cdot 2951.69249 \cdot \begin{pmatrix} 0.15678 & 0.09030  \\ 0.09030 &  0.15943 \end{pmatrix}=
\begin{pmatrix} 231.38908 & 133.26139  \\ 133.26139 &  235.29515 \end{pmatrix}, $ \\[1ex]
\textbf{ii)} \; $\bar{m} = m=-4.71071$, $\bar{M}=M=12.91724$, then \\[1ex] $ \widetilde{\Delta}_2=\beta \delta_{f} \widetilde{A}= 0.5 \cdot 27766.07963 \cdot \begin{pmatrix} 0.36022 & 0.03573  \\ 0.03573 &  0.36155 \end{pmatrix}
= \begin{pmatrix} 5000.89860 & 496.04498  \\ 496.04498 &  5019.50711 \end{pmatrix}, $ \\[1ex]
\textbf{iii)} \;  $\bar{m} = -1$, $\bar{M}=10$, then \\[1ex] $ \widetilde{\Delta}_3= \beta \delta_{f} \widetilde{A}= 0.5 \cdot 9180.875 \cdot \begin{pmatrix} 0.28203 & 0.08975  \\ 0.08975 &  0.27557 \end{pmatrix}
= \begin{pmatrix} 1294.66 & 411.999  \\ 411.999 &  1265.
\end{pmatrix}. $
\\[1ex]
New, we obtain the following improvement of \eqref{ex1-eq1} (see
\eqref{ex1-eq}):
\\[1ex]
\textbf{i)} \; \quad $A_{f}  <  A_{f}+ \widetilde{\Delta}_1
 = \begin{pmatrix} 1220.39299 & 796.73014  \\
796.73014 &  761.42406
\end{pmatrix} $

$ \qquad <  C_{f} < \begin{pmatrix} 134965.89217 & 96159.23861  \\
96159.23861 &  101909.36110
\end{pmatrix}
= B_{f} - \widetilde{\Delta}_1
 < B_{f},$ \\[1ex]
\textbf{ii)} \; \quad $A_{f}  <  A_{f}+ \widetilde{\Delta}_2
 =
 \begin{pmatrix} 5989.90251 & 1159.51373  \\
1159.51373 &  5545.63601
\end{pmatrix} $

$ \qquad < C_{f} < \begin{pmatrix} 130196.38265 & 95796.45502  \\
95796.45502 &  97125.14914
\end{pmatrix}
= B_{f} - \widetilde{\Delta}_2
 < B_{f},$ \\[1ex]
\textbf{iii)} \quad $A_{f}  <  A_{f}+ \widetilde{\Delta}_3
 =
 \begin{pmatrix} 2283.66362 & 1075.46746  \\
1075.46746 &  1791.12874
\end{pmatrix} $

$ \qquad < C_{f} < \begin{pmatrix} 133902.62153 & 95880.50129  \\
95880.50129 &  100879.65641
\end{pmatrix}
= B_{f} - \widetilde{\Delta}_3
 < B_{f}.$
\\[1ex]
\end{example}

Using Theorem~\ref{jmt1} we get the following result.

\begin{corollary}\label{jmc2}
Let the assumptions of Theorem~\ref{jmt1} hold. Then
\begin{equation}\label{jmc2-eq1}
 \frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f(A_i))  \leq  \frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f(A_i)) + \gamma_1 \delta_f \widetilde{A} \leq \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f(A_i))
\end{equation}
and
\begin{equation}\label{jmc2-eq2}
 \frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f(A_i))  \leq  \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f(A_i)) - \gamma_2 \delta_f \widetilde{A}  \leq \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f(A_i))
\end{equation}
holds for every $\gamma_1, \gamma_2$ in the close interval joining
$\alpha$ and $\beta$, where $\delta_f$ and $\widetilde{A}$ are
defined by \eqref{t1-const}.
\end{corollary}

\begin{proof}
 Adding  $\alpha  \delta_f \widetilde{A}$ in \eqref{jmt1-eq} and noticing $ \delta_f \widetilde{A}\geq 0$, we obtain
$$
  \frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f(A_i)) \leq  \frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f(A_i)) + \alpha \delta_f \widetilde{A} \leq \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f(A_i)). \label{jmc2-eq3}
$$
Taking into account the above inequality and the left hand side of
\eqref{jmt1-eq} we obtain \eqref{jmc2-eq1}.

Similarly, subtracting $\beta  \delta_f \widetilde{A}$ in
\eqref{jmt1-eq} we obtain \eqref{jmc2-eq2}.
\end{proof}

\begin{remark} \label{jmr1}
\textsl{Let the assumptions of Theorem~\ref{jmt1}  be valid.}


\vspace {1mm}

\noindent 1) \; We observe that the following inequality
\begin{equation*}\label{jmr1-eq}
f \left(\frac{1}{\beta} \sum_{i=n_1+1}^{n}\Phi_i(A_i) \right) \leq
\frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f(A_i))-\delta_f
\widetilde{A}_{\beta} \leq
\frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f(A_i)),
\end{equation*}
holds for every continuous convex function $f:I  \rightarrow
{\mathbb{R}}$ provided that the interval $I$ contains all $m_i,M_i$,
$i=1,\ldots,n$, where $\delta_f$ is defined by \eqref{t1-const},
$$  \widetilde{A}_{\beta}\equiv \widetilde{A}_{\beta,A,\Phi,n_1}(\bar{m},\bar{M}) =\frac{1}{2} 1_K - \frac{1}{ \bar{M}-\bar{m}}  \left|\frac{1}{\beta}\sum_{i=n_1 +1}^{n} \Phi_i A_i- \frac{\bar{m}+\bar{M}}{2} 1_K \right|
$$
and \; $\bar{m} \in [m,m_L]$, $\bar{M} \in [M_R,M]$, $\bar{m}<
\bar{M}$, \; are arbitrary numbers.

Indeed, by the assumptions of Theorem~\ref{jmt1} we have
$$ m_L \alpha 1_H \leq \sum_{i=1}^{n_1}  \Phi_i(A_i) \leq  M_R \alpha 1_H \quad \text{and} \quad \frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(A_i)=
\frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(A_i)$$ which implies
$$ m_L 1_H \leq \frac{1}{\beta }\sum_{i=n_1+1}^{n} \Phi_i(A_i) \leq  M_R 1_H .$$
Also $(m_L,M_R) \cap [m_i,M_i]= {\O}$ for  $ i=n_1+1,\ldots,n$ and
$\sum_{i=n_1+1}^{n} \frac{1}{\beta } \Phi_i(1_H)=1_K$ hold.
 So we can apply Theorem~B on operators $A_{n_1+1},\ldots, A_n$ and mappings $\frac{1}{\beta}\Phi_i$. We obtain the desired inequality.

\vspace {1mm}

\noindent 2) \; We denote by $m_C$ and $M_C$  the bounds of
$C=\sum_{i=1}^{n}\Phi_{i}(A_i)$.
 If   $(m_C,M_C) \cap [m_i,M_i]= {\O}$,  $i=1,\ldots,n_1$,
 then series inequality \eqref{jmt1-eq} can be extended from the left side if we use refined Jensen's operator inequality \eqref{t1-eq0}
 \begin{eqnarray*}
  && f\left(\sum_{i=1}^{n}\Phi_{i}(A_i)\right) = f\left(\frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_{i}(A_i)\right) \leq  \frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f(A_i))- \delta_f \widetilde{A}_{\alpha}
    \\  &\leq& \frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f(A_i)) \leq \frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f(A_i)) + \beta \delta_f \widetilde{A} \leq \sum_{i=1}^{n}\Phi_i(f(A_i)) \\  &\leq& \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f(A_i)) - \alpha \delta_f \widetilde{A} \leq \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f(A_i)),
 \end{eqnarray*}
 where $\delta_f$ and $\widetilde{A}$ are defined by \eqref{t1-const},
$$  \widetilde{A}_{\alpha}\equiv \widetilde{A}_{\alpha,A,\Phi,n_1}(\bar{m},\bar{M}) =\frac{1}{2} 1_K - \frac{1}{ \bar{M}-\bar{m}}  \left|\frac{1}{\alpha}\sum_{i=n_1 +1}^{n} \Phi_i A_i- \frac{\bar{m}+\bar{M}}{2} 1_K \right|
$$
\end{remark}

\begin{remark} \label{rem-2}
We obtain the equivalent inequalities to the ones in
Theorem~\ref{jmt1} in the case when  $\sum_{i=1}^{n}\Phi_i
(1_H)=\gamma \, 1_K$, for some positive scalar $\gamma$. If $\alpha
+ \beta = \gamma$  and one of two equalities
$$\frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(A_i)=
\frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(A_i)=\frac{1}{\gamma}\sum_{i=1}^{n}\Phi_i(A_i)$$
is valid,
 then
\begin{eqnarray*}
  \frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f(A_i))  & \leq &  \frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f(A_i)) + \frac{\beta}{\gamma} \delta_f \widetilde{A} \leq \frac{1}{\gamma} \sum_{i=1}^{n}\Phi_i(f(A_i))  \nonumber \\
   & \leq & \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f(A_i)) - \frac{\alpha}{\gamma} \delta_f \widetilde{A} \leq \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f(A_i)), \label{rem2-eq}
\end{eqnarray*}
holds for every continuous convex function  $f:I  \rightarrow
{\mathbb{R}}$ provided that the interval $I$ contains all $m_i,M_i$,
$i=1,\ldots,n$, where $\delta_f$ and $\widetilde{A}$ are defined by
\eqref{t1-const}.
\end{remark}

With respect to Remark~\ref{rem-2}, we obtain the following obvious
corollary of Theorem~\ref{jmt1} with the convex combination of
operators $A_i$, $i=1,\ldots,n$.

\begin{corollary}\label{jmc3}
Let  $(A_1,\ldots,A_n)$ be an $n-$tuple of self-adjoint operators
$A_i \in B(H)$ with the bounds $m_i$ and $M_i$,  $m_i \leq M_i$,
$i=1,\ldots,n$.  Let $(p_1,\ldots,p_n)$ be an $n-$tuple  of
non-negative numbers such that $0<\sum_{i=1}^{n_1}
p_i=\mathbf{p_{n_1}}< \mathbf{p_n}=\sum_{i=1}^{n} p_i$, where $1\leq
n_1<n$. Let
\\$m_L= \min \{m_1,\ldots,m_{n_1} \}$, $M_R= \max \{M_1,\ldots,M_{n_1}\}$ and
$$ \begin{array}{rcl}
      m&=&\left\{  \begin{array}{l} m_L, \qquad\text{\rm if} \;  \left\{ M_i \colon M_i \leq m_L, i\in \{n_1+1,\ldots,n\} \right\}={\O}, \\ \max\left\{ M_i \colon M_i \leq m_L, i\in \{n_1+1,\ldots,n\} \right\}, \quad \text{\rm otherwise}, \end{array} \right.\\
     M&=& \left\{  \begin{array}{l} M_R, \qquad\text{\rm if} \;  \left\{m_i \colon m_i \geq M_R, i\in \{n_1+1,\ldots,n\} \right\}={\O}, \\ \min\left\{m_i \colon m_i \geq M_R, i\in \{n_1+1,\ldots,n\} \right\}, \quad \text{\rm otherwise}.\end{array} \right.
   \end{array}
$$
If
$$ (m_L,M_R) \cap [m_i,M_i]= {\O} \quad \text{for} \quad
i=n_1+1,\ldots,n, \quad  \quad m<M,$$ and one of two equalities
$$\frac{1}{\mathbf{p_{n_1}}}\sum_{i=1}^{n_1}p_i A_i=
\frac{1}{\mathbf{p_{n}}}\sum_{i=1}^{n}p_i
A_i=\frac{1}{\mathbf{p_{n}}-\mathbf{p_{n_1}}}\sum_{i=n_1 +1}^{n}p_i
A_i$$ is valid, then
\begin{equation}\label{jmc3-eq}
    \begin{array}{r}
 \displaystyle \frac{1}{\mathbf{p_{n_1}}}\sum_{i=1}^{n_1}p_i f(A_i)   \leq   \frac{1}{\mathbf{p_{n_1}}}\sum_{i=1}^{n_1}p_i f(A_i) +\left(1-\frac{\mathbf{p_{n_1}}}{\mathbf{p_{n}}} \right) \delta_f \widetilde{A} \leq \frac{1}{\mathbf{p_{n}}} \sum_{i=1}^{n}p_i f(A_i)   \\
 \displaystyle   \leq  \frac{1}{\mathbf{p_{n}}-\mathbf{p_{n_1}}}\sum_{i=n_1+1}^{n}p_i f(A_i) -\frac{\mathbf{p_{n_1}}}{\mathbf{p_{n}}} \delta_f \widetilde{A} \leq \frac{1}{\mathbf{p_{n}}-\mathbf{p_{n_1}}}\sum_{i=n_1+1}^{n}p_i f(A_i),
\end{array}
\end{equation}
holds for every continuous convex function  $f:I  \rightarrow
{\mathbb{R}}$ provided that the interval $I$ contains all $m_i,M_i$,
$i=1,\ldots,n$, where where $\delta_f$ is defined by
\eqref{t1-const},
\begin{equation*}\label{c3-const}
  \widetilde{A}\equiv \widetilde{A}_{A,p,n_1}(\bar{m},\bar{M}) = \frac{1}{2} 1_H - \frac{1}{\mathbf{p_{n_1}} (\bar{M}-\bar{m})}  \sum_{i=1}^{n_1} p_i
   \left( \left|A_i- \frac{\bar{m}+\bar{M}}{2} 1_H \right| \right)
\end{equation*}
and \; $\bar{m} \in [m,m_L]$, $\bar{M} \in [M_R,M]$, $\bar{m}<
\bar{M}$, \; are arbitrary numbers.

\bigskip

If $f:I \rightarrow {\mathbb{R}}$ is concave, then the reverse
inequality is valid in \eqref{jmc3-eq}.
\end{corollary}

As a special case of Corollary~\ref{jmc3} we obtain an extension of
\cite[Corollary~6]{MPPeric1}.

\begin{corollary} \label{jmc4}
 Let $(A_1,\ldots,A_n)$ be an $n-$tuple of self-adjoint operators
$A_i \in B(H)$ with the bounds $m_i$ and $M_i$,  $m_i \leq M_i$,
$i=1,\ldots,n$. Let $(p_1,\ldots,p_n)$ be an $n-$tuple  of
non-negative numbers such that $\sum_{i=1}^{n} p_i=1$. Let
$$(m_A,M_A) \cap [m_i,M_i]= {\O} \quad \text{for}~
i=1,\ldots,n,\qquad \text{and} \qquad m<M,$$ where $m_A$ and $M_A$,
$m_A \leq M_A$, are the bounds of $A=\sum_{i=1}^{n} p_{i} A_i$ and
$$m=\max \left\{M_i \leq m_A, i\in \{1,\ldots,n\} \right\}, \; M= \min \left\{m_i \geq M_A, i\in \{1,\ldots,n\} \right\}.
$$
If $f:I\to {\mathbb{R}}$ is a continuous convex function provided
that the interval $I$ contains all $m_i,M_i$, then
\begin{equation}\label{jmc4-eq}
\begin{array}{c}
 \displaystyle f ( \sum_{i=1}^{n} p_{i}A_i )   \leq    f(\sum_{i=1}^{n} p_{i}A_i )  + \frac{1}{2}\delta_f \tilde{\tilde{A}} \leq \frac{1}{2} f(\sum_{i=1}^{n} p_{i}A_i ) +  \frac{1}{2} \sum_{i=1}^{n} p_{i}f(A_i) \\
 \displaystyle     \leq   \sum_{i=1}^{n}p_{i}f(A_i) - \frac{1}{2}\delta_f \tilde{\tilde{A}} \leq  \sum_{i=1}^{n}p_{i}f(A_i),
   \end{array}
\end{equation}
holds, where $\delta_f$ is defined by \eqref{t1-const},
$\tilde{\tilde{A}}= \frac{1}{2} 1_H - \frac{1}{\bar{M}-\bar{m}}
\left|\sum_{i=1}^{n} p_{i} A_i - \frac{\bar{m}+\bar{M}}{2} 1_H
\right|$ and $\bar{m} \in [m,m_A]$, $\bar{M} \in [M_A,M]$, $\bar{m}<
\bar{M}$, are arbitrary numbers.

If $f:I \rightarrow {\mathbb{R}}$ is concave, then the reverse
inequality is valid in \eqref{jmc4-eq}.
\end{corollary}

\begin{proof}
We prove only the convex case.

We define $(n+1)-$tuple of operators $(B_1,\ldots,B_{n+1})$, $B_i
\in B(H)$, by $B_1=A=\sum_{i=1}^{n} p_i A_i$ and $B_{i}=A_{i-1}$,
$i=2,\ldots,n+1$. Then $m_{B_1}=m_A$, $M_{B_1}=M_A$ are the bounds
of $B_1$ and $m_{B_i}=m_{i-1}$, $M_{B_i}=M_{i-1}$ are the ones of
$B_{i}$, $i=2,\ldots,n+1$. Also, we define $(n+1)-$tuple  of
non-negative numbers $(q_1,\ldots,q_{n+1})$ by $q_1=1$ and
$q_i=p_{i-1}$, $i=2,\ldots,n+1$. We have that $\sum_{i=1}^{n+1}
q_i=2$ and \begin{equation}\label{proof-1} (m_{B_1},M_{B_1}) \cap
[m_{B_i},M_{B_i}]= {\O},\;\; \text{for $i=2,\ldots,n+1$  \quad and
\quad $m<M$}\end{equation} holds. Since
$$\sum_{i=1}^{n+1}q_iB_i=B_1+\sum_{i=2}^{n+1}q_iB_i=
\sum_{i=1}^{n}p_{i}A_i +  \sum_{i=1}^{n}p_{i}A_i = 2B_1,$$ then
\begin{equation}\label{proof-2}
 q_1 B_1 =\frac{1}{2}\sum_{i=1}^{n+1}q_i B_i = \sum_{i=2}^{n+1}q_i B_i.
\end{equation}

Taking into account \eqref{proof-1} and \eqref{proof-2}, we can
apply Corollary~\ref{jmc3} for $n_1=1$ and  $B_i$, $q_i$  as above,
and we get
$$
 q_1 f(B_1)  \leq  q_1 f(B_1)  +  \frac{1}{2}\delta_f \widetilde{B} \leq   \frac{1}{2} \sum_{i=1}^{n+1}q_i f(B_i)
   \leq  \sum_{i=2}^{n+1}q_i f(B_i) -   \frac{1}{2}\delta_f \widetilde{B} \leq \sum_{i=2}^{n+1}q_i f(B_i),
$$
where $\widetilde{B}= \frac{1}{2} 1_H - \frac{1}{\bar{M}-\bar{m}}
   \left|B_1- \frac{\bar{m}+\bar{M}}{2} 1_H \right|, $
which gives the desired inequality \eqref{jmc4-eq}.
\end{proof}



\medskip


\section{Quasi-arithmetic means}\label{aqa}

In this section we study an application of Theorem~\ref{jmt1} to the
quasi-arithmetic mean with weight.

For a subset $\{A_{n_1},\ldots,A_{n_2}\}$ of
$\{A_{1},\ldots,A_{n}\}$,  we denote the quasi-arithmetic mean by
\begin{equation}\label{quasi-arithmetic-mean}
\mathcal{M}_{\varphi}(\gamma,\mathbf{A},\mathbf{\Phi},n_1,n_2)=\varphi^{-1}
\left(
 \frac{1}{\gamma} \sum_{i=n_1}^{n_2} \Phi_i \left( \varphi(A_i) \right) \right),
\end{equation}
where $(A_{n_1},\ldots,A_{n_2})$ are self-adjoint operators in
$\mathcal{B}(H)$ with the spectra in $I$,
$(\Phi_{n_1},\ldots,\Phi_{n_2})$ are positive linear mappings
$\Phi_i :\mathcal{B}(H) \rightarrow \mathcal{B}(K)$ such that

\noindent $\sum_{i=n_1}^{n_2} \Phi_i (1_H) = \gamma \, 1_K$, and
$\varphi:I \rightarrow {\mathbb{R}}$ is a continuous strictly
monotone function.

Under the same conditions, for convenience we introduce the
following denotations
\begin{equation}\label{delta-A}
\begin{array}{rcl}
 \delta_{\varphi,\psi} (m,M) &=& \psi(m)+\psi(M) - 2 \psi \circ \varphi ^{-1}\left( \frac{\varphi(m)+\varphi(M)}{2}\right),\\[0.5ex]
\widetilde{A}_{\varphi,n_1,\gamma} (m,M)&=& \frac{1}{2} 1_K -
\frac{1}{\gamma (M-m)}  \sum_{i=1}^{n_1} \Phi_i
   \left( \left|\varphi(A_i)- \frac{\varphi(M)+\varphi(m)}{2} 1_H \right| \right),
\end{array}
\end{equation}
where  $\varphi,\psi:I \rightarrow {\mathbb{R}}$ are continuous
strictly monotone functions and $m,M \in I$, $m<M$. Of course, we
include implicitly that $\widetilde{A}_{\varphi,n_1,\gamma}
(m,M)\equiv \widetilde{A}_{\varphi, A,\Phi,n_1,\gamma} (m,M)$.

\vspace{5mm}

The following theorem  is an extension of \cite[Theorem~7]{MPPeric1}
and a refinement of \cite[Theorem~3.1]{MPP3}.

\begin{theorem}\label{jmt2}
Let  $(A_1,\ldots,A_n)$ be an $n-$tuple of self-adjoint operators
$A_i \in B(H)$ with the bounds $m_i$ and $M_i$,  $m_i \leq M_i$,
$i=1,\ldots,n$. Let $\varphi,\psi:I \rightarrow {\mathbb{R}}$ be
continuous strictly monotone functions on an interval $I$ which
contains all $m_i,M_i$. Let $(\Phi_1,\ldots,\Phi_n)$ be an $n-$tuple
of po\-si\-tive linear mappings $\Phi_i:B(H) \rightarrow B(K)$,
 such that $\sum_{i=1}^{n_1}\Phi_i
(1_H)=\alpha\, 1_K$, $\sum_{i=n_1+1}^{n}\Phi_i (1_H)=\beta\, 1_K$,
where $1\leq n_1<n$, $\alpha, \beta>0$ and $\alpha+\beta=1$. Let one
of two equalities
\begin{equation}\label{jmt2-uvjet}
\mathcal{M}_{\varphi}(\alpha, \mathbf{A},\mathbf{\Phi},1,n_1)=
\mathcal{M}_{\varphi}(1,\mathbf{A},\mathbf{\Phi},1,n)=
\mathcal{M}_{\varphi}(\beta, \mathbf{A},\mathbf{\Phi},n_1+1,n)
\end{equation}
be valid and let $$ (m_L,M_R) \cap [m_i,M_i]= {\O} \quad \text{for}
\quad i=n_1+1,\ldots,n, \quad  \quad m<M,$$
 where $m_L= \min \{m_1,\ldots,m_{n_1} \}$, $M_R= \max \{M_1,\ldots,M_{n_1}\}$,
$$ \begin{array}{rcl}
      m&=&\left\{  \begin{array}{l} m_L, \qquad\text{\rm if} \;  \left\{ M_i \colon M_i \leq m_L, i\in \{n_1+1,\ldots,n\} \right\}={\O}, \\ \max\left\{ M_i \colon M_i \leq m_L, i\in \{n_1+1,\ldots,n\} \right\}, \quad \text{\rm otherwise}, \end{array} \right.\\
     M&=& \left\{  \begin{array}{l} M_R, \qquad\text{\rm if} \;  \left\{m_i \colon m_i \geq M_R, i\in \{n_1+1,\ldots,n\} \right\}={\O}, \\ \min\left\{m_i \colon m_i \geq M_R, i\in \{n_1+1,\ldots,n\} \right\}, \quad \text{\rm otherwise}.\end{array} \right.
   \end{array}
$$

\vspace{2 mm} $\mathrm{(i)}$ \, If $\psi \circ \varphi ^{-1}$ is
convex and $\psi ^{-1}$ is operator monotone, then
\begin{align}\label{jmt2-eq}
\mathcal{M}_{\psi}(\alpha, \mathbf{A}&,\mathbf{\Phi},1,n_1) \leq
 \psi ^{-1} \left( \frac{1}{\alpha}\sum_{i=1}^{n_1} \Phi_i \left( \psi(A_i) \right) +\beta \delta_{\varphi,\psi} \widetilde{A}_{\varphi,n_1,\alpha} \right)\nonumber\\
 &\leq \mathcal{M}_{\psi}(1,\mathbf{A},\mathbf{\Phi},1,n) \leq \psi ^{-1} \left( \frac{1}{\beta}\sum_{i=n_1+1}^n \Phi_i \left( \psi(A_i) \right) -\alpha \delta_{\varphi,\psi} \widetilde{A}_{\varphi,n_1,\alpha} \right)\nonumber\\
 &\leq \mathcal{M}_{\psi}(\beta,\mathbf{A},\mathbf{\Phi},n_1+1,n)
\end{align}
holds, where $\delta_{\varphi,\psi} \geq 0$ and
$\widetilde{A}_{\varphi,n_1,\alpha} \geq 0$.


\vspace{2 mm} $\mathrm{(i')}$ \, If $\psi \circ \varphi ^{-1}$ is
convex and  $- \psi ^{-1}$ is operator monotone, then the reverse
inequality is valid in \eqref{jmt2-eq}, where
$\delta_{\varphi,\psi} \geq 0$ and $
\widetilde{A}_{\varphi,n_1,\alpha} \geq 0$.

\vspace{2 mm} $\mathrm{(ii)}$ \, If $\psi \circ \varphi ^{-1}$ is
concave and $-\psi ^{-1}$ is operator monotone, then \eqref{jmt2-eq}
holds, where   $\delta_{\varphi,\psi}  \leq 0$ and $
\widetilde{A}_{\varphi,n_1,\alpha} \geq 0$.


\vspace{2 mm} $\mathrm{(ii')}$ \, If $\psi \circ \varphi ^{-1}$ is
concave and  $ \psi ^{-1}$ is operator monotone, then the reverse
inequality is valid in \eqref{jmt2-eq}, where $\delta_{\varphi,\psi}
\leq 0$ and $ \widetilde{A}_{\varphi,n_1,\alpha} \geq 0$.

\vspace{2 mm} \noindent In all the above cases, we assume that
$\delta_{\varphi,\psi} \equiv
\delta_{\varphi,\psi}(\bar{m},\bar{M})$,
$\widetilde{A}_{\varphi,n_1,\alpha} \equiv
\widetilde{A}_{\varphi,n_1,\alpha}(\bar{m},\bar{M})$  are defined by
\eqref{delta-A} and $\bar{m} \in [m,m_{L}]$, $\bar{M} \in
[M_{R},M]$, $\bar{m}< \bar{M}$,  are arbitrary numbers.
\end{theorem}

\begin{proof}
We only prove the case (i). Suppose that $\varphi$ is a strictly
increasing function. Then
$$(m_L,M_R) \cap [m_i,M_i]= {\O} \qquad \text{for} \quad
i=n_1+1,\ldots,n$$ implies
\begin{equation}\label{jmt2-eq0}
(\varphi(m_L),\varphi(M_R)) \cap [\varphi(m_i),\varphi(M_i)]= {\O}
\quad \text{ for $i=n_1+1,\ldots,n$}.
\end{equation}
Also, by using \eqref{jmt2-uvjet}, we have
$$\frac{1}{\alpha} \sum_{i=1}^{n_1} \Phi_i \left( \varphi(A_i)
\right) = \sum_{i=1}^{n} \Phi_i \left( \varphi(A_i) \right) =
\frac{1}{\beta} \sum_{i=n_1+1}^{n} \Phi_i \left( \varphi(A_i)
\right).$$ Taking into account \eqref{jmt2-eq0} and the above double
equality, we obtain by Theorem~\ref{jmt1}
\begin{equation}\label{jmt2-eq1}
\begin{array}{c}
\displaystyle
\frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f\left(\varphi(A_i)\right))  \leq   \frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f\left(\varphi(A_i)\right)) + \beta \delta_f \widetilde{A}_{\varphi,n_1,\alpha} \leq \sum_{i=1}^{n}\Phi_i(f\left(\varphi(A_i)\right)) \\
 \displaystyle  \leq  \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f\left(\varphi(A_i)\right)) - \alpha \delta_f \widetilde{A}_{\varphi,n_1,\alpha} \leq \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(f\left(\varphi(A_i)\right)),
\end{array}
\end{equation}
for every continuous convex function $f:J \rightarrow {\mathbb{R}}$
on an interval $J$ which contains all $[\varphi(m_i),\varphi(M_i)] =
\varphi([m_i,M_i])$, $i=1,\ldots,n$, where $\delta_f
=f(\varphi(m))+f(\varphi(M)) - 2 f\left(
\frac{\varphi(m)+\varphi(M)}{2}\right)$.

 Also, if $\varphi$
is strictly decreasing, then we check that \eqref{jmt2-eq1} holds
for convex $f:J \rightarrow {\mathbb{R}}$ on  $J$ which contains all
$[\varphi(M_i),\varphi(m_i)] =  \varphi([m_i,M_i])$.


\vspace{2mm} Putting $f=\psi \circ \varphi ^{-1}$ in
\eqref{jmt2-eq1}, we obtain
$$\begin{array}{c}
\displaystyle
\frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i\left(\psi(A_i)\right)  \leq   \frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i\left(\psi(A_i)\right) + \beta \delta_{\varphi,\psi} \widetilde{A}_{\varphi,n_1,\alpha} \leq \sum_{i=1}^{n}\Phi_i\left(\psi(A_i)\right) \\
 \displaystyle  \leq  \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i\left(\psi(A_i)\right) - \alpha \delta_{\varphi,\psi} \widetilde{A}_{\varphi,n_1,\alpha} \leq \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i\left(\psi(A_i)\right).
\end{array}
$$
Applying an operator monotone function $\psi^{-1}$ on the above
double inequality, we obtain the desired inequality \eqref{jmt2-eq}.
\end{proof}

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

We now give some particular results of interest that can be derived
from Theorem~\ref{jmt2}, which are an extension of
\cite[Corollary~8, Corollary~~10]{MPPeric1} and a refinement of
\cite[Corollary~3.3]{MPP3}.
%
\begin{corollary}\label{jmc5}
Let $(A_1,\ldots,A_n)$ and $(\Phi_1,\ldots,\Phi_n)$, $m_i$, $M_i$,
$m$, $M$, $m_L$, $M_R$, $\alpha$ and $\beta$ be as in
Theorem~\ref{jmt2}. Let $I$ be an interval which contains all
$m_i,M_i$ and
$$ (m_L,M_R) \cap [m_i,M_i]= {\O} \quad \text{for} \quad
i=n_1+1,\ldots,n, \quad  \quad m<M.$$

\bigskip

\noindent \textbf{I)} \quad If one of two equalities
\begin{equation*}\label{jmc5-uvjet}
\mathcal{M}_{\varphi}(\alpha, \mathbf{A},\mathbf{\Phi},1,n_1)=
\mathcal{M}_{\varphi}(1,\mathbf{A},\mathbf{\Phi},1,n)=
\mathcal{M}_{\varphi}(\beta, \mathbf{A},\mathbf{\Phi},n_1+1,n)
\end{equation*}
is valid, then
\begin{equation}\label{jmc5-eq}
\begin{array}{c}
\displaystyle
\frac{1}{\alpha}\sum_{i=1}^{n_1} \Phi_i(A_i)  \leq   \frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(A_i) + \beta \delta_{\varphi^{-1}} \widetilde{A}_{\varphi,n_1,\alpha} \leq \sum_{i=1}^{n}\Phi_i(A_i) \\
 \displaystyle  \leq  \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(A_i) - \alpha \delta_{\varphi^{-1}} \widetilde{A}_{\varphi,n_1,\alpha} \leq \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i\Phi_i(A_i).
\end{array}
\end{equation}
holds for every continuous strictly monotone function $\varphi:I
\rightarrow {\mathbb{R}}$ such that $\varphi ^{-1}$ is convex on
$I$, where $\delta_{\varphi^{-1}} = \bar{m}+\bar{M} - 2 \ \varphi
^{-1}\left( \frac{\varphi(\bar{m})+\varphi(\bar{M})}{2}\right)\geq
0$,  $\widetilde{A}_{\varphi,n_1,\alpha} =\frac{1}{2} 1_K -
\frac{1}{\alpha (\bar{M}-\bar{m})}  \sum_{i=1}^{n_1} \Phi_i
   \left( \left|\varphi(A_i)- \frac{\varphi(\bar{M})+\varphi(\bar{m})}{2} 1_H \right| \right)$  and $\bar{m} \in [m,m_{L}]$, $\bar{M} \in [M_{R},M]$, $\bar{m}< \bar{M}$,  are arbitrary numbers.

But, if $\varphi ^{-1}$ is concave, then the reverse inequality is
valid in \eqref{jmc5-eq} for $\delta_{\varphi^{-1}}\leq0$.

\bigskip

\noindent \textbf{II)} \quad  If one of two equalities
\begin{equation*}\label{jmc5b-uvjet}
\frac{1}{\alpha} \sum_{i=1}^{n_1} \Phi_i(A_i)  = \sum_{i=1}^{n}
\Phi_i(A_i) = \frac{1}{\beta} \sum_{i=n_1+1}^{n} \Phi_i(A_i)
\end{equation*}
is valid, then
\begin{equation}\label{jmc5b-eq}
\begin{array}{c}
\displaystyle
 \mathcal{M}_{\varphi}(\alpha, \mathbf{A},\mathbf{\Phi},1,n_1) \leq
 \varphi ^{-1} \left( \frac{1}{\alpha}\sum_{i=1}^{n_1} \Phi_i \left( \varphi(A_i) \right) +\beta \delta_{\varphi} \widetilde{A}_{n_1} \right) \leq \mathcal{M}_{\varphi}(1,\mathbf{A},\mathbf{\Phi},1,n) \\
  \displaystyle  \leq
 \varphi ^{-1} \left( \frac{1}{\beta}\sum_{i=n_1+1}^n \Phi_i \left( \varphi(A_i) \right) -\alpha \delta_{\varphi} \widetilde{A}_{n_1} \right) \leq \mathcal{M}_{\varphi}(\beta,\mathbf{A},\mathbf{\Phi},n_1+1,n)
 \end{array}
\end{equation}
holds for every continuous strictly monotone function $\varphi:I
\rightarrow {\mathbb{R}}$  such that one of the following conditions
\begin{itemize}
\item [(i)] $\varphi$ is convex and $\varphi ^{-1}$ is
operator monotone,
\item [(i')]  $\varphi$ is
concave  and $ -\varphi ^{-1}$ is operator monotone,
\end{itemize}
 is satisfied, where $\delta_{\varphi}=\varphi(\bar{m})+ \varphi(\bar{M})
- 2 \varphi\left( \frac{\bar{m}+\bar{M}}{2}\right)$,
$\widetilde{A}_{n_1}=\frac{1}{2} 1_K - \frac{1}{\alpha
(\bar{M}-\bar{m})} $ $ \times \sum_{i=1}^{n_1} \Phi_i
   \left( \left|A_i- \frac{\bar{m}+\bar{M}}{2} 1_H \right| \right)$ and $\bar{m} \in [m,m_{L}]$, $\bar{M} \in [M_{R},M]$, $\bar{m}< \bar{M}$,  are arbitrary numbers.

But, if one of the following conditions
\begin{itemize}
\item [(ii)] $\varphi$ is concave  and $\varphi ^{-1}$ is operator monotone,
\item [(ii')]  $\varphi$ is convex  and $-\varphi ^{-1}$ is operator monotone,
\end{itemize}
 is satisfied, then the reverse inequality is valid in \eqref{jmc5b-eq}.
\end{corollary}

\begin{proof}
The inequalities \eqref{jmc5-eq} follows from Theorem~\ref{jmt2} by
replacing $\psi$ with the identity function, while the inequalities
\eqref{jmc5b-eq} follows  by replacing $\varphi$ with the identity
function and $\psi$ with $\varphi$.
 \end{proof}

\begin{remark} \label{jmr2}
\textsl{Let the assumptions of Theorem~\ref{jmt2}  be valid.}


\vspace {1mm}

\noindent 1) \; We observe that if one of the following conditions
\begin{itemize}
\item [(i)] $\psi \circ \varphi ^{-1}$ is convex and $\psi ^{-1}$ is
operator monotone,
\item [(i')]  $\psi \circ \varphi ^{-1}$ is
concave  and $ -\psi ^{-1}$ is operator monotone,
\end{itemize}
 is satisfied, then
the following obvious inequality (see Remark~\ref{jmr1}.1))
\begin{eqnarray*}
\mathcal{M}_{\varphi}(\beta,\mathbf{A},\mathbf{\Phi},n_1+1,n) &\leq&
\psi ^{-1} \left( \frac{1}{\beta}\sum_{i=n_1+1}^{n}\Phi_i(\psi(A_i))
- \delta_{\varphi} \widetilde{A}_{\beta} \right)\\
 &\leq& \mathcal{M}_{\psi}(\beta,\mathbf{A},\mathbf{\Phi},n_1+1,n),
 \end{eqnarray*}
holds, $\delta_{\varphi}=\varphi(\bar{m})+ \varphi(\bar{M}) - 2
\varphi\left( \frac{\bar{m}+\bar{M}}{2}\right)$,
$\widetilde{A}_{\beta} =\frac{1}{2} 1_K - \frac{1}{ \bar{M}-\bar{m}}
\left|\frac{1}{\beta}\sum_{i=n_1 +1}^{n} \Phi_i A_i-
\frac{\bar{m}+\bar{M}}{2} 1_K \right|$ and $\bar{m} \in [m,m_{L}]$,
$\bar{M} \in [M_{R},M]$, $\bar{m}< \bar{M}$,  are arbitrary numbers.



\vspace {1mm}



\noindent 2) \; We denote by $m_{\varphi}$ and $M_{\varphi}$ the
bounds of $\mathcal{M}_{\varphi}(1,\mathbf{A},\mathbf{\Phi},1,n)$.
 If   $(m_{\varphi},M_{\varphi}) \cap [m_i,M_i]= {\O}$,
 $i=1,\ldots,n_1$, and one of two following conditions
\begin{itemize}
\item [(i)] $\psi \circ \varphi ^{-1}$ is convex and $\psi ^{-1}$
is operator monotone \item [(ii)]  $\psi \circ \varphi ^{-1}$ is
concave and $-\psi ^{-1}$ is operator monotone
\end{itemize}
is satisfied, then the double inequality \eqref{jmt2-eq} can be
extended from the left side as follows
 \begin{align*}
&\mathcal{M}_{\varphi}(1,\mathbf{A},\mathbf{\Phi},1,n)=\mathcal{M}_{\varphi}(1,\mathbf{A},\mathbf{\Phi},1,n_1)
\leq  \psi ^{-1} \left(
\frac{1}{\alpha}\sum_{i=1}^{n_1}\Phi_i(f(A_i))-
\delta_{\varphi,\psi} \widetilde{A}_{\alpha} \right)
    \\  &\leq \mathcal{M}_{\psi}(\alpha, \mathbf{A},\mathbf{\Phi},1,n_1)  \leq
    \psi ^{-1} \left( \frac{1}{\alpha}\sum_{i=1}^{n_1} \Phi_i \left( \psi(A_i) \right) +\beta \delta_{\varphi,\psi} \widetilde{A}_{\varphi,n_1,\alpha}
    \right)\\
   &\leq \mathcal{M}_{\psi}(1,\mathbf{A},\mathbf{\Phi},1,n) \leq \psi ^{-1} \left( \frac{1}{\beta}\sum_{i=n_1+1}^n \Phi_i \left( \psi(A_i) \right) -\alpha \delta_{\varphi,\psi} \widetilde{A}_{\varphi,n_1,\alpha} \right)\\
   & \leq \mathcal{M}_{\psi}(\beta,\mathbf{A},\mathbf{\Phi},n_1+1,n),
 \end{align*}
 where $\delta_{\varphi,\psi}$ and $\widetilde{A}_{\varphi,n_1,\alpha} $ are defined by \eqref{delta-A},
$$  \widetilde{A}_{\alpha} =\frac{1}{2} 1_K - \frac{1}{ \bar{M}-\bar{m}}  \left|\frac{1}{\alpha}\sum_{i=n_1 +1}^{n} \Phi_i A_i- \frac{\bar{m}+\bar{M}}{2} 1_K \right|.
$$
\end{remark}



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

\bigskip

As a special case of the quasi-arithmetic mean
\eqref{quasi-arithmetic-mean}  we can study the weighted power mean
as follows. For a subset $\{A_{p_1},\ldots,A_{p_2}\}$ of
$\{A_{1},\ldots,A_{n}\}$  we denote this mean by
\begin{equation*}\label{power-mean2}
M^{[r]}(\gamma,\mathbf{A},\mathbf{\Phi},p_1,p_2)=\left\{
\begin{array}
[c]{ll}%
\displaystyle \left( \frac{1}{\gamma}
 \sum_{i=p_1}^{p_2} \Phi_i \left( A_i^{r} \right) \right)^{1/r}, &\qquad r \in
{\mathbb{R}} \backslash \{0 \},\\
\displaystyle \exp\left( \frac{1}{\gamma} \sum_{i=p_1}^{p_2} \Phi_i
\left( \ln\left( A_{i}\right) \right) \right), & \qquad r=0,
\end{array}
\right.
\end{equation*}
where $(A_{p_1},\ldots,A_{p_2})$ are strictly positive operators,
$(\Phi_{p_1},\ldots,\Phi_{p_2})$ are positive li\-near mappings
$\Phi_i :\mathcal{B}(H) \rightarrow \mathcal{B}(K)$ such that
$\sum_{i=p_1}^{p_2}\Phi_i (1_H)=\gamma \, 1_K$.

Under the same conditions, for convenience we introduce denotations
as a special case of \eqref{delta-A} as follows
\begin{equation}\label{delta-A-rs}
\begin{array}{rcl}
 \delta_{r,s}(m,M)  &=& \left\{ \begin{array}{ll}
                               m^s+M^s - 2 \left( \frac{m^r+M^r}{2}\right)^{s/r}, & r\neq 0, \\
                                m^s + M^s - 2  \left( m M \right)^{s/2}, & r=0,
                             \end{array}
                           \right. \\ [1.5em]
\widetilde{A}_r(m,M)  &=& \left\{ \begin{array}{ll}
\frac{1}{2} 1_K - \frac{1}{|M^r-m^r|} \left|\sum_{i=1}^{n}\Phi_{i}(A_i^r) - \frac{M^r+m^r}{2} 1_K \right|, & r\neq 0, \\
\frac{1}{2} 1_K - |\ln \left(\frac{M}{m} \right)|^{-1}
\left|\sum_{i=1}^{n} \Phi_{i}(\ln A_i) - \ln \sqrt{M m} 1_K \right|,
& r=0,
                             \end{array}
                     \right.
\end{array}
\end{equation}
where $m,M \in {\mathbb{R}}$, $0<m<M$  and $r, s \in {\mathbb{R}}$,
$r\leq s$. Of course, we include implicitly that
$\widetilde{A}_r(m,M)\equiv \widetilde{A}_{r,A} (m,M)$, where
$A=\sum_{i=1}^{n}\Phi_{i}(A_i^r)$ for $r \neq 0$ and
$A=\sum_{i=1}^{n} \Phi_{i}(\ln A_i) $ for $r = 0$.


\medskip

 We obtain the following corollary by applying Theorem~\ref{jmt2} to the above mean. This is an extension of \cite[Corollary~13]{MPPeric1} and a refinement of \cite[Corollary~3.4]{MPP3}.

\begin{corollary}\label{jmc6}
Let  $(A_1,\ldots,A_n)$ be an $n-$tuple of self-adjoint operators
$A_i \in B(H)$ with the bounds $m_i$ and $M_i$,  $m_i \leq M_i$,
$i=1,\ldots,n$.  Let $(\Phi_1,\ldots,\Phi_n)$ be an $n-$tuple  of
po\-si\-tive linear mappings $\Phi_i:B(H) \rightarrow B(K)$,
 such that $\sum_{i=1}^{n_1}\Phi_i(1_H)=\alpha\, 1_K$, $\sum_{i=n_1+1}^{n}\Phi_i (1_H)=\beta\, 1_K$,
where $1\leq n_1<n$, $\alpha, \beta>0$ and $\alpha+\beta=1$. Let
$$ (m_L,M_R) \cap [m_i,M_i]= {\O} \quad \text{for} \quad
i=n_1+1,\ldots,n, \quad  \quad m<M,$$ where $m_L= \min
\{m_1,\ldots,m_{n_1} \}$, $M_R= \max \{M_1,\ldots,M_{n_1}\}$ and
$$ \begin{array}{rcl}
      m&=&\left\{  \begin{array}{l} m_L, \qquad\text{\rm if} \;  \left\{ M_i \colon M_i \leq m_L, i\in \{n_1+1,\ldots,n\} \right\}={\O}, \\ \max\left\{ M_i \colon M_i \leq m_L, i\in \{n_1+1,\ldots,n\} \right\}, \quad \text{\rm otherwise}, \end{array} \right.\\
     M&=& \left\{  \begin{array}{l} M_R, \qquad\text{\rm if} \;  \left\{m_i \colon m_i \geq M_R, i\in \{n_1+1,\ldots,n\} \right\}={\O}, \\ \min\left\{m_i \colon m_i \geq M_R, i\in \{n_1+1,\ldots,n\} \right\}, \quad \text{\rm otherwise}.\end{array} \right.
   \end{array}
$$

$\mathrm{(i)}$ If either $r \leq s$, $s\geq1$ or $r \leq s \leq -1$
and also one of two equalities
\begin{equation*}\label{jmc6-uvjet1} \mathcal{M}^{[r]}(\alpha,\mathbf{A},\mathbf{\Phi},1,n_1)=\mathcal{M}^{[r]}(1,\mathbf{A},\mathbf{\Phi},1,n)=
\mathcal{M}^{[r]}(\beta,\mathbf{A},\mathbf{\Phi},n_1+1,n)
\end{equation*} is valid, then
\begin{equation*}\label{jmc6-eq2}
\begin{array}{c}
\displaystyle \mathcal{M}^{[s]}(\alpha,
\mathbf{A},\mathbf{\Phi},1,n_1) \leq
 \left( \frac{1}{\alpha}\sum_{i=1}^{n_1} \Phi_i \left(A_i^{s} \right) +\beta \delta_{r,s} \widetilde{A}_{s,n_1,\alpha} \right)^{1/s} \leq \mathcal{M}^{[s]}(1,\mathbf{A},\mathbf{\Phi},1,n) \\
  \displaystyle  \leq
 \left( \frac{1}{\beta}\sum_{i=n_1+1}^n \Phi_i \left(A_i^{s} \right) -\alpha \delta_{r,s} \widetilde{A}_{s,n_1,\alpha} \right)^{1/s} \leq \mathcal{M}^{[s]}(\beta,\mathbf{A},\mathbf{\Phi},n_1+1,n)
 \end{array}
\end{equation*}
holds, where $\delta_{r,s} \geq 0$ and $\widetilde{A}_{s,n_1,\alpha}
\geq 0$.

\vspace{1 mm}
 In this case, we assume that $\delta_{r,s} \equiv \delta_{r,s}(\bar{m},\bar{M})$, $\widetilde{A}_{s,n_1,\alpha} \equiv \widetilde{A}_{s,n_1,\alpha}(\bar{m},\bar{M})$  are defined by \eqref{delta-A-rs} and $\bar{m} \in [m,m_{L}]$, $\bar{M} \in [M_{R},M]$, $\bar{m}< \bar{M}$,  are arbitrary numbers.

 \vspace{4 mm}

$\mathrm{(ii)}$   If either $r \leq s$, $r\leq -1$ or $1 \leq r \leq
s$ and also one of two equalities
\begin{equation*}\label{jmc6-uvjet2}\mathcal{M}^{[s]}(\alpha,\mathbf{A},\mathbf{\Phi},1,n_1)=\mathcal{M}^{[s]}(1,\mathbf{A},\mathbf{\Phi},1,n)=
\mathcal{M}^{[s]}(\beta,\mathbf{A},\mathbf{\Phi},n_1+1,n)\end{equation*}
 is valid, then
 \begin{equation*}\label{jmc6-eq3}
\begin{array}{c}
\displaystyle \mathcal{M}^{[r]}(\alpha,
\mathbf{A},\mathbf{\Phi},1,n_1) \geq
 \left( \frac{1}{\alpha}\sum_{i=1}^{n_1} \Phi_i \left(A_i^{r} \right) +\beta \delta_{s,r} \widetilde{A}_{r,n_1,\alpha} \right)^{1/r} \geq \mathcal{M}^{[r]}(1,\mathbf{A},\mathbf{\Phi},1,n) \\
  \displaystyle  \geq
  \left( \frac{1}{\beta}\sum_{i=n_1+1}^n \Phi_i \left(A_i^{r} \right) -\alpha \delta_{s,r} \widetilde{A}_{r,n_1,\alpha} \right)^{1/r} \geq \mathcal{M}^{[r]}(\beta,\mathbf{A},\mathbf{\Phi},n_1+1,n)
 \end{array}
\end{equation*}
holds, where $\delta_{s,r} \leq 0$ and $\widetilde{A}_{s,n_1,\alpha}
\geq 0$.

\vspace{1 mm}
 In this case, we assume that $\delta_{s,r} \equiv \delta_{s,r}(\bar{m},\bar{M})$, $\widetilde{A}_{r,n_1,\alpha} \equiv \widetilde{A}_{r,n_1,\alpha}(\bar{m},\bar{M})$  are defined by \eqref{delta-A-rs} and $\bar{m} \in [m,m_{L}]$, $\bar{M} \in [M_{R},M]$, $\bar{m}< \bar{M}$,  are arbitrary numbers.
\end{corollary}

\begin{proof}
In the case (i) we put  $\psi(t)=t^s$ and $\varphi(t)=t^r$  if
$r\neq0$ or $\varphi(t)=\ln t$ if $r\neq0$ in Theorem~\ref{jmt2}. In
the case (ii) we put  $\psi(t)=t^r$ and $\varphi(t)=t^s$  if
$s\neq0$ or $\varphi(t)=\ln t$ if $s\neq0$. We omit the details.
\end{proof}



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