# Generalized Converses of Operator Jensen’s Inequalities with Applications to
Hypercomplex Function Approximations and Bounds Algebra

Shih-Yu Chang Shih Yu Chang is with the Department of Applied Data Science,
San Jose State University, San Jose, CA, U. S. A. (e-mail:
shihyu.chang@sjsu.edu).

###### Abstract

Mond and Pecaric proposed a powerful method, namd as MP method, to deal with
operator inequalities. However, this method requires a real-valued function to
be convex or concave, and the normalized positive linear map between Hilbert
spaces. The objective of this study is to extend the MP method by allowing
non-convex or non-concave real-valued functions and nonlinear mapping between
Hilbert spaces. The Stone–Weierstrass theorem and Kantorovich function are
fundamental components employed in generalizing the MP method inequality in
this context. Several examples are presented to demonstrate the inequalities
obtained from the conventional MP method by requiring convex function with a
normalized positive linear map. Various new inequalities regarding
hypercomplex functions, i.e., operator-valued functions with operators as
arguments, are derived based on the proposed MP method. These inequalities are
applied to approximate hypercomplex functions using ratio criteria and
difference criteria. Another application of these new inequalities is to
establish bounds for hypercomplex functions algebra, i.e., an abelian monoid
for the addition or multiplication of hypercomplex functions, and to derive
tail bounds for random tensors ensembles addition or multiplication
systematically.

Index terms— Operator inequality, Jensen’s inequality, hypercomplex function,
Stone–Weierstrass theorem, Kantorovich function.

##  1 Introduction

Operator Jensen’s inequality extends the concept of conventional Jensen’s
inequality from real value argument function to operator argument function and
is generally used in optimization, especially in convex optimization involving
operator variables, e.g., matrices, tensors. The inequality states that for an
operator convex function $f$, we have

$\displaystyle f(\Phi(\bm{A}))\leq\Phi(f(\bm{A})),$ (1)

where $\bm{A}$ is a self-adjoint operator, $\Phi$ is a normalized positive
linear map and $\leq$ is by Loewner ordering sense [1, 2]. Operator Jensen’s
inequality is a powerful device in diverse areas, along with machine learning,
signal processing, and control principle. In gadget learning, for example, it
is used inside the analysis of convex optimization problems related to
matrices, consisting of matrix factorization and matrix completion. In signal
processing, it finds applications in the design and analysis of convex
optimization algorithms for jobs like blind source separation and beamforming.
In the field of control science, it is carried out in the evaluation and
synthesis of convex control structures regarding matrix variables. Operator
Jensen’s inequality gives a way to generalize the idea of convex features to
operator variables and presents insights into the function convexity related
to matrices or tensors, that are frequently encountered in many mathematical
and engineering tasks [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15].

Mond and Pečarić demonstrated multiple extensions of Kantorovich-type operator
inequalities concerning normalized positive linear maps. They highlighted that
determining upper bounds for the difference and ratio in Jensen’s inequality
can be simplified to solving a single-variable maximization or minimization
problem by leveraging the concavity of a real-valued function $f$. Building on
this approach, they established complementary inequalities to the Hölder-
McCarthy inequality and Kantorovich-type inequalities, provided estimates for
the difference and ratio of operator means, and explored various converses of
Jensen’s inequality applicable to normalized positive linear maps. In cases
where $f$ exhibits concavity, they derived the dual problem. This method,
known as the Mond-Pečarić (MP) method, has proven to be highly fruitful in the
realm of operator inequalities, offering valuable insights and results [1, 2].

The purpose of this work is to extend MP method by allowing $f$ in Eq. (1) to
be non-convex or non-concave and the mapping $\Phi$ in Eq. (1) to be a
nonlinear mapping. The Stone–Weierstrass theorem and Kantorovich function
serve as the primary ingredients employed in generalizing MP method inequality
in this context. Several examples are shown here to have those inequalities
obtained from the conventional MP method by setting $f$ to be a convex
function with $\Phi$ as a normalized positive linear map. Various new
inequalities about hypercomplex functions, i.e., operator-valued functions
with operators as arguments, are derived based on the proposed MP method.
These new inequalities are applied to approximate hypercomplex functions by
ratio criteria and by difference criteria. The other application of these new
inequalities is to establish hypercomplex functions lower/upper bounds
algebra, i.e., these bounds form a _abelian monoid_ under the addition or the
multiplication of hypercomplex functions. Besides, we also can derive random
tensors tail bounds for the addition or the multiplication of random tensors
ensembles by an uniform way.

The remainder of this paper is organized as follows. In Section 2, fundamental
inequalities about hypercomplex functions are established. In Section 3,
generalized converses of operator Jensen’s inequalities for ratio kinds with
nonlinear $\Phi$ are established. On the other hand, generalized converses of
operator Jensen’s inequalities for different kinds with nonlinear $\Phi$ are
established in Section 4. The first application about applying newly derived
inequalities to hypercomplex functions approximation is studied in Section 5.
The second application about applying newly derived inequalities to
investigate lower and upper bounds algebra is presented in Section 6.

Nomenclature: Inequalities $\geq,>,\leq,$ and $<$, when applied to operators,
follow the Loewner ordering. The symbol $\Lambda(\bm{A})$ denotes the spectrum
of the operator $\bm{A}$, i.e., the set of eigenvalues of $\bm{A}$. If
$\Lambda(\bm{A})$ consists of real numbers, $\min(\Lambda(\bm{A}))$ and
$\max(\Lambda(\bm{A}))$ represent the minimum and maximum values within
$\Lambda(\bm{A})$, respectively. For given $M>m>0$ and any $r\in\mathbb{R}$
where $r\neq 1$, the Kantorovich function with respect to $m$, $M$, and $r$ is
defined as follows:

$\displaystyle\mathscr{K}(m,M,r)$ $\displaystyle=$
$\displaystyle\frac{(mM^{r}-Mm^{r})}{(r-1)(M-m)}\left[\frac{(r-1)(M^{r}-m^{r})}{r(mM^{r}-Mm^{r})}\right]^{r}.$
(2)

##  2 Fundamental Inequalities for Hypercomplex Functions with Nonlinear Map

###  2.1 Upper Bound and Lower Bound of Function $f$

Let’s start with the Stone-Weierstrass Theorem, which asserts that any
continuous real-valued function $f(x)$ defined on the closed interval $[m,M]$,
where $m,M\in\mathbb{R}$ and $M>m$, can be uniformly approximated by a
polynomial $p(x)$. This approximation ensures that the absolute difference
between $f(x)$ and $p(x)$ is less than any given positive value $\epsilon$
across the entire interval $[m,M]$. Mathematically, this difference is bounded
by the supremum norm, denoted as $\left\|f-p\right\|_{\infty}$, which remains
less than $\epsilon$ [16].

Given a continuous real-valued function $f(x)$ and a positive error bound
$\epsilon$, we can employ the Lagrange polynomial interpolation method,
grounded in the Stone-Weierstrass Theorem, to ascertain both an upper
polynomial $p_{\mathscr{U}}(x)\geq f(x)$ and a lower polynomial
$p_{\mathscr{L}}(x)\leq f(x)$ over the interval $[m,M]$. These polynomials are
guaranteed to satisfy the following inequalities:

$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
p_{\mathscr{U}}(x)-f(x)\leavevmode\nobreak\ \leavevmode\nobreak\
\leq\leavevmode\nobreak\ \leavevmode\nobreak\ \epsilon,$ $\displaystyle 0$
$\displaystyle\leq$ $\displaystyle f(x)-p_{\mathscr{L}}(x)\leavevmode\nobreak\
\leavevmode\nobreak\ \leq\leavevmode\nobreak\ \leavevmode\nobreak\ \epsilon,$
(3)

Additionally, in this paper, we assume that $f(\bm{A})$ is a self-adjoint
operator if $\bm{A}$ is a self-adjoint operator.

###  2.2 Polynomial Map $\Phi$

Consider two Hilbert spaces $\mathfrak{H}$ and $\mathfrak{K}$.
$\mathbb{B}(\mathfrak{H})$ and $\mathbb{B}(\mathfrak{K})$ represent the semi-
algebras comprising all bounded linear operators on these respective spaces.
In Choi-Davis-Jensen’s Inequality, the mapping
$\Phi:\mathbb{B}(\mathfrak{H})\rightarrow\mathbb{B}(\mathfrak{K})$ is defined
as a normalized positive linear map. Such a map is precisely defined by
Definition 1 below [1, 2].

###### Definition 1

A map $\Phi:\mathbb{B}(\mathfrak{H})\rightarrow\mathbb{B}(\mathfrak{K})$ is
considered a normalized positive linear map if it satisfies the following
conditions:

  * •

Linearity: $\Phi(a\bm{X}+b\bm{Y})=a\Phi(\bm{X})+b\Phi(\bm{Y})$ for any
$a,b\in\mathbb{C}$ and any $\bm{X},\bm{Y}\in\mathbb{B}(\mathfrak{H})$.

  * •

Positivity: If $\bm{X}\geq\bm{Y}$, then $\Phi(\bm{X})\geq\Phi(\bm{Y})$.

  * •

Normalization: $\Phi(\bm{I}_{\mathfrak{H}})=\bm{I}_{\mathfrak{K}}$, where
$\bm{I}_{\mathfrak{H}}$ and $\bm{I}_{\mathfrak{K}}$ are the identity operators
of the Hilbert spaces $\mathfrak{H}$ and $\mathfrak{K}$, respectively.

In this study, we will explore a broader class of $\Phi$ by defining $\Phi$ as
follows:

$\displaystyle\Phi(\bm{X})$ $\displaystyle=$
$\displaystyle\bm{V}^{\ast}\left(\sum\limits_{i=0}^{I}a_{i}\bm{X}^{i}\right)\bm{V}$
(4) $\displaystyle=$ $\displaystyle\bm{V}^{\ast}\left(\sum\limits_{i_{+}\in
S_{I_{+}}}a_{i_{+}}\bm{X}^{i_{+}}+\sum\limits_{i_{-}\in
S_{I_{-}}}a_{i_{-}}\bm{X}^{i_{-}}\right)\bm{V},$

where, $\bm{V}$ stands as an isometry within $\mathfrak{H}$, adhering to
$\bm{V}^{\ast}\bm{V}=\bm{I}_{\mathfrak{H}}$. In $a_{i}$, $a_{i_{+}}$ signifies
the nonnegative coefficients, while $a_{i_{-}}$ denotes the negative
coefficients. The set of indices corresponding to positive coefficients is
denoted as $S_{I_{+}}$, and those for negative coefficients form $S_{I_{-}}$.
It’s noteworthy that no constraints regarding linearity, positivity, or
normalization are imposed on $\Phi$ as outlined in Eq. (4). With this premise,
the conventional notion of a normalized positive linear map, delineated in
Definition 1, emerges as a special case. This is accomplished by configuring
the polynomial $\sum\limits_{i=0}^{I}a_{i}\bm{X}^{i}$ as the identity map,
wherein all coefficients $a_{i}$ except $a_{1}$ are zero.

Contents from Section 2.1 and Section 2.2 are based on [4], however, we
present them here again for self-contained presentation purpose.

###  2.3 Fundamental Inequalities

In this section, general converses of operator Jensen’s inequalities for any
polynomial map $\Phi$ will be obtained. Let us recall Lemma 2 from [4].

###### Lemma 1

Given a self-adjoint operator $\bm{A}$ with $\Lambda(\bm{A})$, such that

$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
p_{\mathscr{U}}(x)-f(x)\leavevmode\nobreak\ \leavevmode\nobreak\
\leq\leavevmode\nobreak\ \leavevmode\nobreak\ \epsilon,$ $\displaystyle 0$
$\displaystyle\leq$ $\displaystyle f(x)-p_{\mathscr{L}}(x)\leavevmode\nobreak\
\leavevmode\nobreak\ \leq\leavevmode\nobreak\ \leavevmode\nobreak\ \epsilon,$
(5)

for $x\in[\min(\Lambda(\bm{A})),\max(\Lambda(\bm{A}))]$ with polynomials
$p_{\mathscr{L}}(x)$ and $p_{\mathscr{U}}(x)$ expressed by

$\displaystyle
p_{\mathscr{L}}(x)=\sum\limits_{k=0}^{n_{\mathscr{L}}}\alpha_{k}x^{k},\leavevmode\nobreak\
\leavevmode\nobreak\
p_{\mathscr{U}}(x)=\sum\limits_{j=0}^{n_{\mathscr{U}}}\beta_{j}x^{j}.$ (6)

Under the definition of $\Phi$ provided by Eq. (4), we have

$\displaystyle\Phi(f(\bm{A}))$ $\displaystyle\leq$
$\displaystyle\bm{V}^{\ast}\left\\{\sum\limits_{i_{+}\in
S_{I_{+}}}a_{i_{+}}\mathscr{K}(\min(\Lambda(p_{\mathscr{U}}(\bm{A}))),\max(\Lambda(p_{\mathscr{U}}(\bm{A}))),i_{+})p^{i_{+}}_{\mathscr{U}}(\bm{A})\right.$
(7) $\displaystyle\left.+\sum\limits_{i_{-}\in
S_{I_{-}}}a_{i_{-}}\mathscr{K}^{-1}(\min(\Lambda(p_{\mathscr{L}}(\bm{A}))),\max(\Lambda(p_{\mathscr{L}}(\bm{A}))),i_{-})p^{i_{-}}_{\mathscr{L}}(\bm{A})\right\\}\bm{V}$
$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A})\bm{V},$

where $\Lambda(p_{\mathscr{U}}(\bm{A})$ and $\Lambda(p_{\mathscr{L}}(\bm{A})$
are spectrms of operators $p_{\mathscr{U}}(\bm{A}$ and
$p_{\mathscr{L}}(\bm{A}$, respectively. On the other hand, we also have

$\displaystyle\Phi(f(\bm{A}))$ $\displaystyle\geq$
$\displaystyle\bm{V}^{\ast}\left\\{\sum\limits_{i_{+}\in
S_{I_{+}}}a_{i_{+}}\mathscr{K}^{-1}(\min(\Lambda(p_{\mathscr{L}}(\bm{A}))),\max(\Lambda(p_{\mathscr{L}}(\bm{A}))),i_{+})p^{i_{+}}_{\mathscr{L}}(\bm{A})\right.$
(8) $\displaystyle\left.+\sum\limits_{i_{-}\in
S_{I_{-}}}a_{i_{-}}\mathscr{K}(\min(\Lambda(p_{\mathscr{U}}(\bm{A}))),\max(\Lambda(p_{\mathscr{U}}(\bm{A}))),i_{-})p^{i_{-}}_{\mathscr{U}}(\bm{A})\right\\}\bm{V}$
$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A})\bm{V}.$

Given $\Lambda(\bm{A})\in[m,M]$, the value range for the spectrum
$\Lambda(\mbox{Poly}_{f,\mathscr{L}}(\bm{A}))$ is represented by
$\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m,M)$. Similarly, the value range for
the spectrum $\Lambda(\mbox{Poly}_{f,\mathscr{U}}(\bm{A}))$ is represented by
$\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m,M)$.

The main theorem of this work is presented below. Theorem 1 will give operator
inequalties, lower and uppber bounds, of functional with respect to
$\Phi(f(\bm{A}))$.

###### Theorem 1

Let $\bm{A}_{j}$ be self-adjoint operator with
$\Lambda(\bm{A}_{j})\in[m_{j},M_{j}]$ for real scalars $m_{j}<M_{j}$. The
mapping $\Phi:\mathbb{B}(\mathfrak{H})\rightarrow\mathbb{B}(\mathfrak{K})$ is
defined by Eq. (4). The index $j$ is in the range of $1,2,\cdots,k$, and we
have a probability vector $\bm{w}=[w_{1},w_{2},\cdots,w_{k}]$ with the
dimension $k$, i.e., $\sum\limits_{j=1}^{k}w_{j}=1$. Let $f$ be any real
valued continous functions defined on the range
$\bigcup\limits_{j=1}^{k}[m_{j},M_{j}]$, represented by
$f\in\mathcal{C}(\bigcup\limits_{j=1}^{k}[m_{j},M_{j}])$. Besides, we assume
that the function $f$ satisfes thsoe conditions provided by Eq. (1) and Eq.
(6). The function $g$ is also a real valued continous function defined on the
range
$\left(\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})\right)\bigcup\left(\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})\right)$.
We also have a real valued function $F(u,v)$ with operator monotone on the
first variable $u$ defined on $U\times V$ such that
$f(\bigcup\limits_{j=1}^{k}[m_{j},M_{j}])\subset U$, and
$g\left(\left(\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})\right)\bigcup\left(\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})\right)\right)\subset
V$.

Then, we have the following upper bound:

$\displaystyle
F\left(\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j})),g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)\right)\leq\max\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})}F(x,g(x))\bm{1}_{\mathfrak{K}}.$
(9)

Similarly, we also have the following lower bound:

$\displaystyle
F\left(\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j})),g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)\right)\geq\min\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})}F(x,g(x))\bm{1}_{\mathfrak{K}}.$
(10)

Proof: We begin with the proof for the upper bound provided by Eq. (9). From
Lemma 1, we have

$\displaystyle\Phi(f(\bm{A}))\leq\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A})\bm{V}.$
(11)

By replacing $\bm{A}$ with $\bm{A}_{j}$ in Eq. (11) and applying $w_{j}$ with
respect to each $\bm{A}_{j}$, we have

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\leq\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}.$
(12)

According to the function $F(u,v)$ condition, we have

$\displaystyle
F\left(\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j})),g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)\right)$
(13) $\displaystyle\leq$ $\displaystyle
F\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V},g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)\right)$
$\displaystyle\leq$
$\displaystyle\max\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})}F(x,g(x))\bm{1}_{\mathfrak{K}},$

where the last inequality comes from that the spectrum
$\Lambda\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)$
is in the range of
$\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})$.
The desired inequality provided by Eq. (9) is established.

Now, we will prove the lower bound provided by Eq. (10). From Lemma 1, we have

$\displaystyle\Phi(f(\bm{A}))\geq\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A})\bm{V}.$
(14)

By replacing $\bm{A}$ with $\bm{A}_{j}$ in Eq. (11) and applying $w_{j}$ with
respect to each $\bm{A}_{j}$, we have

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\geq\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}.$
(15)

From the function $F(u,v)$ condition, we also have

$\displaystyle
F\left(\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j})),g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)\right)$
(16) $\displaystyle\geq$ $\displaystyle
F\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V},g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)\right)$
$\displaystyle\geq$
$\displaystyle\min\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})}F(x,g(x))\bm{1}_{\mathfrak{K}},$

where the last inequality comes from that the spectrum
$\Lambda\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)$
is in the range of
$\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})$.
The desired inequality provided by Eq. (10) is established. $\hfill\Box$

By applying the function $F(u,v)$ with the following format:

$\displaystyle F(u,v)$ $\displaystyle=$ $\displaystyle u-\alpha v,$ (17)

where $\alpha\in\mathbb{R}$, we can have the following Theorem 2. Theorem 2
will provide generalized converses of operator Jensen’s inequalities with
ratio and difference kinds in coming sections.

###### Theorem 2

Let $\bm{A}_{j}$ be self-adjoint operator with
$\Lambda(\bm{A}_{j})\in[m_{j},M_{j}]$ for real scalars $m_{j}<M_{j}$. The
mapping $\Phi:\mathbb{B}(\mathfrak{H})\rightarrow\mathbb{B}(\mathfrak{K})$ is
defined by Eq. (4). The index $j$ is in the range of $1,2,\cdots,k$, and we
have a probability vector $\bm{w}=[w_{1},w_{2},\cdots,w_{k}]$ with the
dimension $k$, i.e., $\sum\limits_{j=1}^{k}w_{j}=1$. Let $f$ be any real
valued continous functions defined on the range
$\bigcup\limits_{j=1}^{k}[m_{j},M_{j}]$, represented by
$f\in\mathcal{C}(\bigcup\limits_{j=1}^{k}[m_{j},M_{j}])$. Besides, we assume
that the function $f$ satisfes thsoe conditions provided by Eq. (1) and Eq.
(6). The function $g$ is also a real valued continous function defined on the
range
$\left(\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})\right)\bigcup\left(\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})\right)$.
We also have a real valued function $F(u,v)$ defined as Eq. (17) with support
domain on $U\times V$ such that
$f(\bigcup\limits_{j=1}^{k}[m_{j},M_{j}])\subset U$, and
$g\left(\left(\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})\right)\bigcup\left(\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})\right)\right)\subset
V$.

Then, we have the following upper bound:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\leq\alpha
g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)+\max\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})}(x-\alpha
g(x))\bm{1}_{\mathfrak{K}}.$ (18)

Similarly, we also have the following lower bound:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\geq\alpha
g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)+\min\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})}(x-\alpha
g(x))\bm{1}_{\mathfrak{K}}.$ (19)

Proof: By setting $F(u,v)=u-\alpha v$ in Eq. (9) in Theorem 1, we have the
desired inequality provided by Eq. (18). Similarly, By setting
$F(u,v)=u-\alpha v$ in Eq. (10) in Theorem 1, we have the desired inequality
provided by Eq. (19). $\hfill\Box$

Corollary 1 below is provided to give upper and lower bounds for special types
of the function $g$ by applying Theorem 2.

###### Corollary 1

Let $\bm{A}_{j}$ be self-adjoint operator with
$\Lambda(\bm{A}_{j})\in[m_{j},M_{j}]$ for real scalars $m_{j}<M_{j}$. The
mapping $\Phi:\mathbb{B}(\mathfrak{H})\rightarrow\mathbb{B}(\mathfrak{K})$ is
defined by Eq. (4). The index $j$ is in the range of $1,2,\cdots,k$, and we
have a probability vector $\bm{w}=[w_{1},w_{2},\cdots,w_{k}]$ with the
dimension $k$, i.e., $\sum\limits_{j=1}^{k}w_{j}=1$. Let $f$ be any real
valued continous functions defined on the range
$\bigcup\limits_{j=1}^{k}[m_{j},M_{j}]$, represented by
$f\in\mathcal{C}(\bigcup\limits_{j=1}^{k}[m_{j},M_{j}])$. Besides, we assume
that the function $f$ satisfes thsoe conditions provided by Eq. (1) and Eq.
(6).

(I) If $g(x)=x^{q}$, where $q\in\mathbb{R}$ and
$\left(\left(\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})\right)\bigcup\left(\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})\right)\right)\geq
0$, we have the upper bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\leq\alpha\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)^{q}+\max\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})}(x-\alpha
x^{q})\bm{1}_{\mathfrak{K}},$ (20)

and we have the lower bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\geq\alpha\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)^{q}+\min\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})}(x-\alpha
x^{q})\bm{1}_{\mathfrak{K}}.$ (21)

(II) If $g(x)=\log(x)$ and
$\left(\left(\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})\right)\bigcup\left(\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})\right)\right)>0$
, we have the upper bound for $\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\leq\alpha\log\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)+\max\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})}(x-\alpha\log(x)\bm{1}_{\mathfrak{K}},$
(22)

and we have the lower bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\geq\alpha\log\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)+\min\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})}(x-\alpha\log(x))\bm{1}_{\mathfrak{K}}.$
(23)

(III) If $g(x)=\exp(x)$ , we have the upper bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\leq\alpha\exp\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)+\max\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})}(x-\alpha\exp(x))\bm{1}_{\mathfrak{K}},$
(24)

and we have the lower bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\geq\alpha\exp\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)+\min\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})}(x-\alpha\exp(x))\bm{1}_{\mathfrak{K}}.$
(25)

In the following Example 1, we will assume that the function $f$ is bounded by
linear functions and derive related inequalties given by Theorem 1, Theorem 2,
and Corollary 1.

###### Example 1

In this example, we assume that the function $f$ satsifies the following:

$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\overbrace{(\beta_{0}+\beta_{1}x)}^{p_{\mathscr{U}}(x)}-f(x)\leavevmode\nobreak\
\leavevmode\nobreak\ \leq\leavevmode\nobreak\ \leavevmode\nobreak\ \epsilon,$
$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
f(x)-\overbrace{(\alpha_{0}+\alpha_{1}x)}^{p_{\mathscr{L}}(x)}\leavevmode\nobreak\
\leavevmode\nobreak\ \leq\leavevmode\nobreak\ \leavevmode\nobreak\ \epsilon,$
(26)

where $x\in[m,M]$ with positive $\alpha_{0},\alpha_{1},\beta_{0}$ and
$\beta_{1}$. We also assume that $\Lambda(\bm{A}_{j})\in[m,M]$ and
$f(\bm{A}_{j})\geq\bm{0}$ for all $j\in 1,2,\cdots,k$. Moreover, the mapping
$\Phi$ defined by Eq. (4) has all coefficients $a_{i}\geq 0$. Then, from Lemma
1, we have

$\displaystyle\Phi(f(\bm{A}))$ $\displaystyle\leq$
$\displaystyle\bm{V}^{\ast}\left\\{\sum\limits_{i=0}^{I}a_{i}\mathscr{K}(\beta_{0}+\beta_{1}m,\beta_{0}+\beta_{1}M,i)(\beta_{0}+\beta_{1}\bm{A})^{i}\right\\}\bm{V}.$
(27)

Besides, we also have

$\displaystyle\Phi(f(\bm{A}))$ $\displaystyle\geq$
$\displaystyle\bm{V}^{\ast}\left\\{\sum\limits_{i=0}^{I}a_{i}\mathscr{K}^{-1}(\alpha_{0}+\alpha_{1}m,\alpha_{0}+\alpha_{1}M,i)(\alpha_{0}+\alpha_{1}\bm{A})^{i}\right\\}\bm{V}.$
(28)

From Eq. (27), we can express $\mbox{Poly}_{f,\mathscr{U}}(\bm{A})$ as:

$\displaystyle\mbox{Poly}_{f,\mathscr{U}}(\bm{A})$ $\displaystyle=$
$\displaystyle\sum\limits_{i=0}^{I}c_{i}\bm{A}^{i},$ (29)

where coeffcients $c_{i}$ are given by

$\displaystyle c_{i}$ $\displaystyle=$
$\displaystyle\sum\limits_{j=0}^{I-i}{i+j\choose
i}a_{i+j}\mathscr{K}(\beta_{0}+\beta_{1}m,\beta_{0}+\beta_{1}M,i+j)\beta^{i}_{1}\beta^{j}_{0}.$
(30)

Similarly, from Eq. (28), we can express $\mbox{Poly}_{f,\mathscr{L}}(\bm{A})$
as:

$\displaystyle\mbox{Poly}_{f,\mathscr{L}}(\bm{A})$ $\displaystyle=$
$\displaystyle\sum\limits_{i=0}^{I}d_{i}\bm{A}^{i},$ (31)

where coeffcients $c_{i}$ are given by

$\displaystyle d_{i}$ $\displaystyle=$
$\displaystyle\sum\limits_{j=0}^{I-i}{i+j\choose
i}a_{i+j}\mathscr{K}^{-1}(\alpha_{0}+\alpha_{1}m,\alpha_{0}+\alpha_{1}M,i+j)\alpha^{i}_{1}\alpha^{j}_{0}.$
(32)

Suppose conditions provided by Theorem 1 are valid, from polynomials given by
Eq. (29) and Eq. (31), we have

$\displaystyle
F\left(\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j})),g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)\right)\leq\max\limits_{x\in\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m,M)}F(x,g(x))\bm{1}_{\mathfrak{K}}.$
(33)

Similarly, we also have the following lower bound:

$\displaystyle
F\left(\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j})),g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)\right)\geq\min\limits_{x\in\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m,M)}F(x,g(x))\bm{1}_{\mathfrak{K}}.$
(34)

If $F(u,v)=u-\alpha v$, we can apply Theorem 2 to the function $f$ and $\Phi$
provided in this example to obtain the following:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\leq\alpha
g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)+\max\limits_{x\in\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m,M)}(x-\alpha
g(x))\bm{1}_{\mathfrak{K}}.$ (35)

Similarly, we also have the following lower bound:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\geq\alpha
g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)+\min\limits_{x\in\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m,M)}(x-\alpha
g(x))\bm{1}_{\mathfrak{K}}.$ (36)

Finally, the application of Corollary 1 to the function $f$ and $\Phi$
provided in this example will get:

(I) If $g(x)=x^{q}$, where $q\in\mathbb{R}$ and
$\left(\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m,M)\bigcup\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m,M)\right)\geq
0$, we have the upper bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\leq\alpha\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)^{q}+\max\limits_{x\in\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m,M)}(x-\alpha
x^{q})\bm{1}_{\mathfrak{K}},$ (37)

and we have the lower bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\geq\alpha\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)^{q}+\min\limits_{x\in\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m,M)}(x-\alpha
x^{q})\bm{1}_{\mathfrak{K}}.$ (38)

(II) If $g(x)=\log(x)$ and
$\left(\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m,M)\bigcup\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m,M)\right)>0$
, we have the upper bound for $\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\leq\alpha\log\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)+\max\limits_{x\in\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m,M)}(x-\alpha\log(x)\bm{1}_{\mathfrak{K}},$
(39)

and we have the lower bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\geq\alpha\log\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)+\min\limits_{x\in\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m,M)}(x-\alpha\log(x))\bm{1}_{\mathfrak{K}}.$
(40)

(III) If $g(x)=\exp(x)$ , we have the upper bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\leq\alpha\exp\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)+\max\limits_{x\in\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m,M)}(x-\alpha\exp(x))\bm{1}_{\mathfrak{K}},$
(41)

and we have the lower bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\geq\alpha\exp\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)+\min\limits_{x\in\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m,M)}(x-\alpha\exp(x))\bm{1}_{\mathfrak{K}}.$
(42)

##  3 Generalized Converses of Operator Jensen’s Inequalities: Ratio Kind

In this section, we will derive the lower and upper bounds for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$ in terms of ratio criteria
related to the function $g$.

###### Theorem 3

Let $\bm{A}_{j}$ be self-adjoint operator with
$\Lambda(\bm{A}_{j})\in[m_{j},M_{j}]$ for real scalars $m_{j}<M_{j}$. The
mapping $\Phi:\mathbb{B}(\mathfrak{H})\rightarrow\mathbb{B}(\mathfrak{K})$ is
defined by Eq. (4). The index $j$ is in the range of $1,2,\cdots,k$, and we
have a probability vector $\bm{w}=[w_{1},w_{2},\cdots,w_{k}]$ with the
dimension $k$, i.e., $\sum\limits_{j=1}^{k}w_{j}=1$. Let $f$ be any real
valued continous functions defined on the range
$\bigcup\limits_{j=1}^{k}[m_{j},M_{j}]$, represented by
$f\in\mathcal{C}(\bigcup\limits_{j=1}^{k}[m_{j},M_{j}])$. Besides, we assume
that the function $f$ satisfes thsoe conditions provided by Eq. (1) and Eq.
(6). The function $g$ is also a real valued continous function defined on the
range
$\left(\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})\right)\bigcup\left(\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})\right)$
and $g(x)\neq 0$ for
$x\in\left(\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})\right)\bigcup\left(\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})\right)$.

(I) If we also assume that
$g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)>\bm{O}$
and
$g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)>\bm{O}$,
then, we have the following upper bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$
$\displaystyle\leq$
$\displaystyle\left[\max\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})}xg^{-1}(x)\right]g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right);$
(43)

and, the following lower bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$
$\displaystyle\geq$
$\displaystyle\left[\min\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})}xg^{-1}(x)\right]g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right).$
(44)

(II) If we also assume that
$g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)<\bm{O}$
and
$g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)<\bm{O}$,
then, we have the following upper bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$
$\displaystyle\leq$
$\displaystyle\left[\min\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})}xg^{-1}(x)\right]g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right);$
(45)

and, the following lower bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$
$\displaystyle\geq$
$\displaystyle\left[\max\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})}xg^{-1}(x)\right]g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right).$
(46)

Proof: For part (I), we will apply $F(u,v)$ as

$\displaystyle F(u,v)$ $\displaystyle=$ $\displaystyle v^{-1/2}uv^{-1/2},$
(47)

to Eq. (9) in Theorem 1, then, we will obtain

$\displaystyle\left(g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)\right)^{-1/2}\left(\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\right)\left(g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)\right)^{-1/2}$
$\displaystyle\leq\left[\max\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})}xg^{-1}(x)\right]\bm{1}_{\mathfrak{K}}.$
(48)

By multiplying
$\left(g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)\right)^{1/2}$
at both sides of Eq. (3), we obtain the desired inequality provided by Eq.
(43). By applying $F(u,v)$ with Eq. (47) again to Eq. (10) in Theorem 1, then,
we will obtain

$\displaystyle\left(g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)\right)^{-1/2}\left(\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\right)\left(g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)\right)^{-1/2}$
$\displaystyle\geq\left[\min\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})}xg^{-1}(x)\right]\bm{1}_{\mathfrak{K}}.$
(49)

By multiplying
$\left(g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)\right)^{-1/2}$
at both sides of Eq. (3), we obtain the desired inequality provided by Eq.
(44).

The proof of Part (II) is immediate obtained by setting $g(x)$ as $-g(x)$ in
Eq. (3) and Eq. (3). $\hfill\Box$

Next Corollary 2 is obtained by applying Theorem 3 to special types of
function $g$.

###### Corollary 2

Let $\bm{A}_{j}$ be self-adjoint operator with
$\Lambda(\bm{A}_{j})\in[m_{j},M_{j}]$ for real scalars $m_{j}<M_{j}$. The
mapping $\Phi:\mathbb{B}(\mathfrak{H})\rightarrow\mathbb{B}(\mathfrak{K})$ is
defined by Eq. (4). The index $j$ is in the range of $1,2,\cdots,k$, and we
have a probability vector $\bm{w}=[w_{1},w_{2},\cdots,w_{k}]$ with the
dimension $k$, i.e., $\sum\limits_{j=1}^{k}w_{j}=1$. Let $f$ be any real
valued continous functions defined on the range
$\bigcup\limits_{j=1}^{k}[m_{j},M_{j}]$, represented by
$f\in\mathcal{C}(\bigcup\limits_{j=1}^{k}[m_{j},M_{j}])$. Besides, we assume
that the function $f$ satisfes thsoe conditions provided by Eq. (1) and Eq.
(6).

(I) If we also assume that
$\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)^{q}>\bm{O}$
and
$\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)^{q}>\bm{O}$
for $q\in\mathbb{R}$, then, we have the following upper bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$
$\displaystyle\leq$
$\displaystyle\left[\max\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})}x^{1-q}\right]\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)^{q};$
(50)

and, the following lower bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$
$\displaystyle\geq$
$\displaystyle\left[\min\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})}x^{1-q}\right]\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)^{q}.$
(51)

(II) If we also assume that
$\log\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)>\bm{O}$
and
$\log\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)>\bm{O}$
for $q\in\mathbb{R}$, then, we have the following upper bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$
$\displaystyle\leq$
$\displaystyle\left[\max\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})}\frac{x}{\log
x}\right]\log\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right);$
(52)

and, the following lower bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$
$\displaystyle\geq$
$\displaystyle\left[\min\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})}\frac{x}{\log
x}\right]\log\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right).$
(53)

(II’) If we also assume that
$\log\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)<\bm{O}$
and
$\log\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)<\bm{O}$,
then, we have the following upper bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$
$\displaystyle\leq$
$\displaystyle\left[\min\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})}\frac{x}{\log
x}\right]\log\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right);$
(54)

and, the following lower bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$
$\displaystyle\geq$
$\displaystyle\left[\max\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})}\frac{x}{\log
x}\right]\log\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right).$
(55)

(III) Since we have
$\exp\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)>\bm{O}$
and
$\exp\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)>\bm{O}$,
then, we have the following upper bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$
$\displaystyle\leq$
$\displaystyle\left[\max\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})}\frac{x}{\exp
x}\right]\exp\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right);$
(56)

and, the following lower bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$
$\displaystyle\geq$
$\displaystyle\left[\min\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})}\frac{x}{\exp
x}\right]\exp\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right).$
(57)

Proof: Part (I) of this Corollary is proved by applying Theorem 3 Part (I)
with the function $g$ as $g(x)=x^{q}$. Part (II) of this Corollary is proved
by applying Theorem 3 Part (I) with the function $g$ as $g(x)=\log(x)$, where
$\log(x)>0$. Part (II’) of this Corollary is proved by applying Theorem 3 Part
(II) with the function $g$ as $g(x)=\log(x)$, where $\log(x)<0$. Finally, Part
(III) of this Corollary is proved by applying Theorem 3 Part (I) with the
function $g$ as $g(x)=\exp(x)$. $\hfill\Box$

##  4 Generalized Converses of Operator Jensen’s Inequalities: Difference Kind

In this section, we will derive the lower and upper bounds for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$ in terms of difference
criteria related to the function $g$.

###### Theorem 4

Let $\bm{A}_{j}$ be self-adjoint operator with
$\Lambda(\bm{A}_{j})\in[m_{j},M_{j}]$ for real scalars $m_{j}<M_{j}$. The
mapping $\Phi:\mathbb{B}(\mathfrak{H})\rightarrow\mathbb{B}(\mathfrak{K})$ is
defined by Eq. (4). The index $j$ is in the range of $1,2,\cdots,k$, and we
have a probability vector $\bm{w}=[w_{1},w_{2},\cdots,w_{k}]$ with the
dimension $k$, i.e., $\sum\limits_{j=1}^{k}w_{j}=1$. Let $f$ be any real
valued continous functions defined on the range
$\bigcup\limits_{j=1}^{k}[m_{j},M_{j}]$, represented by
$f\in\mathcal{C}(\bigcup\limits_{j=1}^{k}[m_{j},M_{j}])$. Besides, we assume
that the function $f$ satisfes thsoe conditions provided by Eq. (1) and Eq.
(6). The function $g$ is also a real valued continous function defined on the
range
$\left(\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})\right)\bigcup\left(\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})\right)$.
We also have a real valued function $F(u,v)$ defined as Eq. (17) with support
domain on $U\times V$ such that
$f(\bigcup\limits_{j=1}^{k}[m_{j},M_{j}])\subset U$, and
$g\left(\left(\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})\right)\bigcup\left(\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})\right)\right)\subset
V$.

Then, we have the following upper bound:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))-g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)\leq\max\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})}(x-g(x))\bm{1}_{\mathfrak{K}}.$
(58)

Similarly, we also have the following lower bound:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))-g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)\geq\min\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})}(x-g(x))\bm{1}_{\mathfrak{K}}.$
(59)

Proof: The upper bound of this theorem is proved by setting $\alpha=1$ in Eq.
(18) from Theorem 2 and rearrangement of the term
$g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)$
to obtain Eq. (58).

Similarly, the lower bound of this theorem is proved by setting $\alpha=1$ in
Eq. (19) from Theorem 2 and rearrangement of the term
$g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)$
to obtain Eq. (59). $\hfill\Box$

Next Corollary 3 is obtained by applying Theorem 4 to special types of
function $g$.

###### Corollary 3

Let $\bm{A}_{j}$ be self-adjoint operator with
$\Lambda(\bm{A}_{j})\in[m_{j},M_{j}]$ for real scalars $m_{j}<M_{j}$. The
mapping $\Phi:\mathbb{B}(\mathfrak{H})\rightarrow\mathbb{B}(\mathfrak{K})$ is
defined by Eq. (4). The index $j$ is in the range of $1,2,\cdots,k$, and we
have a probability vector $\bm{w}=[w_{1},w_{2},\cdots,w_{k}]$ with the
dimension $k$, i.e., $\sum\limits_{j=1}^{k}w_{j}=1$. Let $f$ be any real
valued continous functions defined on the range
$\bigcup\limits_{j=1}^{k}[m_{j},M_{j}]$, represented by
$f\in\mathcal{C}(\bigcup\limits_{j=1}^{k}[m_{j},M_{j}])$. Besides, we assume
that the function $f$ satisfes thsoe conditions provided by Eq. (1) and Eq.
(6).

(I) If we also assume that $g(x)=x^{q}$, where $q\in\mathbb{R}$, then, we have
the following upper bound for $\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))-\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)^{q}\leq\max\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})}(x-x^{q})\bm{1}_{\mathfrak{K}};$
(60)

and, the following lower bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))-\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)^{q}\geq\min\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})}(x-x^{q})\bm{1}_{\mathfrak{K}}.$
(61)

(II) If we also assume that $g(x)=\log(x)$, then, we have the following upper
bound for $\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))-\log\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)\leq\max\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})}(x-\log(x))\bm{1}_{\mathfrak{K}};$
(62)

and, the following lower bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))-\log\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)\geq\min\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})}(x-\log(x))\bm{1}_{\mathfrak{K}}.$
(63)

(III) If we also assume that $g(x)=\exp(x)$, then, we have the following upper
bound for $\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))-\exp\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A}_{j})\bm{V}\right)\leq\max\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})}(x-\exp(x))\bm{1}_{\mathfrak{K}};$
(64)

and, the following lower bound for
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))-\exp\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A}_{j})\bm{V}\right)\geq\min\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})}(x-\exp(x))\bm{1}_{\mathfrak{K}}.$
(65)

Proof: For Part (I), we will use $g(x)=x^{q}$ in Eq. (58) in Theorem 4 to
obtain Eq. (60). We will also use $g(x)=x^{q}$ in Eq. (59) in Theorem 4 to
obtain Eq. (61).

For Part (II), we will use $g(x)=\log(x)$ in Eq. (58) in Theorem 4 to obtain
Eq. (62). We will also use $g(x)=\log(x)$ in Eq. (59) in Theorem 4 to obtain
Eq. (63).

For Part (III), we will use $g(x)=\exp(x)$ in Eq. (58) in Theorem 4 to obtain
Eq. (64). We will also use $g(x)=\exp(x)$ in Eq. (59) in Theorem 4 to obtain
Eq. (65). $\hfill\Box$

##  5 Hypercomplex Function Approximation

In this section, we will consider hypercomplex function approximation problem
in terms of the ratio error discussed in Section 5.1 and the difference error
in Section 5.2.

###  5.1 Ratio Type Approximation

Let $\bm{A}$ be a self-adjoint operator with $\Lambda(\bm{A})\in[m,M]$ for
real scalars $m<M$. The mapping
$\Phi:\mathbb{B}(\mathfrak{H})\rightarrow\mathbb{B}(\mathfrak{K})$ is defined
by Eq. (4). Let $f$ be any real valued continuous functions defined on the
range $[m,M]$, represented by $f\in\mathcal{C}([m,M])$. The _ratio type
approximation_ problem is to find the function $g$ and the polynomial function
$p_{1}$ to satisfy the following:

$\displaystyle\frac{\Phi(f(\bm{A}))}{g(\bm{V}^{\ast}p_{1}(\bm{A})\bm{V})}\leq\alpha_{1}\bm{1}_{\mathfrak{K}},$
(66)

where $\alpha_{1}$ is some specified positive real number. Similarly, we also
can find the polynomial function $p_{2}$ to satisfy the following:

$\displaystyle\frac{\Phi(f(\bm{A}))}{g(\bm{V}^{\ast}p_{2}(\bm{A})\bm{V})}\geq\alpha_{2}\bm{1}_{\mathfrak{K}},$
(67)

where $\alpha_{2}$ is some specified positive real number.

From Theorem 3, if
$g\left(\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A})\bm{V}\right)>\bm{O}$,
Eq. (66) can be established by setting

$\displaystyle\alpha_{1}$ $\displaystyle\geq$
$\displaystyle\left[\max\limits_{x\in\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m,M)}xg^{-1}(x)\right].$
(68)

Similarly, if
$g\left(\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A})\bm{V}\right)>\bm{O}$,
Eq. (67) can be established by setting

$\displaystyle\alpha_{2}$ $\displaystyle\leq$
$\displaystyle\left[\min\limits_{x\in\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m,M)}xg^{-1}(x)\right].$
(69)

On the other hand, from Theorem 3, if
$g\left(\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A})\bm{V}\right)<\bm{O}$,
Eq. (66) can be established by setting

$\displaystyle\alpha_{1}$ $\displaystyle\geq$
$\displaystyle\left[\min\limits_{x\in\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m,M)}xg^{-1}(x)\right].$
(70)

Similarly, if
$g\left(\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A})\bm{V}\right)<\bm{O}$,
Eq. (67) can be established by setting

$\displaystyle\alpha_{2}$ $\displaystyle\leq$
$\displaystyle\left[\max\limits_{x\in\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m,M)}xg^{-1}(x)\right].$
(71)

For $k$ terms, let $\bm{A}_{j}$ be self-adjoint operator with
$\Lambda(\bm{A}_{j})\in[m_{j},M_{j}]$ for real scalars $m_{j}<M_{j}$. The
mapping $\Phi:\mathbb{B}(\mathfrak{H})\rightarrow\mathbb{B}(\mathfrak{K})$ is
defined by Eq. (4). The index $j$ is in the range of $1,2,\cdots,k$, and we
have a probability vector $\bm{w}=[w_{1},w_{2},\cdots,w_{k}]$ with the
dimension $k$, i.e., $\sum\limits_{j=1}^{k}w_{j}=1$. Let $f$ be any real
valued continuous functions defined on the range
$\bigcup\limits_{j=1}^{k}[m_{j},M_{j}]$, represented by
$f\in\mathcal{C}(\bigcup\limits_{j=1}^{k}[m_{j},M_{j}])$. Besides, we assume
that the function $f$ satisfies the conditions provided by Eq. (1) and Eq.
(6). The _ratio type approximation_ problem is to find the function $g$ and
the polynomial functions $p_{1,1}(x),\cdots,p_{1,k}(x)$ to satisfy the
following:

$\displaystyle\frac{\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))}{g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}p_{1,j}(\bm{A}_{j})\bm{V}\right)}\leq\alpha_{1}\bm{1}_{\mathfrak{K}},$
(72)

where $\alpha_{1}$ is some specfied positive real number. Similarly, we also
can find the polynomial function $p_{2,1}(x),\cdots,p_{2,k}(x)$ to satisfy the
following:

$\displaystyle\frac{\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))}{g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}p_{2,j}(\bm{A}_{j})\bm{V}\right)}\geq\alpha_{2}\bm{1}_{\mathfrak{K}},$
(73)

where $\alpha_{2}$ is some specfied positive real number.

From Theorem 3, if
$g\left(\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A})\bm{V}\right)>\bm{O}$,
Eq. (72) can be established by setting

$\displaystyle\alpha_{1}$ $\displaystyle\geq$
$\displaystyle\left[\max\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})}xg^{-1}(x)\right].$
(74)

Similarly, if
$g\left(\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A})\bm{V}\right)>\bm{O}$,
Eq. (73) can be established by setting

$\displaystyle\alpha_{2}$ $\displaystyle\leq$
$\displaystyle\left[\min\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})}xg^{-1}(x)\right].$
(75)

On the other hand, from Theorem 3, if
$g\left(\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{U}}(\bm{A})\bm{V}\right)<\bm{O}$,
Eq. (72) can be established by setting

$\displaystyle\alpha_{1}$ $\displaystyle\geq$
$\displaystyle\left[\min\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m_{j},M_{j})}xg^{-1}(x)\right].$
(76)

Similarly, if
$g\left(\bm{V}^{\ast}\mbox{Poly}_{f,\mathscr{L}}(\bm{A})\bm{V}\right)<\bm{O}$,
Eq. (73) can be established by setting

$\displaystyle\alpha_{2}$ $\displaystyle\leq$
$\displaystyle\left[\max\limits_{x\in\bigcup\limits_{j=1}^{k}w_{j}\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m_{j},M_{j})}xg^{-1}(x)\right].$
(77)

###### Remark 1

Given requirements of Eq. (66), Eq. (67), Eq. (72), and Eq. (73), we
conjecture the existence of function $g$ and polynomials
$p_{1}(x),p_{2}(x),p_{1,1}(x),\cdots,p_{1,k}(x)$, and
$p_{2,1}(x),\cdots,p_{2,k}(x)$ for any given $\alpha_{1}$ or $\alpha_{2}$. If
existence, how to find these functions?

The following Example 2 is provided to evaluate the upper ratio bound given by
Eq. (72) and evaluate the lower ratio bound given by Eq. (73).

###### Example 2

Let $\bm{A}_{j}$ be self-adjoint operator with $\Lambda(\bm{A}_{j})\in[m,M]$
for real scalars $m<M$. The mapping
$\Phi:\mathbb{B}(\mathfrak{H})\rightarrow\mathbb{B}(\mathfrak{K})$ is defined
as $\Phi(\bm{X})=\bm{V}^{\ast}(\bm{X})\bm{X}$. The index $j$ is in the range
of $1,2,\cdots,k$, and we have a probability vector
$\bm{w}=[w_{1},w_{2},\cdots,w_{k}]$ with the dimension $k$, i.e.,
$\sum\limits_{j=1}^{k}w_{j}=1$. Let $f$ be a convex and differentiable
function such that $ax+b^{\prime}\leq f(x)\leq ax+b$ for $x\in[m,M]$, where

$\displaystyle a$ $\displaystyle=$ $\displaystyle\frac{f(M)-f(m)}{M-m},$
$\displaystyle b$ $\displaystyle=$ $\displaystyle\frac{Mf(m)-mf(M)}{M-m},$
$\displaystyle b^{\prime}$ $\displaystyle=$ $\displaystyle
f(x_{0})-\frac{f(M)-f(m)}{M-m}x_{0},$ (78)

where $f^{\prime}(x_{0})=\frac{f(M)-f(m)}{M-m}$.

If we require $g(x)>0$ for all $x\in[m,M]$, from Theorem 3, we have

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\leq\max\limits_{m<x<M}\frac{ax+b}{g(x)}g\left(\sum\limits_{j=1}^{k}w_{j}\Phi(\bm{A}_{j})\right),$
(79)

and

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\geq\min\limits_{m<x<M}\frac{ax+b^{\prime}}{g(x)}g\left(\sum\limits_{j=1}^{k}w_{j}\Phi(\bm{A}_{j})\right).$
(80)

Let us consider several special cases of the function $g(x)$. If $g(x)=x^{q}$,
where $q\in\mathbb{R}$, and $m>0$, we have

$\displaystyle\alpha_{1}$ $\displaystyle\geq$
$\displaystyle\max\limits_{m<x<M}\frac{ax+b}{x^{q}},$
$\displaystyle\alpha_{2}$ $\displaystyle\leq$
$\displaystyle\min\limits_{m<x<M}\frac{ax+b^{\prime}}{x^{q}}.$ (81)

If $g(x)=\log(x)$ and $m>1$, we have

$\displaystyle\alpha_{1}$ $\displaystyle\geq$
$\displaystyle\max\limits_{m<x<M}\frac{ax+b}{\log(x)},$
$\displaystyle\alpha_{2}$ $\displaystyle\leq$
$\displaystyle\min\limits_{m<x<M}\frac{ax+b^{\prime}}{\log(x)}.$ (82)

If $g(x)=\exp(x)$, we have

$\displaystyle\alpha_{1}$ $\displaystyle\geq$
$\displaystyle\max\limits_{m<x<M}\frac{ax+b}{\exp(x)},$
$\displaystyle\alpha_{2}$ $\displaystyle\leq$
$\displaystyle\min\limits_{m<x<M}\frac{ax+b^{\prime}}{\exp(x)}.$ (83)

If we require $g(x)<0$ for all $x\in[m,M]$, from Theorem 3, we also have

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\leq\min\limits_{m<x<M}\frac{ax+b}{g(x)},$
(84)

and

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\geq\max\limits_{m<x<M}\frac{ax+b^{\prime}}{g(x)},$
(85)

Let us consider several special cases of the function $g(x)$. If
$g(x)=-x^{q}$, where $q\in\mathbb{R}$, and $m>0$, we have

$\displaystyle\alpha_{1}$ $\displaystyle\geq$
$\displaystyle\min\limits_{m<x<M}-\frac{ax+b}{x^{q}},$
$\displaystyle\alpha_{2}$ $\displaystyle\leq$
$\displaystyle\max\limits_{m<x<M}-\frac{ax+b^{\prime}}{x^{q}}.$ (86)

If $g(x)=\log(x)$ and $0<m<M<1$, we have

$\displaystyle\alpha_{1}$ $\displaystyle\geq$
$\displaystyle\min\limits_{m<x<M}\frac{ax+b}{\log(x)},$
$\displaystyle\alpha_{2}$ $\displaystyle\leq$
$\displaystyle\max\limits_{m<x<M}\frac{ax+b^{\prime}}{\log(x)}.$ (87)

###  5.2 Difference Type Approximation

Let $\bm{A}$ be a self-adjoint operator with $\Lambda(\bm{A})\in[m,M]$ for
real scalars $m<M$. The mapping
$\Phi:\mathbb{B}(\mathfrak{H})\rightarrow\mathbb{B}(\mathfrak{K})$ is defined
by Eq. (4). Let $f$ be any real valued continuous functions defined on the
range $[m,M]$, represented by $f\in\mathcal{C}([m,M])$. The _difference type
approximation_ problem is to find the function $g$ and the polynomial function
$p_{1}(x)$ to satisfy the following:

$\displaystyle\Phi(f(\bm{A}))-g(\bm{V}^{\ast}p_{1}(\bm{A})\bm{V})\leq\beta_{1}\bm{1}_{\mathfrak{K}},$
(88)

where $\beta_{1}$ is some specified real number. Similarly, we also can find
the polynomial function $p_{2}(x)$ to satisfy the following:

$\displaystyle\Phi(f(\bm{A}))-g(\bm{V}^{\ast}p_{2}(\bm{A})\bm{V})\geq\beta_{2}\bm{1}_{\mathfrak{K}},$
(89)

where $\beta_{2}$ is some specified real number.

From Theorem 4, Eq. (88) can be established by setting

$\displaystyle\beta_{1}$ $\displaystyle\geq$
$\displaystyle\left[\max\limits_{x\in\widetilde{\mbox{Poly}}_{f,\mathscr{U}}(m,M)}x-g(x)\right].$
(90)

Similarly, Eq. (89) can be established by setting

$\displaystyle\beta_{2}$ $\displaystyle\leq$
$\displaystyle\left[\min\limits_{x\in\widetilde{\mbox{Poly}}_{f,\mathscr{L}}(m,M)}x-g(x)\right].$
(91)

For $k$ terms, let $\bm{A}_{j}$ be self-adjoint operator with
$\Lambda(\bm{A}_{j})\in[m_{j},M_{j}]$ for real scalars $m_{j}<M_{j}$. The
mapping $\Phi:\mathbb{B}(\mathfrak{H})\rightarrow\mathbb{B}(\mathfrak{K})$ is
defined by Eq. (4). The index $j$ is in the range of $1,2,\cdots,k$, and we
have a probability vector $\bm{w}=[w_{1},w_{2},\cdots,w_{k}]$ with the
dimension $k$, i.e., $\sum\limits_{j=1}^{k}w_{j}=1$. Let $f$ be any real
valued continuous functions defined on the range
$\bigcup\limits_{j=1}^{k}[m_{j},M_{j}]$, represented by
$f\in\mathcal{C}(\bigcup\limits_{j=1}^{k}[m_{j},M_{j}])$. Besides, we assume
that the function $f$ satisfies the conditions provided by Eq. (1) and Eq.
(6). The _difference type approximation_ problem is to find the function $g$
and the polynomial functions $p_{1,1}(x),\cdots,p_{1,k}(x)$ to satisfy the
following:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))-g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}p_{1,j}(\bm{A}_{j})\bm{V}\right)\leq\beta_{1}\bm{1}_{\mathfrak{K}},$
(92)

where $\beta_{1}$ is some specfied real number. Similarly, we also can find
the polynomial function $p_{2,1}(x),\cdots,p_{2,k}(x)$ to satisfy the
following:

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))-g\left(\sum\limits_{j=1}^{k}w_{j}\bm{V}^{\ast}p_{2,j}(\bm{A}_{j})\bm{V}\right)\geq\beta_{2}\bm{1}_{\mathfrak{K}},$
(93)

where $\beta_{2}$ is some specfied real number.

The following Example 3 is provided to evaluate the upper difference bound
given by Eq. (92) and evaluate the lower difference bound given by Eq. (93).

###### Example 3

Let $\bm{A}_{j}$ be self-adjoint operator with $\Lambda(\bm{A}_{j})\in[m,M]$
for real scalars $m<M$. The mapping
$\Phi:\mathbb{B}(\mathfrak{H})\rightarrow\mathbb{B}(\mathfrak{K})$ is defined
as $\Phi(\bm{X})=\bm{V}^{\ast}(\bm{X})\bm{X}$. The index $j$ is in the range
of $1,2,\cdots,k$, and we have a probability vector
$\bm{w}=[w_{1},w_{2},\cdots,w_{k}]$ with the dimension $k$, i.e.,
$\sum\limits_{j=1}^{k}w_{j}=1$. Let $f$ be a convex and differentiable
function such that $ax+b^{\prime}\leq f(x)\leq ax+b$ for $x\in[m,M]$, where

$\displaystyle a$ $\displaystyle=$ $\displaystyle\frac{f(M)-f(m)}{M-m},$
$\displaystyle b$ $\displaystyle=$ $\displaystyle\frac{Mf(m)-mf(M)}{M-m},$
$\displaystyle b^{\prime}$ $\displaystyle=$ $\displaystyle
f(x_{0})-\frac{f(M)-f(m)}{M-m}x_{0},$ (94)

where $f^{\prime}(x_{0})=\frac{f(M)-f(m)}{M-m}$.

From Theorem 4, we have

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\leq\max\limits_{m<x<M}(ax+b-g(x)),$
(95)

and

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\geq\min\limits_{m<x<M}(ax+b^{\prime}-g(x)),$
(96)

Let us consider several special cases of the function $g(x)$. If $g(x)=x^{q}$,
where $q\in\mathbb{R}$ and $m>0$, we have

$\displaystyle\beta_{1}$ $\displaystyle\geq$
$\displaystyle\max\limits_{m<x<M}(ax+b-x^{q}),$ $\displaystyle\beta_{2}$
$\displaystyle\leq$ $\displaystyle\min\limits_{m<x<M}(ax+b^{\prime}-x^{q}).$
(97)

If $g(x)=\log(x)$ and $m>0$, we have

$\displaystyle\beta_{1}$ $\displaystyle\geq$
$\displaystyle\max\limits_{m<x<M}(ax+b-\log(x)),$ $\displaystyle\beta_{2}$
$\displaystyle\leq$ $\displaystyle\min\limits_{m<x<M}(ax+b^{\prime}-\log(x)).$
(98)

If $g(x)=\exp(x)$, we have

$\displaystyle\beta_{1}$ $\displaystyle\geq$
$\displaystyle\max\limits_{m<x<M}(ax+b-\exp(x)),$ $\displaystyle\beta_{2}$
$\displaystyle\leq$ $\displaystyle\min\limits_{m<x<M}(ax+b^{\prime}-\exp(x)).$
(99)

###### Remark 2

Given requirements of Eq. (88), Eq. (89), Eq. (92), and Eq. (93), we
conjecture the existence of function $g$ and polynomials
$p_{1}(x),p_{2}(x),p_{1,1}(x),\cdots,p_{1,k}(x)$, and
$p_{2,1}(x),\cdots,p_{2,k}(x)$ for any given $\beta_{1}$ or $\beta_{2}$. If
existence, how to find these functions?

##  6 Bounds Algebra

In this section, we will apply results from Section 3 to build bounds algebra
for addition and multiplication of hypercomplex functions
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$ and
$\sum\limits_{j=1}^{k}w_{j}\Phi(h(\bm{A}_{j}))$, where functions $f$ and $h$
share some common properties.

###  6.1 Hypercomplex Function Bounds Algebra

Let $\bm{A}_{j}$ be self-adjoint operator with $\Lambda(\bm{A}_{j})\in[m,M]$
for real scalars $m<M$. The mapping
$\Phi:\mathbb{B}(\mathfrak{H})\rightarrow\mathbb{B}(\mathfrak{K})$ is defined
as $\Phi(\bm{X})=\bm{V}^{\ast}(\bm{X})\bm{X}$. The index $j$ is in the range
of $1,2,\cdots,k$, and we have a probability vector
$\bm{w}=[w_{1},w_{2},\cdots,w_{k}]$ with the dimension $k$, i.e.,
$\sum\limits_{j=1}^{k}w_{j}=1$. Let $f,h$ be two convex and differentiable
functions defined in $[m,M]$ such that

$\displaystyle ax+b^{\prime}\leq f(x)\leq ax+b,$ $\displaystyle
cx+d^{\prime}\leq h(x)\leq cx+d,$ (100)

where $x\in[m,M]$. If we require $g(x)>0$ for all $x\in[m,M]$, from Theorem 3,
we have

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\leq\underbrace{\max\limits_{m<x<M}\frac{ax+b}{g(x)}}_{:=\alpha_{f,\mathscr{U}}}g\left(\sum\limits_{j=1}^{k}w_{j}\Phi(\bm{A}_{j})\right),$
(101)

and

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\geq\underbrace{\min\limits_{m<x<M}\frac{ax+b^{\prime}}{g(x)}}_{:=\alpha_{f,\mathscr{L}}}g\left(\sum\limits_{j=1}^{k}w_{j}\Phi(\bm{A}_{j})\right).$
(102)

Similarly, from Theorem 3, we also have

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(h(\bm{A}_{j}))\leq\underbrace{\max\limits_{m<x<M}\frac{cx+d}{g(x)}}_{:=\alpha_{h,\mathscr{U}}}g\left(\sum\limits_{j=1}^{k}w_{j}\Phi(\bm{A}_{j})\right),$
(103)

and

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(h(\bm{A}_{j}))\geq\underbrace{\min\limits_{m<x<M}\frac{cx+d^{\prime}}{g(x)}}_{:=\alpha_{h,\mathscr{L}}}g\left(\sum\limits_{j=1}^{k}w_{j}\Phi(\bm{A}_{j})\right).$
(104)

Consider the addition between $\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$
and $\sum\limits_{j=1}^{k}w_{j}\Phi(h(\bm{A}_{j}))$, from Eq. (101) and Eq.
(103), then, we have

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))+\sum\limits_{j=1}^{k}w_{j}\Phi(h(\bm{A}_{j}))\leq(\alpha_{f,\mathscr{U}}+\alpha_{h,\mathscr{U}})g\left(\sum\limits_{j=1}^{k}w_{j}\Phi(\bm{A}_{j})\right),$
(105)

and, from Eq. (102) and Eq. (104), we also have

$\displaystyle\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))+\sum\limits_{j=1}^{k}w_{j}\Phi(h(\bm{A}_{j}))\geq(\alpha_{f,\mathscr{L}}+\alpha_{h,\mathscr{L}})g\left(\sum\limits_{j=1}^{k}w_{j}\Phi(\bm{A}_{j})\right).$
(106)

If any two functions $f,h$ satisfying Eq. (6.1), from Eq. (6.2) and Eq. (106),
we can bound the addition between
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$ and
$\sum\limits_{j=1}^{k}w_{j}\Phi(h(\bm{A}_{j}))$ by coefficients of
$g\left(\sum\limits_{j=1}^{k}w_{j}\Phi(\bm{A}_{j})\right)$ with the term
$g\left(\sum\limits_{j=1}^{k}w_{j}\Phi(\bm{A}_{j})\right)$, which is
independent of the functions $f$ and $h$. Therefore, the bounding coefficients
of $g\left(\sum\limits_{j=1}^{k}w_{j}\Phi(\bm{A}_{j})\right)$ form an
_algebraic system of interval numbers_ , which is an _abelian monoid_ with
respect to the opration _addition_ accoding to Theorem 2.14 [17].

Consider the multiplication between
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$ and
$\sum\limits_{j=1}^{k}w_{j}\Phi(h(\bm{A}_{j}))$, from Eq. (101) and Eq. (103)
with assumptions of positive
$\alpha_{f,\mathscr{U}},\alpha_{h,\mathscr{U}},\alpha_{f,\mathscr{L}}$ and
$\alpha_{h,\mathscr{L}}$ and
$g\left(\sum\limits_{j=1}^{k}w_{j}\Phi(\bm{A}_{j})\right)>\bm{O}$ , then, we
have

$\displaystyle\left(\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\right)\times\left(\sum\limits_{j=1}^{k}w_{j}\Phi(h(\bm{A}_{j}))\right)\leq(\alpha_{f,\mathscr{U}}\times\alpha_{h,\mathscr{U}})\left(g\left(\sum\limits_{j=1}^{k}w_{j}\Phi(\bm{A}_{j})\right)\right)^{2},$
(107)

and, from Eq. (102) and Eq. (104), we also have

$\displaystyle\left(\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\right)\times\left(\sum\limits_{j=1}^{k}w_{j}\Phi(h(\bm{A}_{j}))\right)\geq(\alpha_{f,\mathscr{L}}\times\alpha_{h,\mathscr{L}})\left(g\left(\sum\limits_{j=1}^{k}w_{j}\Phi(\bm{A}_{j})\right)\right)^{2}.$
(108)

Analogly, if any two functions $f,h$ satisfying Eq. (6.1), from Eq. (107) and
Eq. (108), we can bound the multiplication between
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$ and
$\sum\limits_{j=1}^{k}w_{j}\Phi(h(\bm{A}_{j}))$ by coefficients of
$\left(g\left(\sum\limits_{j=1}^{k}w_{j}\Phi(\bm{A}_{j})\right)\right)^{2}$
with the term
$\left(g\left(\sum\limits_{j=1}^{k}w_{j}\Phi(\bm{A}_{j})\right)\right)^{2}$,
which is independent of the functions $f$ and $h$. Therefore, the bounding
coefficients of
$\left(g\left(\sum\limits_{j=1}^{k}w_{j}\Phi(\bm{A}_{j})\right)\right)^{2}$
form an _algebraic system of interval numbers_ , which is an _abelian monoid_
with respect to the opration _multiplication_ accoding to Theorem 2.14 [17].

###### Remark 3

The norms of the addition (or multiplication) between
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$ and
$\sum\limits_{j=1}^{k}w_{j}\Phi(h(\bm{A}_{j}))$ can also be bounded by
coefficients of $g\left(\sum\limits_{j=1}^{k}w_{j}\Phi(\bm{A}_{j})\right)$ (or
$\left(g\left(\sum\limits_{j=1}^{k}w_{j}\Phi(\bm{A}_{j})\right)\right)^{2}$)
with abelian monoid algebraic structure and norms of
$g\left(\sum\limits_{j=1}^{k}w_{j}\Phi(\bm{A}_{j})\right)$ (or
$\left(g\left(\sum\limits_{j=1}^{k}w_{j}\Phi(\bm{A}_{j})\right)\right)^{2}$).

###  6.2 Random Tensor Tail Bounds Algebra

The purpose of this section is to show a method to obtain tail bound for the
addition (or multiplication) between two random tensors
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$ and
$\sum\limits_{j=1}^{k}w_{j}\Phi(h(\bm{A}_{j}))$ via Theorem 3.

Let $\bm{A}_{j}$ be random Hermitian tensor with $\Lambda(\bm{A}_{j})\in[m,M]$
for real scalars $m<M$. The mapping
$\Phi:\mathbb{B}(\mathfrak{H})\rightarrow\mathbb{B}(\mathfrak{K})$ is defined
as $\Phi(\bm{X})=\bm{X}$. The index $j$ is in the range of $1,2,\cdots,k$, and
we have a probability vector $\bm{w}=[w_{1},w_{2},\cdots,w_{k}]$ with the
dimension $k$, i.e., $\sum\limits_{j=1}^{k}w_{j}=1$. Let $f,h$ be two convex
and differentiable functions defined in $[m,M]$ such that

$\displaystyle f(x)\leq ax+b,$ $\displaystyle h(x)\leq cx+d,$ (109)

where $x\in[m,M]$. If we require $g(x)>0$ for all $x\in[m,M]$ and any positive
number $\theta$, from Theorem 3, we have

$\displaystyle\sum\limits_{j=1}^{k}w_{j}f(\bm{A}_{j})\leq\underbrace{\max\limits_{m<x<M}\frac{ax+b}{g(x)}}_{:=\alpha_{f,\mathscr{U}}}g\left(\sum\limits_{j=1}^{k}w_{j}\bm{A}_{j}\right).$
(110)

Similarly, from Theorem 3, we also have

$\displaystyle\sum\limits_{j=1}^{k}w_{j}h(\bm{A}_{j})\leq\underbrace{\min\limits_{m<x<M}\frac{cx+d}{g(x)}}_{:=\alpha_{h,\mathscr{U}}}g\left(\sum\limits_{j=1}^{k}w_{j}\bm{A}_{j}\right).$
(111)

Consider tail bounds for the addition between
$\sum\limits_{j=1}^{k}w_{j}f(\bm{A}_{j})$ and
$\sum\limits_{j=1}^{k}w_{j}h(\bm{A}_{j})$, from Eq. (110) and Eq. (111), then,
we have

$\displaystyle\mathrm{Pr}\left(\left\|\left(\sum\limits_{j=1}^{k}w_{j}f(\bm{A}_{j})\right)+\left(\sum\limits_{j=1}^{k}w_{j}h(\bm{A}_{j})\right)\right\|_{\ell}\geq\theta\right)$
$\displaystyle\leq\mathrm{Pr}\left(\left\|(\alpha_{f,\mathscr{U}}+\alpha_{h,\mathscr{U}})g\left(\sum\limits_{j=1}^{k}w_{j}\bm{A}_{j}\right)\right\|_{\ell}\geq\theta\right),$
(112)

where $\left\|\cdot\right\|_{(\ell)}$ is Ky Fan $\ell$-norm. R.H.S. of Eq.
(6.2), where the random tensors summation part is independent of functions $f$
and $h$ can be upper bounded by those theorems in Section IV in [18].

Consider the multiplication between
$\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))$ and
$\sum\limits_{j=1}^{k}w_{j}\Phi(h(\bm{A}_{j}))$, from Eq. (110) and Eq. (111)
with assumptions of positive $\alpha_{f,\mathscr{U}}$ and
$\alpha_{h,\mathscr{U}}$ and
$g\left(\sum\limits_{j=1}^{k}w_{j}\bm{A}_{j}\right)>\bm{O}$ , then, we have

$\displaystyle\mathrm{Pr}\left(\left\|\left(\sum\limits_{j=1}^{k}w_{j}\Phi(f(\bm{A}_{j}))\right)\times\left(\sum\limits_{j=1}^{k}w_{j}\Phi(h(\bm{A}_{j}))\right)\right\|_{\ell}\geq\theta\right)$
$\displaystyle\leq\mathrm{Pr}\left(\left\|(\alpha_{f,\mathscr{U}}\times\alpha_{h,\mathscr{U}})\left(g\left(\sum\limits_{j=1}^{k}w_{j}\bm{A}_{j}\right)\right)^{2}\right\|_{\ell}\geq\theta\right)$
$\displaystyle=\mathrm{Pr}\left(\left\|g\left(\sum\limits_{j=1}^{k}w_{j}\bm{A}_{j}\right)\right\|_{\ell}\geq\sqrt{\frac{\theta}{(\alpha_{f,\mathscr{U}}\times\alpha_{h,\mathscr{U}})}}\right),$
(113)

where the last term of Eq. (6.2) can be upper bounded by those theorems in
Section IV in [18].

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