# Multivariate Mond-Pecaric Method with Applications to Hypercomplex Function
Sobolev Embedding

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 introduced a method to simplify the determination of
complementary inequalities for Jensen’s inequality by converting it into a
single-variable maximization or minimization problem of continuous functions.
This principle has significantly enriched the field of operator inequalities.
Our contribution lies in extending the Mond-Pecaric method from single-input
operators to multiple-input operators. We commence by defining normalized
positive linear maps, accompanied by illustrative examples. Subsequently, we
employ the Mond-Pecaric method to derive fundamental inequalities for
multivariate hypercomplex functions bounded by linear functions. These
foundational inequalities serve as the basis for establishing several
multivariate hypercomplex function inequalities, focusing on ratio
relationships. Additionally, we present similar results based on difference
relationships. Finally, we apply the derived multivariate hypercomplex
function inequalities to establish Sobolev embedding via Sobolev inequality
for hypercomplex functions with operator inputs.

Index terms— Operator inequality, Jensen’s inequality, hypercomplex function,
Mond-Pecaric method, Sobolev embedding.

##  1 Introduction

The Kantorovich inequality states that when $\bm{A}$ is a positive operator on
the Hilbert space $\mathfrak{H}$ and satisfies the conditions
$m\bm{I}_{\mathfrak{H}}\leq\bm{A}\leq M\bm{I}_{\mathfrak{H}}$ for positive
scalars $m,M$ with $m<M$ and the identity operator $\bm{I}_{\mathfrak{H}}$ in
the Hilber space $\mathfrak{H}$, then

$\displaystyle\langle\bm{A}\bm{x},\bm{x}\rangle\langle\bm{A}^{-1}\bm{x},\bm{x}\rangle\leq\frac{(M+m)^{2}}{4Mm},$
(1)

where $\bm{x}\in\mathfrak{H}$ with $\left\|\bm{x}\right\|=1$. Several authors
have explored extensions of the Kantorovich inequality. Notably, Mond and
Pecaric have conducted an extensive research series in this area [1, 2]. They
developed a method that generates complementary inequalities to Jensen’s type
inequalities associated with convex functions. Consequently, they provided
complementary inequalities to the Holder-McCarthy inequality and extensions of
the Kantorovich type. Furuta [3] further extended the work by Ky Fan [4] and
the Mond-Pecaric generalizations of the Kantorovich inequality by
incorporating the ideas of Ky Fan, Mond, and Pecaric. Additionally, in
integral expressions, S.-E. Takahasi et al. [5] introduced another formula for
a complementary inequality to Jensen’s inequality, which includes the
Kantorovich inequality as a special case. By amalgamating the concepts of
Furuta and Takahasi, new insights into the method established by Mond and
Pecaric are revealed, allowing for the derivation of complementary
inequalities to Jensen’s inequality for convex functions.

Mond and Pecaric proposed a method to transform the complementary inequality
determination of Jensen’s inequality to a single-variable maximization or
minimization problem of continuous functions. Employing this approach, we can
address general complementary inequalities to Jensen’s inequalities for convex
functions. This formulation simplifies the concept of complementary
inequalities and enhances clarity in notions and proofs. This viewpoint proves
valuable for exploring topics such as the Hadamard product, positive linear
maps, operator means, and order-preserving operator inequalities. Referred to
as the Mond-Pecaric method, this principle yields substantial results in the
realm of operator inequalities [1, 2]. For instance, the operator Jensen’s
inequality extends the concept of convexity to operator variables, offering
insights into function convexity concerning matrices or tensors, which are
commonly encountered in various mathematical and engineering tasks [6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18].

The contribution of this work is to extend the Mond-Pecaric method from single
input operator to multiple input operators. We begin by presenting the
definition of normalized positive linear maps along with illustrative
examples. Then, the Mond-Pečarić method is utilized to derive fundamental
inequalities for multivariate hypercomplex functions bounded by linear
functions. These fundamental inequalities are used to establish several
multivariate hypercomplex function inequalities based on ratio relationships.
Besides, we also present analogous results but based on difference
relationships. Finally, we apply the derived multivariate hypercomplex
function inequalities to establish Sobolev inequality for hypercomplex
functions with operator inputs.

The definition about the normalized positive linear map and its examples are
provided by Section 2. Fundamental inequalities for multivariate hypercomplex
functions bounded by linear functions are derived by Mond-Pecaric method in
Section 3. In Section 4, several multivariate hypercomplex functions
inequalities are established based on ratio relationship. On the other hand,
in Section 5, several multivariate hypercomplex functions inequalities are
established based on difference relationships. Finally, the application of
derived multivariate hypercomplex function inequalities to established Sobolev
inequality for hypercomplex function with operator inputs is discussed in
Section 6.

Nomenclature: Inequalities $\geq,>,\leq,$ and $<$, when applied to operators,
follow the Loewner order. 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. In this work, a hypercomplex function
indicates a function with operators/matrices/tensors as input variables. We
assume that all input operators of multivariate functions discussed in this
work are self-adjoint, and the output operator is self-adjoint too. The
summation of polynomials with respect to each self-adjoint operator is a class
of multivariate functions with self-adjoint operators as their outputs with
self-adjoint input operators.

##  2 Normalized Positive Linear Maps

In this section, we will introduce the notion about _Normalized Positive
Linear Maps_. Let us set up several symbols for later usage.
$\mathscr{B}(\mathfrak{H})$ represents semi-algebra of all bounded linear
operators on a Hilbert space from $\mathfrak{H}$ to $\mathfrak{H}$.
$\bm{I}_{\mathfrak{H}}$ is the identity operator in $\mathfrak{H}$.
$\mathscr{B}_{sa}(\mathfrak{H})$ represents semi-algebra of all self-adjoint
bounded linear operators on a Hilbert space from $\mathfrak{H}$ to
$\mathfrak{H}$.

In the following definition 1, we define a normalized positive linear map [1,
2].

###### Definition 1

A map $\Phi:\mathscr{B}(\mathfrak{H})\rightarrow\mathscr{B}(\mathfrak{K})$ 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 $\bm{X},\bm{Y}\in\mathscr{B}(\mathfrak{H})$.

  * •

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.

Given a set of operators $\bm{V}_{i}\in\mathscr{B}(\mathfrak{H})$ for
$i=1,2,\ldots,n$ satisfying

$\displaystyle\sum\limits_{i=1}^{n}\bm{V}_{i}^{\ast}\bm{V}_{i}$
$\displaystyle=$ $\displaystyle\bm{I}_{\mathfrak{H}},$ (2)

where ∗ represents the self-adjoint operation. Then the map
$\Phi:\mathscr{B}(\mathfrak{H})\rightarrow\mathscr{B}(\mathfrak{H})$ defined
below is a normalized positive linear map.

$\displaystyle\Phi(\bm{X})$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{n}\bm{V}_{i}^{\ast}\bm{X}\bm{V}_{i}.$ (3)

##  3 Fundamental Inequalities for Multivariate Hypercomplex Functions

In this section, we will establish two fundamental inequalities for
multivariate hypercomplex functions.

###### Theorem 1

Let $\bm{A}_{j_{i}}$ be self-adjoint operators with
$\Lambda(\bm{A}_{j_{i}})\in[m_{i},M_{i}]$ for real scalars $m_{i}<M_{i}$. The
mappings
$\Phi_{j_{1},\ldots,j_{n}}:\mathscr{B}(\mathfrak{H})\rightarrow\mathscr{B}(\mathfrak{K})$
are normalized positive linear maps, where $j_{i}=1,2,\ldots,k_{i}$ for
$i=1,2,\ldots,n$. We have $n$ probability vectors
$\bm{w}_{i}=[w_{i,1},w_{i,2},\cdots,w_{i,k_{i}}]$ with the dimension $k_{i}$
for $i=1,2,\ldots,n$, i.e., $\sum\limits_{\ell=1}^{k_{i}}w_{i,\ell}=1$. Let
$f(x_{1},x_{2},\ldots,x_{n})$ be any real valued continuous functions with $n$
variables defined on the range
$\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]\in\mathbb{R}^{n}$, where $\times$ is
the Cartesian product. Besides, we assume that the function $f$ satisfies the
following:

$\displaystyle\sum\limits_{i=1}^{n}a_{i}x_{i}+b\leq
f(x_{1},x_{2},\ldots,x_{n})\leq\sum\limits_{i=1}^{n}c_{i}x_{i}+d,$ (4)

for $[x_{1},x_{2},\ldots,x_{n}]\in\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]$.
The function $g(x_{1},x_{2},\ldots,x_{n})$ is also a real valued continuous
function with $n$ variables defined on the range
$\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]$. 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 $f(x_{1},x_{2},\ldots,x_{n})\subset U$, and
$g(x_{1},x_{2},\ldots,x_{n})\subset V$. Then, we have the following upper
bound:

$\displaystyle
F\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})),\leavevmode\nobreak\
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\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle
g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}\Bigg{)}$
$\displaystyle\leq\max\limits_{m_{i}\leq x_{i}\leq
M_{i}}F\Big{(}\sum\limits_{i=1}^{n}c_{i}x_{i}+d,g(x_{1},x_{2},\ldots,x_{n})\Big{)}\bm{I}_{\mathfrak{K}}.$
(5)

Similarly, we also have the following lower bound:

$\displaystyle
F\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})),\leavevmode\nobreak\
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\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle
g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}\Bigg{)}$
$\displaystyle\geq\min\limits_{m_{i}\leq x_{i}\leq
M_{i}}F\Big{(}\sum\limits_{i=1}^{n}a_{i}x_{i}+b,g(x_{1},x_{2},\ldots,x_{n})\Big{)}\bm{I}_{\mathfrak{K}}.$
(6)

Proof: We will prove Eq. (1) first. Due to
$f(x_{1},x_{2},\ldots,x_{n})\leq\sum\limits_{i=1}^{n}c_{i}x_{i}+d$ for
$[x_{1},x_{2},\ldots,x_{n}]\in\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]$ from
Eq. (4), we have

$\displaystyle
f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})\leq\sum\limits_{i=1}^{n}c_{i}\bm{A}_{j_{i}}+d\bm{I}_{\mathfrak{H}},$
(7)

where $j_{i}=1,2,\ldots,k_{i}$ for $i=1,2,\ldots,n$. Since
$\Phi_{j_{1},\ldots,j_{n}}$ is a normalized positive linear map, we also have

$\displaystyle\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
$\displaystyle\leq$
$\displaystyle\Phi_{j_{1},\ldots,j_{n}}\Big{(}\sum\limits_{i=1}^{n}c_{i}\bm{A}_{j_{i}}+d\bm{I}_{\mathfrak{H}}\Big{)}$
(8) $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{n}c_{i}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})+d\bm{I}_{\mathfrak{K}},$

where $j_{i}=1,2,\ldots,k_{i}$ for $i=1,2,\ldots,n$. By multiplying
$w_{j_{1}}\ldots w_{j_{n}}$ to both sides of Eq. (8) and summing items with
respect to $w_{j_{1}}\ldots w_{j_{n}}$, we have

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(9) $\displaystyle\leq$
$\displaystyle\sum\limits_{i=1}^{n}c_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})+d\bm{I}_{\mathfrak{K}}.$

Because $m_{i}\bm{I}_{\mathfrak{H}}\leq\bm{A}_{j_{i}}\leq
M_{i}\bm{I}_{\mathfrak{H}}$ where $j_{i}=1,2,\ldots,k_{i}$ for
$i=1,2,\ldots,n$, we have

$\displaystyle\Lambda\left(\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right)\in[m_{i},M_{i}].$
(10)

By utilizing the operator monotonicity of $F(u,v)$ with respect to the
variable $u$, we have

$\displaystyle
F\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})),\leavevmode\nobreak\
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\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle
g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}\Bigg{)}$
$\displaystyle\leq_{1}F\Bigg{(}\sum\limits_{i=1}^{n}c_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})+d\bm{I}_{\mathfrak{K}},\leavevmode\nobreak\
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\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle
g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}\Bigg{)}$
$\displaystyle\leq_{2}\max\limits_{m_{i}\leq x_{i}\leq
M_{i}}F\Big{(}\sum\limits_{i=1}^{n}c_{i}x_{i}+d,g(x_{1},x_{2},\ldots,x_{n})\Big{)}\bm{I}_{\mathfrak{K}},\leavevmode\nobreak\
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\leavevmode\nobreak\ \leavevmode\nobreak\ $ (11)

where the inequality $\leq_{1}$ comes from the operator monotonicity of
$F(u,v)$ with respect to the variable $u$, and the inequality $\leq_{2}$ comes
from Eq. (10). Then, we have the desired inequality provided by Eq. (1).

Now, we will prove Eq. (1). Since
$f(x_{1},x_{2},\ldots,x_{n})\geq\sum\limits_{i=1}^{n}a_{i}x_{i}+b$ for
$[x_{1},x_{2},\ldots,x_{n}]\in\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]$ from
Eq. (4), we have

$\displaystyle
f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})\geq\sum\limits_{i=1}^{n}a_{i}\bm{A}_{j_{i}}+b\bm{I}_{\mathfrak{H}},$
(12)

where $j_{i}=1,2,\ldots,k_{i}$ for $i=1,2,\ldots,n$. Since
$\Phi_{j_{1},\ldots,j_{n}}$ is a normalized positive linear map, we also have

$\displaystyle\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
$\displaystyle\geq$
$\displaystyle\Phi_{j_{1},\ldots,j_{n}}\Big{(}\sum\limits_{i=1}^{n}a_{i}\bm{A}_{j_{i}}+b\bm{I}_{\mathfrak{H}}\Big{)}$
(13) $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{n}a_{i}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})+b\bm{I}_{\mathfrak{K}},$

where $j_{i}=1,2,\ldots,k_{i}$ for $i=1,2,\ldots,n$. By multiplying
$w_{j_{1}}\ldots w_{j_{n}}$ to both sides of Eq. (13) and summing items with
respect to $w_{j_{1}}\ldots w_{j_{n}}$, we have

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(14) $\displaystyle\geq$
$\displaystyle\sum\limits_{i=1}^{n}c_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})+d\bm{I}_{\mathfrak{K}}.$

Because $m_{i}\bm{I}_{\mathfrak{H}}\leq\bm{A}_{j_{i}}\leq
M_{i}\bm{I}_{\mathfrak{H}}$ where $j_{i}=1,2,\ldots,k_{i}$ for
$i=1,2,\ldots,n$, we have

$\displaystyle\Lambda\left(\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right)\in[m_{i},M_{i}].$
(15)

By utilizing the operator monotonicity of $F(u,v)$ with respect to the
variable $u$, we have

$\displaystyle
F\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle
g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}\Bigg{)}$
$\displaystyle\geq_{1}F\Bigg{(}\sum\limits_{i=1}^{n}c_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})+d\bm{I}_{\mathfrak{K}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle
g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}\Bigg{)}$
$\displaystyle\geq_{2}\min\limits_{m_{i}\leq x_{i}\leq
M_{i}}F\Big{(}\sum\limits_{i=1}^{n}c_{i}x_{i}+d,g(x_{1},x_{2},\ldots,x_{n})\Big{)}\bm{I}_{\mathfrak{K}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (16)

where the inequality $\geq_{1}$ comes from the operator monotonicity of
$F(u,v)$ with respect to the variable $u$, and the inequality $\geq_{2}$ comes
from Eq. (15). Then, we have the desired inequality provided by Eq. (1).
$\hfill\Box$

By setting the function $F(u,v)$ in the following format:

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

where $\alpha\in\mathbb{R}$, we establish Theorem 2. This theorem lays out
multivariate hypercomplex functions inequalities with both ratio and
difference variations, which will be elaborated upon in subsequent sections.

###### Theorem 2

Let $\bm{A}_{j_{i}}$ be self-adjoint operators with
$\Lambda(\bm{A}_{j_{i}})\in[m_{i},M_{i}]$ for real scalars $m_{i}<M_{i}$. The
mappings
$\Phi_{j_{1},\ldots,j_{n}}:\mathscr{B}(\mathfrak{H})\rightarrow\mathscr{B}(\mathfrak{K})$
are normalized positive linear maps, where $j_{i}=1,2,\ldots,k_{i}$ for
$i=1,2,\ldots,n$. We have $n$ probability vectors
$\bm{w}_{i}=[w_{i,1},w_{i,2},\cdots,w_{i,k_{i}}]$ with the dimension $k_{i}$
for $i=1,2,\ldots,n$, i.e., $\sum\limits_{\ell=1}^{k_{i}}w_{i,\ell}=1$. Let
$f(x_{1},x_{2},\ldots,x_{n})$ be any real valued continuous functions with $n$
variables defined on the range
$\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]\in\mathbb{R}^{n}$, where $\times$ is
the Cartesian product. Besides, we assume that the function $f$ satisfies the
following:

$\displaystyle\sum\limits_{i=1}^{n}a_{i}x_{i}+b\leq
f(x_{1},x_{2},\ldots,x_{n})\leq\sum\limits_{i=1}^{n}c_{i}x_{i}+d,$ (18)

for $[x_{1},x_{2},\ldots,x_{n}]\in\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]$.
The function $g(x_{1},x_{2},\ldots,x_{n})$ is also a real valued continuous
function with $n$ variables defined on the range
$\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]$. We also have a real valued function
$F(u,v)$ defined by Eq. (17) with operator monotone on the first variable $u$
defined on $U\times V$ such that
$f(\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}])\subset U$, and
$g(\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}])\subset V$. Then, we have the
following upper bound:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(19) $\displaystyle\leq$ $\displaystyle\alpha
g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}$
$\displaystyle+\max\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\Big{(}\sum\limits_{i=1}^{n}c_{i}x_{i}+d-\alpha
g(x_{1},x_{2},\ldots,x_{n})\Big{)}\bm{I}_{\mathfrak{K}}.$

Similarly, we also have the following lower bound:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(20) $\displaystyle\geq$ $\displaystyle\alpha
g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}$
$\displaystyle+\min\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\Big{(}\sum\limits_{i=1}^{n}a_{i}x_{i}+b-\alpha
g(x_{1},x_{2},\ldots,x_{n})\Big{)}\bm{I}_{\mathfrak{K}}.$

Proof: By setting $F(u,v)=u-\alpha v$ in Eq. (1) in Theorem 1, we have the
desired inequality provided by Eq. (19). Similarly, By setting
$F(u,v)=u-\alpha v$ in Eq. (1) in Theorem 1, we have the desired inequality
provided by Eq. (20). $\hfill\Box$

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 $\bm{A}_{j_{i}}$ be self-adjoint operators with
$\Lambda(\bm{A}_{j_{i}})\in[m_{i},M_{i}]$ for real scalars $m_{i}<M_{i}$. The
mappings
$\Phi_{j_{1},\ldots,j_{n}}:\mathscr{B}(\mathfrak{H})\rightarrow\mathscr{B}(\mathfrak{K})$
are normalized positive linear maps, where $j_{i}=1,2,\ldots,k_{i}$ for
$i=1,2,\ldots,n$. We have $n$ probability vectors
$\bm{w}_{i}=[w_{i,1},w_{i,2},\cdots,w_{i,k_{i}}]$ with the dimension $k_{i}$
for $i=1,2,\ldots,n$, i.e., $\sum\limits_{\ell=1}^{k_{i}}w_{i,\ell}=1$. Let
$f(x_{1},x_{2},\ldots,x_{n})$ be any real valued continuous functions with $n$
variables defined on the range
$\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]\in\mathbb{R}^{n}$, where $\times$ is
the Cartesian product. Besides, we assume that the function $f$ satisfies the
following:

$\displaystyle\sum\limits_{i=1}^{n}a_{i}x_{i}+b\leq
f(x_{1},x_{2},\ldots,x_{n})\leq\sum\limits_{i=1}^{n}c_{i}x_{i}+d,$ (21)

for $[x_{1},x_{2},\ldots,x_{n}]\in\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]$.

(I) If
$g(x_{1},\ldots,x_{n})=\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}^{q}$,
where $\beta_{i}\geq 0$, $q\in\mathbb{R}$ and $m_{i}\geq 0$, we have the upper
bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(22) $\displaystyle\leq$
$\displaystyle\alpha\left(\sum\limits_{i=1}^{n}\beta_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right)^{q}$
$\displaystyle+\max\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\Big{[}\sum\limits_{i=1}^{n}c_{i}x_{i}+d-\alpha\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}^{q}\Big{]}\bm{I}_{\mathfrak{K}},$

and we have the lower bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(23) $\displaystyle\geq$
$\displaystyle\alpha\left(\sum\limits_{i=1}^{n}\beta_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right)^{q}$
$\displaystyle+\min\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\Big{[}\sum\limits_{i=1}^{n}a_{i}x_{i}+b-\alpha\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}^{q}\Big{]}\bm{I}_{\mathfrak{K}}.$

(II) If
$g(x_{1},\ldots,x_{n})=\log\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}$,
where $\beta_{i}\geq 0$, and $m_{i}>0$, we have the upper bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(24) $\displaystyle\leq$
$\displaystyle\alpha\log\left(\sum\limits_{i=1}^{n}\beta_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right)$
$\displaystyle+\max\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\Big{[}\sum\limits_{i=1}^{n}c_{i}x_{i}+d-\alpha\log\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}\Big{]}\bm{I}_{\mathfrak{K}},$

and we have the lower bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(25) $\displaystyle\geq$
$\displaystyle\alpha\log\left(\sum\limits_{i=1}^{n}\beta_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right)$
$\displaystyle+\min\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\Big{[}\sum\limits_{i=1}^{n}a_{i}x_{i}+b-\alpha\log\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}\Big{]}\bm{I}_{\mathfrak{K}}.$

(III) If
$g(x_{1},\ldots,x_{n})=\exp\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}$,
we have the upper bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(26) $\displaystyle\leq$
$\displaystyle\alpha\exp\left(\sum\limits_{i=1}^{n}\beta_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right)$
$\displaystyle+\max\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\Big{[}\sum\limits_{i=1}^{n}c_{i}x_{i}+d-\alpha\exp\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}\Big{]}\bm{I}_{\mathfrak{K}},$

and we have the lower bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(27) $\displaystyle\geq$
$\displaystyle\alpha\exp\left(\sum\limits_{i=1}^{n}\beta_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right)$
$\displaystyle+\min\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\Big{[}\sum\limits_{i=1}^{n}a_{i}x_{i}+b-\alpha\exp\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}\Big{]}\bm{I}_{\mathfrak{K}}.$

Proof: This corollary is obtained by applying Theprem 2 directly to functions
$g$ as
$g(x_{1},\ldots,x_{n})=\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}^{q}$,
$g(x_{1},\ldots,x_{n})=\log\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}$
and
$g(x_{1},\ldots,x_{n})=\exp\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}$,
respectivey. $\hfill\Box$

##  4 Multivariate Hypercomplex Functions Inequalities: Ratio Kind

This section will explore the determination of lower and upper bounds for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
based on ratio criteria associated with the function
$g(x_{1},x_{2},\ldots,x_{n})$.

###### Theorem 3

Let $\bm{A}_{j_{i}}$ be self-adjoint operators with
$\Lambda(\bm{A}_{j_{i}})\in[m_{i},M_{i}]$ for real scalars $m_{i}<M_{i}$. The
mappings
$\Phi_{j_{1},\ldots,j_{n}}:\mathscr{B}(\mathfrak{H})\rightarrow\mathscr{B}(\mathfrak{K})$
are normalized positive linear maps, where $j_{i}=1,2,\ldots,k_{i}$ for
$i=1,2,\ldots,n$. We have $n$ probability vectors
$\bm{w}_{i}=[w_{i,1},w_{i,2},\cdots,w_{i,k_{i}}]$ with the dimension $k_{i}$
for $i=1,2,\ldots,n$, i.e., $\sum\limits_{\ell=1}^{k_{i}}w_{i,\ell}=1$. Let
$f(x_{1},x_{2},\ldots,x_{n})$ be any real valued continuous functions with $n$
variables defined on the range
$\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]\in\mathbb{R}^{n}$, where $\times$ is
the Cartesian product. Besides, we assume that the function $f$ satisfies the
following:

$\displaystyle\sum\limits_{i=1}^{n}a_{i}x_{i}+b\leq
f(x_{1},x_{2},\ldots,x_{n})\leq\sum\limits_{i=1}^{n}c_{i}x_{i}+d,$ (28)

for $[x_{1},x_{2},\ldots,x_{n}]\in\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]$.

(I) If we also assume that $g(x_{1},\ldots,x_{n})>0$ for
$[x_{1},\ldots,x_{n}]\in\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]$, then, we
have the following upper bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(29) $\displaystyle\leq$
$\displaystyle\left[\max\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq
M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\frac{\sum\limits_{i=1}^{n}c_{i}x_{i}+d}{g(x_{1},\ldots,x_{n})}\right]$
$\displaystyle\times
g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)};$

and, the following lower bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(30) $\displaystyle\geq$
$\displaystyle\left[\min\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq
M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\frac{\sum\limits_{i=1}^{n}a_{i}x_{i}+b}{g(x_{1},\ldots,x_{n})}\right]$
$\displaystyle\times
g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}.$

(II) If we also assume that $g(x_{1},\ldots,x_{n})<0$ for
$[x_{1},\ldots,x_{n}]\in\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]$, then, we
have the following upper bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(31) $\displaystyle\leq$
$\displaystyle\left[\min\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq
M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\frac{\sum\limits_{i=1}^{n}c_{i}x_{i}+d}{g(x_{1},\ldots,x_{n})}\right]$
$\displaystyle\times
g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)};$

and, the following lower bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(32) $\displaystyle\geq$
$\displaystyle\left[\max\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq
M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\frac{\sum\limits_{i=1}^{n}a_{i}x_{i}+b}{g(x_{1},\ldots,x_{n})}\right]$
$\displaystyle\times
g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}.$

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

$\displaystyle F(u,v)$ $\displaystyle=$ $\displaystyle v^{-1/2}uv^{-1/2},$
(33)

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

$\displaystyle
g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}^{-1/2}$ (34)
$\displaystyle\times\left(\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))\right)$
$\displaystyle\times
g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}^{-1/2}$
$\displaystyle\leq$
$\displaystyle\left[\max\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq
M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\frac{\sum\limits_{i=1}^{n}c_{i}x_{i}+d}{g(x_{1},\ldots,x_{n})}\right]\bm{1}_{\mathfrak{K}}.$

By multiplying
$g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}^{1/2}$ at both
sides of Eq. (34), we obtain the desired inequality provided by Eq. (29). By
applying $F(u,v)$ with Eq. (33) again to Eq. (1) in Theorem 1, then, we will
obtain

$\displaystyle
g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}^{-1/2}$ (35)
$\displaystyle\times\left(\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))\right)$
$\displaystyle\times
g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}^{-1/2}$
$\displaystyle\geq$
$\displaystyle\left[\min\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq
M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\frac{\sum\limits_{i=1}^{n}a_{i}x_{i}+b}{g(x_{1},\ldots,x_{n})}\right]\bm{1}_{\mathfrak{K}}.$

By multiplying
$g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}^{1/2}$ at both
sides of Eq. (35), we obtain the desired inequality provided by Eq. (30).

The proof of Part (II) is immediate obtained by setting
$g(x_{1},\ldots,x_{n})$ as $-g(x_{1},\ldots,x_{n})$ in Eq. (34) and Eq. (35).
$\hfill\Box$

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

###### Corollary 2

Let $\bm{A}_{j_{i}}$ be self-adjoint operators with
$\Lambda(\bm{A}_{j_{i}})\in[m_{i},M_{i}]$ for real scalars $m_{i}<M_{i}$. The
mappings
$\Phi_{j_{1},\ldots,j_{n}}:\mathscr{B}(\mathfrak{H})\rightarrow\mathscr{B}(\mathfrak{K})$
are normalized positive linear maps, where $j_{i}=1,2,\ldots,k_{i}$ for
$i=1,2,\ldots,n$. We have $n$ probability vectors
$\bm{w}_{i}=[w_{i,1},w_{i,2},\cdots,w_{i,k_{i}}]$ with the dimension $k_{i}$
for $i=1,2,\ldots,n$, i.e., $\sum\limits_{\ell=1}^{k_{i}}w_{i,\ell}=1$. Let
$f(x_{1},x_{2},\ldots,x_{n})$ be any real valued continuous functions with $n$
variables defined on the range
$\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]\in\mathbb{R}^{n}$, where $\times$ is
the Cartesian product. Besides, we assume that the function $f$ satisfies the
following:

$\displaystyle\sum\limits_{i=1}^{n}a_{i}x_{i}+b\leq
f(x_{1},x_{2},\ldots,x_{n})\leq\sum\limits_{i=1}^{n}c_{i}x_{i}+d,$ (36)

for $[x_{1},x_{2},\ldots,x_{n}]\in\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]$.

(I) If we also assume that
$g(x_{1},\ldots,x_{n})=\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}^{q}$,
where $\beta_{i}\geq 0$, $q\in\mathbb{R}$ and $m_{i}\geq 0$, then, we have the
following upper bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(37) $\displaystyle\leq$
$\displaystyle\left[\max\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq
M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\frac{\sum\limits_{i=1}^{n}c_{i}x_{i}+d}{\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}^{q}}\right]\left(\sum\limits_{i=1}^{n}\beta_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right)^{q};$

and, the following lower bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(38) $\displaystyle\geq$
$\displaystyle\left[\min\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq
M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\frac{\sum\limits_{i=1}^{n}a_{i}x_{i}+b}{\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}^{q}}\right]\left(\sum\limits_{i=1}^{n}\beta_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right)^{q}.$

(II) If
$g(x_{1},\ldots,x_{n})=\log\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}$
with $\log\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}>0$ for
$[x_{1},\ldots,x_{n}]\in\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]$, we have the
upper bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(39) $\displaystyle\leq$
$\displaystyle\left[\max\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq
M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\frac{\sum\limits_{i=1}^{n}c_{i}x_{i}+d}{\log\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}}\right]\log\left(\sum\limits_{i=1}^{n}\beta_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right);$

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

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(40) $\displaystyle\geq$
$\displaystyle\left[\min\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq
M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\frac{\sum\limits_{i=1}^{n}a_{i}x_{i}+b}{\log\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}}\right]\log\left(\sum\limits_{i=1}^{n}\beta_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right).$

(II’) If
$g(x_{1},\ldots,x_{n})=\log\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}$
with $\log\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}<0$ for
$[x_{1},\ldots,x_{n}]\in\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]$, we have the
upper bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(41) $\displaystyle\leq$
$\displaystyle\left[\min\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq
M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\frac{\sum\limits_{i=1}^{n}c_{i}x_{i}+d}{\log\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}}\right]\log\left(\sum\limits_{i=1}^{n}\beta_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right);$

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

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(42) $\displaystyle\geq$
$\displaystyle\left[\max\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq
M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\frac{\sum\limits_{i=1}^{n}a_{i}x_{i}+b}{\log\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}}\right]\log\left(\sum\limits_{i=1}^{n}\beta_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right).$

(III) If
$g(x_{1},\ldots,x_{n})=\exp\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}$,
we have the upper bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(43) $\displaystyle\leq$
$\displaystyle\left[\max\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq
M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\frac{\sum\limits_{i=1}^{n}c_{i}x_{i}+d}{\exp\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}}\right]\exp\left(\sum\limits_{i=1}^{n}\beta_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right);$

and, the following lower bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(44) $\displaystyle\geq$
$\displaystyle\left[\min\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq
M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\frac{\sum\limits_{i=1}^{n}a_{i}x_{i}+b}{\exp\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}}\right]\exp\left(\sum\limits_{i=1}^{n}\beta_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right).$

Proof: Part (I) of this Corollary is proved by applying Theorem 3 Part (I)
with the function $g$ as
$g(x_{1},\ldots,x_{n})=\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}^{q}$.
Part (II) of this Corollary is proved by applying Theorem 3 Part (I) with the
function $g$ as
$g(x_{1},\ldots,x_{n})=\log\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}$,
where $\log\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}>0$. Part (II’) of
this Corollary is proved by applying Theorem 3 Part (II) with the function $g$
as
$g(x_{1},\ldots,x_{n})=\log\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}$,
where $\log\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}<0$. Finally, Part
(III) of this Corollary is proved by applying Theorem 3 Part (I) with the
function $g$ as
$g(x_{1},\ldots,x_{n})=\exp\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}$.
$\hfill\Box$

##  5 Multivariate Hypercomplex Functions Inequalities: Difference Kind

This section will focus on deriving the lower and upper bounds for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
using difference criteria associated with the function $g$.

###### Theorem 4

Let $\bm{A}_{j_{i}}$ be self-adjoint operators with
$\Lambda(\bm{A}_{j_{i}})\in[m_{i},M_{i}]$ for real scalars $m_{i}<M_{i}$. The
mappings
$\Phi_{j_{1},\ldots,j_{n}}:\mathscr{B}(\mathfrak{H})\rightarrow\mathscr{B}(\mathfrak{K})$
are normalized positive linear maps, where $j_{i}=1,2,\ldots,k_{i}$ for
$i=1,2,\ldots,n$. We have $n$ probability vectors
$\bm{w}_{i}=[w_{i,1},w_{i,2},\cdots,w_{i,k_{i}}]$ with the dimension $k_{i}$
for $i=1,2,\ldots,n$, i.e., $\sum\limits_{\ell=1}^{k_{i}}w_{i,\ell}=1$. Let
$f(x_{1},x_{2},\ldots,x_{n})$ be any real valued continuous functions with $n$
variables defined on the range
$\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]\in\mathbb{R}^{n}$, where $\times$ is
the Cartesian product. Besides, we assume that the function $f$ satisfies the
following:

$\displaystyle\sum\limits_{i=1}^{n}a_{i}x_{i}+b\leq
f(x_{1},x_{2},\ldots,x_{n})\leq\sum\limits_{i=1}^{n}c_{i}x_{i}+d,$ (45)

for $[x_{1},x_{2},\ldots,x_{n}]\in\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]$.

Then, we have the following upper bound:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(46)
$\displaystyle-g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}$
$\displaystyle\leq$ $\displaystyle\max\limits_{\begin{subarray}{c}m_{i}\leq
x_{i}\leq M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\Big{(}\sum\limits_{i=1}^{n}c_{i}x_{i}+d-g(x_{1},x_{2},\ldots,x_{n})\Big{)}\bm{I}_{\mathfrak{K}}.$

Similarly, we also have the following lower bound:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(47)
$\displaystyle-g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}$
$\displaystyle\geq$ $\displaystyle\min\limits_{\begin{subarray}{c}m_{i}\leq
x_{i}\leq M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\Big{(}\sum\limits_{i=1}^{n}a_{i}x_{i}+b-g(x_{1},x_{2},\ldots,x_{n})\Big{)}\bm{I}_{\mathfrak{K}}.$

Proof: The upper bound of this theorem is proved by setting $\alpha=1$ in Eq.
(19) from Theorem 2 and rearrangement of the term
$g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}$ to obtain Eq.
(46).

Similarly, the lower bound of this theorem is proved by setting $\alpha=1$ in
Eq. (20) from Theorem 2 and rearrangement of the term
$g\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}$ to obtain Eq.
(47). $\hfill\Box$

Next Corollary 3 is obtained by applying Theorem 4 to special types of
function $g(x_{1},x_{2},\ldots,x_{n})$.

###### Corollary 3

Let $\bm{A}_{j_{i}}$ be self-adjoint operators with
$\Lambda(\bm{A}_{j_{i}})\in[m_{i},M_{i}]$ for real scalars $m_{i}<M_{i}$. The
mappings
$\Phi_{j_{1},\ldots,j_{n}}:\mathscr{B}(\mathfrak{H})\rightarrow\mathscr{B}(\mathfrak{K})$
are normalized positive linear maps, where $j_{i}=1,2,\ldots,k_{i}$ for
$i=1,2,\ldots,n$. We have $n$ probability vectors
$\bm{w}_{i}=[w_{i,1},w_{i,2},\cdots,w_{i,k_{i}}]$ with the dimension $k_{i}$
for $i=1,2,\ldots,n$, i.e., $\sum\limits_{\ell=1}^{k_{i}}w_{i,\ell}=1$. Let
$f(x_{1},x_{2},\ldots,x_{n})$ be any real valued continuous functions with $n$
variables defined on the range
$\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]\in\mathbb{R}^{n}$, where $\times$ is
the Cartesian product. Besides, we assume that the function $f$ satisfies the
following:

$\displaystyle\sum\limits_{i=1}^{n}a_{i}x_{i}+b\leq
f(x_{1},x_{2},\ldots,x_{n})\leq\sum\limits_{i=1}^{n}c_{i}x_{i}+d,$ (48)

for $[x_{1},x_{2},\ldots,x_{n}]\in\bigtimes\limits_{i=1}^{n}[m_{i},M_{i}]$.

(I) If we also assume that
$g(x_{1},\ldots,x_{n})=\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}^{q}$,
where $\beta_{i}\geq 0$, $q\in\mathbb{R}$ and $m_{i}\geq 0$, then, we have the
following upper bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(49)
$\displaystyle-\left(\sum\limits_{i=1}^{n}\beta_{i}x_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right)^{q}$
$\displaystyle\leq$ $\displaystyle\max\limits_{\begin{subarray}{c}m_{i}\leq
x_{i}\leq M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\Big{(}\sum\limits_{i=1}^{n}c_{i}x_{i}+d-\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}^{q}\Big{)}\bm{I}_{\mathfrak{K}};$

and, the following lower bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(50)
$\displaystyle-\left(\sum\limits_{i=1}^{n}\beta_{i}x_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right)^{q}$
$\displaystyle\geq$ $\displaystyle\min\limits_{\begin{subarray}{c}m_{i}\leq
x_{i}\leq M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\Big{(}\sum\limits_{i=1}^{n}a_{i}x_{i}+b-\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}^{q}\Big{)}\bm{I}_{\mathfrak{K}}.$

(II) If
$g(x_{1},\ldots,x_{n})=\log\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}$
with $\beta_{i}\geq 0$ and $m_{i}>0$, we have the upper bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(51)
$\displaystyle-\log\left(\sum\limits_{i=1}^{n}\beta_{i}x_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right)$ $\displaystyle\leq$
$\displaystyle\max\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\Big{(}\sum\limits_{i=1}^{n}c_{i}x_{i}+d-\log\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}\Big{)}\bm{I}_{\mathfrak{K}};$

and, the following lower bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(52)
$\displaystyle-\log\left(\sum\limits_{i=1}^{n}\beta_{i}x_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right)$ $\displaystyle\geq$
$\displaystyle\min\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\Big{(}\sum\limits_{i=1}^{n}a_{i}x_{i}+b-\log\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}\Big{)}\bm{I}_{\mathfrak{K}}.$

(III) If
$g(x_{1},\ldots,x_{n})=\exp\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}$,
we have the upper bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(53)
$\displaystyle-\exp\left(\sum\limits_{i=1}^{n}\beta_{i}x_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right)$ $\displaystyle\leq$
$\displaystyle\max\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\Big{(}\sum\limits_{i=1}^{n}c_{i}x_{i}+d-\exp\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}\Big{)}\bm{I}_{\mathfrak{K}};$

and, the following lower bound for
$\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$:

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}}))$
(54)
$\displaystyle-\exp\left(\sum\limits_{i=1}^{n}\beta_{i}x_{i}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{i}})\right)$ $\displaystyle\geq$
$\displaystyle\min\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\Big{(}\sum\limits_{i=1}^{n}a_{i}x_{i}+b-\exp\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}\Big{)}\bm{I}_{\mathfrak{K}}.$

Proof: For Part (I), we will use
$g(x_{1},\ldots,x_{n})=\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}^{q}$
in Eq. (46) in Theorem 4 to obtain Eq. (49). We will also use $g(x)=x^{q}$ in
Eq. (47) in Theorem 4 to obtain Eq. (50).

For Part (II), we will use
$g(x_{1},\ldots,x_{n})=\log\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}$
in Eq. (46) in Theorem 4 to obtain Eq. (51). We will also use $g(x)=\log(x)$
in Eq. (47) in Theorem 4 to obtain Eq. (52).

For Part (III), we will use
$g(x_{1},\ldots,x_{n})=\exp\Big{(}\sum\limits_{i=1}^{n}\beta_{i}x_{i}\Big{)}$
in Eq. (46) in Theorem 4 to obtain Eq. (53). We will also use $g(x)=\exp(x)$
in Eq. (47) in Theorem 4 to obtain Eq. (54). $\hfill\Box$

##  6 Applications

In this section, we will apply derived multivariate hypercomplex function
inequalities to established Sobolev inequality for hypercomplex function.
These inequalities will be used to characterize embedding relationships for
multivariate hypercomplex functions. In Section 6.1, Sobolev inequality for
hypercomplex function is derived by the original input operators. On the other
hand, in Section 6.2, Sobolev inequality for hypercomplex function is derived
based on the arithmetic mean of input operators.

###  6.1 Hypercomplex Function Sobolev Embedding By Original Arguments

Before presenting the main result of this subsection, we have to introduce
definitions about $|f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|$
and $|f^{\prime}(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|$.

We define $|f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|$ by

$\displaystyle|f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\sqrt{f^{\ast}(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})}.$
(55)

Note that $|f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|$ is a
self-adjoint operator.

We define $|f^{\prime}(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|$
by

$\displaystyle|f^{\prime}(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\sqrt{\sum\limits_{i=1}^{n}[f^{(i)}(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})]^{2}}.$
(56)

where $f^{(i)}(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})$ is
expressed by

$\displaystyle
f^{(i)}(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})=\frac{\partial
f(x_{1},\ldots,x_{n})}{\partial
x_{i}}\Bigg{|}_{x_{1}=\bm{A}_{j_{1}},\ldots,x_{n}=\bm{A}_{j_{n}}}.$ (57)

Note that $f^{(i)}(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})$ is a
self-adjoint operator by our assumption.

We also need Lemma 1 and Lemma 2 below in order to present our main theorem in
this seciton.

###### Lemma 1

Given any integer $m>1$, $p$ be any real number within $1<p<m$, and $q$ be
another real number defined by $q=\frac{mp}{m-p}$. Besides, we are also
provided by $n$ self-adjoint operators
$\bm{X}_{1},\bm{X}_{2},\ldots,\bm{X}_{n}$ with the spectrum region
$\bigtimes_{i=1}^{n}\Lambda(\bm{X}_{i})\in\mathbb{R}^{n}$. If
$\Lambda(\bm{X}_{i})$ is a compact region for $i=1,2,\ldots,n$, and the
continuous multivaraite function with $h(x_{1},x_{2},\ldots,x_{n})>0$ for
$[x_{1},x_{2},\ldots,x_{n}]\in\bigtimes_{i=1}^{n}\Lambda(\bm{X}_{i})$, we can
find a positive contant $C_{2}$ such that

$\displaystyle C_{2}h(\bm{X}_{1},\bm{X}_{2},\ldots,\bm{X}_{n})\leq
h^{p/q}(\bm{X}_{1},\bm{X}_{2},\ldots,\bm{X}_{n}).$ (58)

Proof: Since $h(x_{1},x_{2},\ldots,x_{n})>0$ for
$[x_{1},x_{2},\ldots,x_{n}]\in\bigtimes_{i=1}^{n}\Lambda(\bm{X}_{i})$, we can
find a positive $C_{2}$ such that

$\displaystyle C_{2}h(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\leq
h^{p/q}(\lambda_{1},\lambda_{2},\ldots,\lambda_{n}),$ (59)

where $\lambda_{i}\in\Lambda(\bm{X}_{i})$, by selection any positive $C_{2}$
less then the minimum value of
$h^{\frac{p-q}{q}}(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})$. Such minimum
value exists as a continuous function on a compact set has an absolute
minimum. This Lemma follows from spectrum mapping theorem. $\hfill\Box$

Note that the positive contant $C_{2}$ depends only on $p$ and $q$ for any
multivariate function $h$ bounded from below.

###### Lemma 2

Given any integer $m>1$, $p$ be any real number within $1<p<m$, and $q$ be
another real number defined by $q=\frac{mp}{m-p}$. Besides, we are also
provided by $n$ self-adjoint operators
$\bm{X}_{1},\bm{X}_{2},\ldots,\bm{X}_{n}$ with the spectrum region
$\bigtimes_{i=1}^{n}\Lambda(\bm{X}_{i})\in\mathbb{R}^{n}$. If
$\Lambda(\bm{X}_{i})$ is a compact region for $i=1,2,\ldots,n$, and the
multivaraite function with $|f^{\prime}(x_{1},x_{2},\ldots,x_{n})|$ bounded
below from zero for
$[x_{1},x_{2},\ldots,x_{n}]\in\bigtimes_{i=1}^{n}\Lambda(\bm{X}_{i})$, we can
find a positive contant $C_{3}$ such that

$\displaystyle|f(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})|^{p}\leq
C_{3}|f^{\prime}(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})|^{q},$ (60)

where $\lambda_{i}\in\Lambda(\bm{X}_{i})$.

Proof: Because $|f(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})|$ is a
continuous function over a compact region, the maximum value of
$|f(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})|^{p}$ is bounded. On the other
hand, the multivariate function with $|f^{\prime}(x_{1},x_{2},\ldots,x_{n})|$
bounded below from zero for
$[x_{1},x_{2},\ldots,x_{n}]\in\bigtimes_{i=1}^{n}\Lambda(\bm{X}_{i})$, we also
can fine the minimum value of
$|f^{\prime}(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})|^{q}$, which is a
positive number. Therefore, the positive constant $C_{3}$ can be determined by
finding any number greater or equal than the ratio between the maximum value
of $|f(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})|^{p}$ and the minimum value
of $|f^{\prime}(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})|^{q}$.
$\hfill\Box$

Note that the positive contant $C_{3}$ depends only on $p$ and $q$ for any
multivariate function $f$ satisfying the conditions provided by Lemma 2.

######  Theorem 5 (Sobolev Inequality for Hypercomplex Function)

Given any integer $m>1$, $p$ be any real number within $1<p<m$, and $q$ be
another real number defined by $q=\frac{mp}{m-p}$. Let $\bm{A}_{j_{i}}$ be
self-adjoint operators with $\Lambda(\bm{A}_{j_{i}})\in[m_{i},M_{i}]$ for real
scalars $m_{i}<M_{i}$. The mappings
$\Phi_{j_{1},\ldots,j_{n}}:\mathscr{B}(\mathfrak{H})\rightarrow\mathscr{B}(\mathfrak{K})$
are normalized positive linear maps, where $j_{i}=1,2,\ldots,k_{i}$ for
$i=1,2,\ldots,n$. We have $n$ probability vectors
$\bm{w}_{i}=[w_{i,1},w_{i,2},\cdots,w_{i,k_{i}}]$ with the dimension $k_{i}$
for $i=1,2,\ldots,n$, i.e., $\sum\limits_{\ell=1}^{k_{i}}w_{i,\ell}=1$. If the
multivaraite function with $|f^{\prime}(x_{1},x_{2},\ldots,x_{n})|$ bounded
below from zero for
$[x_{1},x_{2},\ldots,x_{n}]\in\bigtimes_{i=1}^{n}[m_{i},M_{i}]$. We have the
following inequality:

$\displaystyle\Big{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(|f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|^{p})\Big{)}^{1/p}$
(61) $\displaystyle\leq$ $\displaystyle
C_{1}\Big{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(|f^{\prime}(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|^{q})\Big{)}^{1/q},$

where $C_{1}$ is a constant indepedent of the function
$f(x_{1},x_{2},\ldots,x_{n})$.

Proof: From Lemma 2 and spectrum mapping theorem, we have

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(|f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|^{p})$
(62) $\displaystyle\leq$ $\displaystyle
C_{3}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(|f^{\prime}(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|^{q}).$

By rasing the power $1/q$ at both sides of Eq. (62), which is less than $1$,
we have

$\displaystyle\Big{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(|f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|^{p})\Big{)}^{1/q}$
(63) $\displaystyle\leq$ $\displaystyle
C^{\prime}_{3}\Big{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(|f^{\prime}(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|^{q})\Big{)}^{1/q},$

where the inequality $\leq$ comes from Lowner-Heinz inequality [1, 2].

The L.H.S. of Eq. (63) can be expressed by

$\displaystyle\Big{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(|f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|^{p})\Big{)}^{1/q}$
(64) $\displaystyle=$
$\displaystyle\Big{(}\Big{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(|f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|^{p})\Big{)}^{1/p}\Big{)}^{p/q}$

From Lemma 1. we have

$\displaystyle
C_{2}\Big{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(|f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|^{p})\Big{)}^{1/p}$
(65) $\displaystyle\leq$
$\displaystyle\Big{(}\Big{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(|f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|^{p})\Big{)}^{1/p}\Big{)}^{p/q}.$

Finally, this Theorem is provoed by combing Eq. (63), Eq. (64), Eq. (65), and
setting $C_{1}=C^{\prime}_{3}/C_{2}$. $\hfill\Box$

We define
$\mathscr{W}_{\Phi}^{1,q}\left(\bigtimes_{i=1}^{n}[m_{i},M_{i}]\right)$ as the
hypercomplex function space such that

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi(|f^{\prime}(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|^{q})\Big{)}^{1/q}$
(66)

is a bounded and positive self-adjoint operator for spectrum in the range of
$\bigtimes_{i=1}^{n}[m_{i},M_{i}]$. In addition, we define
$\mathscr{L}^{p}_{\Phi}\left(\bigtimes_{i=1}^{n}[m_{i},M_{i}]\right)$ as the
hypercomplex function space such that

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi(|f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|^{p})\Big{)}^{1/p}$
(67)

is a bounded and positive self-adjoint operator for spectrum in the range of
$\bigtimes_{i=1}^{n}[m_{i},M_{i}]$. From Theorem 5, we have hypercomplex
function Sobolev embedding as

$\displaystyle\mathscr{W}_{\Phi}^{1,q}\left(\bigtimes_{i=1}^{n}[m_{i},M_{i}]\right)\subset\mathscr{L}^{p}_{\Phi}\left(\bigtimes_{i=1}^{n}[m_{i},M_{i}]\right).$
(68)

###  6.2 Hypercomplex Function Sobolev Embedding By Arguments Mean

######  Theorem 6 (Sobolev Inequality for Hypercomplex Function with Operator
Mean)

Given any integer
$m>1$, $p$ be any real number within $1<p<m$, and $q$ be another real number
defined by $q=\frac{mp}{m-p}$. Let $\bm{A}_{j_{i}}$ be self-adjoint operators
with $\Lambda(\bm{A}_{j_{i}})\in[m_{i},M_{i}]$ for real scalars $m_{i}<M_{i}$.
The mappings
$\Phi_{j_{1},\ldots,j_{n}}:\mathscr{B}(\mathfrak{H})\rightarrow\mathscr{B}(\mathfrak{K})$
are normalized positive linear maps, where $j_{i}=1,2,\ldots,k_{i}$ for
$i=1,2,\ldots,n$. We have $n$ probability vectors
$\bm{w}_{i}=[w_{i,1},w_{i,2},\cdots,w_{i,k_{i}}]$ with the dimension $k_{i}$
for $i=1,2,\ldots,n$, i.e., $\sum\limits_{\ell=1}^{k_{i}}w_{i,\ell}=1$.

Besides, we assume that the function $f$ satisfies the following:

$\displaystyle|f(x_{1},x_{2},\ldots,x_{n})|^{p}\leq\sum\limits_{i=1}^{n}c_{i}x_{i}+d,$
(69)

where $[x_{1},x_{2},\ldots,x_{n}]\in\bigtimes_{i=1}^{n}[m_{i},M_{i}]$. In
addition, we also assume that the function $|f^{\prime}|^{q}$ satisfies the
following:

$\displaystyle|f^{\prime}(x_{1},x_{2},\ldots,x_{n})|^{q}>0,$ (70)

where $[x_{1},x_{2},\ldots,x_{n}]\in\bigtimes_{i=1}^{n}[m_{i},M_{i}]$. We have
the following inequality:

$\displaystyle\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(|f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|^{p})\Bigg{)}^{1/p}\leq$
$\displaystyle
C_{4}\Bigg{(}\Bigg{|}f^{\prime}\Big{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}\Bigg{|}^{q}\Bigg{)}^{1/q},$

where $C_{4}$ is a positive scalar related to $f$, $f^{\prime}$, $c_{i}$, $d$,
and the domain $\bigtimes_{i=1}^{n}[m_{i},M_{i}]$.

Proof: By setting $f=|f(x_{1},x_{2},\ldots,x_{n})|^{p}$ and
$g=|f^{\prime}(x_{1},x_{2},\ldots,x_{n})|^{q}$ in Part (I) of Theorem 3, we
have

$\displaystyle\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(|f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|^{p})$
(72) $\displaystyle\leq$
$\displaystyle\left[\max\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq
M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\frac{\sum\limits_{i=1}^{n}c_{i}x_{i}+d}{|f^{\prime}(x_{1},x_{2},\ldots,x_{n})|^{q}}\right]$
$\displaystyle\times\Bigg{|}f^{\prime}\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}\Bigg{|}^{q}.$

By rasing the power $1/p$ at both sides of Eq. (72), which is less than $1$,
we have

$\displaystyle\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(|f(\bm{A}_{j_{1}},\bm{A}_{j_{2}},\ldots,\bm{A}_{j_{n}})|^{p})\Bigg{)}^{1/p}$
(73) $\displaystyle\leq_{1}$
$\displaystyle\Bigg{(}\left[\max\limits_{\begin{subarray}{c}m_{i}\leq
x_{i}\leq M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\frac{\sum\limits_{i=1}^{n}c_{i}x_{i}+d}{|f^{\prime}(x_{1},x_{2},\ldots,x_{n})|^{q}}\right]$
$\displaystyle\times\Bigg{|}f^{\prime}\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}\Bigg{|}^{q}\Bigg{)}^{1/p}$
$\displaystyle\leq_{2}$ $\displaystyle
C^{\prime}_{4}\Bigg{(}\left[\max\limits_{\begin{subarray}{c}m_{i}\leq
x_{i}\leq M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\frac{\sum\limits_{i=1}^{n}c_{i}x_{i}+d}{|f^{\prime}(x_{1},x_{2},\ldots,x_{n})|^{q}}\right]$
$\displaystyle\times\Bigg{|}f^{\prime}\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}\Bigg{|}^{q}\Bigg{)}^{1/q},$

where the inequality $\leq_{1}$ comes from Lowner-Heinz inequality, and the
inequality $\leq_{2}$ comes from a positive scalar such that
$C^{\prime}_{4}\lambda^{p/q}>\lambda$ for any eigenvalues $\lambda$ of the
following term:

$\displaystyle\Bigg{|}f^{\prime}\Bigg{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi_{j_{1},\ldots,j_{n}}(\bm{A}_{j_{n}})\Bigg{)}\Bigg{|}^{q}.$ (74)

Finally, this Theorem is proved by setting $C_{4}$ as

$\displaystyle
C_{4}=C^{\prime}_{4}\left[\max\limits_{\begin{subarray}{c}m_{i}\leq x_{i}\leq
M_{i}\\\
i=1,2,\ldots,n\end{subarray}}\frac{\sum\limits_{i=1}^{n}c_{i}x_{i}+d}{|f^{\prime}(x_{1},x_{2},\ldots,x_{n})|^{q}}\right]^{1/q}.$
(75)

$\hfill\Box$

We define
$\overline{\mathscr{W}}_{\Phi}^{1,q}\left(\bigtimes_{i=1}^{n}[m_{i},M_{i}]\right)$
as the hypercomplex function space in terms of operator mean such that

$\displaystyle\Bigg{(}\Bigg{|}f^{\prime}\Big{(}\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi(\bm{A}_{j_{1}}),\ldots,\sum\limits_{j_{1}=1,\ldots,j_{n}=1}^{k_{1},\ldots,k_{n}}w_{j_{1}}\ldots
w_{j_{n}}\Phi(\bm{A}_{j_{n}})\Bigg{)}\Bigg{|}^{q}\Bigg{)}^{1/q}$ (76)

is a bounded and positive self-adjoint operator for spectrum in the range of
$\bigtimes_{i=1}^{n}[m_{i},M_{i}]$. From Theorem 6, we have hypercomplex
function Sobolev embedding in terms of operator mean as

$\displaystyle\overline{\mathscr{W}}_{\Phi}^{1,q}\left(\bigtimes_{i=1}^{n}[m_{i},M_{i}]\right)\subset\mathscr{L}^{p}_{\Phi}\left(\bigtimes_{i=1}^{n}[m_{i},M_{i}]\right).$
(77)

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