# Generalized Choi-Davis-Jensen’s Operator Inequalities and Their Applications

Shih Yu Chang and Yimin Wei 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). Yimiin Wei (Corresponding author) is with the School
of Mathematical Sciences and Key Laboratory of Mathematics for Nonlinear
Sciences, Fudan University, Shanghai, 200433, P. R. China. This author is
supported by the Shanghai Municipal Science and Technology Commission under
grant 23WZ2501400 and the Ministry of Science and Technology of China under
grant G2023132005L (e-mail<EMAIL_ADDRESS>yimin.wei@gmail.com).

###### Abstract

The original Choi-Davis-Jensen’s inequality, with its wide-ranging
applications in diverse scientific and engineering fields, has motivated
researchers to explore generalizations. In this study, we extend Davis-Choi-
Jensen’s inequality by considering a nonlinear map instead of a normalized
linear map and generalize operator convex function to any continuous function
defined in a compact region. The Stone–Weierstrass theorem and Kantorovich
function are instrumental in formulating and proving generalized Choi-Davis-
Jensen’s inequalities. Additionally, we present an application of this
generalized inequality in the context of statistical physics.

Index terms— Choi-Davis-Jensen’s inequality, Stone–Weierstrass theorem,
Kantorovich function, Loewner ordering.

##  1 Introduction

Choi [1] and Davis [2] demonstrated that for a normalized positive linear map
$\Phi$ and an operator convex function $f$ defined on an interval $I$, the
following inequality known as Choi-Davis-Jensen’s inequality holds for any
self-adjoint operator $\bm{A}$:

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

where the operator inequality $\leq$ is defined as Loewner ordering, i.e.,
$\bm{B}\leq\bm{A}$ if $\bm{A}-\bm{B}$ is a positive-definite operator. Choi-
Davis-Jensen’s inequality is the extension of classical Jensen’s inequality to
the context of operators in functional analysis. Choi-Davis-Jensen’s
inequality finds applications in various areas, particularly in functional
analysis, information theory, quantum mechanics and random operator theory. In
functional analysis, Choi-Davis-Jensen’s inequality is used to establish norm
inequalities for operators acting on Banach spaces by providing bounds on the
norm of certain functions of operators [3]. In quantum mechanics, operator
Jensen inequality is applied to derive bounds on various entanglement
measures. These measures quantify the amount of entanglement in quantum
systems [4]. In matrix analysis, Choi-Davis-Jensen’s inequality is used to
obtain bounds on the spectral radius of matrices. This is essential in the
study of eigenvalues and spectral properties of matrices [5]. In information
theory, Choi-Davis-Jensen’s inequality plays a role in deriving bounds on the
quantum Fisher information, a measure of statistical distinguishability in
quantum parameter estimation [6]. In statistical machine learning, especially
in multivariate analysis, Choi-Davis-Jensen’s inequality is used to establish
bounds on expectations of certain matrix-valued functions, providing insights
into statistical properties [7]. In noncommutative probability, Choi-Davis-
Jensen’s inequality is employed in the study of noncommutative probability
spaces, providing inequalities for the expectation values of noncommutative
random variables and the tail bounds of random operators, e.g., matrices and
tensors, ensemble [8, 9, 10, 11, 12, 13, 14]. In summary, Choi-Davis-Jensen’s
inequality is a powerful tool in analyzing the properties of operators in
various mathematical and physical contexts. They allow for the extension of
classical results to the realm of functional analysis, providing insights into
the behavior of operators and their applications in diverse fields.

As Choi-Davis-Jensen’s inequality has many applications in science and
engineering, many authors have tried to generalize this inequality. In [15],
the auhtos present an inequality of Choi-Davis-Jensen type without relying on
convexity. Utilizing their principal findings, they establish new inequalities
that enhance existing results and demonstrate their generalized Choi-Davis-
Jensen inequalities applicable to relative operator entropies and quantum
mechanical entropies. Recently, the authors in [16] first provide a better
estimate of the second inequality in Hermite-Hadamard inequality and apply
this to obtain the reverse of the celebrated Davis-Choi-Jensen’s inequality.
Other works in studing the generalization of Choi-Davis-Jensen’s inequality
can also be found at references in [15, 16]. We generalize Davis-Choi-Jensen’s
inequality by allowing the function $f$ in Eq. (1) to be continous at a
compact region and the mapping $\Phi$ to be a nonlinear map, instead of
normalized linear map. Stone–Weierstrass theorem and Kantorovich function are
two main ingredits used here to establish generalized Choi-Davis-Jensen’s
inequality.

The remainder of this paper is organized as follows. In Section 2, generalized
Choi-Davis-Jensen’s inequality is derived based on Stone-Weierstrass Theorem
and Kantorovich function. The application of the new generalized Choi-Davis-
Jensen’s inequality to statistical physics is presented in Section 3.

Nomenclature:

Inequalities $\geq,>,\leq$ and $<$ besides operators are based on Loewner
ordering. The symbol $\Lambda(\bm{(}A))$ represents the spectrum of the
operator $\bm{A}$, i.e., the set of eigenvalues of the operator $\bm{A}$. If
$\Lambda(\bm{(}A))$ are composed by real numbers, we use
$\min(\Lambda(\bm{(}A)))$ and $\max(\Lambda(\bm{(}A)))$ to represent the
minimum and the maximum numbers within $\Lambda(\bm{(}A))$. Given $M>m>0$ with
any $r\in\mathbb{R}$ and $r\neq 1$, Kantorovich function with respect to
$m,M,r$ is defined by

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

##  2 Generalized Choi-Davis-Jensen’s Inequalities with Error Control

In this section, we will establish generalized Choi-Davis-Jensen’s inequality.
Let us recall Stone-Weierstrass Theorem. Consider a continuous real-valued
function, denoted as $f(x)$, defined on the closed real interval $[m,M]$,
where $m,M\in\mathbb{R}$ and $M>m$. For any given positive $\epsilon$, there
exists a polynomial denoted as $p(x)$, such that for all $x$ within the
interval $[m,M]$, the absolute difference between $f(x)$ and $p(x)$ is less
than $\epsilon$. In other words, the supremum norm of the function difference,
denoted as $\left\|f-p\right\|_{\infty}$, is less than $\epsilon$ [17]. Given
a continuous real-valued function $f(x)$ and error bound $\epsilon>0$, we can
apply Lagrange polynomial interpolation method based on Stone-Weierstrass
Theorem to find an upper polynomial $p_{\mathscr{U}}(x)\geq f(x)$ and a lower
polynomial $p_{\mathscr{L}}(x)\leq f(x)$ with respect to $x\in[m,M]$ such
that:

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

In this paper, we also require (assumption) that $f(\bm{A})$ is a self-adjoint
operator if $\bm{A}$ is a self-adjoint operator.

Given two Hilbert spaces $\mathfrak{H}$ and $\mathfrak{K}$,
$\mathbb{B}(\mathfrak{H})$ and $\mathbb{B}(\mathfrak{K})$ represent semi-
algebra of all bounded linear operators on Hilbert spaces $\mathfrak{H}$ and
$\mathfrak{K}$, respectively. Recall that
$\Phi:\mathbb{B}(\mathfrak{H})\rightarrow\mathbb{B}(\mathfrak{K})$ is a
normalized positive linear map at the original settings of Choi-Davis-Jensen’s
Inequality. A normalized positive linear map is defined by the following
Definition 1 [18, 19].

###### Definition 1

A map $\Phi:\mathbb{B}(\mathfrak{H})\rightarrow\mathbb{B}(\mathfrak{K})$ is a
normalized positive linear map if we have the three conditions satisfied
jointly:

  * •

The map $\Phi$ is a _linear_ map, i,e,
$\Phi(a\bm{X}+b\bm{Y})=a\Phi(\bm{X})+b\Phi(\bm{Y})$ for any $a,b\in\mathbb{C}$
and any $\bm{X},\bm{Y}\in\mathbb{B}(\mathfrak{H})$.

  * •

The linear map $\Phi$ is a _positive_ map if the operator order (Loewner
order) is preserved, i,e, $\bm{X}\geq\bm{Y}$ implies
$\Phi(\bm{X})\geq\Phi(\bm{Y})$.

  * •

The linear map $\Phi$ is a _normalized_ map if the identity operator is
preserved, i,e, $\Phi(\bm{I}_{\mathfrak{H}})=\bm{I}_{\mathfrak{K}}$, where
$\bm{I}_{\mathfrak{H}}$ and $\bm{I}_{\mathfrak{K}}$ are identity operators of
the Hilbert spaces $\mathfrak{H}$ and $\mathfrak{K}$, respectively.

In this work, we will consider more general $\Phi$ by assuming that

$\displaystyle\Phi(\bm{X})$ $\displaystyle=$
$\displaystyle\bm{V}^{\ast}\left(\sum\limits_{i=0}^{I}a_{i}\bm{X}^{i}\right)\bm{V}$
(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 $\bm{V}$ is isometry in $\mathfrak{H}$ such that
$\bm{V}^{\ast}\bm{V}=\bm{I}_{\mathfrak{H}}$, $a_{i_{+}}$ represent those
nonnegative coefficients in $a_{i}$, and $a_{i_{-}}$ represent those negative
coefficients in $a_{i}$. The collection of $i_{+}$ forms the set $S_{I_{+}}$,
and the collection of $i_{-}$ forms the set $S_{I_{-}}$. Note that we do not
require linear map, positive map, and normalized map for $\Phi$ defined by Eq.
(4). Under the assumption provided by Eq. (4), the conventional normalized
positive linear map provided by Definition 1 is a special case by setting the
polynomial $\sum\limits_{i=0}^{I}a_{i}\bm{X}^{i}$ as the identity map, i.e.,
all $a_{i}=0$ except $a_{1}=1$.

We require the following Lemma 1 to provide the lower and upper bounds for
$f(\bm{A})$ if $\bm{A}$ is a self-adjoint operator.

###### 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)

Further assume that $p_{\mathscr{L}}(\bm{A})\geq\bm{0}$, we have

$\displaystyle\mathscr{K}^{-1}(\min(\Lambda(p_{\mathscr{L}}(\bm{A}))),\max(\Lambda(p_{\mathscr{L}}(\bm{A}))),i_{+})p^{i_{+}}_{\mathscr{L}}(\bm{A})\leq
f^{i_{+}}(\bm{A}),$ (7)

and

$\displaystyle
f^{i_{+}}(\bm{A})\leq\mathscr{K}(\min(\Lambda(p_{\mathscr{U}}(\bm{A}))),\max(\Lambda(p_{\mathscr{U}}(\bm{A}))),i_{+})p^{i_{+}}_{\mathscr{U}}(\bm{A}),$
(8)

where Kantorovich functions
$\mathscr{K}^{-1}(\min(\Lambda(p_{\mathscr{L}}(\bm{A}))),\max(\Lambda(p_{\mathscr{L}}(\bm{A}))),i_{+})$
and
$\mathscr{K}(\min(\Lambda(p_{\mathscr{U}}(\bm{A}))),\max(\Lambda(p_{\mathscr{U}}(\bm{A}))),i_{+})$
are defined by Eq. (2). Moreover, we also have

$\displaystyle\mathscr{K}(\min(\Lambda(p_{\mathscr{U}}(\bm{A}))),\max(\Lambda(p_{\mathscr{U}}(\bm{A}))),i_{-})p^{i_{-}}_{\mathscr{U}}(\bm{A})\geq
f^{i_{-}}(\bm{A}),$ (9)

and

$\displaystyle\mathscr{K}^{-1}(\min(\Lambda(p_{\mathscr{L}}(\bm{A}))),\max(\Lambda(p_{\mathscr{L}}(\bm{A}))),i_{-})p^{i_{-}}_{\mathscr{L}}(\bm{A})\leq
f^{i_{-}}(\bm{A}),$ (10)

where Kantorovich function
$\mathscr{K}(\min(\Lambda(p_{\mathscr{U}}(\bm{A}))),\max(\Lambda(p_{\mathscr{U}}(\bm{A}))),i_{-})p^{i_{-}}_{\mathscr{U}}(\bm{A})$
and
$\mathscr{K}^{-1}(\min(\Lambda(p_{\mathscr{L}}(\bm{A}))),\max(\Lambda(p_{\mathscr{L}}(\bm{A}))),i_{-})p^{i_{-}}_{\mathscr{L}}(\bm{A})$
are defined by Eq. (2).

Proof: From Eq. (1) and spectrum mapping theorem of the self-adjoint operator
$\bm{A}$, we have

$\displaystyle f(\bm{A})\leq p_{\mathscr{U}}(\bm{A}),$ (11)

and

$\displaystyle p_{\mathscr{L}}(\bm{A})\leq f(\bm{A}).$ (12)

From Theorem 8.3 in [18], we have Eq. (7) from Eq. (12). Similarly, we have
Eq. (8) from Eq. (11).

Again, from Theorem 8.3 in [18], we have Eq. (9) from Eq. (11). Similarly, we
have Eq. (10) from Eq. (12). $\hfill\Box$

###### Remark 1

Note that in Eq. (7) and Eq. (8), we use the Kantorovich function
$\mathscr{K}^{-1}(\min(\Lambda(p_{\mathscr{L}}(\bm{A}))),\max(\Lambda(p_{\mathscr{L}}(\bm{A}))),i_{+})$
and
$\mathscr{K}(\min(\Lambda(p_{\mathscr{U}}(\bm{A}))),\max(\Lambda(p_{\mathscr{U}}(\bm{A}))),i_{+})$
instead of using
$\mathscr{K}(\min(\Lambda(f(\bm{A}))),\max(\Lambda(f(\bm{A}))),i_{+})$ because
$\Lambda(f(\bm{A}))$ may not within real domain. Same reason applies to Eq.
(9) and Eq. (10).

Our next Lemma 2 is about the upper and the lower bounds for $\Phi(f(\bm{A}))$
based on Lemma 1.

###### Lemma 2

Under the definition of $\Phi$ provided by Eq. (4) and same conditions
provided by Lemma 1, we have

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

On the other hand, we also have

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

Proof: This Lemma is proved by applying Lemma 1 to the definition of $\Phi$
provided by Eq. (4). $\hfill\Box$

The following Lemma 3 is to provide the upper and lower bounds for
$f(\Phi(\bm{A}))$.

###### Lemma 3

Given the mapping $\Phi$ defined by Eq. (4) and a self-adjoint operator
$\bm{A}$, we will have the spectrum for $\Phi(\bm{A})$ denoted by
$\Lambda(\Phi(\bm{A}))$. 111Note that the spectrum $\Lambda(\Phi(\bm{A}))$ is
composed by real numbers only from $\Phi$ defined by Eq. (4). From Stone-
Weierstrass Theorem, we have

$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\tilde{p}_{\mathscr{U}}(x)-f(x)\leavevmode\nobreak\
\leavevmode\nobreak\ \leq\leavevmode\nobreak\ \leavevmode\nobreak\ \epsilon,$
$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
f(x)-\tilde{p}_{\mathscr{L}}(x)\leavevmode\nobreak\ \leavevmode\nobreak\
\leq\leavevmode\nobreak\ \leavevmode\nobreak\ \epsilon,$ (15)

for $x\in[\min(\Lambda(\Phi(\bm{A}))),\max(\Lambda(\Phi(\bm{A})))]$. The
polynomials $\tilde{p}_{\mathscr{L}}(x)$ and $\tilde{p}_{\mathscr{U}}(x)$ can
be expressed by

$\displaystyle\tilde{p}_{\mathscr{L}}(x)=\sum\limits_{k=0}^{\tilde{n}_{\mathscr{L}}}\tilde{\alpha}_{k}x^{k},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{p}_{\mathscr{U}}(x)=\sum\limits_{j=0}^{\tilde{n}_{\mathscr{U}}}\tilde{\beta}_{j}x^{j}.$
(16)

Then, we have

$\displaystyle f(\Phi(\bm{A}))$ $\displaystyle\leq$
$\displaystyle\sum\limits_{j=0}^{\tilde{n}_{\mathscr{U}}}\tilde{\beta}_{j}\Phi^{j}(\bm{A}),$
(17)

and

$\displaystyle f(\Phi(\bm{A}))$ $\displaystyle\geq$
$\displaystyle\sum\limits_{k=0}^{\tilde{n}_{\mathscr{L}}}\tilde{\alpha}_{k}\Phi^{k}(\bm{A}).$
(18)

Proof: From $\Phi$ defined by Eq. (4), the operator $\Phi(\bm{A})$ is a self-
adjoint operator as the integer power of any self-adjoint operator $\bm{A}$ is
a self-adjoint opeartor again. Then, we have Eq. (17) from the spectrum
mapping theorem of the self-adjoint operator $\Phi(\bm{A})$ and the first
condition provided by Eq. (3). Similarly, we also have Eq. (18) from the
spectrum mapping theorem of the self-adjoint operator $\Phi(\bm{A})$ and the
second condition provided by Eq. (3). $\hfill\Box$

We are ready to present the main theorem of this work about generalized
operator Jensen’s inequality without requirements of the function $f$
convexity and the mapping $\Phi$ to be a normalized positive linear map.

###### Theorem 1

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

$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
p_{\mathscr{U}}(x)-f(x)\leavevmode\nobreak\ \leavevmode\nobreak\
\leq\leavevmode\nobreak\ \leavevmode\nobreak\ \epsilon,$ $\displaystyle 0$
$\displaystyle\leq$ $\displaystyle f(x)-p_{\mathscr{L}}(x)\leavevmode\nobreak\
\leavevmode\nobreak\ \leq\leavevmode\nobreak\ \leavevmode\nobreak\ \epsilon,$
(19)

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

$\displaystyle
p_{\mathscr{L}}(x)=\sum\limits_{k=0}^{n_{\mathscr{L}}}\alpha_{k}x^{k},\leavevmode\nobreak\
\leavevmode\nobreak\
p_{\mathscr{U}}(x)=\sum\limits_{j=0}^{n_{\mathscr{U}}}\beta_{j}x^{j}.$ (20)

We assume that $p_{\mathscr{L}}(\bm{A})\geq\bm{0}$ and the mapping
$\Phi:\mathbb{B}(\mathfrak{H})\rightarrow\mathbb{B}(\mathfrak{K})$ is defined
by Eq. (4). From such $\Phi$, we have

$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\tilde{p}_{\mathscr{U}}(x)-f(x)\leavevmode\nobreak\
\leavevmode\nobreak\ \leq\leavevmode\nobreak\ \leavevmode\nobreak\ \epsilon,$
$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
f(x)-\tilde{p}_{\mathscr{L}}(x)\leavevmode\nobreak\ \leavevmode\nobreak\
\leq\leavevmode\nobreak\ \leavevmode\nobreak\ \epsilon,$ (21)

for $x\in[\min(\Lambda(\Phi(\bm{A}))),\max(\Lambda(\Phi(\bm{A})))]$. The
polynomals $\tilde{p}_{\mathscr{L}}(x)$ and $\tilde{p}_{\mathscr{U}}(x)$ can
be expressed by

$\displaystyle\tilde{p}_{\mathscr{L}}(x)=\sum\limits_{k=0}^{\tilde{n}_{\mathscr{L}}}\tilde{\alpha}_{k}x^{k},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{p}_{\mathscr{U}}(x)=\sum\limits_{j=0}^{\tilde{n}_{\mathscr{U}}}\tilde{\beta}_{j}x^{j}.$
(22)

We use the following notation simplifications:

$\displaystyle\frac{d}{e}\times W(\bm{A})$
$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}$
$\displaystyle\bm{V}^{\ast}\left\\{\sum\limits_{i_{+}\in
S_{I_{+}}}a_{i_{+}}\mathscr{K}(\min(\Lambda(p_{\mathscr{U}}(\bm{A}))),\max(\Lambda(p_{\mathscr{U}}(\bm{A}))),i_{+})p^{i_{+}}_{\mathscr{U}}(\bm{A})\right.$
(23) $\displaystyle\left.+\sum\limits_{i_{-}\in
S_{I_{-}}}a_{i_{-}}\mathscr{K}^{-1}(\min(\Lambda(p_{\mathscr{L}}(\bm{A}))),\max(\Lambda(p_{\mathscr{L}}(\bm{A}))),i_{-})p^{i_{-}}_{\mathscr{L}}(\bm{A})\right\\}\bm{V},$

where $d,e$ are any positive real numbers and $W(\bm{A})$ is an operator-
valued function with $\bm{A}$ as its argument.

$\displaystyle\frac{c}{e}\times X(\bm{A})$
$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}$
$\displaystyle\bm{V}^{\ast}\left\\{\sum\limits_{i_{+}\in
S_{I_{+}}}a_{i_{+}}\mathscr{K}^{-1}(\min(\Lambda(p_{\mathscr{L}}(\bm{A}))),\max(\Lambda(p_{\mathscr{L}}(\bm{A}))),i_{+})p^{i_{+}}_{\mathscr{L}}(\bm{A})\right.$
(24) $\displaystyle\left.+\sum\limits_{i_{-}\in
S_{I_{-}}}a_{i_{-}}\mathscr{K}(\min(\Lambda(p_{\mathscr{U}}(\bm{A}))),\max(\Lambda(p_{\mathscr{U}}(\bm{A}))),i_{-})p^{i_{-}}_{\mathscr{U}}(\bm{A})\right\\}\bm{V},$

where $c,e$ are any positive real numbers and $X(\bm{A})$ is an operator-
valued function with $\bm{A}$ as its argument.

$\displaystyle c\times Y(\bm{A})$
$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}$
$\displaystyle\sum\limits_{j=0}^{\tilde{n}_{\mathscr{U}}}\tilde{\beta}_{j}\Phi^{j}(\bm{A}),$
(25)

where $Y(\bm{A})$ is an operator-valued function with $\bm{A}$ as its
argument.

$\displaystyle d\times Z(\bm{A})$
$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}$
$\displaystyle\sum\limits_{k=0}^{\tilde{n}_{\mathscr{L}}}\tilde{\alpha}_{k}\Phi^{k}(\bm{A}),$
(26)

where $Z(\bm{A})$ is an operator-valued function with $\bm{A}$ as its
argument. Then, we have

$\displaystyle e\Phi(f(\bm{A}))-d(W(\bm{A})-Z(\bm{A}))\leq f(\Phi(\bm{A}))\leq
e\Phi(f(\bm{A}))-c(X(\bm{A})-Y(\bm{A})).$ (27)

Proof: From Lemma 2 and Eq. (23) with Eq. (24), we have

$\displaystyle
c\left(\frac{X(\bm{A})+Y(\bm{A})}{2}+\frac{X(\bm{A})-Y(\bm{A})}{2}\right)\leq
e\Phi(f(\bm{A}))\leq
d\left(\frac{W(\bm{A})+Z(\bm{A})}{2}+\frac{W(\bm{A})-Z(\bm{A})}{2}\right).$
(28)

Moreover, from Lemma 3 and Eq. (25) with Eq. (26), we have

$\displaystyle
d\left(\frac{W(\bm{A})+Z(\bm{A})}{2}-\frac{W(\bm{A})-Z(\bm{A})}{2}\right)\leq
f(\Phi(\bm{A}))\leq
c\left(\frac{X(\bm{A})+Y(\bm{A})}{2}-\frac{X(\bm{A})-Y(\bm{A})}{2}\right).$
(29)

Finally, this theorem is proved by rearrange related terms in Eq. (28) and Eq.
(29). $\hfill\Box$

From Theorem 1, we can have the following corollary about eigenvalues
majorization inequality immediatedly.

###### Corollary 1

Given $\bm{A}$ as an $n\times n$ Hermitan matrix, and let
$\left(\lambda_{\mathscr{L},1},\lambda_{\mathscr{L},2},\cdots,\lambda_{\mathscr{L},n}\right)$,
$\left(\lambda_{1},\lambda_{2},\cdots,\lambda_{n}\right)$, and
$\left(\lambda_{\mathscr{R},1},\lambda_{\mathscr{R},2},\cdots,\lambda_{\mathscr{R},n}\right)$,
be eigenvalues of $e\Phi(f(\bm{A}))-d(W(\bm{A})-Z(\bm{A}))$, $f(\Phi(\bm{A}))$
and $e\Phi(f(\bm{A}))-c(X(\bm{A})-Y(\bm{A}))$ respectively. Then, we have

$\displaystyle\left(\lambda_{\mathscr{L},1},\lambda_{\mathscr{L},2},\cdots,\lambda_{\mathscr{L},n}\right)\preceq_{wk}\left(\lambda_{1},\lambda_{2},\cdots,\lambda_{n}\right)\preceq_{wk}\left(\lambda_{\mathscr{R},1},\lambda_{\mathscr{R},2},\cdots,\lambda_{\mathscr{R},n}\right),$
(30)

where $\preceq_{wk}$ represents the weakly majorization relation between
entries of two vectors.

Proof: Given two $n\times n$ Hermitan matrices with $\bm{A}\leq\bm{B}$, from
Loewner Theorem (1934) [18, 19], we have

$\displaystyle\lambda_{\bm{A},i}\leq\lambda_{\bm{B},i},$ (31)

where $\lambda_{\bm{A},i}$ and $\lambda_{\bm{B},i}$ are eigenvalues of
Hermitan matrices $\bm{A}$ and $\bm{B}$, respectively. Therefore, Eq. (31) can
be expressed by
$\left(\lambda_{\bm{A},1},\lambda_{\bm{A},2},\cdots,\lambda_{\bm{A},n}\right)\preceq_{wk}\left(\lambda_{\bm{B},1},\lambda_{\bm{B},2},\cdots,\lambda_{\bm{B},n}\right)$.
This corollary is proved by applying Loewner Theorem (1934) to Theorem 1.
$\hfill\Box$

We will provide the following examples about Theorem 1. Example 1 will show
that the work about Choi-Davis-Jensen’s inequality without convexity studied
in [15] is a special case of Theorem 1.

###### Example 1

Let us repeat Theorem 2.2 from [15] here again with their notations for easy
comparison with our work. If $\Phi$ is a normalized positive linear map, and
$f:[m,M]\rightarrow(0,\infty)$ is a continuous twice differentiable function
such that $\alpha\leq f^{\prime\prime}$ on $[m,M]$, we have

$\displaystyle\frac{1}{K(m,M,f)}\Phi(f(\bm{A}))+\frac{\alpha}{2K(m,M,f)}\left[(M+m)\Phi(\bm{A})-Mm-\Phi(\bm{A}^{2})\right]$
$\displaystyle\leq f(\Phi(\bm{A}))\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\
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\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\ $ $\displaystyle\leq
K(m,M,f)\Phi(f(\bm{A}))-\frac{\alpha}{2}\left[(M+m)\Phi(\bm{A})-Mm-\Phi(\bm{A})^{2}\right],$
(32)

where $K(m,M,f)$ is defined by

$\displaystyle K(m,M,f)$ $\displaystyle=$
$\displaystyle\max\limits_{x\in[m,M]}\left(\frac{1}{f(x)}\left(\frac{M-x}{M-m}f(m)+\frac{x-m}{M-m}f(M)\right)\right).$
(33)

Note that $K(m,M,f)$ is greater than 1. As $K(m,M,f)>1$, we have

$\displaystyle\frac{1}{K(m,M,f)}\Phi(f(\bm{A}))\leq K(m,M,f)\Phi(f(\bm{A})),$
(34)

therefore, the following inequality implies Eq. (1):

$\displaystyle
K(m,M,f)\Phi(f(\bm{A}))+\frac{\alpha}{2K(m,M,f)}\left[(M+m)\Phi(\bm{A})-Mm\bm{I}-\Phi(\bm{A})^{2}\right]$
$\displaystyle\leq f(\Phi(\bm{A}))\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\ \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\ $ $\displaystyle\leq
K(m,M,f)\Phi(f(\bm{A}))-\frac{\alpha}{2}\left[(M+m)\Phi(\bm{A})-Mm\bm{I}-\Phi(\bm{A})^{2}\right].$
(35)

If we set the following parameters for Theorem 1 with $\alpha\geq 0$:

$\displaystyle c$ $\displaystyle=$ $\displaystyle\frac{\alpha}{2},$
$\displaystyle d$ $\displaystyle=$ $\displaystyle\frac{\alpha}{2K(m,M,f)},$
$\displaystyle e$ $\displaystyle=$ $\displaystyle K(m,M,f),$ $\displaystyle
W(\bm{A})$ $\displaystyle=$
$\displaystyle\Phi(\bm{A}^{2})\geq_{1}\Phi(\bm{A})^{2},$ $\displaystyle
X(\bm{A})$ $\displaystyle=$ $\displaystyle(M+m)\Phi(\bm{A})-Mm\bm{I},$
$\displaystyle Y(\bm{A})$ $\displaystyle=$ $\displaystyle\Phi^{2}(\bm{A}),$
$\displaystyle Z(\bm{A})$ $\displaystyle=$
$\displaystyle(M+m)\Phi(\bm{A})-Mm\bm{I},$ (36)

where we use $\Phi$ as a normalized positive linear map in $\geq_{1}$. We will
obtain the result provided by Eq. (1) with $\alpha\geq 0$ which implies Eq.
(1). By comparing with the lower bounds of $f(\Phi(\bm{A}))$, our method can
provide tighter lower bound compared to Theorem 2.2 from [15].

On the other hand, if $\alpha<0$ and $\Phi(\bm{A})=\bm{V}^{\ast}\bm{A}\bm{V}$,
we can set the following parameters to obtain the result provided by Eq. (33)
which implies Eq. (1):

$\displaystyle c$ $\displaystyle=$ $\displaystyle\frac{-\alpha}{2},$
$\displaystyle d$ $\displaystyle=$ $\displaystyle\frac{-\alpha}{2K(m,M,f)},$
$\displaystyle e$ $\displaystyle=$ $\displaystyle K(m,M,f),$ $\displaystyle
W(\bm{A})$ $\displaystyle=$ $\displaystyle(M+m)\Phi(\bm{A})-Mm\bm{I},$
$\displaystyle X(\bm{A})$ $\displaystyle=$
$\displaystyle\Phi^{2}(\bm{A})=_{1}\Phi(\bm{A}^{2}),$ $\displaystyle
Y(\bm{A})$ $\displaystyle=$ $\displaystyle(M+m)\Phi(\bm{A})-Mm\bm{I},$
$\displaystyle Z(\bm{A})$ $\displaystyle=$ $\displaystyle\Phi(\bm{A})^{2},$
(37)

where we use $\Phi(\bm{A})=\bm{V}^{\ast}\bm{A}\bm{V}$ in $=_{1}$.

In the follownig example, we will consider the situation that $f$ is bounded
by quadratic functions and $\Phi$ is also being generated by quadratic
function.

###### Example 2

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

$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
p_{\mathscr{U}}(x)-f(x)\leavevmode\nobreak\ \leavevmode\nobreak\
\leq\leavevmode\nobreak\ \leavevmode\nobreak\ \epsilon,$ $\displaystyle 0$
$\displaystyle\leq$ $\displaystyle f(x)-p_{\mathscr{L}}(x)\leavevmode\nobreak\
\leavevmode\nobreak\ \leq\leavevmode\nobreak\ \leavevmode\nobreak\ \epsilon,$
(38)

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

$\displaystyle
p_{\mathscr{L}}(x)=\sum\limits_{k=0}^{2}\alpha_{k}x^{k},\leavevmode\nobreak\
\leavevmode\nobreak\ p_{\mathscr{U}}(x)=\sum\limits_{j=0}^{2}\beta_{j}x^{j}.$
(39)

We assume that $p_{\mathscr{L}}(\bm{A})\geq\bm{0}$ and the mapping
$\Phi:\mathbb{B}(\mathfrak{H})\rightarrow\mathbb{B}(\mathfrak{K})$ is defined
as:

$\displaystyle\Phi(\bm{A})$ $\displaystyle=$
$\displaystyle\bm{V}^{\ast}\left(a_{0}+a_{1}\bm{A}+a_{2}\bm{A}^{2}\right)\bm{V},$
(40)

where we assume that $a_{0},a_{2}>0$ and $a_{1}<0$.

From such $\Phi$, we have

$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\tilde{p}_{\mathscr{U}}(x)-f(x)\leavevmode\nobreak\
\leavevmode\nobreak\ \leq\leavevmode\nobreak\ \leavevmode\nobreak\ \epsilon,$
$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle
f(x)-\tilde{p}_{\mathscr{L}}(x)\leavevmode\nobreak\ \leavevmode\nobreak\
\leq\leavevmode\nobreak\ \leavevmode\nobreak\ \epsilon,$ (41)

for $x\in[\min(\Lambda(\Phi(\bm{A}))),\max(\Lambda(\Phi(\bm{A})))]$ with . The
polynomals $\tilde{p}_{\mathscr{L}}(x)$ and $\tilde{p}_{\mathscr{U}}(x)$ are
assumed to be

$\displaystyle\tilde{p}_{\mathscr{L}}(x)=p_{\mathscr{L}}(x),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{p}_{\mathscr{U}}(x)=p_{\mathscr{U}}(x)$ (42)

If we select real numbers $c,d,e$ as 1, we have the following notation
simplifications:

$\displaystyle W(\bm{A})$ $\displaystyle\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}$
$\displaystyle\bm{V}^{\ast}\left\\{a_{0}\bm{I}+a_{2}\mathscr{K}(\min(\Lambda(p_{\mathscr{U}}(\bm{A}))),\max(\Lambda(p_{\mathscr{U}}(\bm{A}))),2)p^{2}_{\mathscr{U}}(\bm{A})\right.$
(43)
$\displaystyle\left.+a_{1}\mathscr{K}^{-1}(\min(\Lambda(p_{\mathscr{L}}(\bm{A}))),\max(\Lambda(p_{\mathscr{L}}(\bm{A}))),1)p_{\mathscr{L}}(\bm{A})\right\\}\bm{V},$

and

$\displaystyle X(\bm{A})$ $\displaystyle\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}$
$\displaystyle\bm{V}^{\ast}\left\\{a_{0}\bm{I}+a_{2}\mathscr{K}^{-1}(\min(\Lambda(p_{\mathscr{L}}(\bm{A}))),\max(\Lambda(p_{\mathscr{L}}(\bm{A}))),2)p^{2}_{\mathscr{L}}(\bm{A})\right.$
(44)
$\displaystyle\left.+a_{1}\mathscr{K}(\min(\Lambda(p_{\mathscr{U}}(\bm{A}))),\max(\Lambda(p_{\mathscr{U}}(\bm{A}))),1)p^{i_{-}}_{\mathscr{U}}(\bm{A})\right\\}\bm{V},$

and

$\displaystyle Y(\bm{A})$ $\displaystyle\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}$
$\displaystyle\beta_{0}\bm{I}+\beta_{1}\bm{V}^{\ast}\left(a_{0}\bm{I}+a_{1}\bm{A}+a_{2}\bm{A}^{2}\right)\bm{V}$
(45)
$\displaystyle+\beta_{2}\bm{V}^{\ast}\left(a^{2}_{0}\bm{I}+2a_{0}a_{1}\bm{A}+a^{2}_{1}\bm{A}^{2}+2a_{0}a_{2}\bm{A}^{2}+2a_{1}a_{2}\bm{A}^{3}+a^{2}_{2}\bm{A}^{4}\right)\bm{V},$

and

$\displaystyle Z(\bm{A})$ $\displaystyle\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}$
$\displaystyle\alpha_{0}\bm{I}+\alpha_{1}\bm{V}^{\ast}\left(a_{0}+a_{1}\bm{A}+a_{2}\bm{A}^{2}\right)\bm{V}$
(46)
$\displaystyle+\alpha_{2}\bm{V}^{\ast}\left(a^{2}_{0}\bm{I}+2a_{0}a_{1}\bm{A}+a^{2}_{1}\bm{A}^{2}+2a_{0}a_{2}\bm{A}^{2}+2a_{1}a_{2}\bm{A}^{3}+a^{2}_{2}\bm{A}^{4}\right)\bm{V}.$

Then, we have

$\displaystyle\Phi(f(\bm{A}))-(W(\bm{A})-Z(\bm{A}))\leq
f(\Phi(\bm{A}))\leq\Phi(f(\bm{A}))-(X(\bm{A})-Y(\bm{A})).$ (47)

##  3 Applications

In this section, we will apply proposed generalized Choi-Davis-Jensen’s
operator inequalities to statistical physics. Tsallis entropy for single
random variable was introduced by C.Tsallis as
$T_{q}(X)=-\sum\limits_{x}p(x)^{q}f_{q}(p(x))$, incorporating a single
parameter $q$ to extend the concept of Shannon entropy. Here, the
$q$-logarithm is defined as $f_{q}(x)=\frac{x^{1-q}-1}{1-q}$ for any
nonnegative real numbers $q$ and $x$, while $p(x)$ represents the probability
distribution of the given random variable $X$. It is evident that the Tsallis
entropy $T_{q}(X)$ converges to the Shannon entropy $-\sum\limits_{x}p(x)\log
p(x)$ as $q$ approaches $1$, given that the $q$-logarithm uniformly converges
to the natural logarithm as $q$ approaches 1. Tsallis entropy plays a crucial
role in nonextensive statistics, often referred to as Tsallis statistics [20].

###  3.1 Tsallis Relative Entropy Without $\Phi$

For Tsallis relative entropy for self-adjoint operators $\bm{A}>\bm{0}$ and
$\bm{B}$, denoted by $T_{q}(\bm{A}\parallel\bm{B})$, it is defined as [15]:

$\displaystyle T_{q}(\bm{A}\parallel\bm{B})$
$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}$
$\displaystyle\frac{\bm{A}\\#_{q}\bm{B}-\bm{A}}{q},$ (48)

where $-1\leq q\leq 1$ with $q\neq 0$, and
$\bm{A}\\#_{q}\bm{B}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\bm{A}^{1/2}\left(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}\right)^{q}\bm{A}^{1/2}$.
We have the following Lemma 4 about $T_{q}(\bm{A}\parallel\bm{B})$ bounds.

###### Lemma 4

Given two positive self-adjoint operators $\bm{A}$ and $\bm{B}$ satisfying
$m\bm{A}\leq\bm{B}\leq M\bm{A}$, where $0<m<M$ with $m\geq 2$ and $M\geq 5m$,
we define following relations:

$\displaystyle\Gamma(\bm{A},\bm{B},q)$
$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}$
$\displaystyle-\frac{(\bm{B}-m\bm{A})(1-M^{q})+(M\bm{A}-\bm{B})(1-m^{q})}{q(M-m)},$
$\displaystyle\Psi(\bm{A},\bm{B})$
$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}$
$\displaystyle\bm{A}\\#_{2}\bm{B}-(M+m)\bm{B}+Mm\bm{A},$
$\displaystyle\Omega(\bm{A},q)$
$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}$
$\displaystyle\frac{m^{q}\bm{A}}{(M+m)^{Mm}}.$ (49)

Further, if we have $0<q\leq 1$ and $\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}$ assumed
to be self-adjoint, then,

$\displaystyle T_{q}(\bm{A}\parallel\bm{B})$ $\displaystyle\geq$
$\displaystyle\Gamma(\bm{A},\bm{B},q)-\frac{(1-q)M^{q-2}}{2}\Psi(\bm{A},\bm{B})+\Omega(\bm{A},q),$
$\displaystyle T_{q}(\bm{A}\parallel\bm{B})$ $\displaystyle\leq$
$\displaystyle\Gamma(\bm{A},\bm{B},q)-\frac{(1-q)m^{q-2}}{2}\Psi(\bm{A},\bm{B})-\Omega(\bm{A},q).$
(50)

Proof: Because we have the following inequalites with respect to the function
$\frac{1-x^{q}}{q}$ given $0<m<M$ with $m\geq 2$ and $M\geq 5m$:

$\displaystyle\frac{1-x^{q}}{q}$ $\displaystyle\geq$
$\displaystyle\frac{(x-m)(1-M^{q})+(M-x)(1-m^{q})}{q(M-m)}+\frac{(1-q)m^{q-2}}{2}\left(x^{2}-(M+m)x+Mm\right)$
$\displaystyle+\frac{m^{q}}{(M+m)^{Mm}},$ $\displaystyle\frac{1-x^{q}}{q}$
$\displaystyle\leq$
$\displaystyle\frac{(x-m)(1-M^{q})+(M-x)(1-m^{q})}{q(M-m)}+\frac{(1-q)M^{q-2}}{2}\left(x^{2}-(M+m)x+Mm\right)$
(51) $\displaystyle-\frac{m^{q}}{(M+m)^{Mm}}.$

This lemma is proved by setting $x$ with the positive self-adjoint operator
$\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}$ in Eq. (3.1) followed by multiplying both
sides by $\bm{A}^{1/2}$. $\hfill\Box$

From Lemma 4, we have the following lemma immediately about tighter bounds of
relative operator entropy provided by [15] by taking $q\rightarrow 0$.

###### Lemma 5

The relative operator entropy with respect to operators $\bm{A}$ and $\bm{B}$,
denoted by $S(\bm{A}\parallel\bm{B})$, is given by

$\displaystyle S(\bm{A}\parallel\bm{B})$
$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}$
$\displaystyle\bm{A}^{1/2}\log\left(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}\right)\bm{A}^{1/2}.$
(52)

Under same conditions provided by Lemma 4, we have

$\displaystyle S(\bm{A}\parallel\bm{B})$ $\displaystyle\geq$
$\displaystyle\frac{(\bm{B}-m\bm{A})\log M+(M\bm{A}-\bm{B})\log
m}{(M-m)}-\frac{1}{2M^{2}}\Psi(\bm{A},\bm{B})+\Omega(\bm{A},0),$
$\displaystyle S(\bm{A}\parallel\bm{B})$ $\displaystyle\leq$
$\displaystyle\frac{(\bm{B}-m\bm{A})\log M+(M\bm{A}-\bm{B})\log
m}{(M-m)}-\frac{1}{2m^{2}}\Psi(\bm{A},\bm{B})-\Omega(\bm{A},0).$ (53)

Proof: Because we have

$\displaystyle\lim\limits_{q\rightarrow 0}\frac{1-x^{q}}{q}=-\log x,$ (54)

the term $\Gamma(\bm{A},\bm{B},q)$ becomes

$\displaystyle\lim\limits_{q\rightarrow 0}\Gamma(\bm{A},\bm{B},q)$
$\displaystyle=$ $\displaystyle\frac{(\bm{B}-m\bm{A})\log
M+(M\bm{A}-\bm{B})\log m}{(M-m)}.$ (55)

This lemma is proved by applying Eq. (55) to Lemma 4. $\hfill\Box$

###### Remark 2

Lemma 5 provides tighter bounds of relative operator entropy given by by [15]
as $\Omega(\bm{A},0)$ is a positive operator.

###  3.2 Tsallis Relative Entropy With $\Phi$

In section 3.1, operators $\bm{A}$ and $\bm{B}$ in Tsallis relative entropy
are applied directly into Eq. (48). In this section, we will consider applying
$\Phi$ to operators $\bm{A}$, $\bm{B}$ and $T_{q}(\bm{A}\parallel\bm{B})$. The
mapping $\Phi$ is assumed to have the following format:

$\displaystyle\Phi(\bm{X})$ $\displaystyle=$
$\displaystyle\bm{V}^{\ast}\left(a_{0}\bm{I}+a_{1}\bm{X}+a_{2}\bm{X}^{2}\right)\bm{V},$
(56)

where coefficients $a_{0},a_{1}$ and $a_{2}$ are positive real numbers. We
have the following Lemma 6 about Tsallis relative entropy with respect to
$\Phi(\bm{A})$ and $\Phi(\bm{B})$.

###### Lemma 6

Given two positive self-adjoint operators $\bm{A}$ and $\bm{B}$ satisfying
$m\Phi(\bm{A})\leq\Phi(\bm{B})\leq M\Phi(\bm{A})$, where $\Phi$ is given by
Eq. (56) and constants $M,m$ satisfy $0<m<M$ with $m\geq 2$ and $M\geq 5m$, we
define following relations:

$\displaystyle\Gamma(\Phi(\bm{A}),\Phi(\bm{B}),q)$
$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}$
$\displaystyle-\frac{(\Phi(\bm{B})-m\Phi(\bm{A}))(1-M^{q})+(M\Phi(\bm{A})-\Phi(\bm{B}))(1-m^{q})}{q(M-m)},$
$\displaystyle\Psi(\Phi(\bm{A}),\Phi(\bm{B}))$
$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}$
$\displaystyle\Phi(\bm{A})\\#_{2}\Phi(\bm{B})-(M+m)\Phi(\bm{B})+Mm\Phi(\bm{A}),$
$\displaystyle\Omega(\Phi(\bm{A}),q)$
$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}$
$\displaystyle\frac{m^{q}\Phi(\bm{A})}{(M+m)^{Mm}}.$ (57)

Further, if we have $0<q\leq 1$, then,

$\displaystyle T_{q}(\Phi(\bm{A})\parallel\Phi(\bm{B}))$ $\displaystyle\geq$
$\displaystyle\Gamma(\Phi(\bm{A}),\Phi(\bm{B}),q)-\frac{(1-q)M^{q-2}}{2}\Psi(\Phi(\bm{A}),\Phi(\bm{B}))+\Omega(\Phi(\bm{A}),q),$
$\displaystyle T_{q}(\Phi(\bm{A})\parallel\Phi(\bm{B}))$ $\displaystyle\leq$
$\displaystyle\Gamma(\Phi(\bm{A}),\Phi(\bm{B}),q)-\frac{(1-q)m^{q-2}}{2}\Psi(\Phi(\bm{A}),\Phi(\bm{B}))-\Omega(\Phi(\bm{A}),q).$
(58)

Proof: Since we have the following inequalites with respect to the function
$\frac{1-x^{q}}{q}$ given $0<m<M$ with $m\geq 2$ and $M\geq 5m$:

$\displaystyle\frac{1-x^{q}}{q}$ $\displaystyle\geq$
$\displaystyle\frac{(x-m)(1-M^{q})+(M-x)(1-m^{q})}{q(M-m)}+\frac{(1-q)m^{q-2}}{2}\left(x^{2}-(M+m)x+Mm\right)$
$\displaystyle+\frac{m^{q}}{(M+m)^{Mm}},$ $\displaystyle\frac{1-x^{q}}{q}$
$\displaystyle\leq$
$\displaystyle\frac{(x-m)(1-M^{q})+(M-x)(1-m^{q})}{q(M-m)}+\frac{(1-q)M^{q-2}}{2}\left(x^{2}-(M+m)x+Mm\right)$
(59) $\displaystyle-\frac{m^{q}}{(M+m)^{Mm}}.$

This lemma is proved by setting $x$ with the positive self-adjoint operator
$\Phi(\bm{A})^{-1/2}\Phi(\bm{B})\Phi(\bm{A})^{-1/2}$ in Eq. (3.2) followed by
multiplying both sides by $\bm{A}^{1/2}$. $\hfill\Box$

Lemma 7 below is established to provide bounds for
$\Phi(T_{q}(\bm{A}\parallel\bm{B}))$.

###### Lemma 7

Given $\Phi$ defined by Eq. (56), and two positive self-adjoint operators
$\bm{A}$ and $\bm{B}$ satisfying $m\bm{A}\leq\bm{B}\leq M\bm{A}$, where
$0<m<M$ with $m\geq 2$ and $M\geq 5m$, we set up the follownig two polynomials
$p_{\mathscr{L}}(x)$ and $p_{\mathscr{U}}(x)$:

$\displaystyle p_{\mathscr{L}}(x)$ $\displaystyle=$
$\displaystyle-\frac{(x-m)(1-M^{q})+(M-x)(1-m^{q})}{q(M-m)}-\frac{(1-q)M^{q-2}}{2}\left(x^{2}-(M+m)x+Mm\right)$
$\displaystyle+\frac{m^{q}}{(M+m)^{Mm}},$ $\displaystyle p_{\mathscr{U}}(x)$
$\displaystyle=$
$\displaystyle-\frac{(x-m)(1-M^{q})+(M-x)(1-m^{q})}{q(M-m)}-\frac{(1-q)m^{q-2}}{2}\left(x^{2}-(M+m)x+Mm\right)$
(60) $\displaystyle-\frac{m^{q}}{(M+m)^{Mm}},$

where $0<q\leq 1$. Moreover, we also assume that $\bm{A}^{1/2}$ and
$(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2})^{q}$ are commute, then, we have

$\displaystyle\Phi(T_{q}(\bm{A}\parallel\bm{B}))$ $\displaystyle\geq$
$\displaystyle\bm{V}^{\ast}\left\\{a_{0}\bm{I}+a_{1}\mathscr{K}^{-1}(\min(\Lambda(p_{\mathscr{L}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}))),\max(\Lambda(p_{\mathscr{L}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}))),1)\right.$
(61)
$\displaystyle\left.\cdot\bm{A}^{1/2}p_{\mathscr{L}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2})\bm{A}^{1/2}\right.$
$\displaystyle\left.+a_{2}\mathscr{K}^{-1}(\min(\Lambda(p_{\mathscr{L}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}))),\max(\Lambda(p_{\mathscr{L}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}))),2)\right.$
$\displaystyle\left.\cdot\bm{A}p^{2}_{\mathscr{L}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2})\bm{A}\right\\}\bm{V},$
$\displaystyle\Phi(T_{q}(\bm{A}\parallel\bm{B}))$ $\displaystyle\leq$
$\displaystyle\bm{V}^{\ast}\left\\{a_{0}\bm{I}+a_{1}\mathscr{K}(\min(\Lambda(p_{\mathscr{U}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}))),\max(\Lambda(p_{\mathscr{U}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}))),1)\right.$
(62)
$\displaystyle\left.\cdot\bm{A}^{1/2}p_{\mathscr{U}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2})\bm{A}^{1/2}\right.$
$\displaystyle\left.+a_{2}\mathscr{K}(\min(\Lambda(p_{\mathscr{U}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}))),\max(\Lambda(p_{\mathscr{U}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}))),2)\right.$
$\displaystyle\left.\cdot\bm{A}p^{2}_{\mathscr{U}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2})\bm{A}\right\\}\bm{V}.$

Proof: Because we have

$\displaystyle\frac{x^{q}-1}{q}$ $\displaystyle\geq$
$\displaystyle-\frac{(x-m)(1-M^{q})+(M-x)(1-m^{q})}{q(M-m)}-\frac{(1-q)M^{q-2}}{2}\left(x^{2}-(M+m)x+Mm\right)$
(63) $\displaystyle+\frac{m^{q}}{(M+m)^{Mm}}=_{1}p_{\mathscr{L}}(x),$

where $=_{1}$ comes from Eq. (7), by applying
$\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}$ to $x$ in Eq. (63), we also have

$\displaystyle\Phi\left(T_{q}(\bm{A}\parallel\bm{B})\right)$ $\displaystyle=$
$\displaystyle\Phi\left(\bm{A}^{1/2}\left(\frac{(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2})^{q}-\bm{I}}{q}\right)\bm{A}^{1/2}\right)$
(64) $\displaystyle=_{1}$
$\displaystyle\bm{V}^{\ast}\left\\{a_{0}\bm{I}+a_{1}\bm{A}^{1/2}\left(\frac{(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2})^{q}-\bm{I}}{q}\right)\bm{A}^{1/2}\right.$
$\displaystyle+\left.a_{2}\bm{A}\left(\frac{(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2})^{q}-\bm{I}}{q}\right)^{2}\bm{A}\right\\}\bm{V}.$
$\displaystyle\geq_{2}$
$\displaystyle\bm{V}^{\ast}\left\\{a_{0}\bm{I}+a_{1}\mathscr{K}^{-1}(\min(\Lambda(p_{\mathscr{L}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}))),\max(\Lambda(p_{\mathscr{L}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}))),1)\right.$
$\displaystyle\left.\cdot\bm{A}^{1/2}p_{\mathscr{L}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2})\bm{A}^{1/2}\right.$
$\displaystyle\left.+a_{2}\mathscr{K}^{-1}(\min(\Lambda(p_{\mathscr{L}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}))),\max(\Lambda(p_{\mathscr{L}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}))),2)\right.$
$\displaystyle\left.\cdot\bm{A}p^{2}_{\mathscr{L}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2})\bm{A}\right\\}\bm{V},$

where we use that $\bm{A}^{1/2}$ and $(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2})^{q}$
are commutative in $=_{1}$, and Lemma 2 is used to obtain $\geq_{2}$. This
establish Eq. (61).

Similarly, we can apply the following inequality to get Eq. (62) by the same
argument to derive Eq. (61):

$\displaystyle\frac{x^{q}-1}{q}$ $\displaystyle\leq$
$\displaystyle-\frac{(x-m)(1-M^{q})+(M-x)(1-m^{q})}{q(M-m)}-\frac{(1-q)m^{q-2}}{2}\left(x^{2}-(M+m)x+Mm\right)$
(65) $\displaystyle-\frac{m^{q}}{(M+m)^{Mm}}.$

$\hfill\Box$

We are ready to present generalized Choi-Davis-Jensen’s operator inequalities
for Tsallis relative entropy.

###### Theorem 2

As conditions provided by Lemma 6 and Lemma 7, we have the following
inequalities

$\displaystyle\Phi(T_{q}(\bm{A}\parallel\bm{B}))-(W(\bm{A},\bm{B})-Z(\bm{A},\bm{B}))$
$\displaystyle\leq$ $\displaystyle T_{q}(\Phi(\bm{A})\parallel\Phi(\bm{B}))$
(66) $\displaystyle\leq$
$\displaystyle\Phi(T_{q}(\bm{A}\parallel\bm{B}))-(X(\bm{A},\bm{B})-Y(\bm{A},\bm{B})).$

where we have

$\displaystyle W(\bm{A},\bm{B})$ $\displaystyle=$
$\displaystyle\bm{V}^{\ast}\left\\{a_{0}\bm{I}+a_{1}\mathscr{K}(\min(\Lambda(p_{\mathscr{U}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}))),\max(\Lambda(p_{\mathscr{U}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}))),1)\right.$
(67)
$\displaystyle\left.\cdot\bm{A}^{1/2}p_{\mathscr{L}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2})\bm{A}^{1/2}\right.$
$\displaystyle\left.+a_{2}\mathscr{K}(\min(\Lambda(p_{\mathscr{U}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}))),\max(\Lambda(p_{\mathscr{U}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}))),2)\right.$
$\displaystyle\left.\cdot\bm{A}p^{2}_{\mathscr{L}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2})\bm{A}\right\\}\bm{V},$
$\displaystyle X(\bm{A},\bm{B})$ $\displaystyle=$
$\displaystyle\bm{V}^{\ast}\left\\{a_{0}\bm{I}+a_{1}\mathscr{K}^{-1}(\min(\Lambda(p_{\mathscr{L}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}))),\max(\Lambda(p_{\mathscr{L}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}))),1)\right.$
(68)
$\displaystyle\left.\cdot\bm{A}^{1/2}p_{\mathscr{L}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2})\bm{A}^{1/2}\right.$
$\displaystyle\left.+a_{2}\mathscr{K}^{-1}(\min(\Lambda(p_{\mathscr{L}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}))),\max(\Lambda(p_{\mathscr{L}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2}))),2)\right.$
$\displaystyle\left.\cdot\bm{A}p^{2}_{\mathscr{L}}(\bm{A}^{-1/2}\bm{B}\bm{A}^{-1/2})\bm{A}\right\\}\bm{V},$
$\displaystyle Y(\bm{A},\bm{B})$ $\displaystyle=$
$\displaystyle\Gamma(\Phi(\bm{A}),\Phi(\bm{B}),q)-\frac{(1-q)m^{q-2}}{2}\Psi(\Phi(\bm{A}),\Phi(\bm{B}))-\Omega(\Phi(\bm{A}),q),$
(69) $\displaystyle Z(\bm{A},\bm{B})$ $\displaystyle=$
$\displaystyle\Gamma(\Phi(\bm{A}),\Phi(\bm{B}),q)-\frac{(1-q)M^{q-2}}{2}\Psi(\Phi(\bm{A}),\Phi(\bm{B}))+\Omega(\Phi(\bm{A}),q).$
(70)

Proof: By setting $c=d=e=1$ and replacing $W(\bm{A}),X(\bm{A}),Y(\bm{A})$ and
$W(\bm{A})$ with $W(\bm{A},\bm{B}),X(\bm{A},\bm{B})$, $Y(\bm{A},\bm{B})$ and
$Z(\bm{A},\bm{B})$, respectively, obtained from Lemma 6 and Lemma 7 in Theorem
1. $\hfill\Box$

###### Remark 3

All results discussed in this work can also be applied to tensors if tensors
under Einstein products are treated as operators [9, 13].

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