Generalized Choi–Davis–Jensen’s Operator Inequalities and Their Applications

Abstract

The original Choi–Davis–Jensen’s inequality, known for its extensive applications in various scientific and engineering fields, has inspired researchers to pursue its generalizations. In this study, we extend the Choi–Davis–Jensen’s inequality by introducing a nonlinear map instead of a normalized linear map and generalize the concept of operator convex functions to include any continuous function defined within a compact region. Notably, operators can be matrices with structural symmetry, enhancing the scope and applicability of our results. The Stone–Weierstrass theorem and the Kantorovich function play crucial roles in the formulation and proof of these generalized Choi–Davis–Jensen’s inequalities. Furthermore, we demonstrate an application of this generalized inequality in the context of statistical physics.

1. Introduction

Inequalities are fundamental in mathematics because they provide a way to express and analyze relationships where equality is not necessarily exact, reflecting real-world conditions where variables and outcomes are often constrained within certain bounds. They are crucial in optimizing functions, proving theorems, and understanding the behavior of mathematical systems across various fields, from calculus and algebra to economics and engineering. Inequalities enable us to determine limits, estimate values, and establish conditions under which certain results hold, making them essential tools for both theoretical and applied mathematics [1,2].

Choi [3] and Davis [4] 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 $\mathit{A}$:

$$\begin{array}{ccc}\hfill f(\Phi (\mathit{A}\left)\right)& \le & \Phi \left(f\right(\mathit{A}\left)\right),\hfill \end{array}$$

(1)

where the operator inequality ≤ is defined as Loewner ordering, i.e., $\mathit{B}\le \mathit{A}$ if $\mathit{A}-\mathit{B}$ 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 [5]. 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 [6]. 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 [7]. 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 [8]. 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 [9]. 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 [10]. 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 [11], the authors present an inequality of the 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’s inequalities applicable to relative operator entropies and quantum mechanical entropies. Recently, the authors in [12] first provide a better estimate of the second inequality in Hermite-Hadamard inequality and apply this to obtain the reverse of the celebrated Choi–Davis–Jensen’s inequality. Other works in studing the generalization of Choi–Davis–Jensen’sinequality can also be found at references in [11,12]. We generalize Choi–Davis–Jensen’s inequality by allowing the function f in Equation (1) to be continuous 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 ingredients used here to establish generalized Choi–Davis–Jensen’s inequality.

The Stone-Weierstrass theorem is used to approximate functions by polynomials because it guarantees that any continuous function defined on a compact space can be uniformly approximated by polynomials. This is particularly useful in analysis and approximation theory, where polynomials are often easier to work with due to their algebraic properties. Regarding the Löwner-Heinz inequality, it is limited to operator powers between [0, 1] because the inequality relies on the operator monotonicity in this range, meaning the inequality holds only for these specific powers. This limitation motivates the use of the Kantorovich function, which provides a broader framework for dealing with operator inequalities involving powers outside this restricted range, allowing for more general operator power inequalities to be established.

Self-adjoint operators, also known as Hermitian operators, form an essential class of operators in arithmetic and physics due to their extensive-ranging applicability. They generalize the idea of Hermitian matrices, that are square matrices which are equal to their very own conjugate transpose, a property that guarantees actual eigenvalues and orthogonal eigenvectors. This guarantees stability and predictability in physical structures, making Hermitian matrices essential in quantum mechanics. Beyond matrices, self-adjoint operators encompass Hermitian tensors, which amplify those properties to larger-dimensional arrays. Hermitian tensors permit the illustration of more complicated structures and changes in multi-dimensional spaces, consisting of the ones found in quantum field theory and popular relativity. The inclusion of Hermitian matrices and tensors below the umbrella of self-adjoint operators demonstrates their vast applicability, providing critical equipment for modeling and studying a huge type of systems throughout arithmetic, physics, and engineering. Generalized Choi–Davis–Jensen’s operator inequalities derived in this paper should also be applicable to Hermitian matrices and Hermitian tensors.

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. Finally, we discuss several potential future research directions in Section 4.

Notation:

Inequalities $\ge ,>,\le$ and < besides operators are based on Loewner ordering. The symbol $\Lambda \left(\right(\mathit{A}\left)\right)$ represents the spectrum of the operator $\mathit{A}$, i.e., the set of eigenvalues of the operator $\mathit{A}$. If $\Lambda \left(\right(\mathit{A}\left)\right)$ are composed by real numbers, we use $inf(\Lambda (\left(\mathit{A}\right)\left)\right)$ and $sup(\Lambda (\left(\mathit{A}\right)\left)\right)$ to represent the infimum and the supremum numbers within $\Lambda \left(\right(\mathit{A}\left)\right)$. Given $M>m>0$ with any $r\in \mathbb{R}$ and $r\ne 1$, Kantorovich function with respect to $m,M,r$ is defined by

$$\begin{array}{ccc}\hfill \mathcal{K}(m,M,r)& =& {\displaystyle \frac{(m{M}^r-M{m}^r)}{(r-1)(M-m)}}{\left[{\displaystyle \frac{(r-1)(M^r-m^r)}{r(m{M}^r-M{m}^r)}}\right]}^r.\hfill \end{array}$$

(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 $f\left(x\right)$, defined on the closed real interval $[m,M]$, where $m,M\in \mathbb{R}$ and $M>m$. For any given positive $ϵ$, there exists a polynomial denoted as $p\left(x\right)$, such that for all x within the interval $[m,M]$, the absolute difference between $f\left(x\right)$ and $p\left(x\right)$ is less than $ϵ$. In other words, the supremum norm of the function difference, denoted as ${∥f-p∥}_{\infty }$, is less than $ϵ$ [13]. Given a continuous real-valued function $f\left(x\right)$ and error bound $ϵ>0$, we can apply Lagrange polynomial interpolation method based on Stone-Weierstrass Theorem to find an upper polynomial $p_{\mathcal{U}}\left(x\right)\ge f\left(x\right)$ and a lower polynomial $p_{\mathcal{L}}\left(x\right)\le f\left(x\right)$ with respect to $x\in [m,M]$ such that:

$$\begin{array}{ccc}\hfill 0& \le & p_{\mathcal{U}}\left(x\right)-f\left(x\right)\le ϵ,\hfill \\ \hfill 0& \le & f\left(x\right)-p_{\mathcal{L}}\left(x\right)\le ϵ,\hfill \end{array}$$

(3)

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

Given two Hilbert spaces $\mathfrak{H}$ and $\mathfrak{K}$, $\mathbb{B}\left(\mathfrak{H}\right)$ and $\mathbb{B}\left(\mathfrak{K}\right)$ represent semi-algebra of all bounded linear operators on Hilbert spaces $\mathfrak{H}$ and $\mathfrak{K}$, respectively. Recall that $\Phi :\mathbb{B}\left(\mathfrak{H}\right)\to \mathbb{B}\left(\mathfrak{K}\right)$ 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 [14,15].

Definition 1.

A map $\Phi :\mathbb{B}\left(\mathfrak{H}\right)\to \mathbb{B}\left(\mathfrak{K}\right)$ is a normalized positive linear map if we three conditions are jointly satisfied:

• The map Φ is a linear map, i.e., $\Phi (a\mathit{X}+b\mathit{Y})=a\Phi \left(\mathit{X}\right)+b\Phi \left(\mathit{Y}\right)$ for any $a,b\in \mathbb{C}$ and any $\mathit{X},\mathit{Y}\in \mathbb{B}\left(\mathfrak{H}\right)$.
• The linear map Φ is a positive map if the operator order (Loewner order) is preserved, i.e., $\mathit{X}\ge \mathit{Y}$ implies $\Phi \left(\mathit{X}\right)\ge \Phi \left(\mathit{Y}\right)$.
• The linear map Φ is a normalized map if the identity operator is preserved, i.e., $\Phi \left({\mathit{I}}_{\mathfrak{H}}\right)={\mathit{I}}_{\mathfrak{K}}$, where ${\mathit{I}}_{\mathfrak{H}}$ and ${\mathit{I}}_{\mathfrak{K}}$ are the identity operators of the Hilbert spaces $\mathfrak{H}$ and $\mathfrak{K}$, respectively.

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

$$\begin{array}{ccc}\hfill \Phi \left(\mathit{X}\right)& =& {\mathit{V}}^{*}\left(\sum _{i=0}^I{a}_i{\mathit{X}}^i\right)\mathit{V}\hfill \\ & =& {\mathit{V}}^{*}\left(\sum _{i_{+}\in S_{I_{+}}}{a}_{i_{+}}{\mathit{X}}^{i_{+}}+\sum _{i_{-}\in S_{I_{-}}}{a}_{i_{-}}{\mathit{X}}^{i_{-}}\right)\mathit{V},\hfill \end{array}$$

(4)

where $\mathit{V}$ is isometry in $\mathfrak{H}$, such that ${\mathit{V}}^{*}\mathit{V}={\mathit{I}}_{\mathfrak{H}}$, $a_{i_{+}}$ 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 a linear map, positive map, and normalized map for $\Phi$ defined by Equation (4). Under the assumption provided by Equation (4), the conventional normalized positive linear map provided by Definition 1 is a special case by setting the polynomial ${\displaystyle \sum _{i=0}^I}{a}_i{\mathit{X}}^i$ 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 $f\left(\mathit{A}\right)$ if $\mathit{A}$ is a self-adjoint operator.

Lemma 1.

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

$$\begin{array}{ccc}\hfill 0& \le & p_{\mathcal{U}}\left(x\right)-f\left(x\right)\le ϵ,\hfill \\ \hfill 0& \le & f\left(x\right)-p_{\mathcal{L}}\left(x\right)\le ϵ,\hfill \end{array}$$

(5)

for $x\in [inf(\Lambda \left(\mathit{A}\right)),sup(\Lambda \left(\mathit{A}\right)\left)\right]$ with polynomials $p_{\mathcal{L}}\left(x\right)$ and $p_{\mathcal{U}}\left(x\right)$ expressed by

$$\begin{array}{c}\hfill p_{\mathcal{L}}\left(x\right)=\sum _{k=0}^{n_{\mathcal{L}}}{\alpha }_k{x}^k,p_{\mathcal{U}}\left(x\right)=\sum _{j=0}^{n_{\mathcal{U}}}{\beta }_j{x}^j.\end{array}$$

Further, assume that $p_{\mathcal{L}}\left(\mathit{A}\right)>\mathbf{0}$, we have

$$\begin{array}{c}\hfill {\mathcal{K}}^{-1}(inf(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),i_{+})p_{\mathcal{L}}^{i_{+}}\left(\mathit{A}\right)\le f^{i_{+}}\left(\mathit{A}\right),\end{array}$$

(6)

and

$$\begin{array}{c}\hfill f^{i_{+}}\left(\mathit{A}\right)\le \mathcal{K}(inf(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),i_{+})p_{\mathcal{U}}^{i_{+}}\left(\mathit{A}\right),\end{array}$$

(7)

where Kantorovich functions ${\mathcal{K}}^{-1}(inf(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),i_{+})$ and $\mathcal{K}(inf(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),i_{+})$ are defined by Equation (2). Moreover, we also have

$$\begin{array}{c}\hfill \mathcal{K}(inf(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),i_{-})p_{\mathcal{U}}^{i_{-}}\left(\mathit{A}\right)\ge f^{i_{-}}\left(\mathit{A}\right),\end{array}$$

(8)

and

$$\begin{array}{c}\hfill {\mathcal{K}}^{-1}(inf(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),i_{-})p_{\mathcal{L}}^{i_{-}}\left(\mathit{A}\right)\le f^{i_{-}}\left(\mathit{A}\right),\end{array}$$

(9)

where the Kantorovich function $\mathcal{K}(inf(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),i_{-})p_{\mathcal{U}}^{i_{-}}\left(\mathit{A}\right)$, and${\mathcal{K}}^{-1}(inf(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),i_{-})p_{\mathcal{L}}^{i_{-}}\left(\mathit{A}\right)$ are defined by Equation (2).

Proof. 

From Equation (5) and the spectrum mapping theorem of the self-adjoint operator $\mathit{A}$ [14], we have

$$\begin{array}{c}\hfill f\left(\mathit{A}\right)\le p_{\mathcal{U}}\left(\mathit{A}\right),\end{array}$$

(10)

and

$$\begin{array}{c}\hfill p_{\mathcal{L}}\left(\mathit{A}\right)\le f\left(\mathit{A}\right).\end{array}$$

(11)

From Theorem 8.3 in [14], we have Equation (6) from Equation (11). Similarly, we have Equation (7) from Equation (10).

Again, from Theorem 8.3 in [14], we have Equation (8) from Equation (10). Similarly, we have Equation (9) from Equation (11). □

Remark 1.

Note that in Equations (6) and (7), we use the Kantorovich function ${\mathcal{K}}^{-1}(inf(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),i_{+})$ and $\mathcal{K}(inf(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),i_{+})$ instead of using $\mathcal{K}(inf(\Lambda \left(f\left(\mathit{A}\right)\right)),sup(\Lambda \left(f\left(\mathit{A}\right)\right)),i_{+})$ because $\Lambda \left(f\right(\mathit{A}\left)\right)$ may not be within the real domain. Same reason applies to Equations (8) and (9).

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

Lemma 2.

Under the definition of Φ provided by Equation (4) and the same conditions provided by Lemma 1, we have

$$\begin{array}{ccc}\hfill \Phi \left(f\right(\mathit{A}\left)\right)& \le & {\mathit{V}}^{*}\left\{\sum _{i_{+}\in S_{I_{+}}}{a}_{i_{+}}\mathcal{K}(inf(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),i_{+})p_{\mathcal{U}}^{i_{+}}\left(\mathit{A}\right)\right.\hfill \\ & & \left.+\sum _{i_{-}\in S_{I_{-}}}{a}_{i_{-}}{\mathcal{K}}^{-1}(inf(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),i_{-})p_{\mathcal{L}}^{i_{-}}\left(\mathit{A}\right)\right\}\mathit{V}.\hfill \end{array}$$

On the other hand, we also have

$$\begin{array}{ccc}\hfill \Phi \left(f\right(\mathit{A}\left)\right)& \ge & {\mathit{V}}^{*}\left\{\sum _{i_{+}\in S_{I_{+}}}{a}_{i_{+}}{\mathcal{K}}^{-1}(inf(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),i_{+})p_{\mathcal{L}}^{i_{+}}\left(\mathit{A}\right)\right.\hfill \\ & & \left.+\sum _{i_{-}\in S_{I_{-}}}{a}_{i_{-}}\mathcal{K}(inf(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),i_{-})p_{\mathcal{U}}^{i_{-}}\left(\mathit{A}\right)\right\}\mathit{V}.\hfill \end{array}$$

Proof. 

This Lemma is proved by applying Lemma 1 to the definition of $\Phi$ provided by Equation (4). □

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

Lemma 3.

Given the mapping Φ defined by Equation (4) and a self-adjoint operator $\mathit{A}$, we have the spectrum for $\Phi \left(\mathit{A}\right)$, which is denoted by $\Lambda (\Phi (\mathit{A}\left)\right)$. (Note that the spectrum $\Lambda (\Phi (\mathit{A}\left)\right)$ is composed of real numbers only from Φ defined by Equation (4).) From the Stone–Weierstrass Theorem, we have

$$\begin{array}{ccc}\hfill 0& \le & {\tilde{p}}_{\mathcal{U}}\left(x\right)-f\left(x\right)\le ϵ,\hfill \\ \hfill 0& \le & f\left(x\right)-{\tilde{p}}_{\mathcal{L}}\left(x\right)\le ϵ,\hfill \end{array}$$

(12)

for $x\in [inf(\Lambda (\Phi (\mathit{A}\left)\right)),sup(\Lambda (\Phi (\mathit{A}\left)\right)\left)\right]$. The polynomials ${\tilde{p}}_{\mathcal{L}}\left(x\right)$ and ${\tilde{p}}_{\mathcal{U}}\left(x\right)$ can be expressed by

$$\begin{array}{c}\hfill {\tilde{p}}_{\mathcal{L}}\left(x\right)=\sum _{k=0}^{{\tilde{n}}_{\mathcal{L}}}{\tilde{\alpha }}_k{x}^k,{\tilde{p}}_{\mathcal{U}}\left(x\right)=\sum _{j=0}^{{\tilde{n}}_{\mathcal{U}}}{\tilde{\beta }}_j{x}^j.\end{array}$$

(13)

Then, we have

$$\begin{array}{ccc}\hfill f(\Phi (\mathit{A}\left)\right)& \le & \sum _{j=0}^{{\tilde{n}}_{\mathcal{U}}}{\tilde{\beta }}_j{\Phi }^j\left(\mathit{A}\right),\hfill \end{array}$$

(14)

and

$$\begin{array}{ccc}\hfill f(\Phi (\mathit{A}\left)\right)& \ge & \sum _{k=0}^{{\tilde{n}}_{\mathcal{L}}}{\tilde{\alpha }}_k{\Phi }^k\left(\mathit{A}\right).\hfill \end{array}$$

(15)

Proof. 

From $\Phi$, defined by Equation (4), the operator $\Phi \left(\mathit{A}\right)$ is a self-adjoint operator as the integer power of any self-adjoint operator $\mathit{A}$ is a self-adjoint opeartor again. Then, we have Equation (14) from the spectrum mapping theorem of the self-adjoint operator $\Phi \left(\mathit{A}\right)$ and the first condition provided by Equation (12). Similarly, we also have Equation (15) from the spectrum mapping theorem of the self-adjoint operator $\Phi \left(\mathit{A}\right)$ and the second condition provided by Equation (12). □

Here, we present the main theorem of this work about a generalized operator Jensen’s inequality without the need for function f convexity and by mapping $\Phi$ to be a normalized positive linear map.

Theorem 1.

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

$$\begin{array}{ccc}\hfill 0& \le & p_{\mathcal{U}}\left(x\right)-f\left(x\right)\le ϵ,\hfill \\ \hfill 0& \le & f\left(x\right)-p_{\mathcal{L}}\left(x\right)\le ϵ,\hfill \end{array}$$

for $x\in [inf(\Lambda \left(\mathit{A}\right)),sup(\Lambda \left(\mathit{A}\right)\left)\right]$ with polynomials $p_{\mathcal{L}}\left(x\right)$ and $p_{\mathcal{U}}\left(x\right)$, expressed by

$$\begin{array}{c}\hfill p_{\mathcal{L}}\left(x\right)=\sum _{k=0}^{n_{\mathcal{L}}}{\alpha }_k{x}^k,p_{\mathcal{U}}\left(x\right)=\sum _{j=0}^{n_{\mathcal{U}}}{\beta }_j{x}^j.\end{array}$$

We assume that $p_{\mathcal{L}}\left(\mathit{A}\right)\ge \mathbf{0}$ and the mapping $\Phi :\mathbb{B}\left(\mathfrak{H}\right)\to \mathbb{B}\left(\mathfrak{K}\right)$ is defined by Equation (4). From such Φ, we have

$$\begin{array}{ccc}\hfill 0& \le & {\tilde{p}}_{\mathcal{U}}\left(x\right)-f\left(x\right)\le ϵ,\hfill \\ \hfill 0& \le & f\left(x\right)-{\tilde{p}}_{\mathcal{L}}\left(x\right)\le ϵ,\hfill \end{array}$$

for $x\in [inf(\Lambda (\Phi (\mathit{A}\left)\right)),sup(\Lambda (\Phi (\mathit{A}\left)\right)\left)\right]$. The polynomals ${\tilde{p}}_{\mathcal{L}}\left(x\right)$ and ${\tilde{p}}_{\mathcal{U}}\left(x\right)$ can be expressed by

$$\begin{array}{c}\hfill {\tilde{p}}_{\mathcal{L}}\left(x\right)=\sum _{k=0}^{{\tilde{n}}_{\mathcal{L}}}{\tilde{\alpha }}_k{x}^k,{\tilde{p}}_{\mathcal{U}}\left(x\right)=\sum _{j=0}^{{\tilde{n}}_{\mathcal{U}}}{\tilde{\beta }}_j{x}^j.\end{array}$$

We use the following notation simplifications:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d}{e}}\times W\left(\mathit{A}\right)& \stackrel{def}{=}& {\mathit{V}}^{*}\left\{\sum _{i_{+}\in S_{I_{+}}}{a}_{i_{+}}\mathcal{K}(inf(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),i_{+})p_{\mathcal{U}}^{i_{+}}\left(\mathit{A}\right)\right.\hfill \\ & & \left.+\sum _{i_{-}\in S_{I_{-}}}{a}_{i_{-}}{\mathcal{K}}^{-1}(inf(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),i_{-})p_{\mathcal{L}}^{i_{-}}\left(\mathit{A}\right)\right\}\mathit{V},\hfill \end{array}$$

(16)

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

$$\begin{array}{ccc}\hfill {\displaystyle \frac{c}{e}}\times X\left(\mathit{A}\right)& \stackrel{def}{=}& {\mathit{V}}^{*}\left\{\sum _{i_{+}\in S_{I_{+}}}{a}_{i_{+}}{\mathcal{K}}^{-1}(inf(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),i_{+})p_{\mathcal{L}}^{i_{+}}\left(\mathit{A}\right)\right.\hfill \\ & & \left.+\sum _{i_{-}\in S_{I_{-}}}{a}_{i_{-}}\mathcal{K}(inf(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),i_{-})p_{\mathcal{U}}^{i_{-}}\left(\mathit{A}\right)\right\}\mathit{V},\hfill \end{array}$$

(17)

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

$$\begin{array}{ccc}\hfill c\times Y\left(\mathit{A}\right)& \stackrel{def}{=}& \sum _{j=0}^{{\tilde{n}}_{\mathcal{U}}}{\tilde{\beta }}_j{\Phi }^j\left(\mathit{A}\right),\hfill \end{array}$$

(18)

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

$$\begin{array}{ccc}\hfill d\times Z\left(\mathit{A}\right)& \stackrel{def}{=}& \sum _{k=0}^{{\tilde{n}}_{\mathcal{L}}}{\tilde{\alpha }}_k{\Phi }^k\left(\mathit{A}\right),\hfill \end{array}$$

(19)

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

$$\begin{array}{c}\hfill e\Phi \left(f\right(\mathit{A}\left)\right)-d\left(W\right(\mathit{A})-Z(\mathit{A}\left)\right)\le f(\Phi (\mathit{A}\left)\right)\le e\Phi \left(f\right(\mathit{A}\left)\right)-c\left(X\right(\mathit{A})-Y(\mathit{A}\left)\right).\end{array}$$

(20)

Proof. 

From Lemma 2 and Equation (16) with Equation (17), we have

$$\begin{array}{c}\hfill c\left({\displaystyle \frac{X\left(\mathit{A}\right)+Y\left(\mathit{A}\right)}{2}}+{\displaystyle \frac{X\left(\mathit{A}\right)-Y\left(\mathit{A}\right)}{2}}\right)\le e\Phi \left(f\left(\mathit{A}\right)\right)\le d\left({\displaystyle \frac{W\left(\mathit{A}\right)+Z\left(\mathit{A}\right)}{2}}+{\displaystyle \frac{W\left(\mathit{A}\right)-Z\left(\mathit{A}\right)}{2}}\right).\end{array}$$

(21)

Moreover, from Lemma 3 and Equation (18) with Equation (19), we have

$$\begin{array}{c}\hfill d\left({\displaystyle \frac{W\left(\mathit{A}\right)+Z\left(\mathit{A}\right)}{2}}-{\displaystyle \frac{W\left(\mathit{A}\right)-Z\left(\mathit{A}\right)}{2}}\right)\le f(\Phi \left(\mathit{A}\right))\le c\left({\displaystyle \frac{X\left(\mathit{A}\right)+Y\left(\mathit{A}\right)}{2}}-{\displaystyle \frac{X\left(\mathit{A}\right)-Y\left(\mathit{A}\right)}{2}}\right).\end{array}$$

(22)

Finally, this theorem is proven by rearranging related terms in Equations (21) and (22). □

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

Corollary 1.

Given $\mathit{A}$ as an $n\times n$ Hermitan matrix, and let $\left({\lambda }_{\mathcal{L},1},{\lambda }_{\mathcal{L},2},\cdots ,{\lambda }_{\mathcal{L},n}\right)$, $\left({\lambda }_1,{\lambda }_2,\cdots ,\right.$$\left.{\lambda }_n\right)$, and $\left({\lambda }_{\mathcal{R},1},{\lambda }_{\mathcal{R},2},\cdots ,{\lambda }_{\mathcal{R},n}\right)$, be eigenvalues of $e\Phi \left(f\right(\mathit{A}\left)\right)-d\left(W\right(\mathit{A})-Z(\mathit{A}\left)\right)$, $f(\Phi (\mathit{A}\left)\right)$ and $e\Phi \left(f\right(\mathit{A}\left)\right)-c\left(X\right(\mathit{A})-Y(\mathit{A}\left)\right)$ respectively. Then, we have

$$\begin{array}{c}\hfill \left({\lambda }_{\mathcal{L},1},{\lambda }_{\mathcal{L},2},\cdots ,{\lambda }_{\mathcal{L},n}\right){⪯}_{wk}\left({\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_n\right){⪯}_{wk}\left({\lambda }_{\mathcal{R},1},{\lambda }_{\mathcal{R},2},\cdots ,{\lambda }_{\mathcal{R},n}\right),\end{array}$$

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

Proof. 

Given two $n\times n$ Hermitan matrices with $\mathit{A}\le \mathit{B}$, from Loewner Theorem (1934) [14,15], we have

$$\begin{array}{c}\hfill {\lambda }_{\mathit{A},i}\le {\lambda }_{\mathit{B},i},\end{array}$$

(23)

where ${\lambda }_{\mathit{A},i}$ and ${\lambda }_{\mathit{B},i}$ are eigenvalues of Hermitan matrices $\mathit{A}$ and $\mathit{B}$, respectively. Therefore, Equation (23) can be expressed by $\left({\lambda }_{\mathit{A},1},{\lambda }_{\mathit{A},2},\cdots ,{\lambda }_{\mathit{A},n}\right){⪯}_{wk}\left({\lambda }_{\mathit{B},1},{\lambda }_{\mathit{B},2},\cdots ,{\lambda }_{\mathit{B},n}\right)$. This corollary is proved by applying Loewner Theorem (1934) to Theorem 1. □

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 [11] is a special case of Theorem 1.

Example 1.

Let us repeat Theorem 2.2 from [11] here again with their notations for easy comparison with our work. If Φ is a normalized positive linear map, and $f:[m,M]\to (0,\infty )$ is a continuous twice differentiable function, such that $\alpha \le f^{″}$ on $[m,M]$, we have

$$\begin{array}{c}{\displaystyle \frac{1}{K(m,M,f)}}\Phi \left(f\left(\mathit{A}\right)\right)+{\displaystyle \frac{\alpha }{2K(m,M,f)}}\left[(M+m)\Phi \left(\mathit{A}\right)-Mm-\Phi \left({\mathit{A}}^2\right)\right]\hfill \\ \hspace{1em}\hspace{1em}\le f(\Phi (\mathit{A}\left)\right)\hfill \\ \hspace{1em}\hspace{1em}\le K(m,M,f)\Phi \left(f\left(\mathit{A}\right)\right)-{\displaystyle \frac{\alpha }{2}}\left[(M+m)\Phi \left(\mathit{A}\right)-Mm-\Phi {\left(\mathit{A}\right)}^2\right],\hfill \end{array}$$

(24)

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

$$\begin{array}{ccc}\hfill K(m,M,f)& =& \underset{x\in [m,M]}{max}\left({\displaystyle \frac{1}{f\left(x\right)}}\left({\displaystyle \frac{M-x}{M-m}}f\left(m\right)+{\displaystyle \frac{x-m}{M-m}}f\left(M\right)\right)\right).\hfill \end{array}$$

(25)

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

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{K(m,M,f)}}\Phi \left(f\left(\mathit{A}\right)\right)\le K(m,M,f)\Phi \left(f\left(\mathit{A}\right)\right),\end{array}$$

(26)

therefore, the following inequality implies Equation (24):

$$\begin{array}{c}K(m,M,f)\Phi \left(f\left(\mathit{A}\right)\right)+{\displaystyle \frac{\alpha }{2K(m,M,f)}}\left[(M+m)\Phi \left(\mathit{A}\right)-Mm\mathit{I}-\Phi {\left(\mathit{A}\right)}^2\right]\hfill \\ \hspace{1em}\hspace{1em}\le f(\Phi (\mathit{A}\left)\right)\hfill \\ \hspace{1em}\hspace{1em}\le K(m,M,f)\Phi \left(f\left(\mathit{A}\right)\right)-{\displaystyle \frac{\alpha }{2}}\left[(M+m)\Phi \left(\mathit{A}\right)-Mm\mathit{I}-\Phi {\left(\mathit{A}\right)}^2\right].\hfill \end{array}$$

(27)

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

$$\begin{array}{ccc}\hfill c& =& {\displaystyle \frac{\alpha }{2}},\hfill \\ \hfill d& =& {\displaystyle \frac{\alpha }{2K(m,M,f)}},\hfill \\ \hfill e& =& K(m,M,f),\hfill \\ \hfill W\left(\mathit{A}\right)& =& \Phi \left({\mathit{A}}^2\right){\ge }_1\Phi {\left(\mathit{A}\right)}^2,\hfill \\ \hfill X\left(\mathit{A}\right)& =& (M+m)\Phi \left(\mathit{A}\right)-Mm\mathit{I},\hfill \\ \hfill Y\left(\mathit{A}\right)& =& {\Phi }^2\left(\mathit{A}\right),\hfill \\ \hfill Z\left(\mathit{A}\right)& =& (M+m)\Phi \left(\mathit{A}\right)-Mm\mathit{I},\hfill \end{array}$$

(28)

where we use Φ as a normalized positive linear map in ${\ge }_1$. We will obtain the result provided by Equation (27) with $\alpha \ge 0$, which implies Equation (24). In comparison with the lower bounds of $f(\Phi (\mathit{A}\left)\right)$, our method provides a tighter lower bound compared with Theorem 2.2 from [11].

On the other hand, if $\alpha <0$ and $\Phi \left(\mathit{A}\right)={\mathit{V}}^{*}\mathit{A}\mathit{V}$, we can set the following parameters to obtain the result provided by Equation (25), which implies Equation (24):

$$\begin{array}{ccc}\hfill c& =& {\displaystyle \frac{-\alpha }{2}},\hfill \\ \hfill d& =& {\displaystyle \frac{-\alpha }{2K(m,M,f)}},\hfill \\ \hfill e& =& K(m,M,f),\hfill \\ \hfill W\left(\mathit{A}\right)& =& (M+m)\Phi \left(\mathit{A}\right)-Mm\mathit{I},\hfill \\ \hfill X\left(\mathit{A}\right)& =& {\Phi }^2\left(\mathit{A}\right){=}_1\Phi \left({\mathit{A}}^2\right),\hfill \\ \hfill Y\left(\mathit{A}\right)& =& (M+m)\Phi \left(\mathit{A}\right)-Mm\mathit{I},\hfill \\ \hfill Z\left(\mathit{A}\right)& =& \Phi {\left(\mathit{A}\right)}^2,\hfill \end{array}$$

(29)

where we use $\Phi \left(\mathit{A}\right)={\mathit{V}}^{*}\mathit{A}\mathit{V}$ in ${=}_1$.

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

Example 2.

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

$$\begin{array}{ccc}\hfill 0& \le & p_{\mathcal{U}}\left(x\right)-f\left(x\right)\le ϵ,\hfill \\ \hfill 0& \le & f\left(x\right)-p_{\mathcal{L}}\left(x\right)\le ϵ,\hfill \end{array}$$

for $x\in [inf(\Lambda \left(\mathit{A}\right)),sup(\Lambda \left(\mathit{A}\right)\left)\right]$ with polynomials $p_{\mathcal{L}}\left(x\right)$ and $p_{\mathcal{U}}\left(x\right)$ expressed by

$$\begin{array}{c}\hfill p_{\mathcal{L}}\left(x\right)=\sum _{k=0}^2{\alpha }_k{x}^k,p_{\mathcal{U}}\left(x\right)=\sum _{j=0}^2{\beta }_j{x}^j.\end{array}$$

We assume that $p_{\mathcal{L}}\left(\mathit{A}\right)\ge \mathbf{0}$ and the mapping $\Phi :\mathbb{B}\left(\mathfrak{H}\right)\to \mathbb{B}\left(\mathfrak{K}\right)$ is defined as follows:

$$\begin{array}{ccc}\hfill \Phi \left(\mathit{A}\right)& =& {\mathit{V}}^{*}\left(a_0+a_1\mathit{A}+a_2{\mathit{A}}^2\right)\mathit{V},\hfill \end{array}$$

where we assume that $a_0,a_2>0$ and $a_1<0$.

From such Φ, we have

$$\begin{array}{ccc}\hfill 0& \le & {\tilde{p}}_{\mathcal{U}}\left(x\right)-f\left(x\right)\le ϵ,\hfill \\ \hfill 0& \le & f\left(x\right)-{\tilde{p}}_{\mathcal{L}}\left(x\right)\le ϵ,\hfill \end{array}$$

for $x\in [inf(\Lambda (\Phi (\mathit{A}\left)\right)),sup(\Lambda (\Phi (\mathit{A}\left)\right)\left)\right]$. The polynomals ${\tilde{p}}_{\mathcal{L}}\left(x\right)$ and ${\tilde{p}}_{\mathcal{U}}\left(x\right)$ are assumed to be

$$\begin{array}{c}\hfill {\tilde{p}}_{\mathcal{L}}\left(x\right)=p_{\mathcal{L}}\left(x\right),{\tilde{p}}_{\mathcal{U}}\left(x\right)=p_{\mathcal{U}}\left(x\right)\end{array}$$

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

$$\begin{array}{ccc}\hfill W\left(\mathit{A}\right)& \stackrel{def}{=}& {\mathit{V}}^{*}\left\{a_0\mathit{I}+a_2\mathcal{K}(inf(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),2)p_{\mathcal{U}}^2\left(\mathit{A}\right)\right.\hfill \\ & & \left.+a_1{\mathcal{K}}^{-1}(inf(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),1)p_{\mathcal{L}}\left(\mathit{A}\right)\right\}\mathit{V},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill X\left(\mathit{A}\right)& \stackrel{def}{=}& {\mathit{V}}^{*}\left\{a_0\mathit{I}+a_2{\mathcal{K}}^{-1}(inf(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{L}}\left(\mathit{A}\right)\right)),2)p_{\mathcal{L}}^2\left(\mathit{A}\right)\right.\hfill \\ & & \left.+a_1\mathcal{K}(inf(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),sup(\Lambda \left(p_{\mathcal{U}}\left(\mathit{A}\right)\right)),1)p_{\mathcal{U}}^{i_{-}}\left(\mathit{A}\right)\right\}\mathit{V},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill Y\left(\mathit{A}\right)& \stackrel{def}{=}& {\beta }_0\mathit{I}+{\beta }_1{\mathit{V}}^{*}\left(a_0\mathit{I}+a_1\mathit{A}+a_2{\mathit{A}}^2\right)\mathit{V}\hfill \\ & & +{\beta }_2{\mathit{V}}^{*}\left(a_0^2\mathit{I}+2a_0{a}_1\mathit{A}+a_1^2{\mathit{A}}^2+2a_0{a}_2{\mathit{A}}^2+2a_1{a}_2{\mathit{A}}^3+a_2^2{\mathit{A}}^4\right)\mathit{V},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill Z\left(\mathit{A}\right)& \stackrel{def}{=}& {\alpha }_0\mathit{I}+{\alpha }_1{\mathit{V}}^{*}\left(a_0+a_1\mathit{A}+a_2{\mathit{A}}^2\right)\mathit{V}\hfill \\ & & +{\alpha }_2{\mathit{V}}^{*}\left(a_0^2\mathit{I}+2a_0{a}_1\mathit{A}+a_1^2{\mathit{A}}^2+2a_0{a}_2{\mathit{A}}^2+2a_1{a}_2{\mathit{A}}^3+a_2^2{\mathit{A}}^4\right)\mathit{V}.\hfill \end{array}$$

Then, we have

$$\begin{array}{c}\hfill \Phi \left(f\right(\mathit{A}\left)\right)-\left(W\right(\mathit{A})-Z(\mathit{A}\left)\right)\le f(\Phi (\mathit{A}\left)\right)\le \Phi \left(f\right(\mathit{A}\left)\right)-\left(X\right(\mathit{A})-Y(\mathit{A}\left)\right).\end{array}$$

3. Applications

In this section, we will apply the proposed generalized Choi–Davis–Jensen’s operator inequalities to statistical physics. Tsallis entropy for a single random variable was introduced by C.Tsallis as $T_q\left(X\right)=-{\displaystyle \sum _x}p{\left(x\right)}^q{f}_q\left(p\left(x\right)\right)$, incorporating a single parameter q to extend the concept of Shannon entropy. Here, the q-logarithm is defined as $f_q\left(x\right)={\displaystyle \frac{x^{1-q}-1}{1-q}}$ for any nonnegative real numbers q and x, while $p\left(x\right)$ represents the probability distribution of the given random variable X. It is evident that the Tsallis entropy $T_q\left(X\right)$ converges to the Shannon entropy $-{\displaystyle \sum _x}p\left(x\right)logp\left(x\right)$ 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 [16].

3.1. Tsallis Relative Entropy without $\Phi$

For Tsallis relative entropy for self-adjoint operators $\mathit{A}>\mathbf{0}$ and $\mathit{B}$, denoted by $T_q(\mathit{A}\Vert \mathit{B})$, it is defined as [11]:

$$\begin{array}{ccc}\hfill T_q(\mathit{A}\Vert \mathit{B})& \stackrel{def}{=}& {\displaystyle \frac{\mathit{A}{\#}_q\mathit{B}-\mathit{A}}{q}},\hfill \end{array}$$

(30)

where $-1\le q\le 1$ with $q\ne 0$, and $\mathit{A}{\#}_q\mathit{B}\stackrel{def}{=}{\mathit{A}}^{1/2}{\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)}^q{\mathit{A}}^{1/2}$. We have the following Lemma 4 about $T_q(\mathit{A}\Vert \mathit{B})$ bounds.

Lemma 4.

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

$$\begin{array}{ccc}\hfill \Gamma (\mathit{A},\mathit{B},q)& \stackrel{def}{=}& -{\displaystyle \frac{(\mathit{B}-m\mathit{A})(1-M^q)+(M\mathit{A}-\mathit{B})(1-m^q)}{q(M-m)}},\hfill \\ \hfill \Psi (\mathit{A},\mathit{B})& \stackrel{def}{=}& \mathit{A}{\#}_2\mathit{B}-(M+m)\mathit{B}+Mm\mathit{A},\hfill \\ \hfill \Omega (\mathit{A},q)& \stackrel{def}{=}& {\displaystyle \frac{m^q\mathit{A}}{{(M+m)}^{Mm}}}.\hfill \end{array}$$

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

$$\begin{array}{ccc}\hfill T_q(\mathit{A}\Vert \mathit{B})& \ge & \Gamma (\mathit{A},\mathit{B},q)-{\displaystyle \frac{(1-q)M^{q-2}}{2}}\Psi (\mathit{A},\mathit{B})+\Omega (\mathit{A},q),\hfill \\ \hfill T_q(\mathit{A}\Vert \mathit{B})& \le & \Gamma (\mathit{A},\mathit{B},q)-{\displaystyle \frac{(1-q)m^{q-2}}{2}}\Psi (\mathit{A},\mathit{B})-\Omega (\mathit{A},q).\hfill \end{array}$$

Proof. 

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

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1-x^q}{q}}& \ge & {\displaystyle \frac{(x-m)(1-M^q)+(M-x)(1-m^q)}{q(M-m)}}+{\displaystyle \frac{(1-q)m^{q-2}}{2}}\left(x^2-(M+m)x+Mm\right)\hfill \\ & & +{\displaystyle \frac{m^q}{{(M+m)}^{Mm}}},\hfill \\ \hfill {\displaystyle \frac{1-x^q}{q}}& \le & {\displaystyle \frac{(x-m)(1-M^q)+(M-x)(1-m^q)}{q(M-m)}}+{\displaystyle \frac{(1-q)M^{q-2}}{2}}\left(x^2-(M+m)x+Mm\right)\hfill \\ & & -{\displaystyle \frac{m^q}{{(M+m)}^{Mm}}}.\hfill \end{array}$$

This lemma is proven by setting x with the positive self-adjoint operator ${\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}$ in (30) followed by multiplying both sides by ${\mathit{A}}^{1/2}$. □

From Lemma 4, we have the following lemma immediately about tighter bounds of the relative operator entropy provided by [11] by taking $q\to 0$.

Lemma 5.

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

$$\begin{array}{ccc}\hfill S(\mathit{A}\Vert \mathit{B})& \stackrel{def}{=}& {\mathit{A}}^{1/2}log\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right){\mathit{A}}^{1/2}.\hfill \end{array}$$

Under the same conditions provided by Lemma 4, we have

$$\begin{array}{ccc}\hfill S(\mathit{A}\Vert \mathit{B})& \ge & {\displaystyle \frac{(\mathit{B}-m\mathit{A})logM+(M\mathit{A}-\mathit{B})logm}{(M-m)}}-{\displaystyle \frac{1}{2M^2}}\Psi (\mathit{A},\mathit{B})+\Omega (\mathit{A},0),\hfill \\ \hfill S(\mathit{A}\Vert \mathit{B})& \le & {\displaystyle \frac{(\mathit{B}-m\mathit{A})logM+(M\mathit{A}-\mathit{B})logm}{(M-m)}}-{\displaystyle \frac{1}{2m^2}}\Psi (\mathit{A},\mathit{B})-\Omega (\mathit{A},0).\hfill \end{array}$$

Proof. 

Because we have

$$\begin{array}{c}\hfill \underset{q\to 0}{lim}{\displaystyle \frac{1-x^q}{q}}=-logx,\end{array}$$

(31)

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

$$\begin{array}{ccc}\hfill \underset{q\to 0}{lim}\Gamma (\mathit{A},\mathit{B},q)& =& {\displaystyle \frac{(\mathit{B}-m\mathit{A})logM+(M\mathit{A}-\mathit{B})logm}{(M-m)}}.\hfill \end{array}$$

(32)

This lemma is proven by applying Equation (32) to Lemma 4. □

Remark 2.

Lemma 5 provides tighter bounds of the relative operator entropy given by [11] as $\Omega (\mathit{A},0)$ is a positive operator.

3.2. Tsallis Relative Entropy with $\Phi$

In Section 3.1, operators $\mathit{A}$ and $\mathit{B}$ in the Tsallis relative entropy are applied directly into Equation (30). In this section, we consider applying $\Phi$ to operators $\mathit{A}$, $\mathit{B}$ and $T_q(\mathit{A}\Vert \mathit{B})$. The mapping $\Phi$ is assumed to have the following format:

$$\begin{array}{ccc}\hfill \Phi \left(\mathit{X}\right)& =& {\mathit{V}}^{*}\left(a_0\mathit{I}+a_1\mathit{X}+a_2{\mathit{X}}^2\right)\mathit{V},\hfill \end{array}$$

(33)

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 $\Phi \left(\mathit{A}\right)$ and $\Phi \left(\mathit{B}\right)$.

Lemma 6.

Given two positive self-adjoint operators $\mathit{A}$ and $\mathit{B}$ satisfying $m\Phi \left(\mathit{A}\right)\le \Phi \left(\mathit{B}\right)\le M\Phi \left(\mathit{A}\right)$, where Φ is given by Equation (33) and constants $M,m$ satisfy $0<m<M$ with $m\ge 2$ and $M\ge 5m$, we define following relations:

$$\begin{array}{ccc}\hfill \Gamma (\Phi (\mathit{A}),\Phi (\mathit{B}),q)& \stackrel{def}{=}& -{\displaystyle \frac{(\Phi \left(\mathit{B}\right)-m\Phi \left(\mathit{A}\right))(1-M^q)+(M\Phi \left(\mathit{A}\right)-\Phi \left(\mathit{B}\right))(1-m^q)}{q(M-m)}},\hfill \\ \hfill \Psi (\Phi (\mathit{A}),\Phi (\mathit{B}\left)\right)& \stackrel{def}{=}& \Phi \left(\mathit{A}\right){\#}_2\Phi \left(\mathit{B}\right)-(M+m)\Phi \left(\mathit{B}\right)+Mm\Phi \left(\mathit{A}\right),\hfill \\ \hfill \Omega (\Phi (\mathit{A}),q)& \stackrel{def}{=}& {\displaystyle \frac{m^q\Phi \left(\mathit{A}\right)}{{(M+m)}^{Mm}}}.\hfill \end{array}$$

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

$$\begin{array}{ccc}\hfill T_q(\Phi \left(\mathit{A}\right)\Vert \Phi \left(\mathit{B}\right))& \ge & \Gamma (\Phi \left(\mathit{A}\right),\Phi \left(\mathit{B}\right),q)-{\displaystyle \frac{(1-q)M^{q-2}}{2}}\Psi (\Phi \left(\mathit{A}\right),\Phi \left(\mathit{B}\right))+\Omega (\Phi \left(\mathit{A}\right),q),\hfill \\ \hfill T_q(\Phi \left(\mathit{A}\right)\Vert \Phi \left(\mathit{B}\right))& \le & \Gamma (\Phi \left(\mathit{A}\right),\Phi \left(\mathit{B}\right),q)-{\displaystyle \frac{(1-q)m^{q-2}}{2}}\Psi (\Phi \left(\mathit{A}\right),\Phi \left(\mathit{B}\right))-\Omega (\Phi \left(\mathit{A}\right),q).\hfill \end{array}$$

Proof. 

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

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1-x^q}{q}}& \ge & {\displaystyle \frac{(x-m)(1-M^q)+(M-x)(1-m^q)}{q(M-m)}}+{\displaystyle \frac{(1-q)m^{q-2}}{2}}\left(x^2-(M+m)x+Mm\right)\hfill \\ & & +{\displaystyle \frac{m^q}{{(M+m)}^{Mm}}},\hfill \\ \hfill {\displaystyle \frac{1-x^q}{q}}& \le & {\displaystyle \frac{(x-m)(1-M^q)+(M-x)(1-m^q)}{q(M-m)}}+{\displaystyle \frac{(1-q)M^{q-2}}{2}}\left(x^2-(M+m)x+Mm\right)\hfill \\ & & -{\displaystyle \frac{m^q}{{(M+m)}^{Mm}}}.\hfill \end{array}$$

(34)

This lemma is proven by setting x with the positive self-adjoint operator $\Phi {\left(\mathit{A}\right)}^{-1/2}\Phi \left(\mathit{B}\right)$$\Phi {\left(\mathit{A}\right)}^{-1/2}$ in Equation (34) followed by multiplying both sides by ${\mathit{A}}^{1/2}$. □

Lemma 7 below is established to provide the bounds for $\Phi \left(T_q(\mathit{A}\Vert \mathit{B})\right)$.

Lemma 7.

Given Φ defined by Equation (33), and two positive self-adjoint operators $\mathit{A}$ and $\mathit{B}$ satisfying $m\mathit{A}\le \mathit{B}\le M\mathit{A}$, where $0<m<M$ with $m\ge 2$ and $M\ge 5m$, we set up the follownig two polynomials $p_{\mathcal{L}}\left(x\right)$ and $p_{\mathcal{U}}\left(x\right)$:

$$\begin{array}{ccc}\hfill p_{\mathcal{L}}\left(x\right)& =& -{\displaystyle \frac{(x-m)(1-M^q)+(M-x)(1-m^q)}{q(M-m)}}-{\displaystyle \frac{(1-q)M^{q-2}}{2}}\left(x^2-(M+m)x+Mm\right)\hfill \\ & & +{\displaystyle \frac{m^q}{{(M+m)}^{Mm}}},\hfill \\ \hfill p_{\mathcal{U}}\left(x\right)& =& -{\displaystyle \frac{(x-m)(1-M^q)+(M-x)(1-m^q)}{q(M-m)}}-{\displaystyle \frac{(1-q)m^{q-2}}{2}}\left(x^2-(M+m)x+Mm\right)\hfill \\ & & -{\displaystyle \frac{m^q}{{(M+m)}^{Mm}}},\hfill \end{array}$$

(35)

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

$$\begin{array}{ccc}\hfill \Phi \left(T_q(\mathit{A}\Vert \mathit{B})\right)& \ge & {\mathit{V}}^{*}\left\{a_0\mathit{I}+a_1{\mathcal{K}}^{-1}(inf(\Lambda \left(p_{\mathcal{L}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\right)),sup(\Lambda \left(p_{\mathcal{L}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\right)),1)\right.\hfill \\ & & ·{\mathit{A}}^{1/2}{p}_{\mathcal{L}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right){\mathit{A}}^{1/2}\hfill \\ & & +a_2{\mathcal{K}}^{-1}(inf(\Lambda \left(p_{\mathcal{L}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\right)),sup(\Lambda \left(p_{\mathcal{L}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\right)),2)\hfill \\ & & \left.·\mathit{A}{p}_{\mathcal{L}}^2\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\mathit{A}\right\}\mathit{V},\hfill \end{array}$$

(36)

$$\begin{array}{ccc}\hfill \Phi \left(T_q(\mathit{A}\Vert \mathit{B})\right)& \le & {\mathit{V}}^{*}\left\{a_0\mathit{I}+a_1\mathcal{K}(inf(\Lambda \left(p_{\mathcal{U}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\right)),sup(\Lambda \left(p_{\mathcal{U}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\right)),1)\right.\hfill \\ & & ·{\mathit{A}}^{1/2}{p}_{\mathcal{U}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right){\mathit{A}}^{1/2}\hfill \\ & & +a_2\mathcal{K}(inf(\Lambda \left(p_{\mathcal{U}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\right)),sup(\Lambda \left(p_{\mathcal{U}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\right)),2)\hfill \\ & & \left.·\mathit{A}{p}_{\mathcal{U}}^2\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\mathit{A}\right\}\mathit{V}.\hfill \end{array}$$

(37)

Proof. 

Because we have

$$\begin{array}{ccc}\hfill {\displaystyle \frac{x^q-1}{q}}& \ge & -{\displaystyle \frac{(x-m)(1-M^q)+(M-x)(1-m^q)}{q(M-m)}}-{\displaystyle \frac{(1-q)M^{q-2}}{2}}\left(x^2-(M+m)x+Mm\right)\hfill \\ & & +{\displaystyle \frac{m^q}{{(M+m)}^{Mm}}}{=}_1{p}_{\mathcal{L}}\left(x\right),\hfill \end{array}$$

(38)

where ${=}_1$ comes from Equation (35), by applying ${\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}$ to x in Equation (38), we also have

$$\begin{array}{ccc}\hfill \Phi \left(T_q(\mathit{A}\Vert \mathit{B})\right)& =& \Phi \left({\mathit{A}}^{1/2}\left({\displaystyle \frac{{\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)}^q-\mathit{I}}{q}}\right){\mathit{A}}^{1/2}\right)\hfill \\ & {=}_1& {\mathit{V}}^{*}\left\{a_0\mathit{I}+a_1{\mathit{A}}^{1/2}\left({\displaystyle \frac{{\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)}^q-\mathit{I}}{q}}\right){\mathit{A}}^{1/2}\right.\hfill \\ & & +\left.a_2\mathit{A}{\left({\displaystyle \frac{{\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)}^q-\mathit{I}}{q}}\right)}^2\mathit{A}\right\}\mathit{V}.\hfill \\ & {\ge }_2& {\mathit{V}}^{*}\left\{a_0\mathit{I}+a_1{\mathcal{K}}^{-1}(inf(\Lambda \left(p_{\mathcal{L}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\right)),sup(\Lambda \left(p_{\mathcal{L}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\right)),1)\right.\hfill \\ & & ·{\mathit{A}}^{1/2}{p}_{\mathcal{L}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right){\mathit{A}}^{1/2}\hfill \\ & & +a_2{\mathcal{K}}^{-1}(inf(\Lambda \left(p_{\mathcal{L}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\right)),sup(\Lambda \left(p_{\mathcal{L}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\right)),2)\hfill \\ & & \left.·\mathit{A}{p}_{\mathcal{L}}^2\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\mathit{A}\right\}\mathit{V},\hfill \end{array}$$

where ${\mathit{A}}^{1/2}$ and ${\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)}^q$ are commutative in ${=}_1$, and Lemma 2 is used to obtain ${\ge }_2$. This establishes Equation (36).

Similarly, we can apply the following inequality to obtain Equation (37) by the same argument to derive Equation (36):

$$\begin{array}{ccc}\hfill {\displaystyle \frac{x^q-1}{q}}& \le & -{\displaystyle \frac{(x-m)(1-M^q)+(M-x)(1-m^q)}{q(M-m)}}-{\displaystyle \frac{(1-q)m^{q-2}}{2}}\left(x^2-(M+m)x+Mm\right)\hfill \\ & & -{\displaystyle \frac{m^q}{{(M+m)}^{Mm}}}.\hfill \end{array}$$

□

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

Theorem 2.

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

$$\begin{array}{ccc}\hfill \Phi \left(T_q(\mathit{A}\Vert \mathit{B})\right)-(W(\mathit{A},\mathit{B})-Z(\mathit{A},\mathit{B}))& \le & T_q(\Phi \left(\mathit{A}\right)\Vert \Phi \left(\mathit{B}\right))\hfill \\ & \le & \Phi \left(T_q(\mathit{A}\Vert \mathit{B})\right)-(X(\mathit{A},\mathit{B})-Y(\mathit{A},\mathit{B})).\hfill \end{array}$$

where we have

$$\begin{array}{ccc}\hfill W(\mathit{A},\mathit{B})& =& {\mathit{V}}^{*}\left\{a_0\mathit{I}+a_1\mathcal{K}(inf(\Lambda \left(p_{\mathcal{U}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\right)),sup(\Lambda \left(p_{\mathcal{U}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\right)),1)\right.\hfill \\ & & ·{\mathit{A}}^{1/2}{p}_{\mathcal{L}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right){\mathit{A}}^{1/2}\hfill \\ & & +a_2\mathcal{K}(inf(\Lambda \left(p_{\mathcal{U}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\right)),sup(\Lambda \left(p_{\mathcal{U}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\right)),2)\hfill \\ & & \left.·\mathit{A}{p}_{\mathcal{L}}^2\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\mathit{A}\right\}\mathit{V},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill X(\mathit{A},\mathit{B})& =& {\mathit{V}}^{*}\left\{a_0\mathit{I}+a_1{\mathcal{K}}^{-1}(inf(\Lambda \left(p_{\mathcal{L}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\right)),sup(\Lambda \left(p_{\mathcal{L}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\right)),1)\right.\hfill \\ & & ·{\mathit{A}}^{1/2}{p}_{\mathcal{L}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right){\mathit{A}}^{1/2}\hfill \\ & & +a_2{\mathcal{K}}^{-1}(inf(\Lambda \left(p_{\mathcal{L}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\right)),sup(\Lambda \left(p_{\mathcal{L}}\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\right)),2)\hfill \\ & & \left.·\mathit{A}{p}_{\mathcal{L}}^2\left({\mathit{A}}^{-1/2}\mathit{B}{\mathit{A}}^{-1/2}\right)\mathit{A}\right\}\mathit{V},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill Y(\mathit{A},\mathit{B})& =& \Gamma (\Phi \left(\mathit{A}\right),\Phi \left(\mathit{B}\right),q)-{\displaystyle \frac{(1-q)m^{q-2}}{2}}\Psi (\Phi \left(\mathit{A}\right),\Phi \left(\mathit{B}\right))-\Omega (\Phi \left(\mathit{A}\right),q),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill Z(\mathit{A},\mathit{B})& =& \Gamma (\Phi \left(\mathit{A}\right),\Phi \left(\mathit{B}\right),q)-{\displaystyle \frac{(1-q)M^{q-2}}{2}}\Psi (\Phi \left(\mathit{A}\right),\Phi \left(\mathit{B}\right))+\Omega (\Phi \left(\mathit{A}\right),q).\hfill \end{array}$$

Proof. 

By setting $c=d=e=1$ and replacing $W\left(\mathit{A}\right),X\left(\mathit{A}\right),Y\left(\mathit{A}\right)$, and $W\left(\mathit{A}\right)$ with $W(\mathit{A},\mathit{B}),X(\mathit{A},\mathit{B})$, $Y(\mathit{A},\mathit{B})$, and $Z(\mathit{A},\mathit{B})$, respectively, obtained from Lemmas 6 and 7 in Theorem 1. □

Remark 3.

All of the results discussed in this work can also be applied to tensors if under Einstein products they are treated as operators.

4. Conclusions

In conclusion, the Choi–Davis–Jensen’s inequality, renowned for its wide-ranging applications in science and engineering, has led to significant advancements through its generalizations. This study extends the original inequality by incorporating a nonlinear map in place of a normalized linear map and broadens the concept of operator convex functions to encompass any continuous function within a compact region. Our results are particularly relevant to operators, such as matrices with structural symmetry, which enhances the practical utility of these findings. The Stone–Weierstrass theorem and the Kantorovich function were instrumental in the development and proof of these generalized inequalities. Additionally, we illustrate the applicability of this generalized inequality within the realm of statistical physics.

Based on the current work, several future research directions emerge. One potential avenue is the exploration of further generalizations of the Choi–Davis–Jensen’s inequality by considering other types of nonlinear maps or extending the framework to non-compact regions. Another promising direction is to investigate the impact of these generalized inequalities in different scientific and engineering applications, particularly in areas where operator convexity plays a critical role, such as quantum information theory and optimization. Additionally, there is significant potential in studying the dynamic properties of ensembles of operators, especially how the generalized inequalities can be applied to understand the time evolution, stability, and convergence behaviors of operator systems in dynamic environments, which could have implications in fields like statistical physics and control theory.