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
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
:
where the operator inequality ≤ is defined as Loewner ordering, i.e.,
if
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
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
and < besides operators are based on Loewner ordering. The symbol
represents the spectrum of the operator
, i.e., the set of eigenvalues of the operator
. If
are composed by real numbers, we use
and
to represent the infimum and the supremum numbers within
. Given
with any
and
, Kantorovich function with respect to
is defined by
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
, defined on the closed real interval
, where
and
. For any given positive
, there exists a polynomial denoted as
, such that for all
x within the interval
, the absolute difference between
and
is less than
. In other words, the supremum norm of the function difference, denoted as
, is less than
[
13]. Given a continuous real-valued function
and error bound
, we can apply Lagrange polynomial interpolation method based on Stone-Weierstrass Theorem to find an upper polynomial
and a lower polynomial
with respect to
such that:
In this paper, we also require (assumption) that is a self-adjoint operator if is a self-adjoint operator.
Given two Hilbert spaces
and
,
and
represent semi-algebra of all bounded linear operators on Hilbert spaces
and
, respectively. Recall that
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 is a normalized positive linear map if we three conditions are jointly satisfied:
The map Φ is a linear map, i.e., for any and any .
The linear map Φ is a positive map if the operator order (Loewner order) is preserved, i.e., implies .
The linear map Φ is a normalized map if the identity operator is preserved, i.e., , where and are the identity operators of the Hilbert spaces and , respectively.
In this work, we consider a more general
by assuming that
where
is isometry in
, such that
,
represent those nonnegative coefficients in
, and
represent those negative coefficients in
. The collection of
forms the set
, and the collection of
forms the set
. Note that we do not require a linear map, positive map, and normalized map for
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
as the identity map, i.e., all
, except
.
We require the following Lemma 1 to provide the lower and upper bounds for if is a self-adjoint operator.
Lemma 1. Given a self-adjoint operator with , such thatfor with polynomials and expressed by Further, assume that , we haveandwhere Kantorovich functions and are defined by Equation (2). Moreover, we also haveandwhere the Kantorovich function , and are defined by Equation (2). Proof. From Equation (
5) and the spectrum mapping theorem of the self-adjoint operator
[
14], we have
and
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 and instead of using because 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 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 On the other hand, we also have Proof. This Lemma is proved by applying Lemma 1 to the definition of
provided by Equation (
4). □
The following Lemma 3 is used to provide the upper and lower bounds for .
Lemma 3. Given the mapping Φ
defined by Equation (4) and a self-adjoint operator , we have the spectrum for , which is denoted by . (Note that the spectrum is composed of real numbers only from Φ
defined by Equation (4).) From the Stone–Weierstrass Theorem, we have for . The polynomials and can be expressed by Proof. From
, defined by Equation (
4), the operator
is a self-adjoint operator as the integer power of any self-adjoint operator
is a self-adjoint opeartor again. Then, we have Equation (
14) from the spectrum mapping theorem of the self-adjoint operator
and the first condition provided by Equation (
12). Similarly, we also have Equation (
15) from the spectrum mapping theorem of the self-adjoint operator
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 to be a normalized positive linear map.
Theorem 1. Given a self-adjoint operator with , such thatfor with polynomials and , expressed by We assume that and the mapping is defined by Equation (4). From such Φ,
we have for . The polynomals and can be expressed by We use the following notation simplifications:where are any positive real numbers and is an operator-valued function with as its argument.where are any positive real numbers and is an operator-valued function with as its argument.where is an operator-valued function with as its argument.where is an operator-valued function with as its argument. Then, we have Proof. From Lemma 2 and Equation (
16) with Equation (
17), we have
Moreover, from Lemma 3 and Equation (
18) with Equation (
19), we have
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 as an Hermitan matrix, and let , , and , be eigenvalues of , and respectively. Then, we havewhere represents the weakly majorization relation between the entries of two vectors. Proof. Given two
Hermitan matrices with
, from Loewner Theorem (1934) [
14,
15], we have
where
and
are eigenvalues of Hermitan matrices
and
, respectively. Therefore, Equation (
23) can be expressed by
. 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 is a continuous twice differentiable function, such that on , we have where is defined by Note that is greater than 1. As , we havetherefore, the following inequality implies Equation (24): If we set the following parameters for Theorem 1 with :where we use Φ
as a normalized positive linear map in . We will obtain the result provided by Equation (27) with , which implies Equation (24). In comparison with the lower bounds of , our method provides a tighter lower bound compared with Theorem 2.2 from [11]. On the other hand, if and , we can set the following parameters to obtain the result provided by Equation (25), which implies Equation (24):where we use in . In the follownig example, we will consider the situation that f is bounded by quadratic functions and is also generated by quadratic function.
Example 2. Given a self-adjoint operator with , such thatfor with polynomials and expressed by We assume that and the mapping is defined as follows:where we assume that and . From such Φ,
we have for . The polynomals and are assumed to be If we select real numbers as 1, we have the following notation simplifications:andandand 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
, incorporating a single parameter
q to extend the concept of Shannon entropy. Here, the
q-logarithm is defined as
for any nonnegative real numbers
q and
x, while
represents the probability distribution of the given random variable
X. It is evident that the Tsallis entropy
converges to the Shannon entropy
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
For Tsallis relative entropy for self-adjoint operators
and
, denoted by
, it is defined as [
11]:
where
with
, and
. We have the following Lemma 4 about
bounds.
Lemma 4. Given two positive self-adjoint operators and satisfying , where with and , we define the following relations: Further, if we have and assumed to be self-adjoint, then, Proof. Because we have the following inequalites with respect to the function
given
with
and
:
This lemma is proven by setting
x with the positive self-adjoint operator
in (
30) followed by multiplying both sides by
. □
From Lemma 4, we have the following lemma immediately about tighter bounds of the relative operator entropy provided by [
11] by taking
.
Lemma 5. The relative operator entropy with respect to operators and , denoted by , is given by Under the same conditions provided by Lemma 4, we have Proof. Because we have
the term
becomes
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 is a positive operator. 3.2. Tsallis Relative Entropy with
In
Section 3.1, operators
and
in the Tsallis relative entropy are applied directly into Equation (
30). In this section, we consider applying
to operators
,
and
. The mapping
is assumed to have the following format:
where coefficients
and
are positive real numbers. We have the following Lemma 6 about Tsallis relative entropy with respect to
and
.
Lemma 6. Given two positive self-adjoint operators and satisfying , where Φ
is given by Equation (33) and constants satisfy with and , we define following relations: Further, if we have , then, Proof. As we have the following inequalites with respect to the function
given
with
and
:
This lemma is proven by setting
x with the positive self-adjoint operator
in Equation (
34) followed by multiplying both sides by
. □
Lemma 7 below is established to provide the bounds for .
Lemma 7. Given Φ
defined by Equation (33), and two positive self-adjoint operators and satisfying , where with and , we set up the follownig two polynomials and : where . Moreover, we also assume that and are commute, then, we have Proof. Because we have
where
comes from Equation (
35), by applying
to
x in Equation (
38), we also have
where
and
are commutative in
, and Lemma 2 is used to obtain
. This establishes Equation (
36).
Similarly, we can apply the following inequality to obtain Equation (
37) by the same argument to derive Equation (
36):
□
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 inequalitieswhere we have Proof. By setting and replacing , and with , , and , 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.