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Omrani, Zahra; Pourbahri Rahpeyma, Omid; Rahimi, Hamiderza

doi:https://doi.org/10.1155/2022/3836837

Journal of Mathematics

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Research Article | Open Access

Volume 2022 | Article ID 3836837 | https://doi.org/10.1155/2022/3836837

Some Inequalities for Operator -Convex Function

Zahra Omrani,¹Omid Pourbahri Rahpeyma,²and Hamiderza Rahimi¹

Academic Editor: Ji Gao

Received25 Oct 2021

Revised14 Dec 2021

Accepted06 Jan 2022

Published07 Feb 2022

Abstract

In this paper, we introduce operator -convex functions and establish a Hermite–Hadamard inequality for these functions. As application, we obtain several trace and singular value inequalities of operators.

1. Introduction

In recent years, several extensions and generalizations have been considered for classical convexity and the theory of inequalities has made essential contributions to many areas of mathematics.

In 1973, Elliott Lieb published a ground-breaking article on operator inequalities [1]. This and a subsequent article by Lieb and Ruskai [2] have had a profound effect on quantum statistical mechanics and more recently on quantum information theory. Since then, a number of attempts have been made to elucidate and extend these results. Two elegant examples are those of Nielsen and Petz [3] and Ruskai [4], which use the analytic representations for operator convex functions. In addition, Hansen [5] has developed a powerful theory that utilizes geometric means of positive operators. The latter notion was formulated by Pusz and Woronowicz [6] and subsequently investigated by Ando [7] (see the discussion in Section 3) and by Kubo and Ando [8].

We recall some concepts of convexity that is well-known in the literature.

The following inequality holds for any convex function defined on and , with ,

It was firstly discovered by Hermite in 1881 in the journal Mathematics (see [9]) and independetly proved in 1893 by Hadamard in [10]. The inequality (1) is known in the literature as the Hermite–Hadamard inequality.

The Hermite–Hadamard inequality has several applications in nonlinear analysis and the geometry of Banach space (see [11]).

The Hermite–Hadamard inequality has been the subject of intensive research; many applications, generalizations, and improvement of them can be found in the literature (see [12]).

In recent years, many scholars have been interested in modifying and extending the Hermite–Hadamard inequality.

In [13], Dragomir and Fitzpatrick proved the following version of Hermite–Hadamard inequality for -convex functions in the second sense: let be an -convex function, where and , . If , then the following inequalities hold:

The authors of [13] defined the operator s-convex and proved the following inequality for operator s-convex function. He proved that if is an operator -convex function, then the following inequalities hold:

The following inequalities due to the authors [14] give the Hermite–Hadamard inequalities for operator -convex function.

Let be an operator -convex function. Then,for any self-adjoint operators and with spectra in .

Motivated by the above results, we investigate in this paper the operator version of the Hermite–Hadamard inequality for operator -preinvex function.

Let stand for the -algebra of all bounded linear operators on a complex separable Hilbert space . An operator is positive if for all (we write ). A positive invertible operator is naturally denoted by . Let stand for all positive operators . For self-adjoint operators , we write if . A linear map is positive if whenever and is said to be unital if .

Let be a self-adjoint operator in . The Gelfand map establishes a -isometrically isomorphism between the set of all continuous functions defined on the spectrum of denoting and -algebra generated by and the identity operator on as follows:

For any and any , we have(i)(ii) and (iii)(iv) and , where and for all

If is a continuous complex valued function on , the element of is denoted by and we call it the continuous functional calculus for a bounded self-adjoint operator . If is a bounded self-adjoint operator and is a real-valued continuous function on , then for any implies that is a positive operator on . Moreover, if both and are real-valued functions on such that for any , then in the operator order in .

An important and useful class of functions are called operator convex functions. A real-valued continuous function on interval is said to be operator convex (operator concave) ifin the operator order in , for all and for every bounded self-adjoint operators and in whose spectra are contained in .

is -convex set iffor every positive operators with spectra in , and .

Let such that . A function is called super-multiplicative function if for all .

In this paper, we assume that is a -convex set. We introduce operator -convex function. We establish some properties of operator -convex function. This paper is organized as follows: In Section 2, we will show some properties of operator -convex functions. In Section 3, a new refinement of the Hermite–Hadamard type inequality is presented for operator -convex functions. If is an operator -convex function, then

Our results enable us to obtain a new inequality for positive operators on . For example, let and be an unital linear operator positive, thenfor any positive operators and belonging to with spectra in .

In some special cases, we show that our result gives a generalized estimation for operator -convex functions than the corresponding results obtained in [15].

In Section 4, we will show that if is an operator -convex function, then we havefor any self-adjoint operators and . We establish several trace inequalities for positive operator on .

2. Some Properties of Operator -Convex Functions

In order to obtain the main result of this section, we need the following known definitions.

Let be a real-valued continuous function defined on an interval . We say that is of operator -class function on iffor all self-adjoint operators , with spectra in and all .

For some properties of this class of operators, see [16].

Let be a continuous real-valued function defined on an interval . We say that is an operator -class function on iffor all self-adjoint operators , with spectra in and all .

See [17] for some results and inequalities on operator -class function.

A continuous function is said to be operator -convex on iffor all and for every positive operators and in , whose spectra are contained in for some fixed .

For some properties about operator -convex function, see [18].

Let , , and be a nonnegative function nonidentical to 0. We say that a continuous function is said be a operator h-convex function on iffor every whose spectra are contained in for all .

For some results on operator -convex, see [14].

Lemma 1 (see [19]). Let , then if and only if for all nonnegative operator monotone function on .

Lemma 2 (see [20]). Let be a unital positive linear map on and an operator monotone function on , then for every ,

Lemma 3 (Davis-Choi-Jensen’s inequality). Let be a unital positive linear map on and an operator convex function on , then for every ,

The following lemma is the special case of Lemmas 2 and 3.

Lemma 4 (see [20]). Let be a unital positive linear map on , then(i) for every and (ii) for every and or

The following lemma is a consequence of theorem Hansen-Pedersen-Yensen’s inequality.

Lemma 5 (see [21]). Let be a continuous function mapping the positive half-line into itself. Then, is the operator monotone if and only if it is operator concave.

Definition 1 (see [22]). Let and , , and be a nonnegative function nonidentical to 0. A continuous function is said to be operator -convex (concave) iffor any positive operators with spectra in and .
This class contains several well-known classes of nonnegative operator convex function, operator -class function, operator -class function, operator -convex function, and operator -convex functions on . We can see some results operator -convex functions by Dinh and Khue in [22].

Lemma 6 (see [22]). Let be a unital positive linear map on , a positive operator in , and an operator -convex function on such that . Let be a nonnegative and nonzero super-multiplicative function on satisfying . Then,

Now, let us prove some properties of operator -convex functions. In this paper, we suppose and .

Proposition 1. Let and be a nonzero function and be an operator -convex. Therefore, for positive operator with spectra in ,(iIf , then (ii)If , then

Proof. (i)Suppose that is a positive operator with spectra in and in (16). Then, . Therefore, . Hence, and so .(ii)On account of the mentioned, .

Proposition 2. Let be a nonzero function; therefore,(i)Let be an operator -concave. If is an operator monotone and , then is an operator -concave.(ii)Let be an operator -convex and operator monotone. Then, for all , is the operator -convex function.

Proof. (i)Let be an operator -concave and and are positive operators with spectra in , then So, Therefore, is the operator -concave.(ii)If , then is the operator convex. For positive operators and with spectra in , we have . Since , then is the operator monotone. Hence, We have Then, is the operator -convex function.

Proposition 3. Let be a unital positive linear map on , and be positive operators in with spectra in , and be an operator -convex function such that . Let be a super-multiplicative function satisfying for . We have

Proof. By Lemma 7, . If , then

Corollary 1. With conditions, Proposition 3, we have

Proof. Put in inequality (22).

Remark 1. For , inequality (24) reduces to the inequality

Proposition 4. Let and be a nonzero function, then the following statements are equivalent:(i) is an operator -convex function(ii) is an operator -convex function

Proof. : suppose that is an operator -convex. Thus, for each positive operator and with spectra in and . For and , we have . Therefore, is an operator -convex. : the proof is similarly .

Remark 2. Let for be a nonnegative operator monotone function on and . By Lemma 1 for each , we have . Put and , . Hence, . Therefore,Therefore, for is an operator -convex for .
The following example is a characteristic example of operator -convex function.

Example 1. Let . By Remark 2, is an operator -convex function for on . By Proposition 4, is an operator -convex on .
In the following preposition, we eliminate condition and we get the new example of operator -convex.

Proposition 5. Let and and , then is the operator -convex if and only if .

Proof. If is an operator -convex, for each positive operator and with spectra in , . Hence, . The last inequality means that the function is the operator convex, which is equivalent to the condition .

Corollary 2. Let , , and , then we have(i)If , then the function is an operator -convex and operator concave(ii)If , then the function is an operator -convex and operator convex

Proof. Because for is operator concave and for is operator convex, then proof is trivial.

Proposition 6. Let be an operator -convex and be a nonzero function and are two positive operators with spectra in , then defined by is a -convex function on for any with .

Proof. For each , we have

3. Hermite–Hadamard Inequality for Operator -Convex Functions

Now, we are ready to state the main result of this paper.

Theorem 1. Let be a continuous operator -convex function and be a continuous nonzero function. Then, for any with spectra in , we have

Proof. Let in Definition 1, thenPut and . Then, we have . So,Therefore,Asthus (28) holds.

Remark 3. If in inequality (28), then Theorem 1 produces [14].

Corollary 3.3. Let , , and be an unital linear positive map, thenfor any positive operators and with spectra in .

Proof. By Corollary 2, is the operator -convex for and . Put in Theorem 1.
In particular, we have the following,

Corollary 4. Let and be an unital linear positive map, thenfor any positive operators and belonging to with spectra in .

Proof. By Example 1, is the operator -convex for and . Therefore,So,Put and in inequality (36).

Corollary 5. Let and be a positive operator, thenfor each positive operators and in .

Proof. If , then (by Lemma 4), we have

Definition 2. Let be operator -convex functions and self-adjoint operators on Hilbert space with spectra in . We defined real functions and on Hilbert space byandfor any .

Theorem 2. Let and be continuous functions. If is an operator -convex and is an operator -convex function, then for any positive operators and with spectra in and each with , the following inequality holds:

Proof. On account of the operator -convexity of and -convexity of , we givefor each and with . We haveTherefore,The proof is complete.

Corollary 6. In Theorem 2, if , then

Remark 4. If in Theorem 2, thenHence, in this case, Theorem 2 produces [15], Theorem 3.

Theorem 3. Let and be continuous functions. If is an operator -convex and is an operator -convex function, then for any positive operators and with spectra in and each with , the following inequality holds:

Proof. Since , for any with , we haveIntegrating both sides of inequality over , we get the required inequality (48).

Remark 5. In Theorem 3, if , thenTherefore, in this case, Theorem 3 produces [15], Theorem 4.

4. Some Trace Functional Inequalities for Operator

Let be two sided ideal of compact operator . For , let sup denote the usual operator norm of and be the absolute value of . The direct sum denotes the block diagonal matrix defined on (see [20]). It is clear that max .

For any operator , the operator is always positive and its unique positive square root is denoted by .

We remind some basic properties of trace for operators. Let be an orthonormal basis of ; we say that is the trace class if

The definition of does not depend on the choice of the orthonormal basis . We denote the set of trace class operators in by . We define the trace of a trace class operator to bewhere is an orthonormal basis of . is a bounded linear functional on with .

Lemma 7. Let be an operator -convex function and be a nonzero function, then is the -convex on for self-adjoint operators and with spectra in .

Proof. Because trace functional is convexity and monotonicity, then for each and , we obtainTherefore, is -convex.

Theorem 4. Let be an operator -convex function and be a nonzero function, then we havefor each self-adjoint operators and with spectra in .

Proof. is -convex on by Lemma 4. Hence, by inequality (28),Therefore, we have the desired result.

Remark 6. We think that with the new definition operator -convex function for the Jensen’s inequality and operator convex inequalities in [23], generalization or new inequalities can be obtained.

Data Availability

This article is about pure mathematics and the authors of the article express their satisfaction that the use of its data is in accordance with the policies of the journal. The data are available from the corresponding author.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2022 Zahra Omrani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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