Operator (p, )-Convexity and Some Classical Inequalities

Abstract

In this paper, we will introduce the definition of operator -convex functions, we will derive some basic properties for operator -convex function, and also check the conditions under which operations’ function preserves the operator -convexity. Furthermore, we develop famous Hermite–Hadamard, Jensen type, Schur type, and Fejér’s type inequalities for this generalized function.

1. Introduction and Preliminary

Convexity plays an essential part in optimization theory and nonlinear programming. Although, different results have been derived under convexity, most of the real-world problems are nonconvex in nature. So, it is always appreciable to study nonconvex functions, which are near to convex function approximately [1, 2].

In the twentieth century, many famous mathematicians give recognition of the subject of convex functions such as Jensen, Hermite, Holder, and Stolz [3–10]. Throughout the twentieth century, an exceptional research activity was carried out and important results were obtained in convex analysis, geometric functional analysis, and nonlinear programming [11–14]. Among the most important of all the inequalities related to convex function is doubtlessly the Hermite–Hadamard inequality:

The above inequality is very useful in many mathematical contexts and also put up as a tool for demonstrating some interesting estimations, and the literature above inequality is famously known as Hermite–Hadamard inequality [15]. If is concave, then the couple inequalities in (1) hold in reversed direction. For more studies of Hermite–Hadamard-type inequalities, we refer [8, 9, 16]. The weighted version of Hermite–Hadamard inequality is known as Fejér Inequality, and for the famous work on Fejér Inequality, we refer [17–25].

In [6], Dragomir obtained some Hermite–Hadamard inequalities, which hold for convex function of self-adjoint operators in Hilbert spaces and slaked applications for special cases of interest. For interesting works on operator convex functions, we refer [3, 5, 7].

For simplicity, now onward, we will utilize the given notations:  is Hilbert space  is an inner product   is all positive operators in   is a convex subset of   

For .

Also, let be a bifunction for appropriate . Considering self-adjoint , we write, for every , .

If is a function on which is a real-valued continuous function and is a bounded self-adjoint operator, for any , then implies that . Furthermore, if and are both real-valued function on such that for any , then .

Definition 1 (see [6]). Assume be a function, and we call it the operator convex function, iffor all and for every and , which are bounded self-adjoint operators in , and contains spectra of and . The function is called operator concave if the above inequality is reversed.

Definition 2 (see [4]). Considering a function, it is called -convex function if the following inequality holds:where and for all .

Definition 3 (see [26]). Let be a function, and we call it operator -convex function, if the next inequality is maintained,for all and for every and , which are bounded self-adjoint operators in , where contains spectra of and . The above function is called operator -concave function, if the above inequality is reversed.

Remark 1. Equation (4) reduces to the operator convex function for .

Definition 4 (see [27]). Suppose a function , and we call it -convex function, iffor all , , and is a -convex set.

Definition 5. Let be a bifunction for appropriate and be a -convex set; then, we call -convex function, iffor all and .
The paper is organized as follows. Section 2 is devoted for some basic properties, and Section 2.1 is devoted to Schur-type inequality for operator -convexity. However, Sections 2.2–2.4 are devoted for Hermite–Hadamard-, Jensen-, and Fejér-type inequalities, respectively.

2. Basic Properties

Now, we are ready to set forth the definition of operator -convex function.

Definition 6. Considering a function, we call it operator -convex function, if the following inequality is maintained:for all and for every and which are bounded self-adjoint operators in , where contains spectra of and .
The above function in (7) is known as operator -concave function, if the above inequality is reversed.

Example 1. Let be a function, where and also ; then, is operator -convex function.

Proof. TakeHence, is an operator -convex function.

Proposition 1. Considering as two operators convex functions, the following holds:(i)If is additive, then is operator -convex function(ii)If is nonnegatively homogenous, then, for any , is an operator -convex function

Proof. (i)Using operator -convexity, we have for all , D and , where contains the spectra of and . By summing up the above inequalities (9) and (10), implies that is an operator -convex.(ii)Consider implies that is an operator -convex function.

Theorem 1. Assume , , is the nonempty collection of operator -convex functions such that(a)There exist and such that for all C, D whose spectra contained in I(b)For each , exists in ; then, is defined by for each is operator -convex function.

Proof. For any and , we have

2.1. Schur-Type Inequality

Theorem 2. Let be a bifunction for appropriate and let be a function defined on interval I such that is operator -convex function. Then, for all such that and , the following inequality holds:

Proof. Let be an operator -convex function and let be given. Then, we haveInvoking (4), for , , and , we have andAssuming and after the multiplication on the above inequality by , we will obtain inequality (14).

2.2. Hermite–Hadamard-Type Inequalities

Next, we employ the Hermite–Hadmard-type inequality for the above said generalization.

Theorem 3. Assume be operator -convex function for any C and D, whose spectra is contained in with condition ; then, the next estimate holds:

Proof. Take and , which impliesBy definition of operator -convex function, we haveIntegrating the above inequality w.r.t “” on , we will obtainwhich impliesNow,which impliesSimilarly,Summing up (21) and (23) yieldsCombining (21) and (25) and small calculation yields (17).

Remark 2. (17) is the classical Hermite–Hadamard-type inequality for the operator convex function for and .

2.3. Jensen-Type Inequalities

Lemma 1. Suppose be an operator -convex function, for , where contains the spectra of and and , and we haveAlso, when , for , whose spectra is contained in , where and , we have

Now, in the proof of next theorem, we will utilize the above lemma.

Theorem 4 (Jensen-type inequality). Let with and for , whose spectra is contained in . Let be an operator -convex function and be nondecreasing and nonnegatively sublinear in the first variable; then, we have the following inequality:where , alsoand for all whose spectra contained in .

Proof. Since is nondecreasing and nonnegatively sublinear in the first variable, so from the above lemma it yields thatHence, the proof is completed.

Remark 3 (28). is the Jensen-type inequality for operator -convex functions for .

Remark 4 (28). is the Jensen-type inequality for the operator convex function for and .

2.4. Fejér-Type Inequality

Theorem 5. Let u, be nonnegative operator -convex functions such that ; then,where

Proof. Since and are operator -convex functions, we havefor all . Since and are nonnegative, soIntegrating (34) over , we will obtain the following inequality:Setting , we obtainThen,

Remark 5. If we put and in (31), then it reduces for operator convex functions.

3. Conclusion

In this report, we introduced the definition of operator -convex functions and derived some basic properties for operator -convex function. We also gave the conditions under which operations’ function preserves the operator -convexity. Furthermore, we developed famous Hermite–Hadamard, Jensen-type, Schur-type, and Fejér-type inequalities for this generalized function.

Data Availability

All data used in this study are included within this paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors have equally contributed to the article.

Acknowledgments

This work was supported by the Doctoral Program of Guizhou Normal College in 2020 (no. 2020BS001), Higher Education Content and Curriculum System Reform Project of Guizhou Province in 2019 (no. 2019083), Specialized Fund for Science and Technology Platform and Talent Team Project of Guizhou Province (no. QianKeHePingTaiRenCai [2016]5609), the Key Disciplines of Guizhou Province–Computer Science and Technology (ZDXK [2018]007), First-Class C Discipline Project of Guizhou Normal College in 2019 (no. 2019YLXKC02), and the Key Supported Disciplines of Guizhou Province–Computer Application Technology (no. QianXueWeiHeZi ZDXK [2016]20).