Operator (p, η)-Convexity and Some Classical Inequalities

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]. -roughout 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:


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][4][5][6][7][8][9][10]. roughout 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][12][13][14]. Among the most important of all the inequalities related to convex function is doubtlessly the Hermite-Hadamard inequality: e 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 u 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]. e weighted version of Hermite-Hadamard inequality is known as Fejér Inequality, and for the famous work on Fejér Inequality, we refer [17][18][19][20][21][22][23][24][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: { }. Also, let η: C × C ⟶ D be a bifunction for appropriate C, D ⊆ R. Considering self-adjoint C, D ∈ B(H), we write, for every l ∈ H, C ≤ Dif < Cl, l > ≤ < Dl, l > .
If u is a function on Sp(C) which is a real-valued continuous function and S is a bounded self-adjoint operator, for any s ∈ Sp(C), then u(s) ≥ 0 implies that u(C) ≥ 0. Furthermore, if u and v are both real-valued function on Definition 1 (see [6]). Assume u: I ⊆ R ⟶ R be a function, and we call it the operator convex function, if for all s ∈ [0, 1] and for every C and D, which are bounded self-adjoint operators in B(H), and I contains spectra of C and D. e function u is called operator concave if the above inequality is reversed.
Definition 2 (see [4]). Considering u: I ⟶ R a function, it is called η-convex function if the following inequality holds: where s ∈ [0, 1] and for all l, m ∈ I.
Definition 3 (see [26]). Let u: I ⟶ R be a function, and we call it operator η-convex function, if the next inequality is maintained, for all s ∈ [0, 1] and for every C and D, which are bounded self-adjoint operators in B(H), where I contains spectra of C and D. e above function u is called operator η-concave function, if the above inequality is reversed. Definition 4 (see [27]). Suppose a function u: I ⟶ R, and we call it p-convex function, if for all l, m ∈ I, s ∈ [0, 1], and I is a p-convex set.
Definition 5. Let η: C × C ⟶ D be a bifunction for appropriate C, D⊆R and I be a p-convex set; then, we call u: I ⟶ R(p, η)-convex function, if for all l, m ∈ I and s ∈ [0, 1]. e paper is organized as follows. Section 2 is devoted for some basic properties, and Section 2.1 is devoted to Schurtype inequality for operator (p, η)-convexity. However, Sections 2.2-2.4 are devoted for Hermite-Hadamard-, Jensen-, and Fejér-type inequalities, respectively.

Basic Properties
Now, we are ready to set forth the definition of operator (p, η)-convex function.
Definition 6. Considering u: I ⟶ R a function, we call it operator (p, η)-convex function, if the following inequality is maintained: for all s ∈ [0, 1] and for every C and D which are bounded self-adjoint operators in B(H), where I contains spectra of C and D.
e above function u in (7) is known as operator (p, η)-concave function, if the above inequality is reversed.
Proof. Take
(i) Using operator (p, η)-convexity, we have for all C, D and s ∈ [0, 1], where I contains the spectra of C and D.

Journal of Mathematics
By summing up the above inequalities (9) and (10), implies that cu is an operator (p, η)-convex function.  (0, 1), the following inequality holds: Proof. Let u be an operator (p, η)-convex function and let C 1 , C 2 , C 3 ∈ I be given. en, we have Assuming

Hermite-Hadamard-Type
Inequalities. Next, we employ the Hermite-Hadmard-type inequality for the above said generalization.
Theorem 3. Assume u: I ⟶ R be operator (p, η)-convex function for any C and D, whose spectra is contained in I with condition C < D; then, the next estimate holds:

Journal of Mathematics
Proof. Take S p � sC p + (1 − s)D p and T p � (1 − s)C p + sD p , which implies By definition of operator (p, η)-convex function, we have Integrating the above inequality w.r.t "x" on [0, 1], we will obtain which implies which implies

Jensen-Type Inequalities
Lemma 1. Suppose u: I ⟶ R be an operator (p, η)-convex function, for C 1 and C 2 , where I contains the spectra of C and D and α 1 + α 2 � 1, and we have Also, when n > 2, for C 1 , C 2 , . . . , C n , whose spectra is contained in I, where n i�1 α i � 1 and T i � i j�1 α j , we have Now, in the proof of next theorem, we will utilize the above lemma.
Theorem 4 (Jensen-type inequality). Let w 1 , w 2 , . . . , w n ∈ R + with n ≥ 2 and for C 1 , C 2 , . . . , C n , whose spectra is contained in I. Let u: I ⟶ R be an operator (p, η)-convex function and η be nondecreasing and nonnegatively sublinear in the first variable; then, we have the following inequality: where W n � n i�1 w i , also η u C i , C i+1 , . . . , C n � η η l C i , C i+1 , . . . , C n−1 , u C n , and η u (C) � u(C) for all C whose spectra contained in I.

Journal of Mathematics 5
Hence, the proof is completed.
Remark 4 (28) is the Jensen-type inequality for the operator convex function for p � 1 and η(l, m) � l − m.

Fejér-Type Inequality
where Proof. Since u and v are operator (p, η)-convex functions, we have Integrating (34) over (0, 1), we will obtain the following inequality:

Conclusion
In this report, we introduced the definition of operator (p, η)-convex functions and derived some basic properties for operator (p, η)-convex function. We also gave the conditions under which operations' function preserves the operator (p, η)-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
e authors declare that they have no conflicts of interest.