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

Authors: Chuanjun Zhang; Muhammad Shoaib Saleem; Waqas Nazeer; Naqash Shoukat; Yongsheng Rao
Journal: Journal of Mathematics
 (2020)

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## Abstract

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


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## Body

## 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:(1)ua+b2≤1b−a∫abuldl≤ua+ub2.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 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]. 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:H is Hilbert space<.,.> is an inner productBH=C/C:H⟶Hbe bounded linear operatorBH+ is all positive operators in BHK is a convex subset of BH+ρD=λ∈C:D−λE−1∈LXSpD=C/ρDForC,D∈K,C,D:=1−sC+sD:s∈0,1.Also, letη:C×C⟶D be a bifunction for appropriate C,D⊆R. Considering self-adjoint C,D∈BH, we write, for every l∈H, C≤Dif<Cl,l>≤<Dl,l>.Ifu is a function on SpC which is a real-valued continuous function and S is a bounded self-adjoint operator, for any s∈SpC, then us≥0 implies that uC≥0. Furthermore, if u and v are both real-valued function on SpS such that us≤vs for any s∈SpC, then uC≤vC.Definition 1 (see [6]).
Assumeu:I⊆R⟶R be a function, and we call it the operator convex function, if(2)usC+1−sD≤suC+1−suD,for all s∈0,1 and for every C and D, which are bounded self-adjoint operators in BH, and I contains spectra of C and D. The function u is called operator concave if the above inequality is reversed.Definition 2 (see [4]).
Consideringu:I⟶ℝ a function, it is called η-convex function if the following inequality holds:(3)usl+1−sm≤um+sηul,um,where s∈0,1 and for all l,m∈I.Definition 3 (see [26]).
Letu:I⟶ℝ be a function, and we call it operator η-convex function, if the next inequality is maintained,(4)usC+1−sD≤uC+sηuC,uD,for all s∈0,1 and for every C and D, which are bounded self-adjoint operators in BH, where I contains spectra of C and D. The above function u is called operator η-concave function, if the above inequality is reversed.Remark 1.
Equation (4) reduces to the operator convex function for ηl,m=l−m.Definition 4 (see [27]).
Suppose a functionu:I⟶ℝ, and we call it p-convex function, if(5)usp+1−smp1/p≤sul+1−sum,for all l,m∈I, s∈0,1, and I is a p-convex set.Definition 5.
Letη:C×C⟶D be a bifunction for appropriate C,D⊆ℝ and I be a p-convex set; then, we call u:I⟶ℝp,η-convex function, if(6)uslp+1−smp1/p≤um+sηul,vm,for all l,m∈I and s∈0,1.
The paper is organized as follows. Section2 is devoted for some basic properties, and Section 2.1 is devoted to Schur-type inequality for operator p,η-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 operatorp,η-convex function.Definition 6.
Consideringu:I⟶ℝ a function, we call it operator p,η-convex function, if the following inequality is maintained:(7)usCp+1−sDp1/p≤uD+sηuC,uD,for all s∈0,1 and for every C and D which are bounded self-adjoint operators in BH, where I contains spectra of C and D.
The above functionu in (7) is known as operator p,η-concave function, if the above inequality is reversed.Example 1.
Letu:I⟶R be a function, where uC=Cp and ηC,D≥C−D also C≥0; then, u is operator p,η-convex function.Proof.
Take(8)usCp+1−sDp1/p=sCp+1−sDp=sCp+Dp−sDp=Dp+sCp−sDp=Dp+sCp−Dp=Dp+suC−uD≤uD+sηuC,uD.
Hence,u is an operator p,η-convex function.Proposition 1.
Consideringu,v:I⟶R as two operators p,η convex functions, the following holds:(i)
Ifη is additive, then u+v is operator p,η-convex function(ii)
Ifη is nonnegatively homogenous, then, for any c≥0, cu:I⟶R is an operator p,η-convex functionProof.
(i)
Using operatorp,η-convexity, we have(9)usCp+1−sDp1/p≤uC+sηuC,uD,(10)vsCp+1−sDp1/p≤vD+sηvC,vD,for allC, D and s∈0,1, where I contains the spectra of C and D.By summing up the above inequalities (9) and (10),(11)u+vsCp+1−sDp1/p=uD+sηuC,uD+vD+sηvC,vD≤vD+vD+sηuC,uD+ηuC,uD=uD+vD+sηuC+vD,uD+vD=u+vD+ηu+vC,u+vD,implies thatu+v is an operator p,η-convex.(ii)
Consider(12)cusCp+1−sDp1/p=cusCp+1−sDp1/p≤cuD+sηuC,uD=cuD+sηuC,uD=cuD+sηcuC,cuD=cuD+sηcuU,cuD,implies thatcu is an operator p,η-convex function.Theorem 1.
Assumeuj:I⟶R, j∈J, is the nonempty collection of operator p,η-convex functions such that(a)
There existα∈0,∞ and β∈−1,∞ such that ηC,D=αC+βD for all C, D whose spectra contained in I(b)
For eachC∈I, supj∈JujC exists in R; then, u:I⟶R is defined by uC=supj∈JujC for each C∈I is operator p,η-convex function.Proof.
For anyC,D∈I and s∈0,1, we have(13)usCp+1−sDp1/p=supj∈JujsCp+1−sDp1/p≤supj∈JujD+sηujC,ujD=supj∈JujD+sαujC+βujD=supj∈J1+βsujD+αsujC≤1+βssupj∈JujD+αssupj∈JujC=1+βsuD+αsuC=uD+sαuC+βuD=uC+sηuC,uC.


### 2.1. Schur-Type Inequality

Theorem 2.
Letη:C×C⟶B be a bifunction for appropriate C,D⊆R and let u be a function defined on interval I such that l is operator p,η-convex function. Then, for all C1,C2,C3∈I such that C1<C2<C3 and C3p−C1p,C3p−C2p,C2p−C1p∈0,1, the following inequality holds:(14)uC3C3p−C1p−uC2C3p−C1p+C3p−C2pηuC1,uC3≥0.Proof.
Letu be an operator p,η-convex function and let C1,C2,C3∈I be given. Then, we have(15)C3p−C2pC2p−C1p,C2p−C1pC3p−C1p∈0,1C3p−C2pC3p−C1p+C2p−C1pC3p−C1p=1.
Invoking (4), for s=C3p−C2p/C3p−C1p, C=C1, and D=C3, we have C2p=sCp+1−sDp and(16)uC2≤uC3+C3p−C2pC3p−C1pηuC1,uC3.
AssumingC3p−C1p>0 and after the multiplication on the above inequality by C3p−C1p, 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.
Assumeu: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:(17)uCp+Dp21/p−p2Dp−Cp∫abup−1ηuCp+Dp−up1/p,uldl≤pDp−Cp∫ablp−1uldl≤uC+uD2+14ηuC,uD+ηuD,uC.Proof.
TakeSp=sCp+1−sDp and Tp=1−sCp+sDp, which implies(18)Cp+Dp2=Sp+Tp2,uCp+Dp21/p=uSp+Tp21/p.
By definition of operatorp,η-convex function, we have(19)uCp+Dp21/p=u12sCp+1−sDp1/pp+121−sCp+sCp1/pp1/p≤u1−sCp+sVp1/p+12ηusCp+1−sDp1/p,u1−sCp+sVp1/p.
Integrating the above inequality w.r.t “x” on 0,1, we will obtain(20)uCp+Dp21/p≤∫01u1−sCp+sVp1/pds+12∫01ηusCp+1−sDp1/p,u1−sCp+sVp1/pds,which implies(21)uCP+DP21/p−p2Dp−Cp∫ablp−1ηuCp+Dp−lp1/p,uldl≤pDp−Cp∫ablp−1uldl.
Now,(22)∫ablp−1uldl=Dp−Cpp∫01usCp+1−sDp1/pds≤Dp−Cppub+∫01sηuC,uDds,which implies(23)pDp−Cp∫ablp−1uldl≤uD+∫01sηuC,uDds.
Similarly,(24)pDp−Cp∫ablp−1uldl≤uC+∫01sηuD,uCds.
Summing up (21) and (23) yields(25)pDp−Cp∫ablp−1uldl≤uC+uD2+14ηuC,uD+ηuD,uC.
Combining (21) and (25) and small calculation yields (17).Remark 2.
(17) is the classical Hermite–Hadamard-type inequality for the operator convex function for ηl,m=l−m and p=1.


### 2.3. Jensen-Type Inequalities

Lemma 1.
Supposeu:I⟶R be an operator p,η-convex function, for C1andC2, where I contains the spectra of C and D and α1+α2=1, and we have(26)uα1C1+α2C2≤uC2+αηuC1,uC2.
Also, whenn>2, for C1,C2,…,Cn, whose spectra is contained in I, where ∑i=1nαi=1 and Ti=∑j=1iαj, we have(27)u∑i=1nαiCip1/p=uTn−1∑i=1n−1αiTn−1CiP1/p+αnCn≤uCn+Tn−1ηu∑i=1n−1αiTn−1Cip1/p,uCn.Now, in the proof of next theorem, we will utilize the above lemma.Theorem 4 (Jensen-type inequality).
Letw1,w2,…,wn∈R+ with n≥2 and for C1,C2,…,Cn, 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:(28)u1Wn∑i=1nwiCip1/p≤uCn+∑i=1nWiWnηuCi,Ci+1,…,Cn,where Wn=∑i=1nwi, also(29)ηuCi,Ci+1,…,Cn=ηηlCi,Ci+1,…,Cn−1,uCn,and ηuC=uC for all C whose spectra contained in I.Proof.
Sinceη is nondecreasing and nonnegatively sublinear in the first variable, so from the above lemma it yields that(30)u1Wn∑i=1nwiCip1/p=uwnWnCnp+∑i=1n−1wiWnCip1/p=uWn−1Wn∑i=1n−1wiWn−1Cip+wnWnCnp1/p≤uCn+Wn−1Wnηu∑i=1n−1wiWn−1Cip1/p,uCn=uCn+Wn−1WnηuWn−2Wn−1∑i=1n−2wiWn−2Cip+wn−1Wn−1Cn−1p1/p,uCn≤uCn+Wn−1Wnηuun−1+Wn−2Wn−1ηu∑i=1n−2wiWn−2Cip1/p,uCn−1,uCn≤uCn+Wn−1WnηuCn−1,uCn+Wn−2Wnηηu∑i=1n−2wiWn−2Cip1/p,uln−1,uln⋮≤uCn+Wn−1WnηuCn−1,uCn+Wn−2WnηηuCn−2,uCn,uCn+⋯+W1Wnηη⋯ηuC1,uC2,uC3⋯,uCn−1,uCn=uCn+Wn−1wnηuCn−1,Cn+Wn−2WnηuCn−2,Cn−1,Cn+⋯+W1WnηuC1,C2,…,Cn−1,Cn=uCn+∑i=1n−1WiWnηuCi,Ci+1,…,Cn.
Hence, the proof is completed.Remark 3 (28).
is the Jensen-type inequality for operatorη-convex functions for p=1.Remark 4 (28).
is the Jensen-type inequality for the operator convex function forp=1 and ηl,m=l−m.


### 2.4. Fejér-Type Inequality

Theorem 5.
Letu, v be nonnegative operator p,η-convex functions a,b∈IC<D such that uv∈L1a,b; then,(31)pDp−Cp∫abup−1ulvldl≤CC,D+12DC,D,where(32)CC,D=uDCvD+13ηuC,uDηvC,vD,DC,D=vDηvC,vD+vDηuC,uD.Proof.
Sinceu and v are operator p,η-convex functions, we have(33)usUp+1−sDp1/p≤uD+sηuC,uD,vsUp+1−sDp1/p≤vD+sηvC,vD,for all s∈C,D. Since u and v are nonnegative, so(34)usUp+1−sDp1/pvsUp+1−sDp1/p≤uDvD+suDηvC,vD+svDηuC,uD+s2ηuC,uDηvC,vD.
Integrating (34) over 0,1, we will obtain the following inequality:(35)∫01usUp+1−sDp1/pvsUp+1−sDp1/pds≤∫01uDvDds+∫01suDηvC,vDds+∫01svDηuC,uDds+∫01s2ηuC,uDηvC,vDds.
Settingu=sCp+1−sDp1/p, we obtain(36)pDp−Cp∫abup−1ulvldl≤uDvD+12uDηvC,vD+12vDηuC,uD+13ηuC,uDηvC,vD.
Then,(37)pDp−Cp∫abup−1ulgldl≤CC,D+12DC,D.Remark 5.
If we putp=1 and ηl,m=l−m in (31), then it reduces for operator convex functions.


## 2.1. Schur-Type Inequality

Theorem 2.
Letη:C×C⟶B be a bifunction for appropriate C,D⊆R and let u be a function defined on interval I such that l is operator p,η-convex function. Then, for all C1,C2,C3∈I such that C1<C2<C3 and C3p−C1p,C3p−C2p,C2p−C1p∈0,1, the following inequality holds:(14)uC3C3p−C1p−uC2C3p−C1p+C3p−C2pηuC1,uC3≥0.Proof.
Letu be an operator p,η-convex function and let C1,C2,C3∈I be given. Then, we have(15)C3p−C2pC2p−C1p,C2p−C1pC3p−C1p∈0,1C3p−C2pC3p−C1p+C2p−C1pC3p−C1p=1.
Invoking (4), for s=C3p−C2p/C3p−C1p, C=C1, and D=C3, we have C2p=sCp+1−sDp and(16)uC2≤uC3+C3p−C2pC3p−C1pηuC1,uC3.
AssumingC3p−C1p>0 and after the multiplication on the above inequality by C3p−C1p, 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.
Assumeu: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:(17)uCp+Dp21/p−p2Dp−Cp∫abup−1ηuCp+Dp−up1/p,uldl≤pDp−Cp∫ablp−1uldl≤uC+uD2+14ηuC,uD+ηuD,uC.Proof.
TakeSp=sCp+1−sDp and Tp=1−sCp+sDp, which implies(18)Cp+Dp2=Sp+Tp2,uCp+Dp21/p=uSp+Tp21/p.
By definition of operatorp,η-convex function, we have(19)uCp+Dp21/p=u12sCp+1−sDp1/pp+121−sCp+sCp1/pp1/p≤u1−sCp+sVp1/p+12ηusCp+1−sDp1/p,u1−sCp+sVp1/p.
Integrating the above inequality w.r.t “x” on 0,1, we will obtain(20)uCp+Dp21/p≤∫01u1−sCp+sVp1/pds+12∫01ηusCp+1−sDp1/p,u1−sCp+sVp1/pds,which implies(21)uCP+DP21/p−p2Dp−Cp∫ablp−1ηuCp+Dp−lp1/p,uldl≤pDp−Cp∫ablp−1uldl.
Now,(22)∫ablp−1uldl=Dp−Cpp∫01usCp+1−sDp1/pds≤Dp−Cppub+∫01sηuC,uDds,which implies(23)pDp−Cp∫ablp−1uldl≤uD+∫01sηuC,uDds.
Similarly,(24)pDp−Cp∫ablp−1uldl≤uC+∫01sηuD,uCds.
Summing up (21) and (23) yields(25)pDp−Cp∫ablp−1uldl≤uC+uD2+14ηuC,uD+ηuD,uC.
Combining (21) and (25) and small calculation yields (17).Remark 2.
(17) is the classical Hermite–Hadamard-type inequality for the operator convex function for ηl,m=l−m and p=1.


## 2.3. Jensen-Type Inequalities

Lemma 1.
Supposeu:I⟶R be an operator p,η-convex function, for C1andC2, where I contains the spectra of C and D and α1+α2=1, and we have(26)uα1C1+α2C2≤uC2+αηuC1,uC2.
Also, whenn>2, for C1,C2,…,Cn, whose spectra is contained in I, where ∑i=1nαi=1 and Ti=∑j=1iαj, we have(27)u∑i=1nαiCip1/p=uTn−1∑i=1n−1αiTn−1CiP1/p+αnCn≤uCn+Tn−1ηu∑i=1n−1αiTn−1Cip1/p,uCn.Now, in the proof of next theorem, we will utilize the above lemma.Theorem 4 (Jensen-type inequality).
Letw1,w2,…,wn∈R+ with n≥2 and for C1,C2,…,Cn, 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:(28)u1Wn∑i=1nwiCip1/p≤uCn+∑i=1nWiWnηuCi,Ci+1,…,Cn,where Wn=∑i=1nwi, also(29)ηuCi,Ci+1,…,Cn=ηηlCi,Ci+1,…,Cn−1,uCn,and ηuC=uC for all C whose spectra contained in I.Proof.
Sinceη is nondecreasing and nonnegatively sublinear in the first variable, so from the above lemma it yields that(30)u1Wn∑i=1nwiCip1/p=uwnWnCnp+∑i=1n−1wiWnCip1/p=uWn−1Wn∑i=1n−1wiWn−1Cip+wnWnCnp1/p≤uCn+Wn−1Wnηu∑i=1n−1wiWn−1Cip1/p,uCn=uCn+Wn−1WnηuWn−2Wn−1∑i=1n−2wiWn−2Cip+wn−1Wn−1Cn−1p1/p,uCn≤uCn+Wn−1Wnηuun−1+Wn−2Wn−1ηu∑i=1n−2wiWn−2Cip1/p,uCn−1,uCn≤uCn+Wn−1WnηuCn−1,uCn+Wn−2Wnηηu∑i=1n−2wiWn−2Cip1/p,uln−1,uln⋮≤uCn+Wn−1WnηuCn−1,uCn+Wn−2WnηηuCn−2,uCn,uCn+⋯+W1Wnηη⋯ηuC1,uC2,uC3⋯,uCn−1,uCn=uCn+Wn−1wnηuCn−1,Cn+Wn−2WnηuCn−2,Cn−1,Cn+⋯+W1WnηuC1,C2,…,Cn−1,Cn=uCn+∑i=1n−1WiWnηuCi,Ci+1,…,Cn.
Hence, the proof is completed.Remark 3 (28).
is the Jensen-type inequality for operatorη-convex functions for p=1.Remark 4 (28).
is the Jensen-type inequality for the operator convex function forp=1 and ηl,m=l−m.


## 2.4. Fejér-Type Inequality

Theorem 5.
Letu, v be nonnegative operator p,η-convex functions a,b∈IC<D such that uv∈L1a,b; then,(31)pDp−Cp∫abup−1ulvldl≤CC,D+12DC,D,where(32)CC,D=uDCvD+13ηuC,uDηvC,vD,DC,D=vDηvC,vD+vDηuC,uD.Proof.
Sinceu and v are operator p,η-convex functions, we have(33)usUp+1−sDp1/p≤uD+sηuC,uD,vsUp+1−sDp1/p≤vD+sηvC,vD,for all s∈C,D. Since u and v are nonnegative, so(34)usUp+1−sDp1/pvsUp+1−sDp1/p≤uDvD+suDηvC,vD+svDηuC,uD+s2ηuC,uDηvC,vD.
Integrating (34) over 0,1, we will obtain the following inequality:(35)∫01usUp+1−sDp1/pvsUp+1−sDp1/pds≤∫01uDvDds+∫01suDηvC,vDds+∫01svDηuC,uDds+∫01s2ηuC,uDηvC,vDds.
Settingu=sCp+1−sDp1/p, we obtain(36)pDp−Cp∫abup−1ulvldl≤uDvD+12uDηvC,vD+12vDηuC,uD+13ηuC,uDηvC,vD.
Then,(37)pDp−Cp∫abup−1ulgldl≤CC,D+12DC,D.Remark 5.
If we putp=1 and ηl,m=l−m in (31), then it reduces for operator convex functions.


## 3. Conclusion

In this report, we introduced the definition of operatorp,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.



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