In this paper, we will introduce the definition of operator p,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.
Guizhou Normal College2020BS001Higher Education Content and Curriculum System Reform Project of Guizhou Province2019083Specialized Fund for Science and Technology Platform and Talent Team Project of Guizhou Province[2016]5609Key Disciplines of Guizhou Province–Computer Science and Technology[2018]007First-Class C Discipline Project of Guizhou Normal College2019YLXKC02Key Supported Disciplines of Guizhou Province–Computer Application Technology[2016]201. 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 product
BH=C/C:H⟶H be bounded linear operator
BH+ is all positive operators in BH
K is a convex subset of BH+
ρD=λ∈C:D−λE−1∈LX
SpD=C/ρD
For C,D∈K,C,D:=1−sC+sD:s∈0,1.
Also, let η: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>.
If u 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]).
Assume u: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]).
Considering u: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]).
Let u: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 function u: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.
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⟶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. Section 2 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 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:(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 function u in (7) is known as operator p,η-concave function, if the above inequality is reversed.
Example 1.
Let u:I⟶R be a function, where uC=Cp and ηC,D≥C−D also C≥0; then, u is operator p,η-convex function.
implies that cu is an operator p,η-convex function.
Theorem 1.
Assume uj:I⟶R, j∈J, is the nonempty collection of operator p,η-convex functions such that
There exist α∈0,∞ and β∈−1,∞ such that ηC,D=αC+βD for all C, D whose spectra contained in I
For each C∈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 any C,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 InequalityTheorem 2.
Let η: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.
Let u 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.
Assuming C3p−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.
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:(17)uCp+Dp21/p−p2Dp−Cp∫abup−1ηuCp+Dp−up1/p,uldl≤pDp−Cp∫ablp−1uldl≤uC+uD2+14ηuC,uD+ηuD,uC.
Proof.
Take Sp=sCp+1−sDp and Tp=1−sCp+sDp, which implies(18)Cp+Dp2=Sp+Tp2,uCp+Dp21/p=uSp+Tp21/p.
By definition of operator p,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.
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 InequalitiesLemma 1.
Suppose u: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, when n>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).
Let w1,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.
Since η 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 operator η-convex functions for p=1.
Remark 4 (28).
is the Jensen-type inequality for the operator convex function for p=1 and ηl,m=l−m.
2.4. Fejér-Type InequalityTheorem 5.
Let u, 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.
Since u 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.
Setting u=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 put p=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 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
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).
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