# Extended Jensen’s Functional for Diamond Integral via Hermite Polynomial

Authors: Rabia Bibi; Fazilat Bibi; Ammara Nosheen; Josip Pečarić
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, with the help of Hermite interpolating polynomial, extension of Jensen’s functional forn-convex function is deduced from Jensen’s inequality involving diamond integrals. Special Hermite conditions, including Taylor two-point formula and Lagrange’s interpolation, are also deployed to find further extensions of Jensen’s functional. The paper also includes discussion on bounds for Grüss-type inequality, Ostrowski-type inequality, and C˘ebyšev functional associated with newly defined Jensen’s functional.


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## Body

## 1. Introduction

John Jensen proved Jensen’s inequality in [1]. It serves as a tool in discrete and continuous analysis for generating classical inequalities. A discrete variant is as below:
(1)χ∑m=1ngmzm∑m=1ngm≤∑m=1ngmχzm∑m=1ngm,where z1,⋯,zn∈S, S is interval in ℝg1,⋯,gn∈ℝ+n (i.e., nonnegative weights are taken into account in this inequality), and function χ:S⟶ℝ is a convex on S. Steffensen in [2] extended it by using negative weights.Integral representation of Jensen’s inequality in [3] is as follows: If τ∈Ca1,a2,a3,a4 and χ∈Ca3,a4,ℝ are convex, then
(2)χ∫a1a2τsdsa2−a1≤∫a1a2χτsdsa2−a1.The researchers have devised several new functions for refinements of Jensen’s discrete/integral inequalities. For instance, in [4–7], improvements of the operated version of Jensen’s inequality are given. In [8], Aras-Gazic et al. generalized Jensen’s inequality via the Hermite polynomial.Several researchers discussed and applied these inequalities on time scales. In [9], Anwar et al. proved Jensen’s inequality for delta integrals.Supposea1,a2∈T s.t. a1<a2. Let ð∈Crda1,a2T,ℝ assures ∫a1a2∣ðs∣Δs>0. If χ∈CS,ℝ is convex, an interval S⊂ℝ and ℏ∈Crda1,a2T,S, then
(3)χ∫a1a2∣ðs∣ℏsΔs∫a1a2∣ðs∣Δs≤∫a1a2∣ðs∣χℏsΔs∫a1a2∣ðs∣Δs.Under a similar hypothesis, in [10], by replacing the delta integral with the nabla integral, the same results are obtained.Sheng et al. in [11] presented the convex combination of the delta and nabla integrals named as diamond alpha integrals, where α∈0,1. For α=1, we get the usual delta integral and nabla integral for α=0. In [12], following Jensen’s inequality for the diamond alpha integral is given.Suppose a time scaleTa1,a2∈T s.t. a1<a2, and E∈ℝ is an interval ℏ∈Crda1,a2T,E, and ð∈Ca1,a2,ℝ so that
(4)∫a1a2ðs◊αs>0,if χ∈CE,ℝ is convex, then
(5)χ∫a1a2ðsℏs◊αs∫a1a2ðs◊αs≤∫a1a2ðsχℏs◊αs∫a1a2ðs◊αs.In [13], the authors introduced the more generalized variant of diamond-alpha integrals; termed as the diamond integral, those are of special concern even for T=ℝ. These integrals get us nearer in building a true symmetric integral on time scales.In [14], Jensen’s inequality for diamond integrals is proved.Leta1,a2∈T with a1<a2, ð∈Ca1,a2T,ℝ+ and ℏ∈Ca1,a2T,S. Suppose a convex function χ∈CS,ℝ assuring ∫a1a2ðu◊u>0, where S=m1,m2 and m1=mins∈a1,a2Tℏs, m2=maxs∈a1,a2Tℏs, then
(6)χ∫a1a2ðsℏs◊s∫a1a2ðs◊s≤∫a1a2ðsχℏs◊s∫a1a2ðs◊s.Jensen’s-type linear functional defined onT is given as the following.Leta1,a2∈T with a1<a2 and ð∈Ca1,a2T,ℝ, ℏ∈Ca1,a2T,S and ζ∈CS,ℝ satisfying ∫a1a2ðs◊s≠0, where S=m1,m2 and m1=mins∈a1,a2Tℏs, m2=maxs∈a1,a2Tℏs; then,
(7)Jζ=∫a1a2ðsζℏs◊s∫a1a2ðs◊s−ζ∫a1a2ðsℏs◊s∫a1a2ðs◊s.Remark 1.
Inequality (6) implies that Jζ≥0 for the family of convex mappings and Jζ=0 for identity or constant functions.The present study is aimed at extending (7) for the n-convex function with some types of interpolations introduced by Hermite. In the next section, after defining diamond derivative and integral, we recall the Hermite interpolating polynomial along with some of its special forms. Section 3 consists of the paper’s main results, and finally, concluding remarks are given in the last section.


## 2. Preliminaries

### 2.1. Some Essentials Form Diamond Calculus

A time scaleT is a nonempty closed subset of ℝ. It may be connected or not; keeping the time scales’ disconnection under consideration, forward and backward jump operatorsσ,ρ:T⟶T, are defined by σℏ=infs∈T:s>ℏ, and ρℏ=sups∈T:s<ℏ. In general, σℏ≥ÿ and ρℏ≤ℏ. The mappings μ,ν:T⟶0,+∞ defined by μℏ=σℏ−ℏ, and νℏ=ℏ−ρℏ are called in the sequal the forward and backward graininess functions. For the classification of points on time scales, for any ℏεT,
(i)
ifρℏ=ℏ, then ℏ is left dense(ii)
ifσℏ=ℏ, then ℏ is right dense(iii)
ifρℏ<ℏ, then ℏ is left scattered(iv)
ifσℏ>ℏ, then right scattered(v)
ifρℏ=ℏ and σℏ=ℏ, then ℏ is dense(vi)
ifρℏ<ℏ and σℏ>ℏ, then ℏ is isolatedA mappingρ:T⟶ℝ is said to be rd-continuous if
(i)
it is continuous∀ℏ∈T s.t. σℏ=ℏ(ii)
left-sided limit is finite∀ℏ∈T s.t. ρℏ=ℏA set of such functions is denoted byCrd.Definition 2.
LetΛ:T⟶ℝ be a mapping and ℏ∈Tkk define Λ◊ℏ (presumed it as a finite positive number) having characteristic that, for given ε>0, there exists neighbourhood W of ℏ (i.e. W=ℏ−s,ℏ+s)∩T) for some δ>0 such that
(8)Λσℏ−Λs+Λ2ℏ−s−Λρℏ−Λ◊ℏσℏ+2ℏ−2s−ρℏ≤εσℏ+2ℏ−2s−ρℏholds. For all s∈W for which 2ℏ−s∈W. Then Λ◊r is known as diamond derivative of Λ at ℏ.Definition 3.
Letϱ:T⟶ℝ and a1,a2∈T be a function. The diamond integral of ϱ from a1toa2 is given by
(9)∫a1a2ϱℏ◊ℏ≔∫a1a2γℏϱℏΔℏ+∫a1a21−γℏϱℏ∇ℏ,for all ℏ∈T.Letγϱ and 1−γϱ be delta and nabla integrable on r,sT, respectively. It is to be noted that the antiderivative is absent for diamond combined derivatives. For ℏ∈Tkk, ∫bsϱℏ◊◊≠ϱs, in general. The fundamental theorem of calculus also does not hold for diamond integrals.


### 2.2. Results on Hermite Interpolating Polynomial

Let−∞<μ<ν<∞ and μ=a1<⋯<ar=νr≥2 be the given r points. For I∈Cnμ,ν, there is a n−1th degree polynomial ⊑Ht, defined as
(10)⊑Ht=∑v=1r∑u=0kvHuvtζuav.It satisfies the following Hermite condition:(11)⊑Huav=ζuav,0≤u≤kv,1≤v≤r,∑v=1rkv+r=n.Huv represents essential polynomials of the Hermite basis which satisfy the relations:
(12)Huvpad=0,d≠v,p=0,⋯,kd,Huvpav=δup,p=0,⋯,kv,foru=0,⋯,kv,with d,v=1,⋯,r and
(13)δup=1u=p,0u≠p.Huvt is given by
(14)Huvt=1u!ωtt−avkv+1−u∑k=0kv−u1k!dkdtkt−avkv+1ωtt=avt−avk,with
(15)ωt=∏v=1rt−avkv+1.Hermite conditions encompass the following specific cases.


#### 2.2.1. Lagrange Conditions

Letr=n,kv=0 for all v, where 1<v<r. Then, we have Lagrange polynomial ⊑Lt, satisfying
(16)⊑Lav=ζav,1≤v≤n.


#### 2.2.2. Conditions for Typez,n−z

Letr=2,1≤z≤n−1,k1=z−1,k2=n−z−1. Then we have ⊑z,nt polynomial, satisfying
(17)⊑z,nuμ=ζuμ,0≤u≤z−1,⊑z,nuν=ζuν,0≤u≤n−z−1.


#### 2.2.3. Conditions for Taylor’s Two-Point Formula

Forn=2z,r=2,k1=k2=z−1, we have Taylor two-point interpolating polynomial ⊑2Tt, satisfying
(18)⊑2Tiμ=ζuμ,⊑2Tuν=ζuν,0≤u≤z−1.The next theorem is useful for our results and is given in [15].Theorem 4.
Suppose we have−∞<μ<ν<∞andμ=a1<⋯<ar=νr≥2, andI∈Cnμ,ν.Then we have(19)ζt=⊑Ht+RHζ,t,where ⊑Ht is the Hermite interpolating polynomial as defined in (10) and RHζ,t denotes the remainder given by
(20)RHζ,t=∫μνGH,nt,sζnsds;
GH,nt,s is
(21)GH,nt,s=∑v=1b∑u=0kvav−hn−u−1n−u−1!Huvt,s≤t,−∑v=b+1r∑u=0kvav−hn−u−1n−u−1!Huvt,s≥t,for all ab≤s≤ab+1,b=0,⋯,r with a0=μ and ar+1=ν.Remark 5.
By imposing the Lagrange conditions, Theorem4 takes the form
(22)ζt=⊑Lt+RLζ,t.
Here,⊑Lt represents the Lagrange polynomial, which is
(23)⊑Lt=∑v=1n∏k=1k≠vnt−akav−akζav,and RLζ,t is the remainder, defined by
(24)RLζ,t=∫μνGLt,sζnsds,with
(25)GLt,s=1n−1!∑v=1bav−sn−1∏k=1k≠vnt−akav−aks≤t,−∑v=1bav−sn−1∏k=1k≠vnt−akav−aks≤t,
ab≤s≤ab+1, b=1,⋯,n−1 along a1=μ and an=ν.Remark 6.
Similarly, by imposingz,n−z conditions on Theorem 4, one gets
(26)ζt=⊑z,nt+Rz,nζ,t,where
(27)⊑z,nt=∑u=0z−1ξutζuμ+∑u=0n−z−1ηutζuν,with
(28)ξut=1u!t−μut−νμ−νn−z∑k=0z−1−un−z+k−1kt−μν−μk,(29)ηut=1u!t−νut−μν−μz∑k=0n−z−1−uz+k−1kt−νμ−νk.The remainderRz,nζ,t is
(30)Rz,nζ,t=∫μνGm,nt,sζnsds,with
(31)Gz,nt,s=∑v=0z−1∑l=0z−1−vn−z+l−1lt−νν−μl×t−μvμ−sn−v−1v!n−v−1!ν−tν−μn−zμ≤s≤t≤ν,−∑u=0n−z−1∑r=0n−z−u−1z+r−1rν−tν−μr×t−νuν−sn−u−1u!n−u−1!t−μν−μeμ≤t≤s≤ν.Remark 7.
Theorem4 in the form of the Taylor two-point formula becomes
(32)ζt=⊑2Tt+R2Tζ,t,where the Taylor two-point interpolating polynomial ⊑2Tt is defined by
(33)⊑2Tt=∑u=0z−1∑k=0z−1−uz+k−1kt−μuu!t−νμ−νzt−μν−μkζuμ+t−νuu!t−μν−μzt−νμ−νkζuν,and R2Tζ,t is
(34)R2Tζ,t=∫μνG2Tt,sζnsds,with
(35)G2Tt,s=−1z2z−1!lzt,s∑v=0z−1z−1+vvt−sz−1−vrvt,s,s≤t,−1z2z−1!rzt,s∑v=0z−1z−1+vvs−tz−1−vlvt,st≤s,where lt,s=s−μν−t/ν−μ, rt,s=lt,s, for all t,s∈μ,ν.


## 2.1. Some Essentials Form Diamond Calculus

A time scaleT is a nonempty closed subset of ℝ. It may be connected or not; keeping the time scales’ disconnection under consideration, forward and backward jump operatorsσ,ρ:T⟶T, are defined by σℏ=infs∈T:s>ℏ, and ρℏ=sups∈T:s<ℏ. In general, σℏ≥ÿ and ρℏ≤ℏ. The mappings μ,ν:T⟶0,+∞ defined by μℏ=σℏ−ℏ, and νℏ=ℏ−ρℏ are called in the sequal the forward and backward graininess functions. For the classification of points on time scales, for any ℏεT,
(i)
ifρℏ=ℏ, then ℏ is left dense(ii)
ifσℏ=ℏ, then ℏ is right dense(iii)
ifρℏ<ℏ, then ℏ is left scattered(iv)
ifσℏ>ℏ, then right scattered(v)
ifρℏ=ℏ and σℏ=ℏ, then ℏ is dense(vi)
ifρℏ<ℏ and σℏ>ℏ, then ℏ is isolatedA mappingρ:T⟶ℝ is said to be rd-continuous if
(i)
it is continuous∀ℏ∈T s.t. σℏ=ℏ(ii)
left-sided limit is finite∀ℏ∈T s.t. ρℏ=ℏA set of such functions is denoted byCrd.Definition 2.
LetΛ:T⟶ℝ be a mapping and ℏ∈Tkk define Λ◊ℏ (presumed it as a finite positive number) having characteristic that, for given ε>0, there exists neighbourhood W of ℏ (i.e. W=ℏ−s,ℏ+s)∩T) for some δ>0 such that
(8)Λσℏ−Λs+Λ2ℏ−s−Λρℏ−Λ◊ℏσℏ+2ℏ−2s−ρℏ≤εσℏ+2ℏ−2s−ρℏholds. For all s∈W for which 2ℏ−s∈W. Then Λ◊r is known as diamond derivative of Λ at ℏ.Definition 3.
Letϱ:T⟶ℝ and a1,a2∈T be a function. The diamond integral of ϱ from a1toa2 is given by
(9)∫a1a2ϱℏ◊ℏ≔∫a1a2γℏϱℏΔℏ+∫a1a21−γℏϱℏ∇ℏ,for all ℏ∈T.Letγϱ and 1−γϱ be delta and nabla integrable on r,sT, respectively. It is to be noted that the antiderivative is absent for diamond combined derivatives. For ℏ∈Tkk, ∫bsϱℏ◊◊≠ϱs, in general. The fundamental theorem of calculus also does not hold for diamond integrals.


## 2.2. Results on Hermite Interpolating Polynomial

Let−∞<μ<ν<∞ and μ=a1<⋯<ar=νr≥2 be the given r points. For I∈Cnμ,ν, there is a n−1th degree polynomial ⊑Ht, defined as
(10)⊑Ht=∑v=1r∑u=0kvHuvtζuav.It satisfies the following Hermite condition:(11)⊑Huav=ζuav,0≤u≤kv,1≤v≤r,∑v=1rkv+r=n.Huv represents essential polynomials of the Hermite basis which satisfy the relations:
(12)Huvpad=0,d≠v,p=0,⋯,kd,Huvpav=δup,p=0,⋯,kv,foru=0,⋯,kv,with d,v=1,⋯,r and
(13)δup=1u=p,0u≠p.Huvt is given by
(14)Huvt=1u!ωtt−avkv+1−u∑k=0kv−u1k!dkdtkt−avkv+1ωtt=avt−avk,with
(15)ωt=∏v=1rt−avkv+1.Hermite conditions encompass the following specific cases.


### 2.2.1. Lagrange Conditions

Letr=n,kv=0 for all v, where 1<v<r. Then, we have Lagrange polynomial ⊑Lt, satisfying
(16)⊑Lav=ζav,1≤v≤n.


### 2.2.2. Conditions for Typez,n−z

Letr=2,1≤z≤n−1,k1=z−1,k2=n−z−1. Then we have ⊑z,nt polynomial, satisfying
(17)⊑z,nuμ=ζuμ,0≤u≤z−1,⊑z,nuν=ζuν,0≤u≤n−z−1.


### 2.2.3. Conditions for Taylor’s Two-Point Formula

Forn=2z,r=2,k1=k2=z−1, we have Taylor two-point interpolating polynomial ⊑2Tt, satisfying
(18)⊑2Tiμ=ζuμ,⊑2Tuν=ζuν,0≤u≤z−1.The next theorem is useful for our results and is given in [15].Theorem 4.
Suppose we have−∞<μ<ν<∞andμ=a1<⋯<ar=νr≥2, andI∈Cnμ,ν.Then we have(19)ζt=⊑Ht+RHζ,t,where ⊑Ht is the Hermite interpolating polynomial as defined in (10) and RHζ,t denotes the remainder given by
(20)RHζ,t=∫μνGH,nt,sζnsds;
GH,nt,s is
(21)GH,nt,s=∑v=1b∑u=0kvav−hn−u−1n−u−1!Huvt,s≤t,−∑v=b+1r∑u=0kvav−hn−u−1n−u−1!Huvt,s≥t,for all ab≤s≤ab+1,b=0,⋯,r with a0=μ and ar+1=ν.Remark 5.
By imposing the Lagrange conditions, Theorem4 takes the form
(22)ζt=⊑Lt+RLζ,t.
Here,⊑Lt represents the Lagrange polynomial, which is
(23)⊑Lt=∑v=1n∏k=1k≠vnt−akav−akζav,and RLζ,t is the remainder, defined by
(24)RLζ,t=∫μνGLt,sζnsds,with
(25)GLt,s=1n−1!∑v=1bav−sn−1∏k=1k≠vnt−akav−aks≤t,−∑v=1bav−sn−1∏k=1k≠vnt−akav−aks≤t,
ab≤s≤ab+1, b=1,⋯,n−1 along a1=μ and an=ν.Remark 6.
Similarly, by imposingz,n−z conditions on Theorem 4, one gets
(26)ζt=⊑z,nt+Rz,nζ,t,where
(27)⊑z,nt=∑u=0z−1ξutζuμ+∑u=0n−z−1ηutζuν,with
(28)ξut=1u!t−μut−νμ−νn−z∑k=0z−1−un−z+k−1kt−μν−μk,(29)ηut=1u!t−νut−μν−μz∑k=0n−z−1−uz+k−1kt−νμ−νk.The remainderRz,nζ,t is
(30)Rz,nζ,t=∫μνGm,nt,sζnsds,with
(31)Gz,nt,s=∑v=0z−1∑l=0z−1−vn−z+l−1lt−νν−μl×t−μvμ−sn−v−1v!n−v−1!ν−tν−μn−zμ≤s≤t≤ν,−∑u=0n−z−1∑r=0n−z−u−1z+r−1rν−tν−μr×t−νuν−sn−u−1u!n−u−1!t−μν−μeμ≤t≤s≤ν.Remark 7.
Theorem4 in the form of the Taylor two-point formula becomes
(32)ζt=⊑2Tt+R2Tζ,t,where the Taylor two-point interpolating polynomial ⊑2Tt is defined by
(33)⊑2Tt=∑u=0z−1∑k=0z−1−uz+k−1kt−μuu!t−νμ−νzt−μν−μkζuμ+t−νuu!t−μν−μzt−νμ−νkζuν,and R2Tζ,t is
(34)R2Tζ,t=∫μνG2Tt,sζnsds,with
(35)G2Tt,s=−1z2z−1!lzt,s∑v=0z−1z−1+vvt−sz−1−vrvt,s,s≤t,−1z2z−1!rzt,s∑v=0z−1z−1+vvs−tz−1−vlvt,st≤s,where lt,s=s−μν−t/ν−μ, rt,s=lt,s, for all t,s∈μ,ν.


## 2.2.1. Lagrange Conditions

Letr=n,kv=0 for all v, where 1<v<r. Then, we have Lagrange polynomial ⊑Lt, satisfying
(16)⊑Lav=ζav,1≤v≤n.


## 2.2.2. Conditions for Typez,n−z

Letr=2,1≤z≤n−1,k1=z−1,k2=n−z−1. Then we have ⊑z,nt polynomial, satisfying
(17)⊑z,nuμ=ζuμ,0≤u≤z−1,⊑z,nuν=ζuν,0≤u≤n−z−1.


## 2.2.3. Conditions for Taylor’s Two-Point Formula

Forn=2z,r=2,k1=k2=z−1, we have Taylor two-point interpolating polynomial ⊑2Tt, satisfying
(18)⊑2Tiμ=ζuμ,⊑2Tuν=ζuν,0≤u≤z−1.The next theorem is useful for our results and is given in [15].Theorem 4.
Suppose we have−∞<μ<ν<∞andμ=a1<⋯<ar=νr≥2, andI∈Cnμ,ν.Then we have(19)ζt=⊑Ht+RHζ,t,where ⊑Ht is the Hermite interpolating polynomial as defined in (10) and RHζ,t denotes the remainder given by
(20)RHζ,t=∫μνGH,nt,sζnsds;
GH,nt,s is
(21)GH,nt,s=∑v=1b∑u=0kvav−hn−u−1n−u−1!Huvt,s≤t,−∑v=b+1r∑u=0kvav−hn−u−1n−u−1!Huvt,s≥t,for all ab≤s≤ab+1,b=0,⋯,r with a0=μ and ar+1=ν.Remark 5.
By imposing the Lagrange conditions, Theorem4 takes the form
(22)ζt=⊑Lt+RLζ,t.
Here,⊑Lt represents the Lagrange polynomial, which is
(23)⊑Lt=∑v=1n∏k=1k≠vnt−akav−akζav,and RLζ,t is the remainder, defined by
(24)RLζ,t=∫μνGLt,sζnsds,with
(25)GLt,s=1n−1!∑v=1bav−sn−1∏k=1k≠vnt−akav−aks≤t,−∑v=1bav−sn−1∏k=1k≠vnt−akav−aks≤t,
ab≤s≤ab+1, b=1,⋯,n−1 along a1=μ and an=ν.Remark 6.
Similarly, by imposingz,n−z conditions on Theorem 4, one gets
(26)ζt=⊑z,nt+Rz,nζ,t,where
(27)⊑z,nt=∑u=0z−1ξutζuμ+∑u=0n−z−1ηutζuν,with
(28)ξut=1u!t−μut−νμ−νn−z∑k=0z−1−un−z+k−1kt−μν−μk,(29)ηut=1u!t−νut−μν−μz∑k=0n−z−1−uz+k−1kt−νμ−νk.The remainderRz,nζ,t is
(30)Rz,nζ,t=∫μνGm,nt,sζnsds,with
(31)Gz,nt,s=∑v=0z−1∑l=0z−1−vn−z+l−1lt−νν−μl×t−μvμ−sn−v−1v!n−v−1!ν−tν−μn−zμ≤s≤t≤ν,−∑u=0n−z−1∑r=0n−z−u−1z+r−1rν−tν−μr×t−νuν−sn−u−1u!n−u−1!t−μν−μeμ≤t≤s≤ν.Remark 7.
Theorem4 in the form of the Taylor two-point formula becomes
(32)ζt=⊑2Tt+R2Tζ,t,where the Taylor two-point interpolating polynomial ⊑2Tt is defined by
(33)⊑2Tt=∑u=0z−1∑k=0z−1−uz+k−1kt−μuu!t−νμ−νzt−μν−μkζuμ+t−νuu!t−μν−μzt−νμ−νkζuν,and R2Tζ,t is
(34)R2Tζ,t=∫μνG2Tt,sζnsds,with
(35)G2Tt,s=−1z2z−1!lzt,s∑v=0z−1z−1+vvt−sz−1−vrvt,s,s≤t,−1z2z−1!rzt,s∑v=0z−1z−1+vvs−tz−1−vlvt,st≤s,where lt,s=s−μν−t/ν−μ, rt,s=lt,s, for all t,s∈μ,ν.


## 3. Main Results

### 3.1. Extension of Jensen’s Functional via Hermite Polynomial

Here, we prove our key identity regarding the extension of Jensen’s functional.Theorem 8.
Letn∈ℕ and ζ∈Cnμ,ν be a convex function. Then for all t∈μ,ν, we have
(36)Jζ=∑v=1r∑u=0kvJHuvtζuav+∫μνJGH,nt,sζnsds,where JGH,nt,s is defined as
(37)JGH,nt,s=∫a1a2htGH,nft,s◊t∫a1a2ht◊t−GH,n∫a1a2htft◊t∫a1a2ht◊t,s.Proof.
Substitute (19) in (7), then the linearity of J· gives us (36).☐Remark 9.
Instead of using (19) in (7), if we use (22), (26), or (32) in (7), then extended results similar to Theorem 8 are obtained for Lagrange conditions, type z,n−z conditions, and two-point Taylor conditions, respectively, in the following form:
(38)Jζ=∑v=1nζavJ∏k=1k≠vnt−akav−ak+∫μνJGLt,sζnsds,Jζ=∑u=0z−1ζuμJξut+∑u=on−z−1ζuνJηut+∫μνJGz,nt,sζnsds,Jζ=∑u=0z−1∑k=0z−1−uz+k−1kJt−μuu!t−νμ−νzt−μν−μkζuμ+Jt−νuu!t−μν−μzt−νμ−νkζuν+∫μνJG2Tt,sζnsds.Theorem 10.
Presumen be a natural number and GH,n be a Green function given in (21) satisfying
(39)JGH,nt,s≥0,for all t∈μ,ν. Then for every n-convex ζ:μ,ν⟶ℝ,(40)Jζ≥JBholds, where
(41)Bt=∑v=1r∑u=0kvζuavHuvt.Proof.
As the functionζ is n-convex, ζnt≥0 for all t∈μ,ν; hence,
(42)JGH,nt,sζnt≥0.
Substituting (42) in (36), we have
(43)Jζ−∑v=1r∑u=0kvζuavJHuvt≥0.
SinceJ is linear, (43) can be written as
(44)Jζ−JB≥0.
Hence,(45)Jζ≥JB.☐Remark 11.
(40) is reversed if (39) is reversed.Utilization of the Lagrange condition produced the result given below.Corollary 12.
Presumen is a natural number and GL defined as in (25) satisfying
(46)JGLt,s≥0,for all t∈μ,ν. Then for every n-convex function ζ:μ,ν⟶ℝ,(47)Jζ≥JB,where
(48)Bt=∑v=1nζav∏k=1k≠vnt−akav−ak.The use of typez,n−z condition yields the following result.Corollary 13.
Letn be a natural number and Gz,n be defined in (31). Suppose ξu,ηu are defined as in (28) and (29), respectively, satisfying
(49)JGz,nt,s≥0,for all t∈μ,ν. Then for every n-convex function ζ:μ,ν⟶ℝ, we have
(50)Jζ≥JB,where
(51)Bt=∑u=0z−1ζuμξut+∑u=on−z−1ζuνηut.The application of two-point Taylor conditions gives a result as below.Corollary 14.
Letn∈ℕ and G2T be the Green function defined as in (35), satisfying
(52)JG2Tt,s≥0,for all t∈μ,ν. Then for n-convex function ζ:μ,ν⟶ℝ,(53)Jζ≥JB,where
(54)Bt=∑u=0z−1∑k=0z−1−uz+k−1kt−μuu!t−νμ−νzt−μν−μkζuμ+t−νuu!t−μν−μzt−νμ−νkζuν.Remark 15.
Technique to prove Corollaries12–14 is the same as proof of Theorem 10, where we use n convexity of ζ and linearity of J.Theorem 16.
In addition to presumptions of Theorem10, consider h∈Ca1,a2T,ℝ+ and B:μ,ν⟶ℝ is convex, then
(55)Jζt≥0,for all t∈μ,ν.Proof.
AsBt is convex for all t∈μ,ν, so Remark 1, inferred JBt≥0. Hence, (40) implies Jζt≥0.☐Remark 17.
Relation (55) in Theorem 16 is an extension of Jensen’s inequality (6).


### 3.2. Bounds for Identities Associated to Generalization of Jensen’s Functional

Here, we utilizeC˘ebyšev functional and Grüss-type inequalities to present few important results. Let two functions g1,g2:≀,ȷ⟶ℝ be Lebesgue integrable, and the C˘ebyšev functional is
(56)Yg1,g2=1ȷ−≀∫≀ȷg1tg2tdt−1ȷ−≀∫iȷg1tdt·1ȷ−≀∫≀ȷg2tdt.The next two theorems are given in [16].Theorem 18.
Letg1,g2:≀,ȷ⟶ℝ be functions such that g1 is Lebesgue integrable and g2 be the function which is absolutely continuous along ·−≀ȷ−·g2′2∈L≀,ȷ. Then, we have
(57)Yg1,g2≤12Yg1,g11/21j−i∫≀ȷy−≀j−yg2′y2dy1/2.
1/2 in (57) is possibly the best constant.Theorem 19.
Letg1,g2:≀,ȷ⟶ℝ be functions, such that g1 is absolutely continuous with g1′∈L∞≀,ȷ and g2 is monotonic nondecreasing on ≀,ȷ, then
(58)Yg1,g2≤12j−ig1′∞∫≀ȷy−ij−ydg2y.
The best possible constant is1/2 in (58).
We denote(59)ψs=JGH,nx,s.
TheC˘ebyšev functional is given as
(60)Yψ,ψ=1ν−μ∫μνψ2sds−1ν−μ∫μνψsds2.Theorem 20.
Letζ:μ,ν⟶ℝ be such that ζ∈Cnμ,ν for n∈ℕ with ·−μν−·ζn+12∈Lμ,ν and t∈μ,ν and GH,n, ψ and Y are defined in (21), (59), and (60), respectively. Then,
(61)Jζ=∑v=1r∑u=0kvζuavJHuvx+ζn−1ν−ζn−1μν−μ∫μνψsds+Rnμ,ν;ζ,where the remainder satisfies Rnμ,ν;ζ the estimation
(62)Rnμ,ν;ζ≤Yψ,ψ1/2ν−μ2∫μνs−μν−sζn+1s2ds1/2.Proof.
We use Theorem18 for g1⟶ψ and g2⟶ζn, to obtain
(63)1ν−μ∫μνψsζnsds−1ν−μ∫μνψsds·1ν−μ∫μνζnsds≤12Yψ,ψ1/21ν−μ∫μνs−μν−sζn+1s2ds1/2.
Therefore, we have(64)1ν−μ∫μνψsζnsds=1ν−μ2ζn−1ν−ζn−1μ∫μνψsds,∫μνψsζnsds=ζn−1ν−ζn−1μν−μ∫μνψsds+Rμ,ν;ζ,where the remainder Rμ,ν;ζ satisfy the estimation (62). Now, from the identity (36),
(65)Jζ=∑v=1r∑u=0kvζuavJHuvt+∫μνψsζnsds,Jζ=∑v=1r∑u=0kvζuavJHuvt+ζn−1ν−ζn−1μν−μ∫μνψsds+Rμ,ν;ζ.☐The Grüss-type inequality given below can be obtained by using Theorem19.Theorem 21.
Assumeζ:μ,ν⟶ℝ such that ζ∈Cnμ,νn∈ℕandζ2n+1≥0onμ,ν. Suppose ψ and Y are defined in (59) and (60), respectively. Then, we have (61) the remainder Rμ,ν:ζ satisfies the bound
(66)Rμ,ν;ζ≤ν−μψ′∞×ζn−1ν+ζn−1μν−μ−ζn−2ν−ζn−2μν−μ.Proof.
Applying Theorem19 for g1⟶ψ and g2⟶ζn, we have
(67)1ν−μ∫μνψsζnsds−1ν−μ∫μνψsds·1ν−μ∫μνζnsds≤12ν−μψ′∞∫μνs−μν−sζn+1sds.
Since(68)∫μνs−μν−sζn+1sds=∫μν2s−μ+νζnsds,∫μνs−μν−sζn+1sds=ν−μζn−1ν+ζn−1μ−2ζn−2ν+ζn−2μ;using (36) and (67), we get (66).☐Theorem 22.
Let all the presumptions of Theorem8 be satisfied. Suppose i˘,j˘ is a couple of numbers, i.e., 1≤i˘,j˘≤∞,1/i˘+1/j˘=1. Suppose ζni˘:μ,ν⟶ℝ is a function which is R-integrable for some n≥2. Then, we have
(69)Jζ−∑v=1r∑u=0kvζuavJHuvx≤ζni˘∫μνJGHx,sj˘ds1/j.
Fori˘∈1,∞, the constant on the right of (69) is sharp and suitable when i˘ is one.Proof.
Assume℘s=JGnt,s. Hölder’s inequality and identity (36) give us
(70)∣Jζ−∑v=1r∑u=0kvJHuvtζuav∣=∫μν℘sζnsds≤ζni˘∫μν℘tj˘ds1/j˘.
Takingζ for i˘∈1,∞, such that, ζns=sgn℘s℘s1/i˘−1. In case of i˘=∞, take ζns=sgn℘s.
We prove that fori˘=1,
(71)∫μν℘sζnsds≤maxs∈μ,ν℘s∫μνζnsdsis the suitable inequality. Let ∣℘s∣ achieve its maximum at p∈μ,ν. We suppose firstly that ℘d¨>0. We define ζs for small enough δ, by
(72)ζδs≔0,μ≤s≤p,1εn!s−pn,p≤s≤p+δ,1n!s−pn−1,p+δ≤s≤ν.
Then, forδ small enough,
(73)∫μν℘sζnsds=∫pp+δ℘s1δds=1δ∫pp+δ℘sds.
Now, from the inequality (71), we have
(74)1δ∫pp+δ℘sds≤℘p∫pp+δ1δds=℘p.
For℘p<0, we set ζδs as
(75)ζδs≔1n!s−p−δn−1,μ≤s≤p,−1εn!s−p−δn,p≤s≤p+∈,0,p+δ≤s≤ν.
The rest of the proof is the same as above.☐


## 3.1. Extension of Jensen’s Functional via Hermite Polynomial

Here, we prove our key identity regarding the extension of Jensen’s functional.Theorem 8.
Letn∈ℕ and ζ∈Cnμ,ν be a convex function. Then for all t∈μ,ν, we have
(36)Jζ=∑v=1r∑u=0kvJHuvtζuav+∫μνJGH,nt,sζnsds,where JGH,nt,s is defined as
(37)JGH,nt,s=∫a1a2htGH,nft,s◊t∫a1a2ht◊t−GH,n∫a1a2htft◊t∫a1a2ht◊t,s.Proof.
Substitute (19) in (7), then the linearity of J· gives us (36).☐Remark 9.
Instead of using (19) in (7), if we use (22), (26), or (32) in (7), then extended results similar to Theorem 8 are obtained for Lagrange conditions, type z,n−z conditions, and two-point Taylor conditions, respectively, in the following form:
(38)Jζ=∑v=1nζavJ∏k=1k≠vnt−akav−ak+∫μνJGLt,sζnsds,Jζ=∑u=0z−1ζuμJξut+∑u=on−z−1ζuνJηut+∫μνJGz,nt,sζnsds,Jζ=∑u=0z−1∑k=0z−1−uz+k−1kJt−μuu!t−νμ−νzt−μν−μkζuμ+Jt−νuu!t−μν−μzt−νμ−νkζuν+∫μνJG2Tt,sζnsds.Theorem 10.
Presumen be a natural number and GH,n be a Green function given in (21) satisfying
(39)JGH,nt,s≥0,for all t∈μ,ν. Then for every n-convex ζ:μ,ν⟶ℝ,(40)Jζ≥JBholds, where
(41)Bt=∑v=1r∑u=0kvζuavHuvt.Proof.
As the functionζ is n-convex, ζnt≥0 for all t∈μ,ν; hence,
(42)JGH,nt,sζnt≥0.
Substituting (42) in (36), we have
(43)Jζ−∑v=1r∑u=0kvζuavJHuvt≥0.
SinceJ is linear, (43) can be written as
(44)Jζ−JB≥0.
Hence,(45)Jζ≥JB.☐Remark 11.
(40) is reversed if (39) is reversed.Utilization of the Lagrange condition produced the result given below.Corollary 12.
Presumen is a natural number and GL defined as in (25) satisfying
(46)JGLt,s≥0,for all t∈μ,ν. Then for every n-convex function ζ:μ,ν⟶ℝ,(47)Jζ≥JB,where
(48)Bt=∑v=1nζav∏k=1k≠vnt−akav−ak.The use of typez,n−z condition yields the following result.Corollary 13.
Letn be a natural number and Gz,n be defined in (31). Suppose ξu,ηu are defined as in (28) and (29), respectively, satisfying
(49)JGz,nt,s≥0,for all t∈μ,ν. Then for every n-convex function ζ:μ,ν⟶ℝ, we have
(50)Jζ≥JB,where
(51)Bt=∑u=0z−1ζuμξut+∑u=on−z−1ζuνηut.The application of two-point Taylor conditions gives a result as below.Corollary 14.
Letn∈ℕ and G2T be the Green function defined as in (35), satisfying
(52)JG2Tt,s≥0,for all t∈μ,ν. Then for n-convex function ζ:μ,ν⟶ℝ,(53)Jζ≥JB,where
(54)Bt=∑u=0z−1∑k=0z−1−uz+k−1kt−μuu!t−νμ−νzt−μν−μkζuμ+t−νuu!t−μν−μzt−νμ−νkζuν.Remark 15.
Technique to prove Corollaries12–14 is the same as proof of Theorem 10, where we use n convexity of ζ and linearity of J.Theorem 16.
In addition to presumptions of Theorem10, consider h∈Ca1,a2T,ℝ+ and B:μ,ν⟶ℝ is convex, then
(55)Jζt≥0,for all t∈μ,ν.Proof.
AsBt is convex for all t∈μ,ν, so Remark 1, inferred JBt≥0. Hence, (40) implies Jζt≥0.☐Remark 17.
Relation (55) in Theorem 16 is an extension of Jensen’s inequality (6).


## 3.2. Bounds for Identities Associated to Generalization of Jensen’s Functional

Here, we utilizeC˘ebyšev functional and Grüss-type inequalities to present few important results. Let two functions g1,g2:≀,ȷ⟶ℝ be Lebesgue integrable, and the C˘ebyšev functional is
(56)Yg1,g2=1ȷ−≀∫≀ȷg1tg2tdt−1ȷ−≀∫iȷg1tdt·1ȷ−≀∫≀ȷg2tdt.The next two theorems are given in [16].Theorem 18.
Letg1,g2:≀,ȷ⟶ℝ be functions such that g1 is Lebesgue integrable and g2 be the function which is absolutely continuous along ·−≀ȷ−·g2′2∈L≀,ȷ. Then, we have
(57)Yg1,g2≤12Yg1,g11/21j−i∫≀ȷy−≀j−yg2′y2dy1/2.
1/2 in (57) is possibly the best constant.Theorem 19.
Letg1,g2:≀,ȷ⟶ℝ be functions, such that g1 is absolutely continuous with g1′∈L∞≀,ȷ and g2 is monotonic nondecreasing on ≀,ȷ, then
(58)Yg1,g2≤12j−ig1′∞∫≀ȷy−ij−ydg2y.
The best possible constant is1/2 in (58).
We denote(59)ψs=JGH,nx,s.
TheC˘ebyšev functional is given as
(60)Yψ,ψ=1ν−μ∫μνψ2sds−1ν−μ∫μνψsds2.Theorem 20.
Letζ:μ,ν⟶ℝ be such that ζ∈Cnμ,ν for n∈ℕ with ·−μν−·ζn+12∈Lμ,ν and t∈μ,ν and GH,n, ψ and Y are defined in (21), (59), and (60), respectively. Then,
(61)Jζ=∑v=1r∑u=0kvζuavJHuvx+ζn−1ν−ζn−1μν−μ∫μνψsds+Rnμ,ν;ζ,where the remainder satisfies Rnμ,ν;ζ the estimation
(62)Rnμ,ν;ζ≤Yψ,ψ1/2ν−μ2∫μνs−μν−sζn+1s2ds1/2.Proof.
We use Theorem18 for g1⟶ψ and g2⟶ζn, to obtain
(63)1ν−μ∫μνψsζnsds−1ν−μ∫μνψsds·1ν−μ∫μνζnsds≤12Yψ,ψ1/21ν−μ∫μνs−μν−sζn+1s2ds1/2.
Therefore, we have(64)1ν−μ∫μνψsζnsds=1ν−μ2ζn−1ν−ζn−1μ∫μνψsds,∫μνψsζnsds=ζn−1ν−ζn−1μν−μ∫μνψsds+Rμ,ν;ζ,where the remainder Rμ,ν;ζ satisfy the estimation (62). Now, from the identity (36),
(65)Jζ=∑v=1r∑u=0kvζuavJHuvt+∫μνψsζnsds,Jζ=∑v=1r∑u=0kvζuavJHuvt+ζn−1ν−ζn−1μν−μ∫μνψsds+Rμ,ν;ζ.☐The Grüss-type inequality given below can be obtained by using Theorem19.Theorem 21.
Assumeζ:μ,ν⟶ℝ such that ζ∈Cnμ,νn∈ℕandζ2n+1≥0onμ,ν. Suppose ψ and Y are defined in (59) and (60), respectively. Then, we have (61) the remainder Rμ,ν:ζ satisfies the bound
(66)Rμ,ν;ζ≤ν−μψ′∞×ζn−1ν+ζn−1μν−μ−ζn−2ν−ζn−2μν−μ.Proof.
Applying Theorem19 for g1⟶ψ and g2⟶ζn, we have
(67)1ν−μ∫μνψsζnsds−1ν−μ∫μνψsds·1ν−μ∫μνζnsds≤12ν−μψ′∞∫μνs−μν−sζn+1sds.
Since(68)∫μνs−μν−sζn+1sds=∫μν2s−μ+νζnsds,∫μνs−μν−sζn+1sds=ν−μζn−1ν+ζn−1μ−2ζn−2ν+ζn−2μ;using (36) and (67), we get (66).☐Theorem 22.
Let all the presumptions of Theorem8 be satisfied. Suppose i˘,j˘ is a couple of numbers, i.e., 1≤i˘,j˘≤∞,1/i˘+1/j˘=1. Suppose ζni˘:μ,ν⟶ℝ is a function which is R-integrable for some n≥2. Then, we have
(69)Jζ−∑v=1r∑u=0kvζuavJHuvx≤ζni˘∫μνJGHx,sj˘ds1/j.
Fori˘∈1,∞, the constant on the right of (69) is sharp and suitable when i˘ is one.Proof.
Assume℘s=JGnt,s. Hölder’s inequality and identity (36) give us
(70)∣Jζ−∑v=1r∑u=0kvJHuvtζuav∣=∫μν℘sζnsds≤ζni˘∫μν℘tj˘ds1/j˘.
Takingζ for i˘∈1,∞, such that, ζns=sgn℘s℘s1/i˘−1. In case of i˘=∞, take ζns=sgn℘s.
We prove that fori˘=1,
(71)∫μν℘sζnsds≤maxs∈μ,ν℘s∫μνζnsdsis the suitable inequality. Let ∣℘s∣ achieve its maximum at p∈μ,ν. We suppose firstly that ℘d¨>0. We define ζs for small enough δ, by
(72)ζδs≔0,μ≤s≤p,1εn!s−pn,p≤s≤p+δ,1n!s−pn−1,p+δ≤s≤ν.
Then, forδ small enough,
(73)∫μν℘sζnsds=∫pp+δ℘s1δds=1δ∫pp+δ℘sds.
Now, from the inequality (71), we have
(74)1δ∫pp+δ℘sds≤℘p∫pp+δ1δds=℘p.
For℘p<0, we set ζδs as
(75)ζδs≔1n!s−p−δn−1,μ≤s≤p,−1εn!s−p−δn,p≤s≤p+∈,0,p+δ≤s≤ν.
The rest of the proof is the same as above.☐


## 4. Conclusion

Jensen’s functional for diamond integral (7) is generalized for n-convex functions using the Hermite polynomial in the present article. Different conditions of Hermite polynomial are utilized to describe respective refinements of the functional. In seek of applications, bounds for identities associated to constructed functional are also discussed. Moreover, by defining the functional as difference of right and left sides of extended inequality (40) (where B is defined in (41)), it is possible to study n-exponential convexity, exponential convexity, and applications to Stolarsky-type means as discussed by Aras-Gazic et al., in [17] (Sections 5 and 6). This article extends the results of [8] on time scales.



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