# Schur, Hermite-Hadamard, and Fejér Type Inequalities for the Class of Higher-Order Generalized Convex Functions

Authors: Yi Ma; Muhammad Shoaib Saleem; Imran Bashir; Yeliang Xiao
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

The study of convex functions is an interesting area of research due to its huge applications in pure and applied mathematics special in optimization theory. The aim of this paper is to introduce and study a more generalized class of convex functions. We established Schur (S), Hermite-Hadamard (HH), and Fejér (F) type inequalities for introduced class of convex functions. The results presented in this paper extend and generalize many existing results of the literature.


---

## Body

## 1. Introduction

The study of convexity is very simple and ordinary concept and very important due to its massive applications in industry and business; convexity has a great influence on our daily life. In the solution of many real-world problems, the concept of convexity is very decisive. Problems faced in constrained control and estimation are convex. Geometrically, “the convex function is a real-valued function if the line segment joining any two of its points lies on or above the graph of the function in Euclidean space” [1, 2].The inequality theory is a best way to understand and apply convexity and the following HH inequality for a C-functiong is most famous and most studied inequality in recent years;
(1)ga1+a22≤1a2−a1∫a1a2gxdx≤ga1+ga22.There are many interesting generalizations for HH inequality, for example, see [3, 4]. Alp et al. [5] established q–HH inequalities and gave quantum estimates for midpoint type inequalities via convex and quasi–C-functions. Rahman et al. [6] presented certain fractional proportional integral inequalities via C-functions. Liu et al. [7] established the several HH type inequality via C-function for ψ–Riemann–Liouville fractional integrals. In [8], Farid et al. established certain inequalities for Riemann–Liouville fractional integrals in more general form via C-functions and proposed some applications of their established results. Tunc [9] studied h–C-functions and presented Ostrowski-type inequalities via h–C-functions. He also proposed applications of his established Ostrowski-type inequalities to special means. Set et al. [10] established the HH’s inequality for some C-functions via fractional integrals. They also presented some other resulted results in the setting of fractional integrals.Moreover, Avci, Kavurmaci, and Özdemir in [11] established the HH type for the class of s–C-functions in 2nd sense which is more generalized convexity. The exponentially m–C-functions were investigated by Rashid et al., in [12]. Different kinds of inequalities for exponentially m–C-functions were also established in the same paper. In [13], Baloch and Chu investigated harmonic C-functions and established Petrovic-type inequalities. Sun and Chu [14] studied g–C-functions and established inequalities for the generalized weighted mean values. Ali, Khan, and Khurshidi [15] investigated η–C-functions and established HH inequality for fractional integrals. Chu et al. [16] generalized the HH type inequalities for MT–C-functions and a variant of Jensen-type inequality for harmonic C-functions was given in [17].The strongly C-function is one of the important functions of convexity, see [4, 18–21]. A function g:I⟶ℝ is strongly C-function if
(2)gta1+1−ta2≤tga1+1−tga2−μt1−ta1−a22holds ∀a1,a2∈L,t∈0,1 and μ is modulus value.Convexity is being generalized day by day using different methods, In this work, we discuss the convexity of higher order. For the generalization and different application of convexity, see for instance [22–24].Convex functiong:I=a1,a2⊂ℝ⟶ℝ in the classical sense is introduced as
(3)gta1+1−ta2≤tga1+1−tga2,where a1,a2∈I and t∈0,1.A functiong on close set L is said to be a higher-order strongly convex, if there exists a constant μ≥0, such that
(4)gta1+1−ta2≤tga1+1−tga2−μϕta2−a1q,where a1,a2∈L, q≥0, t∈0,1 and
(5)ϕt=t1−t.If we putq=2, then (4) becomes strongly C-function in classical sense with same ϕt as defined in (5).It is noticed that the functionϕ. in (5) is not modified. Characterizations of the higher-order strongly C-function discussed in [25] are not correct. Mohsen et al. [26] modify ϕ. for the higher-order strongly C-function. Noor and Noor [27] generalized the concept of higher-order strongly convex defined in [26] by higher-order strongly generalized C-functions. For more detailed survey, we refer the readers to [28] and references therein.Saleem et al. [29] extended the concept of higher-order strongly generalized C-function by introducing generalized strongly p–C-functions of higher order. The generalized strongly modified p,h–C-functions of higher order are studied in this paper.In this paper, we introduced the generalized strongly modifiedp;h–C-functions of higher order and establish some well-known inequalities. Motivation of this work is to improve and extend the existing results. As stated in the remarks of this paper, many existing results can be derived from our results.This paper is organized as follows: Section2 contains definitions, Section 3 contains some basic results, and Section 4 contains HH, F, and S type inequalities for the generalized strongly modified p,h–C-functions of higher order.


## 2. Definitions and Basic Results

The first part of this paper contains basic definitions of convexity.Definition 1 (Additive).
A functionη is additive when ηx1,y1+ηx2,y2=ηx1+x2,y1+y2∀x1,x2,y1,y2∈ℝ, see [30] for detail.Definition 2 (Super multiplicative function) [31].
A mappingg:I⊂ℝ is super multiplicative function when
(6)ga1a2≥ga1ga2,∀a1,a2∈I,t∈0,1.Definition 3.
Non-negatively homogeneous. A functionη is said to be non-negatively homogeneous if ηλa1,λa2=ληa1,a2∀a1,a2∈ℝ and λ≥0.The definition ofh–C-function is given in [32], which was generalized by Noor et al. in [33] as modified h–C-function. Later on in [34], the class of generalized strongly modified h–C-function. The class of generalized strongly C-functions of higher order was introduced in [27] and generalized strongly p–C-functions of higher order were introduced in [29]. Motivated by these innovations, the extended version of higher order strongly convexity, which is generalized strongly modified p,h–C-functions of higher order, is introduced in this paper as follows:Definition 4.
Generalized strongly modifiedp,h–C-functions of higher order. A function g:I⟶ℝ is said to be generalized strongly modified p,h– convex of higher order if
(7)gta1p+1−ta2p1/p≤ga2+htηga1,ga2−μϕta2p−a1pqholds for all a1,a2∈I, μ≥0, and t∈0,1.The class of all generalized strongly modifiedp,h–C-functions of higher order is denoted by GSMPHCF.Remark 5.
(1)
By takingμ=0, p=1, and ηa,b=a−b in (7), we get the definition of modified h convex function(2)
By takingμ=0,p=1,ηa,b=a−b, and ht=t in (7), we get the definition of C-function(3)
By takinght=t and p=1 in (7), we get the definition of generalized strongly convex function(4)
By takinght=t in (7), we get the definition of generalized strongly p convex function
Hence, in the light of the above remark, our definition is more generalized than all exiting notions.Now, we establish some basic properties.Proposition 6.
Considerη is non-negatively homogeneous and additive and the functions g,h:I⊂ℝ⟶ℝ are in GSMPHCF, then af+bg∀a,b∈ℝ+ is also in GSMPHCF.Proof.
It is cleared thata+bμ is always greater than μ because a,b∈ℝ+ and μ≥0 using this fact we can show that af+bg ∀a,b∈ℝ+ is in GSMPHCF.Proposition 7.
Consider the functionsh1, h2 on L are non-negative and h2t⩽h1t. If g is generalized strongly modified p,h2–C-function of higher order, then g is also generalized strongly modified p,h1–C-function of higher order.Proof.
The proof is straight forward.Proposition 8.
Suppose a class of functionsgj:I⊂ℝ⟶ℝ be in GSMPHCF where λj∈ℝ+ for every j∈ℕ and η is non-negatively homogeneous and additive then their linear combination f:ℝ⟶ℝ is also in GSMPHCF.Proof.
Let the linear combination(8)fx=∑j=1mλjgjx.
Setx=ta1p+1−ta2p1/p, so
(9)fta1p+1−ta2p1/p=∑j=1mλjgjta1p+1−ta2p1/p≤∑j=1mλjgja2+ht∑j=1mλjηgia1,gia2−∑j=1mλjμϕta2p−a1pq≤fa2+htη∑j=1mλjgia1,∑j=1mλjgia2−μϕta2p−a1pq=fa2+htηfa1,fa2−μϕta2p−a1pq.
In this proof, we have used a straight forward result that∑j=1mλjμ is always greater than μ when λj∈ℝ+ for every j∈ℕ.


## 3. Hermite-Hadamard (HH), Fejér (F), and Schur (S) Type Inequalities

In this section, we present Hermite-Hadamard, Fejér, and Schur type inequalities for the new notion of convexity introduced in this paper.


### 3.1. S Type Inequality

Firstly, we present Schur type inequality.Theorem 9.
Letg:I⟶ℝ and h be the super multiplicative function, where η:N×N⟶M be a bi-function for appropriate A,B⊆ℝ. Then, for a1,a2,a3∈L, a1<a2<a3 such that a3−a1,a3−a2,a2−a1∈L, the following inequality holds
(10)ga2ha3p−a1p≤ha3p−a1pga3+ha3p−a2pηga1,ga3−μϕa3p−a2pa3p−a1pa1p−a3pqha3p−a1pif and only if g is in GSMPHCF.Proof.
Leta1,a2,a3∈L⊂ℝ, such that a3−a2/a3−a1∈0,1⊆L,a2−a1/a3−a1∈0,1⊆L, and a3−a2/a3−a1+a2−a1/a3−a1=1, then
(11)ha3p−a1p=ha3p−a1pa3p−a2pa3p−a2p≥ha3p−a1pa3p−a2pha3p−a2p,as h is super multiplicative.
Supposeha3p−a2p≥0, by definition of g, we have
(12)gtxp+1−typ1/p≤gy+htηgx,gy−μϕtyp−xpq.
Insertinga3p−a2p/a3p−a1p=t, x=a1, and y=a3 in inequality (12), we obtain,
(13)ga3p−a2pa3p−a1pa1p+1−a3p−a2pa3p−a1pa3p1/p≤ga3+ha3p−a2pa3p−a1pηga1,ga3−μϕa3p−a2pa3p−a1pa3p−a1pq≤ga3+ha3p−a2pha3p−a1pηga1,ga3−μϕa3p−a2pa3p−a1pa3p−a1pq.
Multiplying (13) by ha3p−a1p and using the fact that h is super multiplicative, we get
(14)ga2ha3p−a1p≤ha3p−a1pga3+ha3p−a2pηga1,ga3−μϕa3p−a2pa3p−a1pa3p−a1pqha3p−a1p
For conversely inserta1=x, a2=txp+1−typ1/p, a3=y, and a3p−a2p/a3p−a1p=t in inequality (10) and using the fact that h is super multiplicative function such that htyp−xp≥hyp−xpht, we get
(15)hyp−xpgtxp+1−typ1/p≤hyp−xpgy+hyp−xphtηgx,gy−μϕtyp−xpqhyp−xp,implies
(16)gtxp+1−typ1/p≤gy+htηgx,gy−μϕtyp−xpq.
The prove is completed.Remark 10.
(1)
Forht=t, (10) reduced to S type inequality for function defined in [29].(2)
Forμ=0, p=1, and ηa,b=a−b, (10) reduced to S type inequality for modified h–C-function, see [33].


### 3.2. HH Type Inequality

Here we present Hermite–Hadamard type inequality.Theorem 11.
Letg:I⟶ℝ in GSMPHCF defined on a1,a2 with a1<a2, then
(17)ga1p+a2p21/p−h12Mη+μϕ12a1p−a2pq1+−1q+12q+11/p≤pa2p−a1p∫a1a2xp−1gxdx≤ga1+ga22+Nη∫01htdt−μa2p−a1pq∫01ϕtdt.Proof.
Choosew=ta1p+1−ta2p1/p and z=1−ta1p+ta2p1/p, then
(18)ga1p+a2p21/p=gwp+zp21/p≤gz+h12ηgw,gz−μϕ12zp−wpq.
Implies(19)ga1p+a2p21/p≤g1−ta1p+ta2p1/p+h12ηgta1p+1−ta2p1/p,g1−ta1p+ta2p1/p−μ2t−1qϕ12a2p−a1pq.
Integrating with respect tot on [0,1], we get
(20)ga1p+a2p21/p≤∫01g1−ta1p+ta2p1/pdt+h12∫01ηgta1p+1−ta2p1/p,g1−ta1p+ta2p1/pdt−μϕ12a2p−a1pq∫012t−1qdt.
Implies(21)ga1p+a2p21/p≤∫01g1−ta1p+ta2p1/pdt+h12Mη−μϕ12a2p−a1pq1+−1q+12q+11/p.
Puttingx=1−ta1+ta2, we get
(22)ga1p+a2p21/p≤pa2p−a1p∫a1a2xp−1gxdx+h12Mη−μϕ12a2p−a1pq1+−1q+12q+11/p.
Implies(23)ga1p+a2p21/p−h12Mη+μϕ12a2p−a1pq1+−1q+12q+11/p≤pa2p−a1p∫a1a2xp−1gxdx.
Now,(24)∫01gta1p+1−ta2p1/pdt≤ga2+∫01htηga1,ga2dt−μa1p−a2pq∫01ϕtdt.
Takingta1p+1−ta2p1/p=x, we get
(25)pa2p−a1p∫a1a2xp−1gxdx≤ga2+ηga1,ga2∫01htdt−μa1p−a2pq∫01ϕtdt=A.
Similarly,(26)pa2p−a1p∫a1a2xp−1gxdx≤ga1+ηga2,ga1∫01htdt−μa1p−a2pq∫01ϕtdt=B.
From inequalities (25) and (26), we have
(27)pa2p−a1p∫a1a2xp−1gxdx≤minA,B.
This implies(28)pa2p−a1p∫a1a2xp−1gxdx≤A+B2.pa2p−a1p∫a1a2xp−1gxdx≤ga1+ga22+ηga1,ga2+ηga2,ga12×∫01htdt−μ2a1p−a2pq∫01ϕtdt.
Implies(29)pa2p−a1p∫a1a2xp−1gxdx≤ga1+ga22+Nη∫01htdt−μa2p−a1pq∫01ϕtdt.
Inequalities (23) and (29) assure to proof.Remark 12.
(1)
Forp=1,μ=0, and ηa,b=a−b, (17) reduced to HH type inequality for modified h–C-function, see [33].(2)
Forht=t, (17) reduced to HH type inequality for function defined in [29].(3)
Forμ=0,p=1, ηa,b=a−b, and ht=t,(17) reduced to HH type inequality for C-function in classical sense


### 3.3. F Type Inequality

This subsection contains Fejér type inequality.Lemma 13.
Considerg be in GSMPHCF and ηx,y=−ηy,x, then
(30)ga1p+a2p−xp1/p≤ga1+ga2−gx∀x∈a1,a2,where x=ta1p+1−ta2p1/p and t∈0,1.Proof.
Letg:I⟶ℝ be in GSMPHCF then for x=ta1p+1−ta2p1/p, we get
(31)ga1p+a2p−xp1/p=g1−ta1p+ta2p1/p≤ga1+htηga2,ga1−μϕta1p−a2pq≤ga1+ga2−ga2−htηga1,ga2−μϕta1p−a2pq+2μϕta1p−a2pq≤ga1+ga2−ga2+htηga1,ga2−μϕta1p−a2pq≤ga1+ga2−gx.Theorem 14.
Letg:a1,a2⟶ℝ be in GSMPHCF and w:a1,a2⟶ℝ be integrable and symmetric w.r.t a1+a2/2, where w≥0, then we have
(32)ga1p+a2p21/p∫a1a2xp−1wxdx+μϕ12∫a1a2xp−1a1p+a2p−2xpq×wxdx−h12∫a1a2xp−1ηga1p+a2p−xp1/p,gxwxdx≤∫a1a2xp−1gxwxdx≤ga1+ga22∫a1a2xp−1wxdx+Mη∫a1a2xp−1hxp−a2pa1p−a2p1/pwxdx−μ∫a1a2xp−1ϕxp−a2pa1p−a2p1/pwxdx,where Mη=ηga1,ga2+ηga2,ga1/2.Proof.
Letg be a generalized strongly modified p,h–C-function of higher order where x=ta1p+1−ta2p1/p and y=ta2p+1−ta1p1/p, then
(33)ga1p+a2p21/p∫a1a2xp−1wxdx=∫a1a2xp−1ga1p+a2p−xp+xp21/pwxdx=∫a1a2xp−1gyp+xp21/pwxdx≤∫a1a2xp−1gxwxdx+h12∫a1a2xp−1ηgy,gxwxdx−μϕ12∫a1a2xp−1xp−ypqwxdx,where yp=a1p+a2p−xp, so
(34)ga1p+a2p21/p∫a1a2xp−1wxdx+μϕ12∫a1a2xp−1a1p+a2p−2xpqwxdx−h12∫a1a2xp−1ηga1p+a2p−xp1/p,gxwxdx≤∫a1a2xp−1gxwxdx.
Now, ifx=ta1p+1−ta2p1/p, then
(35)∫a1a2xp−1gxwxdx=a2p−a1pp∫01gta1p+1−ta2p1/pwta1p+1−ta2p1/pdt.
So,(36)pa2p−a1p∫a1a2xp−1gxwxdx≤∫01ga2wta1p+1−ta2p1/pdt+ηga1,ga2∫01htwta1p+1−ta2p1/pdt−μa2p−a1pq∫01ϕtdt.
Similarly, ifx=ta2p+1−ta1p1/p, then
(37)pa2p−a1p∫a1a2xp−1gxwxdx≤∫01ga1wta2p+1−ta1p1/pdt+ηga2,ga1∫01htwta2p+1−ta1p1/pdt−μa2p−a1qp∫01ϕtwta2p+1−ta1p1/pdt.
Adding inequalities (36) and (37), where w is symmetric, then
(38)2pa2p−a1p∫a1a2xp−1gxwxdx≤ga1+ga2∫01wta1p+1−ta2p1/pdt+ηga1,ga2+ηga2,ga1∫01htwta1p+1−ta2p1/pdt−2μa2p−a1pq∫01ϕtwta1p+1−ta2p1/pdt.
Now, forx=ta1p+1−ta2p1/p, we have
(39)∫a1a2xp−1gxwxdx≤ga1+ga22∫a1a2xp−1wxdx+ηga1,ga2+ηga2,ga12∫a1a2xp−1hxp−a2pa1p−a2p1/pwxdx−μ∫a1a2xp−1ϕxp−a2pa1p−a2p1/pwxdx.
Inequalities (34) and (39) assure to proof.Remark 15.
(1)
Forht=t, this F type inequality reduced to F type inequality for the function defined in [29].(2)
Forμ=0,p=1, and ηa,b=a−b, this F Type inequality reduced to F type inequality for the function defined in [33].(3)
Forμ=0, p=1, ηa,b=a−b, and ht=t, this F type inequality reduced to F type inequality for C-function in classical sense


## 3.1. S Type Inequality

Firstly, we present Schur type inequality.Theorem 9.
Letg:I⟶ℝ and h be the super multiplicative function, where η:N×N⟶M be a bi-function for appropriate A,B⊆ℝ. Then, for a1,a2,a3∈L, a1<a2<a3 such that a3−a1,a3−a2,a2−a1∈L, the following inequality holds
(10)ga2ha3p−a1p≤ha3p−a1pga3+ha3p−a2pηga1,ga3−μϕa3p−a2pa3p−a1pa1p−a3pqha3p−a1pif and only if g is in GSMPHCF.Proof.
Leta1,a2,a3∈L⊂ℝ, such that a3−a2/a3−a1∈0,1⊆L,a2−a1/a3−a1∈0,1⊆L, and a3−a2/a3−a1+a2−a1/a3−a1=1, then
(11)ha3p−a1p=ha3p−a1pa3p−a2pa3p−a2p≥ha3p−a1pa3p−a2pha3p−a2p,as h is super multiplicative.
Supposeha3p−a2p≥0, by definition of g, we have
(12)gtxp+1−typ1/p≤gy+htηgx,gy−μϕtyp−xpq.
Insertinga3p−a2p/a3p−a1p=t, x=a1, and y=a3 in inequality (12), we obtain,
(13)ga3p−a2pa3p−a1pa1p+1−a3p−a2pa3p−a1pa3p1/p≤ga3+ha3p−a2pa3p−a1pηga1,ga3−μϕa3p−a2pa3p−a1pa3p−a1pq≤ga3+ha3p−a2pha3p−a1pηga1,ga3−μϕa3p−a2pa3p−a1pa3p−a1pq.
Multiplying (13) by ha3p−a1p and using the fact that h is super multiplicative, we get
(14)ga2ha3p−a1p≤ha3p−a1pga3+ha3p−a2pηga1,ga3−μϕa3p−a2pa3p−a1pa3p−a1pqha3p−a1p
For conversely inserta1=x, a2=txp+1−typ1/p, a3=y, and a3p−a2p/a3p−a1p=t in inequality (10) and using the fact that h is super multiplicative function such that htyp−xp≥hyp−xpht, we get
(15)hyp−xpgtxp+1−typ1/p≤hyp−xpgy+hyp−xphtηgx,gy−μϕtyp−xpqhyp−xp,implies
(16)gtxp+1−typ1/p≤gy+htηgx,gy−μϕtyp−xpq.
The prove is completed.Remark 10.
(1)
Forht=t, (10) reduced to S type inequality for function defined in [29].(2)
Forμ=0, p=1, and ηa,b=a−b, (10) reduced to S type inequality for modified h–C-function, see [33].


## 3.2. HH Type Inequality

Here we present Hermite–Hadamard type inequality.Theorem 11.
Letg:I⟶ℝ in GSMPHCF defined on a1,a2 with a1<a2, then
(17)ga1p+a2p21/p−h12Mη+μϕ12a1p−a2pq1+−1q+12q+11/p≤pa2p−a1p∫a1a2xp−1gxdx≤ga1+ga22+Nη∫01htdt−μa2p−a1pq∫01ϕtdt.Proof.
Choosew=ta1p+1−ta2p1/p and z=1−ta1p+ta2p1/p, then
(18)ga1p+a2p21/p=gwp+zp21/p≤gz+h12ηgw,gz−μϕ12zp−wpq.
Implies(19)ga1p+a2p21/p≤g1−ta1p+ta2p1/p+h12ηgta1p+1−ta2p1/p,g1−ta1p+ta2p1/p−μ2t−1qϕ12a2p−a1pq.
Integrating with respect tot on [0,1], we get
(20)ga1p+a2p21/p≤∫01g1−ta1p+ta2p1/pdt+h12∫01ηgta1p+1−ta2p1/p,g1−ta1p+ta2p1/pdt−μϕ12a2p−a1pq∫012t−1qdt.
Implies(21)ga1p+a2p21/p≤∫01g1−ta1p+ta2p1/pdt+h12Mη−μϕ12a2p−a1pq1+−1q+12q+11/p.
Puttingx=1−ta1+ta2, we get
(22)ga1p+a2p21/p≤pa2p−a1p∫a1a2xp−1gxdx+h12Mη−μϕ12a2p−a1pq1+−1q+12q+11/p.
Implies(23)ga1p+a2p21/p−h12Mη+μϕ12a2p−a1pq1+−1q+12q+11/p≤pa2p−a1p∫a1a2xp−1gxdx.
Now,(24)∫01gta1p+1−ta2p1/pdt≤ga2+∫01htηga1,ga2dt−μa1p−a2pq∫01ϕtdt.
Takingta1p+1−ta2p1/p=x, we get
(25)pa2p−a1p∫a1a2xp−1gxdx≤ga2+ηga1,ga2∫01htdt−μa1p−a2pq∫01ϕtdt=A.
Similarly,(26)pa2p−a1p∫a1a2xp−1gxdx≤ga1+ηga2,ga1∫01htdt−μa1p−a2pq∫01ϕtdt=B.
From inequalities (25) and (26), we have
(27)pa2p−a1p∫a1a2xp−1gxdx≤minA,B.
This implies(28)pa2p−a1p∫a1a2xp−1gxdx≤A+B2.pa2p−a1p∫a1a2xp−1gxdx≤ga1+ga22+ηga1,ga2+ηga2,ga12×∫01htdt−μ2a1p−a2pq∫01ϕtdt.
Implies(29)pa2p−a1p∫a1a2xp−1gxdx≤ga1+ga22+Nη∫01htdt−μa2p−a1pq∫01ϕtdt.
Inequalities (23) and (29) assure to proof.Remark 12.
(1)
Forp=1,μ=0, and ηa,b=a−b, (17) reduced to HH type inequality for modified h–C-function, see [33].(2)
Forht=t, (17) reduced to HH type inequality for function defined in [29].(3)
Forμ=0,p=1, ηa,b=a−b, and ht=t,(17) reduced to HH type inequality for C-function in classical sense


## 3.3. F Type Inequality

This subsection contains Fejér type inequality.Lemma 13.
Considerg be in GSMPHCF and ηx,y=−ηy,x, then
(30)ga1p+a2p−xp1/p≤ga1+ga2−gx∀x∈a1,a2,where x=ta1p+1−ta2p1/p and t∈0,1.Proof.
Letg:I⟶ℝ be in GSMPHCF then for x=ta1p+1−ta2p1/p, we get
(31)ga1p+a2p−xp1/p=g1−ta1p+ta2p1/p≤ga1+htηga2,ga1−μϕta1p−a2pq≤ga1+ga2−ga2−htηga1,ga2−μϕta1p−a2pq+2μϕta1p−a2pq≤ga1+ga2−ga2+htηga1,ga2−μϕta1p−a2pq≤ga1+ga2−gx.Theorem 14.
Letg:a1,a2⟶ℝ be in GSMPHCF and w:a1,a2⟶ℝ be integrable and symmetric w.r.t a1+a2/2, where w≥0, then we have
(32)ga1p+a2p21/p∫a1a2xp−1wxdx+μϕ12∫a1a2xp−1a1p+a2p−2xpq×wxdx−h12∫a1a2xp−1ηga1p+a2p−xp1/p,gxwxdx≤∫a1a2xp−1gxwxdx≤ga1+ga22∫a1a2xp−1wxdx+Mη∫a1a2xp−1hxp−a2pa1p−a2p1/pwxdx−μ∫a1a2xp−1ϕxp−a2pa1p−a2p1/pwxdx,where Mη=ηga1,ga2+ηga2,ga1/2.Proof.
Letg be a generalized strongly modified p,h–C-function of higher order where x=ta1p+1−ta2p1/p and y=ta2p+1−ta1p1/p, then
(33)ga1p+a2p21/p∫a1a2xp−1wxdx=∫a1a2xp−1ga1p+a2p−xp+xp21/pwxdx=∫a1a2xp−1gyp+xp21/pwxdx≤∫a1a2xp−1gxwxdx+h12∫a1a2xp−1ηgy,gxwxdx−μϕ12∫a1a2xp−1xp−ypqwxdx,where yp=a1p+a2p−xp, so
(34)ga1p+a2p21/p∫a1a2xp−1wxdx+μϕ12∫a1a2xp−1a1p+a2p−2xpqwxdx−h12∫a1a2xp−1ηga1p+a2p−xp1/p,gxwxdx≤∫a1a2xp−1gxwxdx.
Now, ifx=ta1p+1−ta2p1/p, then
(35)∫a1a2xp−1gxwxdx=a2p−a1pp∫01gta1p+1−ta2p1/pwta1p+1−ta2p1/pdt.
So,(36)pa2p−a1p∫a1a2xp−1gxwxdx≤∫01ga2wta1p+1−ta2p1/pdt+ηga1,ga2∫01htwta1p+1−ta2p1/pdt−μa2p−a1pq∫01ϕtdt.
Similarly, ifx=ta2p+1−ta1p1/p, then
(37)pa2p−a1p∫a1a2xp−1gxwxdx≤∫01ga1wta2p+1−ta1p1/pdt+ηga2,ga1∫01htwta2p+1−ta1p1/pdt−μa2p−a1qp∫01ϕtwta2p+1−ta1p1/pdt.
Adding inequalities (36) and (37), where w is symmetric, then
(38)2pa2p−a1p∫a1a2xp−1gxwxdx≤ga1+ga2∫01wta1p+1−ta2p1/pdt+ηga1,ga2+ηga2,ga1∫01htwta1p+1−ta2p1/pdt−2μa2p−a1pq∫01ϕtwta1p+1−ta2p1/pdt.
Now, forx=ta1p+1−ta2p1/p, we have
(39)∫a1a2xp−1gxwxdx≤ga1+ga22∫a1a2xp−1wxdx+ηga1,ga2+ηga2,ga12∫a1a2xp−1hxp−a2pa1p−a2p1/pwxdx−μ∫a1a2xp−1ϕxp−a2pa1p−a2p1/pwxdx.
Inequalities (34) and (39) assure to proof.Remark 15.
(1)
Forht=t, this F type inequality reduced to F type inequality for the function defined in [29].(2)
Forμ=0,p=1, and ηa,b=a−b, this F Type inequality reduced to F type inequality for the function defined in [33].(3)
Forμ=0, p=1, ηa,b=a−b, and ht=t, this F type inequality reduced to F type inequality for C-function in classical sense


## 4. Conclusions

In this paper, we introduced the notion of generalized strongly modifiedp;h–C-functions of higher order and established HH, F, and S type inequalities. Many existing results can be derived from our results. Our results are applicable in pure and applied mathematics especially in optimization theory. For applications point, we refer to the readers [35–38].


## 5. Future Directions

It will be interesting to introduce a new and more generalized version of the convexity. The inequalities established in this paper can be further extended for different versions of fractional integral operators.



---

*Source: 8575563-2022-06-11.xml*