# Integral Inequalities of Integer and Fractional Orders forn–Polynomial Harmonically tgs–Convex Functions and Their Applications

Authors: Artion Kashuri; Soubhagya Kumar Sahoo; Bibhakar Kodamasingh; Muhammad Tariq; Ahmed A. Hamoud; Homan Emadifar; Faraidun K. Hamasalh; Nedal M. Mohammed; Masoumeh Khademi
Journal: Journal of Mathematics
 (2022)

---

## Abstract

The main objective of this article is to introduce the notion ofn–polynomial harmonically tgs–convex function and study its algebraic properties. First, we use this notion to present new variants of the Hermite–Hadamard type inequality and related integral inequalities, as well as their fractional analogues. Further, we prove two interesting integral and fractional identities for differentiable mappings, and, using them as auxiliary results, some refinements of Hermite–Hadamard type integral inequalities for both classical and fractional versions are presented. Finally, in order to show the efficiency of our results, some applications for special means and error estimations are obtained as well.


---

## Body

## 1. Introduction and Preliminaries

The theory of convexity has become an astonishing and profound motivation for researchers in various fields of science. This theory has not just intriguing results in branches of mathematics and engineering but also gives mathematical computations and innovative tools for mathematicians to address and handle numerous issues because of its simplicity and wide scope of significance.The concept of convexity theory is not a new one as it happens in some other structures in determining the orbital length. These days, the utilization of convexity theory can be straightforwardly or in a roundabout way seen in different subjects like geometry, probability theory, mathematical finance, optimization, functional analysis, etc. Convexity theory has shown up as an incredible procedure to solve several real-life issues in different branches of science. Numerous articles have been composed by various mathematicians on different kinds of convex functions for some related inequalities likeH−H inequality, Simpson inequality, Ostrowski inequality, Bullen type inequality, Opial inequality, Olsen type inequality, etc.Inequalities have a lot of applications in the fields of differential equations and applied mathematics (see [1, 2]). This theory also has a wide range of applications in other areas of science, such as convexity theory, mathematical analysis, discrete fractional calculus, mathematical physics, and many more (see [3–7]). Let us start our main discussion with the notion of convexity, which is defined as follows.Definition 1.
A functionϒ:a,b⟶ℝ is called convex, if(1)ϒκξ1+1−κξ2≤κϒξ1+1−κϒξ2,holds for every ξ1,ξ2∈a,b and κ∈0,1.
Toplu et al. [8] very recently determined the H−H inequality for n–polynomial convex function.
Letϒ:K⊆ℝ⟶ℝ be a convex function with ξ1<ξ2 and ξ1,ξ2∈K. Then,(2)ϒξ1+ξ22≤1ξ2−ξ1∫ξ1ξ2ϒzdz≤ϒξ1+ϒξ22.
In the present paper, we have given more emphasis to introduce the concept of a new type ofn–polynomial harmonically convexity with some inspiring algebraic properties. To know detailed knowledge about the new concept, we encourage interested readers to get familiarized with the theory of harmonic convexity [9], tgs–convexity [10], n–polynomial convexity [8], and n–polynomial harmonically convexity [11]. Park et al. in [12] generalized the notion of convexity as n–polynomial p–convexity. To follow some results associated with generalized n–polynomial convex function established via fractional operators, see [13–16], and for coordinates, see [17] and the references therein.Definition 2.
(see [10]). Let K⊂ℝ∖0 be any set and ϒ:K⟶ℝ be a function. Then, the function ϒ is said to be tgs–convex, if(3)ϒκξ1+1−κξ2≤κ1−κϒξ1+ϒξ2,∀ξ1,ξ2∈K,κ∈0,1.
Tunç et al. [10] provided the following results related to tgs–convexity involving classical and fractional operators.(4)2ϒξ1+ξ22≤1ξ2−ξ1∫ξ1ξ2ϒzdz≤ϒξ1+ϒξ26,2ϒξ1+ξ22≤Γα+12ξ2−ξ1αJξ1+αϒξ2+Jξ2−αϒξ1≤αϒξ1+ϒξ2α+1α+2.
Chebyshev’s inequality: ifb1m and b2m are two nonincreasing sequences such that(5)b11≥b12≥⋯≥b1m,b21≥b22≥⋯≥b2m,then(6)1m∑ℓ=1mb1ℓb2ℓ≥1m∑ℓ=1mb1ℓ1m∑ℓ=1mb2ℓ.
We denote in the sequel of the paperℒξ1,ξ2 the space of Lebesgue integrable functions on ξ1,ξ2.Definition 3.
(see [18]). Let ϒ∈ℒξ1,ξ2. The Riemann–Liouville fractional integrals Jξ1+αϒandJξ2−αϒ of order α>0 for a real valued continuous function ϒz are defined by(7)Jξ1+αϒz=1Γα∫ξ1zz−tα−1ϒtdt,Jξ2−αϒz=1Γα∫zξ2t−zα−1ϒtdt,respectively, where Γ⋅ is the well-known gamma function.Definition 4.
(see [19]). Let ϒ∈ℒξ1,ξ2 and k>0. Then, the k–Riemann–Liouville fractional integrals Jξ1+α,kϒandJξ2−α,kϒ of order α>0 for a real valued continuous function ϒz are defined by(8)Jξ1+α,kϒz=1kΓkα∫ξ1zz−tα/k−1ϒtdt,Jξ2−α,kϒz=1kΓkα∫zξ2t−zα/k−1ϒtdt,respectively, where Γk⋅ is the well-known k-gamma function defined by Γkz=∫0∞tz−1exp−tk/kdt.
Convexity has recently gained popularity due to its numerous applications in various fields of science, and fractional calculus has played an important role in this development (see [20]). Sarikaya et al. [18] proposed the Riemann–Liouville fractional integral operator to present the Hermite–Hadamard inequality. The Hermite–Hadamard inequality for the ψ-Riemann–Liouville fractional operator was generalized by Liu et al. [21], which was motivated by Sarikaya’s work. Similarly, generalized proportional fractional integrals were employed by various scholars, including Mumcu et al. [22]. Gürbüz et al. [23] worked on the Caputo–Fabrizio operator, Mohammed et al. [24] used tempered fractional integrals, Sahoo et al. [25] used k−Riemann–Liouville fractional integrals, and Khan et al. [26] established a new version of Hermite–Hadamard inequality employing generalized conformable fractional integrals. Butt and his collaborators, for example, worked on Jensen–Mercer type inequalities using novel fractional operators (see [27–29]), while Set et al. [30] and Fernandez et al. [31] presented the Hermite–Hadamard inequality using the Atangana–Baleanu fractional operator. For some recent generalizations of the Hermite–Hadamard inequality, we suggest interested readers to see [32–36] and the references therein. Recently, Naila et al. [37] presented some results for tgs–convex function via fractional calculus. In [38], the authors worked on exponentially tgs–convex function, and Tunç and Şanal in [39] obtained some perturbed trapezoid type inequalities for tgs–convex function.
We recall the following hypergeometric function [40], which is defined by Euler in integral form:(9)F12ξ1,ξ2;ξ3;z≔1βξ2,ξ3−ξ2∫01κξ2−11−κξ3−ξ2−11−zκ−ξ1dκ,ξ3>ξ2>0,z<1,where β⋅ is the Euler beta function.Definition 5.
(see [8]). Let K⊆0,∞ be an interval and n∈ℕ. A nonnegative real valued function ϒ:K⟶ℝ is called n–polynomial convex, if(10)ϒtξ1+1−tξ2≤1n∑m=1n1−1−tmϒξ1+1n∑m=1n1−tmϒξ2,holds for all ξ1,ξ2∈K.
Utilizing the concept ofn–polynomial convexity and harmonic convexity, then the concept of n–polynomial harmonically convex function is established.Definition 6.
(see [11]). Let K⊆0,∞ be an interval and n∈ℕ. A nonnegative real valued function ϒ:K⟶ℝ is said to be n–polynomial harmonic convex, if(11)ϒξ1ξ2κξ1+1−κξ2≤1n∑m=1n1−1−κmϒδξ1+1n∑m=1n1−κmϒξ2,holds for every ξ1,ξ2∈K.
This paper is organized as follows. After reviewing some preliminary studies about convexity and inequalities, we introduce the notion ofn–polynomial harmonically tgs–convex function in Section 2. In Section 3, we study some related algebraic properties. Section 4 deals with proving new variants of the H−H type inequality for n–polynomial harmonically tgs–convex functions. In Section 5, we present an integral identity and some related inequalities of H−H type for classical integrals. In Section 6, the fractional version of H−H type inequality and its related results are discussed using Riemann–Liouville fractional operators. Some applications for special means and error estimations are derived in Section 7. Section 8 presents a brief conclusion and future scope of this study.


## 2. A New Definition

Definition 7.
LetK⊆0,∞ be an interval and n∈ℕ. A nonnegative real valued function ϒ:K⟶ℝ is said to be n–polynomial tgs–convex, if(12)ϒtξ1+1−tξ2≤∑m=1ntm1−tmϒξ1+ϒξ2,holds for all ξ1,ξ2∈Kandt∈0,1.Definition 8.
LetK⊆0,∞ be an interval and n∈ℕ. A nonnegative real valued function ϒ:K⟶ℝ is called n–polynomial harmonically tgs–convex, if(13)ϒξ1ξ2tξ1+1−tξ2≤∑m=1ntm1−tmϒξ1+ϒξ2,holds for every ξ1,ξ2∈Kandt∈0,1.Remark 1.
(i)
Forn=1, n–polynomial harmonically tgs–convex function reduces to the classical tgs–harmonic convex function.(ii)
Forn=2, we have(14)∑m=12tm1−tm=t1−t+t21−t2=t1−t1+t−t2.
Since1+t−t2>1 as t>t2 for t∈0,1, we get(15)∑m=12tm1−tm≥t1−t.Remark 2.
Everyn–polynomial harmonically tgs–convex function is a h–convex function with ht=∑m=1ntm1−tm.


## 3. Algebraic Properties of the New Class of Functions

In this section, we begin with discussing some algebraic properties in support of the newly developed class of functions.Lemma 1.
The inequality∑m=1ntm1−tm≥t1−t holds for all t∈0,1.Proof.
The proof is evident from Remark1.
In what follows, we discuss some algebraic properties ofn–polynomial harmonically tgs–convex function.Proposition 1.
LetK⊆0,∞ be an interval and n∈ℕ. Every harmonic tgs–convex function on K is an n–polynomial harmonically tgs–convex function.Proof.
Using the definition ofn–polynomial harmonically tgs–convex function and from Lemma 1, we have(16)ϒξ1ξ2tξ1+1−tξ2≤t1−tϒξ1+ϒξ2≤∑m=1ntm1−tmϒξ1+ϒξ2.Proposition 2.
Every decreasing convex functions defined byϒ1z=−lnz,ϒ2z=−z, and ϒ3z=−z are harmonically tgs–convex. Then, by Proposition 1, they are n–polynomial harmonically tgs–convex functions.Proposition 3.
The sum of twon–polynomial harmonically tgs–convex functions is an n–polynomial harmonically tgs–convex function.Proof.
The proof is straightforward and hence left for the readers.Proposition 4.
Letϒ be an n–polynomial harmonically tgs–convex function and c>0, where n∈ℕ; then, cϒ is an n–polynomial harmonically tgs–convex function.Proof.
The proof is straightforward and hence left for the readers.Proposition 5.
Letϒα:K=ξ1,ξ2⟶ℝ be a family of n–polynomial harmonically tgs–convex functions with ϒu<∞ for every u∈ξ1,ξ2, where 0<ξ1<ξ2 and ϒu≔supαϒαu. Then, ϒ is an n–polynomial harmonically tgs–convex function on K.Proof.
Lett∈0,1 and ξ1,ξ2∈K; then, we have(17)ϒξ1ξ2tξ1+1−tξ2=supαϒαξ1ξ2tξ1+1−tξ2≤supα∑m=1ntm1−tmϒαξ1+ϒαξ2≤∑m=1ntm1−tmsupαϒαξ1+supαϒαξ2=∑m=1ntm1−tmϒξ1+ϒξ2.


## 4. Hermite–Hadamard Type Inequalities

In this section, we will establish two Hermite–Hadamard type integral inequalities forn–polynomial harmonically tgs–convex functions.Theorem 1.
Let0<ξ1<ξ2 and ϒ:ξ1,ξ2⟶ℝ be an n–polynomial harmonically tgs–convex function. If ϒ∈ℒξ1,ξ2, then the following inequality holds:(18)12∑m=1n1/4mϒ2ξ1ξ2ξ1+ξ2≤ξ1ξ2ξ2−ξ1∫ξ1ξ2ϒzz2dz≤ϒξ1+ϒξ2∑m=1nβm+1,m+1.Proof.
Sinceϒ is an n–polynomial harmonically tgs–convex function, we have(19)ϒxytx+1−ty≤∑m=1ntm1−tmϒx+ϒy.
Takingt=1/2 and using the change of variable technique as follows:(20)x=ξ1ξ2tξ1+1−tξ2,y=ξ1ξ2tξ2+1−tξ1,we get(21)ϒ2ξ1ξ2ξ1+ξ2≤∑m=1n12m1−12mϒξ1ξ2tξ1+1−tξ2+ϒξ1ξ2tξ2+1−tξ1=∑m=1n14mϒξ1ξ2tξ1+1−tξ2+ϒξ1ξ2tξ2+1−tξ1.
Now, integrating the above inequality with respect tot on 0,1, we obtain(22)12∑m=1n1/4mϒ2ξ1ξ2ξ1+ξ2≤ξ1ξ2ξ2−ξ1∫ξ1ξ2ϒzz2dz.
This completes the proof of the left inequality of (18).
For the right inequality of (18), taking z=ξ1ξ2/tξ1+1−tξ2 and using the property of the n–polynomial harmonically tgs–convex function ϒ, we have(23)ξ1ξ2ξ2−ξ1∫ξ1ξ2ϒzz2dz=∫01ϒξ1ξ2tξ1+1−tξ2dt≤∫01∑m=1ntm1−tmϒξ1+ϒξ2dt=ϒξ1+ϒξ2∑m=1nβm+1,m+1.
From inequalities (22) and (23), we get(24)12∑m=1n1/4mϒ2ξ1ξ2ξ1+ξ2≤ξ1ξ2ξ2−ξ1∫ξ1ξ2ϒzz2dz≤ϒξ1+ϒξ2∑m=1nβm+1,m+1,which completes the proof.Remark 3.
If we putn=1 in Theorem 1, then the following H−H inequality for harmonically tgs–convex function is obtained:(25)2ϒ2ξ1ξ2ξ1+ξ2≤ξ1ξ2ξ2−ξ1∫ξ1ξ2ϒzz2dz≤ϒξ1+ϒξ26.Theorem 2.
Letϒ,Ψ:ξ1,ξ2⊆ℝ⟶ℝ be two n–polynomial harmonically tgs–convex functions on ξ1,ξ2 with 0<ξ1<ξ2. If ϒ,Ψ∈ℒξ1,ξ2, then the following inequality holds:(26)ξ1ξ2ξ2−ξ1∫ξ1ξ2ϒzΨzz2dz≤n∑m=1nβ2m+1,2m+1Uξ1,ξ2+Vξ1,ξ2,where Uξ1,ξ2≔ϒξ1Ψξ1+ϒξ2Ψξ2 and Vξ1,ξ2≔ϒξ1Ψξ2+ϒξ2Ψξ1.Proof.
LetΥ and Ψ be two real valued, n–polynomial harmonically tgs–convex functions on ξ1,ξ2. Then, we have(27)ϒξ1ξ2tξ1+1−tξ2≤∑m=1ntm1−tmϒξ1+ϒξ2,Ψξ1ξ2tξ1+1−tξ2≤∑m=1ntm1−tmΨξ1+Ψξ2.
Multiplying above inequalities, we get(28)ϒξ1ξ2tξ1+1−tξ2Ψξ1ξ2tξ1+1−tξ2≤∑m=1ntm1−tmϒξ1+ϒξ2∑m=1ntm1−tmΨξ1+Ψξ2.
Integrating both sides of above inequality with respect tot over 0,1, due to the fact that tm1−tm is a nonincreasing function, and using Chebyshev’s sum inequality, we obtain(29)∫01ϒξ1ξ2tξ1+1−tξ2Ψξ1ξ2tξ1+1−tξ2dt≤n∑m=1nβ2m+1,2m+1×ϒξ1Ψξ1+ϒξ2Ψξ2+ϒξ1Ψξ2+ϒξ2Ψξ1=n∑m=1nβ2m+1,2m+1Uξ1,ξ2+Vξ1,ξ2.
Takingz=ξ1ξ2/tξ1+1−tξ2, then inequality (29) reduces to desired inequality (26), which completes the proof.


## 5. Refinements of Hermite–Hadamard Type Inequalities

Here, we will establish a new equality, and taking this equality into account, some novel refinements ofH−H type inequalities are discussed.Lemma 2.
Suppose0<r≤1 and consider a function ϒ:ξ1,ξ2/r⟶ℝ, which is differentiable on ξ1,ξ2/r with 0<ξ1<ξ2. If ϒ′∈ℒξ1,ξ2/r, then the following identity holds:(30)ϒξ1+ϒξ2/r2−ξ1ξ2ξ2−rξ1∫ξ1ξ2/rϒzz2dz=ξ1ξ2ξ2−rξ12r2∫011−2ttξ2/r+1−tξ12ϒ′ξ1ξ2/rtξ2/r+1−tξ1dt.Proof.
Let us denote(31)S≔ξ1ξ2ξ2−rξ12r2∫011−2ttξ2/r+1−tξ12ϒ′ξ1ξ2/rtξ2/r+1−tξ1dt.
Integrating by parts, we have(32)S=ξ1ξ2ξ2−rξ12r22t−1tξ2/r+d1−tξ12ϒξ1ξ2/rtξ2/r+1−tξ1r2tξ2/r+1−tξ12ξ1ξ2ξ2−rξ1|01−∫01ϒξ1ξ2/rtξ2/r+1−tξ1dt=2t−12ϒξ1ξ2/rtξ2/r+1−tξ1|01−∫01ϒξ1ξ2/rtξ2/r+1−tξ1dt.
Using change of variable technique, wherez=ξ1ξ2/r/tξ2/r+1−tξ1, we obtain(33)S=ϒξ1+ϒξ2/r2−ξ1ξ2ξ2−rξ1∫ξ1ξ2/rϒzz2dz.
This led us to the desired result.Theorem 3.
Suppose0<r≤1 and consider a function ϒ:ξ1,ξ2/r⟶ℝ, which is differentiable on ξ1,ξ2/r with 0<ξ1<ξ2. If ϒ′∈ℒξ1,ξ2/r and ϒ′q is an n–polynomial harmonically tgs–convex function, then for q>1 and 1/p+1/q=1, the following inequality holds:(34)ϒξ1+ϒξ2/r2−ξ1ξ2ξ2−rξ1∫ξ1ξ2/rϒzz2dz≤ξ1ξ2ξ2−rξ12r21p+11/pϒ′ξ1q+ϒ′ξ2rq∑m=1nS1ξ1,ξ2,r,m,q1/q,where(35)S1ξ1,ξ2,r,m,q≔∫01tm1−tmtξ2/r+1−tξ12qdt=ξ1−2qβm+1,m+1F122q,m+1;2m+2;1−ξ2/rξ1.Proof.
Applying Lemma2 and using Hölder’s inequality, n–polynomial harmonically tgs–convexity of ϒ′q, we have(36)ϒξ1+ϒξ2/r2−ξ1ξ2ξ2−rξ1∫ξ1ξ2/rϒzz2dz≤ξ1ξ2ξ2−rξ12r2∫011−2ttξ2/r+1−tξ12ϒ′ξ1ξ2/rtξ2/r+1−tξ1dt≤ξ1ξ2ξ2−rξ12r2∫011−2tpdt1/p∫011tξ2/r+1−tξ12qϒ′ξ1ϒ′tξ2/r+1−tξ1qdt1/q≤ξ1ξ2ξ2−rξ12r21p+11/p∑m=1n∫01tm1−tmϒ′ξ1q+ϒ′ξ2/rqtξ2/r+1−tξ12qdt1/q=ξ1ξ2ξ2−rξ12r21p+11/pϒ′ξ1q+ϒ′ξ2/rq∑m=1nS1ξ1,ξ2,r,m,q1/q.
This led us to the desired result.Theorem 4.
Suppose0<r≤1 and consider a function ϒ:ξ1,ξ2/r⟶ℝ, which is differentiable on ξ1,ξ2/r with 0<ξ1<ξ2. If ϒ′∈ℒξ1,ξ2/r and ϒ′q is an n–polynomial harmonically tgs–convex function, then for q≥1, the following inequality holds:(37)ϒξ1+ϒξ2/r2−ξ1ξ2ξ2−rξ1∫ξ1ξ2/rϒzz2dz≤ξ1ξ2ξ2−rξ12r2S2ξ1,ξ2,r1−1/q∑m=1nϒ′ξ1q+ϒ′ξ2rqS1ξ1,ξ2,r,m1/q,where(38)S1ξ1,ξ2,r,m≔∫011−2ttm1−tmtξ2/r+1−tξ12dt,S2ξ1,ξ2,r≔∫011−2ttξ2/r+1−tξ12dt.Proof.
Applying Lemma2 and using the well-known power mean inequality, n–polynomial harmonically tgs–convexity of ϒ′q, we have(39)ϒξ1+ϒξ2/r2−ξ1ξ2ξ2−rξ1∫ξ1ξ2/rϒzz2dz≤ξ1ξ2ξ2−rξ12r2∫011−2ttξ2/r+1−tξ12dt1−1/q×∫011−2ttξ2/r+1−tξ12ϒ′ξ1ξ2/rtξ2/r+1−tξ1qdt1/q≤ξ1ξ2ξ2−rξ12r2S2dξ1,ξ2,r1−1/q∑m=1n∫011−2ttm1−tmϒ′ξ1q+ϒ′ξ2/rqtξ2/r+1−tξ12dt1/q=ξ1ξ2ξ2−rξ12r2S2ξ1,ξ2,r1−1/q∑m=1nϒ′ξ1q+ϒ′ξ2/rqS1ξ1,ξ2,r,m1/q.
This led us to the desired result.Theorem 5.
Suppose0<r≤1 and consider a function ϒ:ξ1,ξ2/r⟶ℝ, which is differentiable on ξ1,ξ2/r with 0<ξ1<ξ2. If ϒ′∈ℒξ1,ξ2/r and ϒ′q is an n–polynomial harmonically tgs–convex function, then for q≥1, the following inequality holds:(40)ϒξ1+ϒξ2/r2−ξ1ξ2ξ2−rξ1∫ξ1ξ2/rϒzz2dz≤ξ1ξ2ξ2−rξ12r2S3ξ1,ξ2,r1−1/q∑m=1nϒ′ξ1q+ϒ′ξ2rq×2βm+2,m+1+β1/2m+1,m+1−βm+1,m+1+4β1/2m+2,m+11/q,where(41)S3ξ1,ξ2,r≔∫01dttξ2/r+1−tξ12=ξ1−2F122,1;2;1−ξ2/rξ1,and the incomplete beta function is defined as(42)βxξ1,ξ2≔∫0xtξ1−11−tξ2−1dt,0<x≤1.Proof.
Applying Lemma2 and using the well-known power mean inequality, n–polynomial harmonically tgs–convexity of ϒ′q, we have(43)ϒξ1+ϒξ2/r2−ξ1ξ2ξ2−rξ1∫ξ1ξ2/rϒzz2dz≤ξ1ξ2ξ2−rξ12r2∫01dttξ2/r+1−tξ121−1/q∫011−2tϒ′ξ1ξ2/rtξ2/r+1−tξ1qdt1/q≤ξ1ξ2ξ2−rξ12r2S3ξ1,ξ2,r1−1/qϒ′ξ1q+ϒ′ξ2rq∑m=1n∫011−2ttm1−tmdt1/q=ξ1ξ2ξ2−rξ12r2S3ξ1,ξ2,r1−1/q∑m=1nϒ′ξ1q+ϒ′ξ2rq×2βm+2,m+1+β1/2m+1,m+1−βm+1,m+1+4β1/2m+2,m+11/q.
The proof is completed.Theorem 6.
Suppose0<r≤1 and consider a function ϒ:ξ1,ξ2/r⟶ℝ, which is differentiable on ξ1,ξ2/r with 0<ξ1<ξ2. If ϒ′∈ℒξ1,ξ2/r and ϒ′q is an n–polynomial harmonically tgs–convex function, then for q≥1, the following inequality holds:(44)ϒξ1+ϒξ2/r2−ξ1ξ2ξ2−rξ1∫ξ1ξ2/rϒzz2dz≤ξ1ξ2ξ2−rξ12r2121−1/qϒ′ξ1q+ϒ′ξ2rq∑m=1nS4ξ1,ξ2,r,m1/q,where(45)S4ξ1,ξ2,r,m≔∫01tm1−tmtξ2/r+1−tξ12dt=ξ1−2βm+1,m+1F122,m+1;2m+2;1−ξ2/rξ1.Proof.
Applying Lemma2 and using the well-known power mean inequality, n–polynomial harmonically tgs–convexity of ϒ′q, we have(46)ϒξ1+ϒξ2/r2−ξ1ξ2ξ2−rξ1∫ξ1ξ2/rϒzz2dz≤ξ1ξ2ξ2−rξ12r2∫011−2tdt1−1/q∫011tξ2/r+1−tξ12ϒ′ξ1ξ2/rtξ2/r+1−tξ1qdt1/q≤ξ1ξ2ξ2−rξ12r2∫011−2tdt1−1/q×∫011tξ2/r+1−tξ12∑m=1ntm1−tmϒ′ξ1q+ϒ′ξ2rqdt1/q=ξ1ξ2ξ2−rξ12r2121−1/qϒ′ξ1q+ϒ′ξ2rq∑m=1nS4ξ1,ξ2,r,m1/q.
The proof is completed.


## 6. Fractional Hermite–Hadamard Type Inequalities

Now applyingk–Riemann–Liouville fractional integrals, we obtain the following fractional version of Theorem 1.Theorem 7.
Let0<ξ1<ξ2 and ϒ:ξ1,ξ2⟶ℝ be an n–polynomial harmonically tgs–convex function. If ϒ∈ℒξ1,ξ2, then for α,k>0, the following fractional inequalities hold:(47)12∑m=1n1/4mϒ2ξ1ξ2ξ1+ξ2≤ξ1ξ2α/kΓkα+k2ξ2−ξ1α/kJ1−/ξ1α,kϒ∘Ω1ξ2+J1+/ξ2α,kϒ∘Ω1ξ1≤αϒξ1+ϒξ2k∑m=1nβm+αk,m+1,where Ωx≔1/x.Proof.
Using the definition ofn–polynomial harmonically tgs–convex function, we have(48)ϒxytx+1−ty≤∑m=1ntm1−tmϒx+ϒy.
Takingt=1/2 and then changing the variables, we get(49)ϒ2ξ1ξ2ξ1+ξ2≤∑m=1n12m1−12mϒξ1ξ2tξ1+1−tξ2+ϒξ1ξ2tξ2+1−tξ1=∑m=1n14mϒξ1ξ2tξ1+1−tξ2+ϒξ1ξ2tξ2+1−tξ1.
Now, multiplying both sides of above inequality bytα/k−1 and integrating with respect to t∈0,1, we obtain k/αϒ2ξ1ξ2/ξ1+ξ2≤∑m=1n1/4m∫01ϒξ1ξ2/tξ1+1−tξ2+ϒξ1ξ2/tξ2+1−tξ1tα/k−1dt. Hence, k/α∑m=1n1/4mϒ2ξ1ξ2/ξ1+ξ2≤kΓkαξ1ξ2/ξ2−ξ1α/kJ1−/ξ1α,kϒ∘Ω1/ξ2+J1+/ξ2α,kϒ∘Ω1/ξ1.
Multiplying both sides byα/2k, we have(50)kα∑m=1n1/4mϒ2ξ1ξ2ξ1+ξ2≤Γkα+k2ξ1ξ2ξ2−ξ1α/kJ1−/ξ1α,kϒ∘Ω1ξ2+J1+/ξ2α,kϒ∘Ω1ξ1.
This completes the proof of the left inequality.
For the right inequality, using the property of then–polynomial tgs–convex function ϒ, we have(51)ϒξ1ξ2tξ1+1−tξ2≤∑m=1ntm1−tmϒξ1+ϒξ2,ϒξ1ξ2tξ2+1−tξ1≤∑m=1ntm1−tmϒξ1+ϒξ2.
Adding above inequalities, we get(52)ϒξ1ξ2tξ1+1−tξ2+ϒξ1ξ2tξ2+1−tξ1≤2∑m=1ntm1−tmϒξ1+ϒξ2.
Multiplying both sides of above inequality bytα/k−1 and integrating with respect to t∈0,1, we obtain(53)∫01ϒξ1ξ2tξ1+1−tξ2+ϒξ1ξ2tξ2+1−tξ1tα/k−1dt≤2ϒξ1+ϒξ2∑m=1n∫01tm1−tmtα/k−1dt.
Then, we have(54)kξ1ξ2α/kΓkαξ2−ξ1α/kJ1−/ξ1α,kϒ∘Ω1ξ2+J1+/ξ2α,kϒ∘Ωd1ξ1≤2ϒξ1+ϒξ2∑m=1nβm+αk,m+1.
Multiplying both sides of above inequality byα/2k, we derived the desired result. The proof is completed.Corollary 1.
If we takek=1 in Theorem 7, we have(55)12∑m=1n1/4mϒ2ξ1ξ2ξ1+ξ2≤ξ1ξ2αΓα+12ξ2−ξ1αJ1−/ξ1αϒ∘Ω1ξ2+J1+/ξ2αϒ∘Ω1ξ1≤αϒξ1+ϒξ2∑m=1nβm+α,m+1.


### 6.1. Refinements via Fractional Integral Operator

The following new lemma is very crucial for our next results.Lemma 3.
Suppose0<r≤1 and consider a function ϒ:ξ1,ξ2/r⟶ℝ, which is differentiable on ξ1,ξ2/r with 0<ξ1<ξ2. If ϒ′∈ℒξ1,ξ2/r, then for α>0, the following identity holds:(56)ϒξ1+ϒξ2/r2−Γα+12ξ1ξ2ξ2−rξ1αJ1/ξ1−αϒ∘Ωrξ2−Jr/ξ2+αϒ∘Ω1ξ1=ξ1ξ2ξ2−rξ12r2∫01tα−1−tαtξ1+1−tξ2/r2ϒ′ξ1ξ2/rtξ1+1−tξ2/rdt.Proof.
We denote(57)Fξ1ξ2ξ2−rξ12r2∫01tα−1−tαtξ1+1−tξ2/r2ϒ′ξ1ξ2/rtξ1+1−tξ2/rdt=ξ1ξ2ξ2−rξ12r2∫01tαtξ1+1−tξ2/r2ϒ′ξ1ξ2/rtξ1+1−tξ2/rdt−ξ1ξ2ξ2−rξ12r2∫011−tαtξ1+1−tξ2/r2ϒ′ξ1ξ2/rtξ1+1−tξ2/rdt=F1+F2.
Using integration by parts, we have(58)F1=ξ1ξ2ξ2−rξ12r2∫01tαtξ1+1−tξ2/r2ϒ′ξ1ξ2/rtξ1+1−tξ2/rdt=12ϒξ2r−αξ1ξ2ξ2−rξ1α∫r/ξ21/ξ11ξ1−zα−1ϒ1zdz=12ϒξ2r−Γα+1ξ1ξ2ξ2−rξ1αJr/ξ2+αϒ∘Ω1ξ1.
Similarly, we get(59)F2=12ϒξ1−Γα+1ξ1ξ2ξ2−rξ1αJ1/ξ1−αϒ∘Ωrξ2.
Substituting the values ofF1 and F2 in (57), we obtain the desired result.Theorem 8.
Suppose0<r≤1 and consider a function ϒ:ξ1,ξ2/r⟶ℝ, which is differentiable on ξ1,ξ2/r with 0<ξ1<ξ2. If ϒ′∈ℒξ1,ξ2/r and ϒ′q is an n–polynomial harmonically tgs–convex function, then for α>0 and q≥1, the following inequality holds:(60)ϒξ1+ϒξ2/r2−Γα+12ξ1ξ2ξ2−rξ1αJ1/ξ1−αϒ∘Ωrξ2−Jr/ξ2+αϒ∘Ω1ξ1≤ξ1ξ2ξ2−rξ12r22α+11−12α1−1/qϒ′ξ1q+ϒ′ξ2rqS5ξ1,ξ2,r,m1/q,where(61)S5ξ1,ξ2,r,m≔∫01tm1−tmtξ1+1−tξ2/r2dt=ξ2r−2βm+1,m+1F122,m+1;2m+2;1−rξ1ξ2.Proof.
Applying Lemma3 and using the well-known power mean inequality, n–polynomial harmonically tgs–convexity of ϒ′q, we have(62)ϒξ1+ϒξ2/r2−Γα+12ξ1ξ2ξ2−rξ1αJ1/ξ1−αϒ∘Ωrξ2−Jr/ξ2+αϒ∘Ω1ξ1≤ξ1ξ2ξ2−rξ12r2∫01tα−1−tαdt1−1/q×∫011tξ1+1−tξ2/r2ϒ′ξ1ξ2/rtξ1+1−tξ2/rqdt1/q≤ξ1ξ2ξ2−rξ12r22α+11−12α1/q×ϒ′ξ1q+ϒ′ξ2rq∑m=1n∫01tm1−tmtξ1+1−tξ2/r2dt1/q=ξ1ξ2ξ2−rξ12r22α+11−12α1−1/qϒ′ξ1q+ϒ′ξ2rqS5ξ1,ξ2,r,m1/q.
The proof is completed.Theorem 9.
Suppose0<r≤1 and consider a function ϒ:ξ1,ξ2/r⟶ℝ, which is differentiable on ξ1,ξ2/r with 0<ξ1<ξ2. If ϒ′∈ℒξ1,ξ2/r and ϒ′q is an n–polynomial harmonically tgs–convex function, then for α>0 and q≥1, the following inequality holds:(63)ϒξ1+ϒξ2/r2−Γα+12ξ1ξ2ξ2−rξ1αJ1/ξ1−αϒ∘Ωrξ2−Jr/ξ2+αϒ∘Ω1ξ1≤ξ1ξ2ξ2−rξ12r2C1ξ1,ξ2,r,α1−1/q×ϒ′ξ1q+ξ1ξ2rq∑m=1nS6ξ1,ξ2,r,α,m+S7ξ1,ξ2,r,α,m1/q,where(64)C1ξ1,ξ2,r,α≔∫01tα−1−tαtξ1+1−tξ2/r2dt,S6ξ1,ξ2,r,α,m≔∫01tα+m1−tmtξ1+1−tξ2/r2dt=ξ2r−2βm+α+1,mF122,m+α+1;2m+α+1;1−rξ1ξ2,S7ξ1,ξ2,r,α,m≔∫01tm1−tα+m1−tξ2/r2dt=ξ2r−2βm+α+1,mF122,m+1;2m+α+2;1−rξ1ξ2.Proof.
Applying Lemma3 and using the well-known power mean inequality, n–polynomial harmonically tgs–convexity of ϒ′q, we have(65)ϒξ1+ϒξ2/r2−Γα+12ξ1ξ2ξ2−rξ1αJ1/ξ1−αdϒ∘Ωrξ2−Jr/ξ2+αϒ∘Ω1ξ1≤ξ1ξ2ξ2−rξ12r2∫01tα−1−tαtξ1+1−tξ2/r2dt1−1/q×∫01tα−1−tαtξ1+1−tξ2/r2ϒ′ξ1ξ2/rtξ1+1−tξ2/rqdt1/q.
Using the same technique used by Tunç et al. [10, Theorem 6, page 242], we get(66)≤ξ1ξ2ξ2−rξ12r2C1ξ1,ξ2,r,α1−1/q×∫01tα+1−tαtξ1+1−tξ2/r2∑m=1ntm1−tmϒ′ξ1q+ϒ′ξ2rqdt1/q=ξ1ξ2dξ2−rξ12r2C1ξ1,ξ2,r,α1−1/q×ϒ′ξ1q+ϒ′ξ2rq∑m=1nS6ξ1,ξ2,r,α,m+S7ξ1,ξ2,r,α,m1/q.
The proof is completed.


## 6.1. Refinements via Fractional Integral Operator

The following new lemma is very crucial for our next results.Lemma 3.
Suppose0<r≤1 and consider a function ϒ:ξ1,ξ2/r⟶ℝ, which is differentiable on ξ1,ξ2/r with 0<ξ1<ξ2. If ϒ′∈ℒξ1,ξ2/r, then for α>0, the following identity holds:(56)ϒξ1+ϒξ2/r2−Γα+12ξ1ξ2ξ2−rξ1αJ1/ξ1−αϒ∘Ωrξ2−Jr/ξ2+αϒ∘Ω1ξ1=ξ1ξ2ξ2−rξ12r2∫01tα−1−tαtξ1+1−tξ2/r2ϒ′ξ1ξ2/rtξ1+1−tξ2/rdt.Proof.
We denote(57)Fξ1ξ2ξ2−rξ12r2∫01tα−1−tαtξ1+1−tξ2/r2ϒ′ξ1ξ2/rtξ1+1−tξ2/rdt=ξ1ξ2ξ2−rξ12r2∫01tαtξ1+1−tξ2/r2ϒ′ξ1ξ2/rtξ1+1−tξ2/rdt−ξ1ξ2ξ2−rξ12r2∫011−tαtξ1+1−tξ2/r2ϒ′ξ1ξ2/rtξ1+1−tξ2/rdt=F1+F2.
Using integration by parts, we have(58)F1=ξ1ξ2ξ2−rξ12r2∫01tαtξ1+1−tξ2/r2ϒ′ξ1ξ2/rtξ1+1−tξ2/rdt=12ϒξ2r−αξ1ξ2ξ2−rξ1α∫r/ξ21/ξ11ξ1−zα−1ϒ1zdz=12ϒξ2r−Γα+1ξ1ξ2ξ2−rξ1αJr/ξ2+αϒ∘Ω1ξ1.
Similarly, we get(59)F2=12ϒξ1−Γα+1ξ1ξ2ξ2−rξ1αJ1/ξ1−αϒ∘Ωrξ2.
Substituting the values ofF1 and F2 in (57), we obtain the desired result.Theorem 8.
Suppose0<r≤1 and consider a function ϒ:ξ1,ξ2/r⟶ℝ, which is differentiable on ξ1,ξ2/r with 0<ξ1<ξ2. If ϒ′∈ℒξ1,ξ2/r and ϒ′q is an n–polynomial harmonically tgs–convex function, then for α>0 and q≥1, the following inequality holds:(60)ϒξ1+ϒξ2/r2−Γα+12ξ1ξ2ξ2−rξ1αJ1/ξ1−αϒ∘Ωrξ2−Jr/ξ2+αϒ∘Ω1ξ1≤ξ1ξ2ξ2−rξ12r22α+11−12α1−1/qϒ′ξ1q+ϒ′ξ2rqS5ξ1,ξ2,r,m1/q,where(61)S5ξ1,ξ2,r,m≔∫01tm1−tmtξ1+1−tξ2/r2dt=ξ2r−2βm+1,m+1F122,m+1;2m+2;1−rξ1ξ2.Proof.
Applying Lemma3 and using the well-known power mean inequality, n–polynomial harmonically tgs–convexity of ϒ′q, we have(62)ϒξ1+ϒξ2/r2−Γα+12ξ1ξ2ξ2−rξ1αJ1/ξ1−αϒ∘Ωrξ2−Jr/ξ2+αϒ∘Ω1ξ1≤ξ1ξ2ξ2−rξ12r2∫01tα−1−tαdt1−1/q×∫011tξ1+1−tξ2/r2ϒ′ξ1ξ2/rtξ1+1−tξ2/rqdt1/q≤ξ1ξ2ξ2−rξ12r22α+11−12α1/q×ϒ′ξ1q+ϒ′ξ2rq∑m=1n∫01tm1−tmtξ1+1−tξ2/r2dt1/q=ξ1ξ2ξ2−rξ12r22α+11−12α1−1/qϒ′ξ1q+ϒ′ξ2rqS5ξ1,ξ2,r,m1/q.
The proof is completed.Theorem 9.
Suppose0<r≤1 and consider a function ϒ:ξ1,ξ2/r⟶ℝ, which is differentiable on ξ1,ξ2/r with 0<ξ1<ξ2. If ϒ′∈ℒξ1,ξ2/r and ϒ′q is an n–polynomial harmonically tgs–convex function, then for α>0 and q≥1, the following inequality holds:(63)ϒξ1+ϒξ2/r2−Γα+12ξ1ξ2ξ2−rξ1αJ1/ξ1−αϒ∘Ωrξ2−Jr/ξ2+αϒ∘Ω1ξ1≤ξ1ξ2ξ2−rξ12r2C1ξ1,ξ2,r,α1−1/q×ϒ′ξ1q+ξ1ξ2rq∑m=1nS6ξ1,ξ2,r,α,m+S7ξ1,ξ2,r,α,m1/q,where(64)C1ξ1,ξ2,r,α≔∫01tα−1−tαtξ1+1−tξ2/r2dt,S6ξ1,ξ2,r,α,m≔∫01tα+m1−tmtξ1+1−tξ2/r2dt=ξ2r−2βm+α+1,mF122,m+α+1;2m+α+1;1−rξ1ξ2,S7ξ1,ξ2,r,α,m≔∫01tm1−tα+m1−tξ2/r2dt=ξ2r−2βm+α+1,mF122,m+1;2m+α+2;1−rξ1ξ2.Proof.
Applying Lemma3 and using the well-known power mean inequality, n–polynomial harmonically tgs–convexity of ϒ′q, we have(65)ϒξ1+ϒξ2/r2−Γα+12ξ1ξ2ξ2−rξ1αJ1/ξ1−αdϒ∘Ωrξ2−Jr/ξ2+αϒ∘Ω1ξ1≤ξ1ξ2ξ2−rξ12r2∫01tα−1−tαtξ1+1−tξ2/r2dt1−1/q×∫01tα−1−tαtξ1+1−tξ2/r2ϒ′ξ1ξ2/rtξ1+1−tξ2/rqdt1/q.
Using the same technique used by Tunç et al. [10, Theorem 6, page 242], we get(66)≤ξ1ξ2ξ2−rξ12r2C1ξ1,ξ2,r,α1−1/q×∫01tα+1−tαtξ1+1−tξ2/r2∑m=1ntm1−tmϒ′ξ1q+ϒ′ξ2rqdt1/q=ξ1ξ2dξ2−rξ12r2C1ξ1,ξ2,r,α1−1/q×ϒ′ξ1q+ϒ′ξ2rq∑m=1nS6ξ1,ξ2,r,α,m+S7ξ1,ξ2,r,α,m1/q.
The proof is completed.


## 7. Applications

### 7.1. Application to Special Means

Consider the following special means for different positive real numbersξ1,ξ2, where ξ1<ξ2:(1)
The arithmetic mean:(67)Aξ1,ξ2=ξ1+ξ22.(2)
The geometric mean:(68)Gξ1,ξ2=ξ1ξ2.(3)
The harmonic mean:(69)ℋξ1,ξ2=2ξ1ξ2ξ1+ξ2.(4)
The logarithmic mean:(70)ℒξ1,ξ2=ξ2−ξ1lnξ2−lnξ1.Applying Theorem1, we can derive the following inequalities using above special means.Proposition 6.
Let0<ξ1<ξ2 and n∈ℕ; then,(71)2Alnξ1,lnξ2∑m=1nβm+1,m+1≤ξ2lnξ1−ξ1lnξ2ξ2−ξ1+1≤12∑m=1n1/4mlnℋξ1,ξ2.Proof.
Takingn–polynomial harmonically tgs–convex function ϒz=−lnz, then the assertion follows from Theorem 1.Proposition 7.
Let0<ξ1<ξ2 and n∈ℕ; then,(72)2Aξ1,ξ2∑m=1nβm+1,m+1≤G2ξ1,ξ2ℒξ1,ξ2≤12∑m=1n1/4mℋξ1,ξ2.Proof.
Choosingn–polynomial harmonically tgs–convex function ϒz=−z, then the assertion follows from Theorem 1.Proposition 8.
Let0<ξ1<ξ2 and n∈ℕ; then,(73)2Aξ1,ξ2∑m=1nβm+1,m+1≤Gξ1,ξ2Aξ1,ξ2≤12∑m=1n1/4mℋξ1,ξ2.Proof.
Settingn–polynomial harmonically tgs–convex function ϒz=−z, then the assertion follows from Theorem 1.


### 7.2. Trapezoidal Quadrature Formula

Here we present some error estimation for the trapezoidal quadrature formula by using the results of Section5. We set the parameter r∈0,1. Let(74)X:ξ1=ζ0<ζ1<⋯<ζl−1<ζl=ξ2rbe a partition of the closed interval ξ1,ξ2/r.Let us define(75)TX,ϒ≔∑i=0l−1ϒζi+ϒζi+1/r2ζi+1−rζiζiζi+1,∫ξ1ξ2rϒzz2dz≔TX,ϒ+RX,ϒ,where RX,ϒ is the remainder term.Proposition 9.
Suppose0<r≤1 is fixed real number and consider a function ϒ:ξ1,ξ2/r⟶ℝ, which is differentiable on ξ1,ξ2/r with 0<ξ1<ξ2. If ϒ′∈ℒξ1,ξ2/r and ϒ′q is an n–polynomial harmonically tgs–convex function, then for q>1 and 1/p+1/q=1, the following inequality holds:(76)RX,ϒ≤1p+11/p∑i=0l−1ζi+1−rζi22r2ϒ′ζiq+ϒ′ζi+1rq∑ω=1nS1ζi,ζi+1,r,ω,q1/q,where(77)S1ζi,ζi+1,r,ω,q≔∫01tω1−tωtζi+1/r+1−tζi2qdt=ζi−2qβω+1,ω+1F122q,ω+1;2ω+2;1−ζi+1rζi.Proof.
Employing Theorem3 on the subinterval ζi,ζi+1/r for fixed r∈0,1 of the closed interval ξ1,ξ2/r∀i=0,1,2,…,ℓ−1, we have(78)ϒζi+ϒζi+12ζi+1−rζiζiζi+1−∫ζiζi+1ϒzz2dz≤1p+11/p×ζi+1−rζi22r2ϒ′ζiq+ϒ′ζi+1rq∑ω=1nS1ζi,ζi+1,r,ω,q1/q.
Summing overi from 0 to l−1 and using the property of the modulus, we get the desired inequality.Proposition 10.
Suppose0<r≤1 is fixed real number and consider a function ϒ:ξ1,ξ2/r⟶ℝ, which is differentiable on ξ1,ξ2/r with 0<ξ1<ξ2. If ϒ′∈ℒξ1,ξ2/r and ϒ′q is an n–polynomial harmonically tgs–convex function, then for q≥1, the following inequality holds:(79)RX,ϒ≤∑i=0l−1ζi+1−rζi22r2S2ζi,ζi+1,r1−1/q×∑ω=1nϒ′ζiq+ϒ′ζi+1rqS1ζi,ζi+1,r,ω1/q,where(80)S1ζi,ζi+1,r,ω≔∫011−2ttω1−tωtζi+1/r+1−tζi2dt,S2ζi,ζi+1,r≔∫011−2ttζi+1/r+1−tζi2dt.Proof.
Employing Theorem4 and using the same technique as used in Proposition 9, we have the desired result.Proposition 11.
Suppose0<r≤1 is fixed real number and consider a function ϒ:ξ1,ξ2/r⟶ℝ, which is differentiable on ξ1,ξ2/r with 0<ξ1<ξ2. If ϒ′∈ℒξ1,ξ2/r and ϒ′q is an n–polynomial harmonically tgs–convex function, then for q≥1, the following inequality holds:(81)RX,ϒ≤∑i=0l−1ζi+1−rζi22r2S3ζi,ζi+1,r1−1/q∑ω=1nϒ′ζiq+ϒ′ζi+1rq×2βω+2,ω+1+β1/2ω+1,ω+1−βω+1,ω+1+4β1/2ω+2,ω+11/q,where(82)S3ζi,ζi+1,r≔∫01dttζi+1/r+1−tζi2=ζi−2F122,1;2;1−ζi+1rζi.Proof.
Employing Theorem5 and using the same technique as used in Proposition 9, we have the desired result.Proposition 12.
Suppose0<r≤1 is fixed real number and consider a function ϒ:ξ1,ξ2/r⟶ℝ, which is differentiable on ξ1,ξ2/r with 0<ξ1<ξ2. If ϒ′∈ℒξ1,ξ2/r and ϒ′q is an n–polynomial harmonically tgs–convex function, then for q≥1, the following inequality holds:(83)RX,ϒ≤121−1/q∑i=0l−1ζi+1−rζi22r2ϒ′ζiq+ϒ′ζi+1rq∑ω=1nS4ζi,ζi+1,r,ω1/q,where(84)S4ζi,ζi+1,r,ω≔∫01tω1−tωtζi+1/r+1−tζi2dt=ζi−2βω+1,ω+1F122,ω+1;2ω+2;1−ζi+1rζi.Proof.
Employing Theorem6 and using the same technique as used in Proposition 9, we have the desired result.


## 7.1. Application to Special Means

Consider the following special means for different positive real numbersξ1,ξ2, where ξ1<ξ2:(1)
The arithmetic mean:(67)Aξ1,ξ2=ξ1+ξ22.(2)
The geometric mean:(68)Gξ1,ξ2=ξ1ξ2.(3)
The harmonic mean:(69)ℋξ1,ξ2=2ξ1ξ2ξ1+ξ2.(4)
The logarithmic mean:(70)ℒξ1,ξ2=ξ2−ξ1lnξ2−lnξ1.Applying Theorem1, we can derive the following inequalities using above special means.Proposition 6.
Let0<ξ1<ξ2 and n∈ℕ; then,(71)2Alnξ1,lnξ2∑m=1nβm+1,m+1≤ξ2lnξ1−ξ1lnξ2ξ2−ξ1+1≤12∑m=1n1/4mlnℋξ1,ξ2.Proof.
Takingn–polynomial harmonically tgs–convex function ϒz=−lnz, then the assertion follows from Theorem 1.Proposition 7.
Let0<ξ1<ξ2 and n∈ℕ; then,(72)2Aξ1,ξ2∑m=1nβm+1,m+1≤G2ξ1,ξ2ℒξ1,ξ2≤12∑m=1n1/4mℋξ1,ξ2.Proof.
Choosingn–polynomial harmonically tgs–convex function ϒz=−z, then the assertion follows from Theorem 1.Proposition 8.
Let0<ξ1<ξ2 and n∈ℕ; then,(73)2Aξ1,ξ2∑m=1nβm+1,m+1≤Gξ1,ξ2Aξ1,ξ2≤12∑m=1n1/4mℋξ1,ξ2.Proof.
Settingn–polynomial harmonically tgs–convex function ϒz=−z, then the assertion follows from Theorem 1.


## 7.2. Trapezoidal Quadrature Formula

Here we present some error estimation for the trapezoidal quadrature formula by using the results of Section5. We set the parameter r∈0,1. Let(74)X:ξ1=ζ0<ζ1<⋯<ζl−1<ζl=ξ2rbe a partition of the closed interval ξ1,ξ2/r.Let us define(75)TX,ϒ≔∑i=0l−1ϒζi+ϒζi+1/r2ζi+1−rζiζiζi+1,∫ξ1ξ2rϒzz2dz≔TX,ϒ+RX,ϒ,where RX,ϒ is the remainder term.Proposition 9.
Suppose0<r≤1 is fixed real number and consider a function ϒ:ξ1,ξ2/r⟶ℝ, which is differentiable on ξ1,ξ2/r with 0<ξ1<ξ2. If ϒ′∈ℒξ1,ξ2/r and ϒ′q is an n–polynomial harmonically tgs–convex function, then for q>1 and 1/p+1/q=1, the following inequality holds:(76)RX,ϒ≤1p+11/p∑i=0l−1ζi+1−rζi22r2ϒ′ζiq+ϒ′ζi+1rq∑ω=1nS1ζi,ζi+1,r,ω,q1/q,where(77)S1ζi,ζi+1,r,ω,q≔∫01tω1−tωtζi+1/r+1−tζi2qdt=ζi−2qβω+1,ω+1F122q,ω+1;2ω+2;1−ζi+1rζi.Proof.
Employing Theorem3 on the subinterval ζi,ζi+1/r for fixed r∈0,1 of the closed interval ξ1,ξ2/r∀i=0,1,2,…,ℓ−1, we have(78)ϒζi+ϒζi+12ζi+1−rζiζiζi+1−∫ζiζi+1ϒzz2dz≤1p+11/p×ζi+1−rζi22r2ϒ′ζiq+ϒ′ζi+1rq∑ω=1nS1ζi,ζi+1,r,ω,q1/q.
Summing overi from 0 to l−1 and using the property of the modulus, we get the desired inequality.Proposition 10.
Suppose0<r≤1 is fixed real number and consider a function ϒ:ξ1,ξ2/r⟶ℝ, which is differentiable on ξ1,ξ2/r with 0<ξ1<ξ2. If ϒ′∈ℒξ1,ξ2/r and ϒ′q is an n–polynomial harmonically tgs–convex function, then for q≥1, the following inequality holds:(79)RX,ϒ≤∑i=0l−1ζi+1−rζi22r2S2ζi,ζi+1,r1−1/q×∑ω=1nϒ′ζiq+ϒ′ζi+1rqS1ζi,ζi+1,r,ω1/q,where(80)S1ζi,ζi+1,r,ω≔∫011−2ttω1−tωtζi+1/r+1−tζi2dt,S2ζi,ζi+1,r≔∫011−2ttζi+1/r+1−tζi2dt.Proof.
Employing Theorem4 and using the same technique as used in Proposition 9, we have the desired result.Proposition 11.
Suppose0<r≤1 is fixed real number and consider a function ϒ:ξ1,ξ2/r⟶ℝ, which is differentiable on ξ1,ξ2/r with 0<ξ1<ξ2. If ϒ′∈ℒξ1,ξ2/r and ϒ′q is an n–polynomial harmonically tgs–convex function, then for q≥1, the following inequality holds:(81)RX,ϒ≤∑i=0l−1ζi+1−rζi22r2S3ζi,ζi+1,r1−1/q∑ω=1nϒ′ζiq+ϒ′ζi+1rq×2βω+2,ω+1+β1/2ω+1,ω+1−βω+1,ω+1+4β1/2ω+2,ω+11/q,where(82)S3ζi,ζi+1,r≔∫01dttζi+1/r+1−tζi2=ζi−2F122,1;2;1−ζi+1rζi.Proof.
Employing Theorem5 and using the same technique as used in Proposition 9, we have the desired result.Proposition 12.
Suppose0<r≤1 is fixed real number and consider a function ϒ:ξ1,ξ2/r⟶ℝ, which is differentiable on ξ1,ξ2/r with 0<ξ1<ξ2. If ϒ′∈ℒξ1,ξ2/r and ϒ′q is an n–polynomial harmonically tgs–convex function, then for q≥1, the following inequality holds:(83)RX,ϒ≤121−1/q∑i=0l−1ζi+1−rζi22r2ϒ′ζiq+ϒ′ζi+1rq∑ω=1nS4ζi,ζi+1,r,ω1/q,where(84)S4ζi,ζi+1,r,ω≔∫01tω1−tωtζi+1/r+1−tζi2dt=ζi−2βω+1,ω+1F122,ω+1;2ω+2;1−ζi+1rζi.Proof.
Employing Theorem6 and using the same technique as used in Proposition 9, we have the desired result.


## 8. Conclusion

Recently, establishing new versions of well-known inequalities using different convexities has attracted many mathematicians. As a step forward in this advancement, a new class of convexity,n–polynomial harmonically tgs–convexity, is introduced. Two H−H type integral inequalities for the new convex functions are established. Moreover, two new identities for differentiable mappings are proved, and, using them as auxiliary results, we also discuss some novel refinements of H−H type inequalities. Finally, some applications related to special means and error estimations are derived. In the future, the concepts in this article can be generalized to the coordinates, quantum calculus, time scale calculus, and interval valued analysis, along with fractional calculus. Our results have a deep connection with fractional integral operators.



---

*Source: 2493944-2022-05-31.xml*