# Novel Refinements vian–Polynomial Harmonically s–Type Convex Functions and Application in Special Functions

Authors: Saad Ihsan Butt; Saima Rashid; Muhammad Tariq; Miao-Kun Wang
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

In this work, we introduce the idea ofn–polynomial harmonically s–type convex function. We elaborate the new introduced idea by examples and some interesting algebraic properties. As a result, new Hermite–Hadamard, some refinements of Hermite–Hadamard and Ostrowski type integral inequalities are established, which are the generalized variants of the previously known results for harmonically convex functions. Finally, we illustrate the applicability of this new investigation in special functions (hypergeometric function and special mean of real numbers).


---

## Body

## 1. Introduction

The theory of convexity presents an amazing, fascinating, and captivating field of research and also played significant and important roles in many areas, such as statistics, economics, optimization, management science, finance, engineering, game theory, and mathematical inequalities. In probability theory, a convex function applied to the expected value of a random variable is always bounded above by the expected value of the convex function of the random variable. This result, known as Jensen’s inequality, can be used to deduce inequalities such as the arithmetic-geometric mean inequality and Hölder’s inequality. Due to widespread views, robustness, and a lot of applications, the theory of convexity has become a rich source of motivation and absorbing field for researchers. Using the theory of convexity, mathematicians provided amazing tools and numerical techniques to tackle and to solve a wide class of problems, which arise in pure and applied sciences. This theory has a rich and paramount history and has been the focus and fixed point of intense study for over a century in mathematics. Many generalizations, variants, and extensions for the convexity have attracted the attention of many researchers, see [1–3]. This theory also played a meaningful role in the development of the theory of inequalities. In diverse and opponent research, inequalities have a lot of applications in statistical problems, probability, and numerical quadrature formulas. Integral inequalities have an interesting mathematical model due to its robustness and pivotal applications in fractional integral calculus and mathematical analysis. In approximation theory, integral inequalities explain and brief the growth rate of competing mathematical analysis. For the applications, interested readers refer to [4–6]. Recently, Toplu et al. [7] introduced a generalization form of convexity namely n-polynomial convex functions.It is well known that the harmonic mean is the special case of power mean. It is often used for situations when the average rates are desired and have a lot of applications in different fields of sciences which are statistics, computer science, trigonometry, geometry, probability, finance, and electric circuit theory. Harmonic mean is the most appropriate measure for rates and ratios because it equalizes the weights of each data point. Harmonic mean is used to define the harmonic convex set. In2003, the first harmonic convex set was introduced by Shi and Zhang [8]. Harmonic convex function was first introduced and discussed by Anderson et al. [9]. Awan et al. [10] keeping his work on generalizations introduced a new generalized class of convex function called n–polynomial harmonic convex function. For the literature and attention for the readers about harmonically convex functions, see [11–14]. Motivated and inspired by the ongoing activities and research in the convex analysis field, we find out that there exists a special class of functions known as s–type convex function. Recently, Rashid et al. [15] introduced n–polynomial s–type convex function. Eventually, many mathematicians put an effort, hardworking, and has collaborated with different ideas and concepts in the field of convex analysis. The amazing techniques and remarkable ideas of this article may inspire and motivate for further research in this pivotal, captivating, and valuable field. Before we start, we need the following necessary known definitions and literature.


## 2. Preliminaries

Letψ:I→ℝ be a real-valued function. A function ψ is said to be convex if
(1)ψκϑ1+1−κϑ2≤κψϑ1+1−κψϑ2,holds for all ϑ1,ϑ2∈I and κ∈0,1.Definition 1 (see [16]).
A functionψ:H⊆0,+∞→ℝ is said to be harmonically convex if
(2)ψϑ1ϑ2κϑ2+1−κϑ1≤κψϑ1+1−κψϑ2,holds for all ϑ1,ϑ2∈H, and κ∈0,1.
For the harmonically convex function,İscan [16] provided the Hermite–Hadamard type inequality.Theorem 2 (see [16]).
Letψ:H⊆0,+∞→ℝ be a harmonically convex function. If ψ∈Lϑ1,ϑ2 for all ϑ1,ϑ2∈H with ϑ1<ϑ2, then
(3)ψ2ϑ1ϑ2ϑ1+ϑ2≤ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx≤ψϑ1+ψϑ22.Definition 3 (see [7]).
A nonnegative functionψ:I→ℝ is called n–polynomial convex if
(4)ψκϑ1+1−κϑ2≤1n∑i=1n1−1−κiψϑ1+1n∑i=1n1−κiψϑ2,holds for every ϑ1,ϑ2∈I,n∈ℕ, and κ∈0,1.Definition 4 (see [15]).
A functionψ:I→ℝ is said to be s–type convex function if
(5)ψκϑ1+1−κϑ2≤1−s1−κψϑ1+1−sκψϑ2,holds ∀ϑ1,ϑ2∈I, s∈0,1, and κ∈0,1.Definition 5 (see [15]).
A functionψ:I→ℝ is called n–polynomial s–type convex if
(6)ψκϑ1+1−κϑ2≤1n∑i=1n1−s1−κiψϑ1+1n∑i=1n1−sκiψϑ2,holds for every ϑ1,ϑ2∈I,n∈ℕ, s∈0,1, and κ∈0,1.Motivated by the above results, literature and ongoing activities, and research in this amazing and captivating field, we will give in Section3 the idea and its algebraic properties of n–polynomial harmonically s–type convex function. In Section 4, we will derive the new version of Hermite–Hadamard inequality by using the newly introduced definition. As a result in Sections 5 and 6, we will give some refinements of the Hermite–Hadamard and Ostrowski type inequalities, and in Section 7, we will give some applications for our proposed new definition. Finally, a brief conclusion will be provided as well.


## 3. Newn–Polynomial Harmonically s–Type Convex Function and Its Properties

We are going to introduce a new definition called ann–polynomial harmonically s–type convex function and will study some of their algebraic properties.Definition 6.
A nonnegative real-valued functionψ:H⊆0,+∞→0,+∞ is called an n–polynomial harmonically s–type convex if
(7)ψϑ1ϑ2κϑ2+1−κϑ1≤1n∑i=1n1−s1−κiψϑ1+1n∑i=1n1−sκiψϑ2,holds for every ϑ1,ϑ2∈I,n∈ℕ, s∈0,1, and κ∈0,1.Remark 7.
(i)
Takingn=1 in Definition 6, we obtain the following new definition about harmonically s–type convex function
(8)ψϑ1ϑ2κϑ2+1−κϑ1≤1−s1−κψϑ1+1−sκψϑ2(ii)
Takings=1 in Definition 6, then, we get a definition, namely, n–polynomial harmonically convex function, which is defined by Awan et al. [10](iii)
Takingn=1 and s=1 in Definition 6, then, we get a definition, namely, harmonically convex function, which is introduced by İscan [16](iv)
Takingn=2 and s=1 in Definition 6, we obtain the following new definition about 2-polynomial harmonically convex function
(9)ψϑ1ϑ2κϑ2+1−κϑ1≤3κ−κ22ψϑ1+2−κ−κ22ψϑ2
That is the beauty of this newly introduce definition if we choosing different values ofn and s, as a result, we obtain new amazing integral inequalities and also found some results which connect with previous results.Lemma 8.
The following inequalities1/n∑i=1n1−s1−κi≥κ and 1/n∑i=1n1−sκi≥1−κ are hold. If for all κ∈0,1 and s∈0,1.Proof.
The proof is evident.Proposition 9.
LetI⊂0,+∞ be a harmonically convex set. Every harmonically convex function on a harmonically convex set is an n–polynomial harmonically s–type convex function.Proof.
Using the definition of harmonically convex function and from the Lemma8, since κ≤1/n∑i=1n1−s1−κi and 1−κ≤1/n∑i=1n1−sκi for all κ∈0,1 and s∈0,1, we have
(10)ψϑ1ϑ2κϑ2+1−κϑ1≤κψϑ1+1−κψϑ2≤1n∑i=1n1−s1−κiψϑ1+1n∑i=1n1−sκiψϑ2.Proposition 10.
Everyn–polynomial harmonically convex function is an n–polynomial harmonically s–type convex function.Proof.
Using the definition of ann–polynomial harmonically convex function and from (Remark 1.6, see [15]), we have 1/n∑i=1n1−1−κi≤1/n∑i=1n1−s1−κi and 1/n∑i=1n1−κi≤1/n∑i=1n1−sκi for all κ∈0,1,n∈ℕ, and s∈0,1.
(11)ψϑ1ϑ2κϑ2+1−κϑ1≤1n∑i=1n1−1−κiψϑ1+1n∑i=1n1−κiψϑ2≤1n∑i=1n1−s1−κiψϑ1+1n∑i=1n1−sκiψϑ2.Proposition 11.
Everyn–polynomial harmonically s–type convex function is harmonically h–convex function with hκ=1/n∑i=1n1−s1−κi.Proof.
(12)ψϑ1ϑ2κϑ2+1−κϑ1≤1n∑i=1n1−s1−κiψϑ1+1n∑i=1n1−sκiψϑ2≤hκψϑ1+h1−κψϑ2.
Now, we make some examples via newly introduce definitionn–polynomial harmonically s–type convex function.Example 12.
ψx=ex is an increasing convex function, so it is harmonically convex function (see [17]). By using Proposition 9, it is an n–polynomial harmonically s–type convex function.Example 13.
ψx=lnx is an increasing convex function on 0,∞, so it is harmonically convex function ∀x∈0,∞ (see [17]). By using Proposition 9, it is an n–polynomial harmonically s–type convex function.
These are the clear advantages of the proposed new definition with respect to other known functions on the topic mentioned above. Now, we will study some of their algebraic properties.Theorem 14.
Letψ,φ:H⊆0,+∞→0,+∞.Ifψandφ are two n–polynomial harmonically s–type convex functions, then
(1)
ψ+φ is an n–polynomial harmonically s–type convex function(2)
For nonnegative real numberc,cψ is an n–polynomial harmonically s–type convex functionProof.
(1)
Letψ and φ be an n–polynomial harmonically s–type convex, then
(13)ψ+φϑ1ϑ2κϑ2+1−κϑ1=ψϑ1ϑ2κϑ2+1−κϑ1+φϑ1ϑ2κϑ2+1−κϑ1≤1n∑i=1n1−s1−κiψϑ1+1n∑i=1n1−sκiψϑ2+1n∑i=1n1−s1−κiφϑ1+1n∑i=1n1−sκiφϑ2=1n∑i=1n1−s1−κiψϑ1+φϑ1+1n∑i=1n1−sκiψϑ2+φϑ2=1n∑i=1n1−s1−κiψ+φϑ1+1n∑i=1n1−sκiψ+φϑ2(2)
Letψ be an n–polynomial harmonically s–type convex function, then
(14)cψϑ1ϑ2κϑ2+1−κϑ1≤c1n∑i=1n1−s1−κiψϑ1+1n∑i=1n1−sκiψϑ2=1n∑i=1n1−s1−κicψϑ1+1n∑i=1n1−sκicψϑ2=1n∑i=1n1−s1−κicψϑ1+1n∑i=1n1−sκicψϑ2which completes the proof.Theorem 15.
Letφ:H→0,+∞ be harmonically convex function and ψ:0,+∞→0,+∞ is nondecreasing and n–polynomial s–type convex function. Then, the function ψ∘φ:H→0,+∞ is an n–polynomial harmonically s–type convex.Proof.
For allϑ1,ϑ2∈H, s∈0,1, and κ∈0,1, we have
(15)ψ∘φϑ1ϑ2κϑ2+1−κϑ1=ψφϑ1ϑ2κϑ2+1−κϑ1≤ψκφϑ1+1−κφϑ2≤1n∑i=1n1−s1−κiψφϑ1+1n∑i=1n1−sκiψφϑ2=1n∑i=1n1−s1−κiψ∘φϑ1+1n∑i=1n1−sκiψ∘φϑ2,which completes the proof.Theorem 16.
Let0<ϑ1<ϑ2,ψj:ϑ1,ϑ2→0,+∞ be a family of an n–polynomial harmonically s–type convex functions and ψu=supjψju. Then, ψ is an n–polynomial harmonically s–type convex function, and U=u∈ϑ1,ϑ2:ψu<+∞ is an interval.Proof.
Letϑ1,ϑ2∈U, s∈0,1, and κ∈0,1, then
(16)ψϑ1ϑ2κϑ2+1−κϑ1=supjψjϑ1ϑ2κϑ2+1−κϑ1≤1n∑i=1n1−s1−κisupjψjϑ1+1n∑i=1n1−sκisupjψjϑ2=1n∑i=1n1−s1−κiψϑ1+1n∑i=1n1−sκiψϑ2<+∞,which completes the proof.Remark 17.
Interested readers can find many other nice properties of this new class of functions. We omit here the details.


## 4. Hermite–Hadamard Type Inequality vian–Polynomial Harmonically s–Type Convex Functions

The purpose of this section is to derive a new version of Hermite–Hadamard type usingn–polynomial harmonically s–type convexity.Theorem 18.
Letψ:ϑ1,ϑ2→0,+∞ be an n–polynomial harmonically s–type convex function. If ψ∈Lϑ1,ϑ2, then
(17)n2∑i=1n1−s/2iψ2ϑ1ϑ2ϑ1+ϑ2≤ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx≤ψϑ1+ψϑ2n∑i=1ni+1−sii+1.Proof.
Sinceψ is an n–polynomial harmonically s–type convex function, we have
(18)ψxyκy+1−κx≤1n∑i=1n1−s1−κiψx+1n∑i=1n1−sκiψy,which lead to
(19)ψ2xyx+y≤1n∑i=1n1−s2iψx+1n∑i=1n1−s2iψy.
Using the change of variables, we get(20)ψ2ϑ1ϑ2ϑ1+ϑ2≤1n∑i=1n1−s2i×ψϑ1ϑ2κϑ2+1−κϑ1+ψϑ1ϑ2κϑ1+1−κϑ2,
Integrate the above inequality with respect toκ over [0,1], then
(21)n2∑i=1n1−s/2iψ2ϑ1ϑ2ϑ1+ϑ2≤ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx,which completes the left side inequality. For the right side inequality, changing the variable of integration as x=ϑ1ϑ2/κϑ2+1−κϑ1 and using Definition 6 for the function ψ, we have
(22)ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx=∫01ψϑ1ϑ2κϑ2+1−κϑ1dκ≤∫011n∑i=1n1−s1−κiψϑ1+1n∑i=1n1−sκiψϑ2dκ=ψϑ1n∑i=1n∫011−s1−κidκ+ψϑ2n∑i=1n∫011−sκidκ=ψϑ1+ψϑ2n∑i=1ni+1−sii+1,which completes the proof.Corollary 19.
Choosingn=1 in Theorem 18, then
(23)12−sψ2ϑ1ϑ2ϑ1+ϑ2≤ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx≤2−s2ψϑ1+ψϑ2.Remark 20.
Choosings=1 in Theorem 18, then, we get the Theorem (2.3) in [10].Remark 21.
Choosingn=1 and s=1 in Theorem 18, then we get the Theorem (3) in [18].


## 5. Refinements of (H-H) Type Inequality vian–Polynomial Harmonically s–Type Convex Functions

In order to obtain some refinements of (H-H) type inequality usingn–polynomial harmonically s–type convex functions, we need the following lemma.Lemma 22 (see [10]).
Letψ:ϑ1,ϑ2⊆0,+∞→ℝ be a differentiable function and ϖ,ω∈0,1. Ifψ′∈Lϑ1,ϑ2, then the following identity holds
(24)ϖψϑ1+ωψϑ22+2−ϖ−ω2ψ2ϑ1ϑ2ϑ1+ϑ2−ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx=ϑ1ϑ2ϑ2−ϑ14∫0141−ϖ−κ1−κϑ2+1+κϑ12ψ′2ϑ1ϑ21−κϑ2+1+κϑ1+4ω−κκϑ1+2−κϑ22ψ′2ϑ1ϑ2κϑ1+2−κϑ2dκ.
For simplicity, we denote(25)Aϑ1,ϑ2=1−κϑ2+1+κϑ1andBϑ1,ϑ2=κϑ1+2−κϑ2.
The following special functions will be used in sequel:(26)Γϑ=∫0+∞e−κκϑ−1dκ,ϑ>0,βϑ1,ϑ2=∫01κϑ1−11−κϑ2−1dκ,ϑ1,ϑ2>0,βϑ1,ϑ2=Γϑ1Γϑ2Γϑ1+ϑ2,ϑ1,ϑ2>0,2F1ϑ1,ϑ2;ϑ3;ϑ=1βϑ2,ϑ3−ϑ2∫01κϑ2−11−κϑ3−ϑ2−11−ϑκ−ϑ1dκ,where ϑ3>ϑ2>0 and ∣ϑ∣<1.Theorem 23.
Letψ:ϑ1,ϑ2⊆0,+∞→ℝ be a differentiable function such that ψ′∈Lϑ1,ϑ2 and ϖ,ω∈0,1. If the function ψ′q is an n–polynomial harmonically s–type convex, then for s∈0,1 and p,q>1 with 1/p+1/q=1, we have
(27)ϖψϑ1+ωψϑ22+2−ϖ−ω2ψ2ϑ1ϑ2ϑ1+ϑ2−ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx≤ϑ1ϑ2ϑ2−ϑ1×ρ11/pC1′ψ′ϑ1q+C2′ψ′ϑ2q1/q+ρ21/pC3′ψ′ϑ1q+C4′ψ′ϑ2q1/q,where
(28)ρ1=∫011−ϖ−κpdκ=1−ϖp+1+ϖp+1p+1,ρ2=∫01ω−κpdκ=1−ωp+1+ωp+1p+1,C1′=12n∑i=1nβ1,12ϑ12q2F12q,1;2;1−ϑ2ϑ1−12n∑i=1nsiβ1,i+12ϑ12q2F12q,1;i+2;1−ϑ2ϑ1,C2′=12n∑i=1nβ1,1ϑ1+ϑ22q2F12q,1;2;ϑ2−ϑ1ϑ1+ϑ2−12n∑i=1nsiβi+1,12ϑ12q2F12q,i+1;i+2;ϑ2−ϑ1ϑ1+ϑ2,C3′=1n∑i=1nβ1,12ϑ22q2F12q,1;2;121−ϑ2ϑ1−12n∑i=1nsiβi+1,i+22ϑ22q2F12q,i+1;i+2;121−ϑ2ϑ1,C4′=12n∑i=1nβ1,12ϑ22q2F12q,1;2;121−ϑ2ϑ1−12n∑i=1nsiβ1,i+12ϑ12q2F12q,1;i+2;121−ϑ2ϑ1.and Aϑ1,ϑ2,Bϑ1,ϑ2 are defined from (25).Proof.
From Lemma22, Hölder’s inequality, n–polynomial harmonically s–type convexity of ψ′q, and properties of modulus, we have
(29)ϖψϑ1+ωψϑ22+2−ϖ−ω2ψ2ϑ1ϑ2ϑ1+ϑ2−ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx≤ϑ1ϑ2ϑ2−ϑ14∫0141−ϖ−κ1−κϑ2+1+κϑ12ψ′2ϑ1ϑ21−κϑ2+1+κϑ1dκ+∫014ω−κκϑ1+2−κϑ22ψ′2ϑ1ϑ2κϑ1+2−κϑ2dκ≤ϑ1ϑ2ϑ2−ϑ1∫011−ϖ−κpdκ1/p×∫011Aϑ1,ϑ22q12n∑i=1n1−s1−1−κiψ′ϑ1q+12n∑i=1n1+sκiψ′ϑ2qdκ1/q+∫01ω−κpdκ1/p×∫011Bϑ1,ϑ22q12n∑i=1n2−sκiψ′ϑ1q+12n∑i=1n1−s1−κiψ′ϑ2qdκ1/q=ϑ1ϑ2ϑ2−ϑ1×ρ11/pC1′ψ′ϑ1q+C2′ψ′ϑ2q1/q+ρ21/pC3′ψ′ϑ1q+C4′ψ′ϑ2q1/q,which completes the proof.Corollary 24.
Takingn=1 in Theorem 23, then
(30)ϖψϑ1+ωψϑ22+2−ϖ−ω2ψ2ϑ1ϑ2ϑ1+ϑ2−ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx≤ϑ1ϑ2ϑ2−ϑ1×ρ11/pD1′ψ′ϑ1q+D2′ψ′ϑ2q1/q+ρ21/pD3′ψ′ϑ1q+D4′ψ′ϑ2q1/q,where
(31)D1′=12β1,12ϑ12q2F12q,1;2;1−ϑ2ϑ1−12sβ1,22ϑ12q2F12q,1;3;1−ϑ2ϑ1,D2′=12β1,1ϑ1+ϑ22q2F12q,1;2;ϑ2−ϑ1ϑ1+ϑ2−12sβ2,12ϑ12q2F12q,2;3;ϑ2−ϑ1ϑ1+ϑ2,D3′=β1,12ϑ22q2F12q,1;2;121−ϑ2ϑ1−12sβ2,32ϑ22q2F12q,2;3;121−ϑ2ϑ1,D4′=12β1,12ϑ22q2F12q,1;2;121−ϑ2ϑ1−12sβ1,22ϑ12q2F12q,1;3;121−ϑ2ϑ1.Corollary 25.
Takingϖ=ω in Theorem 23, then
(32)ϖψϑ1+ψϑ22+1−ϖψ2ϑ1ϑ2ϑ1+ϑ2−ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx≤ϑ1ϑ2ϑ2−ϑ1ρ1/p×C1′ψ′ϑ1q+C2′ψ′ϑ2q1/q+C3′ψ′ϑ1q+C4′ψ′ϑ2q1/q,
whereρ1=ρ2=ρ.Corollary 26.
Choosingϖ=ω=0 in Theorem 23, then
(33)ψ2ϑ1ϑ2ϑ1+ϑ2−2ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx≤ϑ1ϑ2ϑ2−ϑ1p+1p×C1′ψ′ϑ1q+C2′ψ′ϑ2q1/q+C3′ψ′ϑ1q+C4′ψ′ϑ2q1/q.Corollary 27.
Choosingϖ=ω=1/2 in Theorem 23, then
(34)ψϑ1+ψϑ22+ψ2ϑ1ϑ2ϑ1+ϑ2−2ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx≤ϑ1ϑ2ϑ2−ϑ14p+1p×C1′ψ′ϑ1q+C2′ψ′ϑ2q1/q+C3′ψ′ϑ1q+C4′ψ′ϑ2q1/q.Corollary 28.
Takingϖ=ω=1/3 in Theorem 23, then
(35)ψϑ1+ψϑ22+2ψ2ϑ1ϑ2ϑ1+ϑ2−3ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx≤3ϑ1ϑ2ϑ2−ϑ142/3p+1+1/3p+1p+1p×C1′ψ′ϑ1q+C2′ψ′ϑ2q1/q+C3′ψ′ϑ1q+C4′ψ′ϑ2q1/q.Corollary 29.
Takingϖ=ω=1 in Theorem 23, then
(36)ψϑ1+ψϑ22−ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx≤ϑ1ϑ2ϑ2−ϑ1p+1p×C1′ψ′ϑ1q+C2′ψ′ϑ2q1/q+C3′ψ′ϑ1q+C4′ψ′ϑ2q1/q.Theorem 30.
Letψ:ϑ1,ϑ2⊆0,+∞→ℝ be a differentiable function such that ψ′∈Lϑ1,ϑ2 and ϖ,ω∈0,1. If the function ψ′q is an n–polynomial harmonically s–type convex, then for s∈0,1 and p,q>1 with 1/p+1/q=1, we have
(37)ϖψϑ1+ωψϑ22+2−ϖ−ω2ψϑ1ϑ2ϑ1+ϑ2−ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx≤ϑ1ϑ2ϑ2−ϑ14×4ϑ1+ϑ222F12p,1;2;ϑ1−ϑ2ϑ1+ϑ21/pC5′ψ′ϑ1q+C6′ψ′ϑ2q1/q+1ϑ122F12p,1;2;ϑ1−ϑ22ϑ11/pC7′ψ′ϑ1q+C8′ψ′ϑ2q1/q,where
(38)C5′=12n∑i=1n∫011−ϖ−ωq1−s1−1−κidκ,C6′=12n∑i=1n∫011−ϖ−ωq1+sκidκ,C7′=12n∑i=1n∫01ω−κq2−sκidκ,C8′=12n∑i=1n∫01ω−κq1−s1−κidκ.Proof.
Applying Lemma22, Hölder’s inequality, n–polynomial harmonically s–type convexity of ψ′q, and properties of modulus, we have
(39)ϖψϑ1+ωψϑ22+2−ϖ−ω2ψ2ϑ1ϑ2ϑ1+ϑ2−ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx≤ϑ1ϑ2ϑ2−ϑ14∫0141−ϖ−κ1−κϑ2+1+κϑ12ψ′2ϑ1ϑ21−κϑ2+1+κϑ1dκ+∫014ω−κtϑ1+2−κϑ22ψ′2ϑ1ϑ2κϑ1+2−κϑ2dκ≤ϑ1ϑ2ϑ2−ϑ144∫011Aϑ1,ϑ22pdκ1/p×∫011−ϖ−ωq12n∑i=1n1−s1−1−κiψ′ϑ1q+12n∑i=1n1+sκiψ′ϑ2qdκ1/q+4∫011Bϑ1,ϑ22pdκ1/p×∫01ω−κq12n∑i=1n2−sκiψ′ϑ1q+12n∑i=1n1−s1−κiψ′ϑ2qdκ1/q=ϑ1ϑ2ϑ2−ϑ14×4ϑ1+ϑ222F12p,1;2;ϑ1−ϑ2ϑ1+ϑ21/pC5′ψ′ϑ1q+C6′ψ′ϑ2q1/q+1ϑ122F12p,1;2;ϑ1−ϑ22ϑ11/pC7′ψ′ϑ1q+C8′ψ′ϑ2q1/q,which completes the proof.Corollary 31.
Takingn=1 in Theorem 30, then
(40)ϖψϑ1+ωψϑ22+2−ϖ−ω2ψ2ϑ1ϑ2ϑ1+ϑ2−ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx≤ϑ1ϑ2ϑ2−ϑ14×4ϑ1+ϑ222F12p,1;2;ϑ1−ϑ2ϑ1+ϑ21/pD5′ψ′ϑ1q+D6′ψ′ϑ2q1/q+1ϑ122F12p,1;2;ϑ1−ϑ22ϑ11/pD7′ψ′ϑ1q+D8′ψ′ϑ2q1/q,where
(41)D5′=12∫011−ϖ−ωq1−s1−1−κdκ,D6′=12∫011−ϖ−ωq1+sκdκ,D7′=12∫01ω−κq2−sκdκ,D8′=12∫01ω−κq1−s1−κdκ.Corollary 32.
Takingϖ=ω in Theorem 30, then
(42)ϖψϑ1+ψϑ22+1−ϖψ2ϑ1ϑ2ϑ1+ϑ2−ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx≤ϑ1ϑ2ϑ2−ϑ14×4ϑ1+ϑ222F12p,1;2;ϑ1−ϑ2ϑ1+ϑ21/pE1′ψ′ϑ1q+E2′ψ′ϑ2q1/q+1ϑ122F12p,1;2;ϑ1−ϑ22ϑ11/pE3′ψ′ϑ1q+E4′ψ′ϑ2q1/q,where
(43)E1′=12n∑i=1n∫011−2ϖq1−s1−1−κidκ,E2′=12n∑i=1n∫011−2ϖq1+sκidκ,E3′=12n∑i=1n∫01ϖ−κq2−sκidκ,E4′=12n∑i=1n∫01ϖ−κq1−s1−κidκ.Corollary 33.
Takingϖ=ω=0 in Theorem 30, then
(44)ψ2ϑ1ϑ2ϑ1+ϑ2−ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx≤ϑ1ϑ2ϑ2−ϑ14×4ϑ1+ϑ222F12p,1;2;ϑ1−ϑ2ϑ1+ϑ21/pE5′ψ′ϑ1q+E6′ψ′ϑ2q1/q+1ϑ122F12p,1;2;ϑ1−ϑ22ϑ11/pE7′ψ′ϑ1q+E8′ψ′ϑ2q1/q,where
(45)E5′=12n∑i=1n∫011−s1−1−κidκ=12n∑i=1ni+1−sii+1,E6′=E6′=12n∑i=1n∫011+sκidκ=12n∑i=1ni+1+sii+1,E7′=12n∑i=1n∫01κq2−sκidκ=12n∑i=1n2q+1−siq+i+1,E8′=12n∑i=1n∫01κq1−s1−κidκ.Corollary 34.
Choosingϖ=ω=1/2 in Theorem 30, then
(46)ψϑ1+ψϑ22+ψ2ϑ1ϑ2ϑ1+ϑ2−2ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx≤ϑ1ϑ2ϑ2−ϑ12×1ϑ122F12p,1;2;ϑ1−ϑ22ϑ11/pE9′ψ′ϑ1q+E10′ψ′ϑ2q1/q,where
(47)E9′=12q+1n∑i=1n∫011−2κq2−sκidκ,E10′=12q+1n∑i=1n∫011−2κq1−s1−κidκ.Corollary 35.
Choosingϖ=ω=1/3 in Theorem 30, then
(48)ψϑ1+ψϑ22+2ψ2ϑ1ϑ2ϑ1+ϑ2−3ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx≤3ϑ1ϑ2ϑ2−ϑ14×4ϑ1+ϑ222F12p,1;2;ϑ1−ϑ2ϑ1+ϑ21/pG1′ψ′ϑ1q+G2′ψ′ϑ2q1/q+1ϑ122F12p,1;2;ϑ1−ϑ22ϑ11/pG3′ψ′ϑ1q+G4′ψ′ϑ2q1/q,where
(49)G1′=13q2n∑i=1n∫011−s1−1−κidκ=13q2n∑i=1ni+1−sii+1,G2′=13q2n∑i=1n∫011+sκidκ=13q2n∑i=1ni+1+sii+1,G3′=13q2n∑i=1n∫011−3κq2−sκidκ,G4′=13q2n∑i=1n∫011−3κq1−s1−κidκ.Corollary 36.
Takingϖ=ω=1 in Theorem 30, then
(50)ψϑ1+ψϑ22−ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψxx2dx≤ϑ1ϑ2ϑ2−ϑ14×4ϑ1+ϑ222F12p,1;2;ϑ1−ϑ2ϑ1+ϑ21/pG5′ψ′ϑ1q+G6′ψ′ϑ2q1/q+1ϑ122F12p,1;2;ϑ1−ϑ22ϑ11/pG7′ψ′ϑ1q+G8′ψ′ϑ2q1/q,where
(51)G5′=12n∑i=1n∫011−s1−1−κidκ=12n∑i=1ni+1−sii+1,G6′=12n∑i=1n∫011+sκidκ=12n∑i=1ni+1+sii+1,G7′=12n∑i=1n∫011−κq2−sκidκ,G8′=12n∑i=1n∫011−κq1−s1−κidκ.


## 6. Ostrowski Type Inequalities vian–Polynomial Harmonically s–Type Convex Functions

In this section, we are going to add some new Ostrowski type inequalities via this newly introduced definition namelyn–polynomial harmonically s–type convex function. In order to obtain the results, we need the following lemma.Lemma 37 (see [19]).
Suppose a mappingψ:I⊆R\0→R is differentiable on Io, where ϑ1,ϑ2∈I with ϑ1<ϑ2.Ifψ′∈Lϑ1,ϑ2, then the following inequality holds:
(52)ψx−ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψuu2du=ϑ1ϑ2ϑ2−ϑ1x−ϑ12∫01κκϑ1+1−κx2ψ′ϑ1xκϑ1+1−κxdκ−ϑ2−x2∫01κκϑ2+1−κx2ψ′ϑ2xκϑ2+1−κxdκ,for each x∈ϑ1,ϑ2.Theorem 38.
Letψ:ϑ1,ϑ2⊆0,+∞→ℝ be a differentiable function on Io, where ϑ1,ϑ2∈I with ϑ1<ϑ2 and ψ′∈Lϑ1,ϑ2. If the function ψ′q is an n–polynomial harmonically s–type convex, for q≥1 and s∈0,1, then
(53)ψx−ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψuu2du≤ϑ1ϑ2ϑ2−ϑ1×x−ϑ12Λ1ψ′xq+Λ2ψ′ϑ1q1/q+ϑ2−x2Λ3ψ′xq+Λ4ψ′ϑ2q1/q,where
(54)Λ1=1n∑i=1nβq+1,1x2q2F12q,q+1;q+2;1−ϑ1x−1n∑i=1nsiβq+1,i+1x2q2F12q,q+1;q+2;1−ϑ1x,Λ2=1n∑i=1nβq+1,1x2q2F12q,q+1;q+2;1−ϑ1x−1n∑i=1nsiβq+i+1,1x2q2F12q,q+i+1;q+i+2;1−ϑ1x,Λ3=1n∑i=1nβq+1,1x2q2F12q,q+1;q+2;1−ϑ2x−1n∑i=1nsiβq+1,i+1x2q2F12q,q+1;q+i+2;1−ϑ2x,Λ4=1n∑i=1nβq+1,1x2q2F12q,q+1;q+2;1−ϑ2x−1n∑i=1nsiβq+i+1,1x2q2F12q,q+i+1;q+i+2;1−ϑ2x.Proof.
Using Lemma37, power mean inequality, n–polynomial harmonically s–type convexity of ψ′q and properties of modulus, we have
(55)ψx−ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψudu≤ϑ1ϑ2ϑ2−ϑ1x−ϑ12∫01κκϑ1+1−κx2ψ′ϑ1xκϑ1+1−κxdκ+ϑ2−x2∫01κκϑ2+1−κx2ψ′ϑ2xκϑ2+1−κxdκ≤ϑ1ϑ2x−ϑ12ϑ2−ϑ1∫011dκ1−1/q∫01κqκϑ1+1−κx2qψ′ϑ1xκϑ1+1−κxqdκ1/q+ϑ1ϑ2ϑ2−x2ϑ2−ϑ1∫011dκ1−1/q∫01κqκϑ2+1−κx2qψ′ϑ2xκϑ2+1−κxqdκ1/q≤ϑ1ϑ2x−ϑ12ϑ2−ϑ1∫01κq1/n∑i=1n1−s1−κiψ′xq+1/n∑i=1n1−sκiψ′ϑ1qtϑ1+1−tx2qdκ1/q+ϑ1ϑ2ϑ2−x2ϑ2−ϑ1∫01κq1/n∑i=1n1−s1−κiψ′xq+1/n∑i=1n1−sκiψ′ϑ2qκϑ2+1−κx2qdκ1/q≤ϑ1ϑ2x−ϑ12ϑ2−ϑ11n∑i=1n∫01κq1−s1−κiκϑ1+1−κx2qdκψ′xq+1n∑i=1n∫01κq1−sκiκϑ1+1−κx2qdκψ′ϑ1q1/q+ϑ1ϑ2ϑ2−x2ϑ2−ϑ11n∑i=1n∫01κq1−s1−κiκϑ2+1−κx2qdκψ′xq+1n∑i=1n∫01κq1−sκiκϑ2+1−κx2qdκψ′ϑ2q1/q≤ϑ1ϑ2ϑ2−ϑ1x−ϑ12Λ1ψ′xq+Λ2ψ′ϑ1q1/q+ϑ2−x2Λ3ψ′xq+Λ4ψ′ϑ2q1/q,which completes the proof.Theorem 39.
Letψ:ϑ1,ϑ2⊆0,+∞→ℝ be a differentiable function on Io, where ϑ1,ϑ2∈I with ϑ1<ϑ2 and ψ′∈Lϑ1,ϑ2. If the function ψ′q is an n–polynomial harmonically s–type convex, for q≥1 and s∈0,1, then for all x∈ϑ1,ϑ2, one has
(56)ψx−ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψuu2du≤ϑ1ϑ2ϑ2−ϑ1121−1/q×x−ϑ12Λ5ψ′xq+Λ6ψ′ϑ1q1/q+ϑ2−x2Λ7ψ′xq+Λ8ψ′ϑ2q1/q,where
(57)Λ5=1n∑i=1nβ2,1x2q2F12q,2;3;1−ϑ1x−1n∑i=1nsiβ2,i+1x2q2F12q,2;i+1;1−ϑ1x,Λ6=1n∑i=1nβ2,1x2q2F12q,2;3;1−ϑ1x−1n∑i=1nsiβi+2,1x2q2F12q,i+2;i+3;1−ϑ1x,Λ7=1n∑i=1nβ2,1x2q2F12q,2;3;1−ϑ2x−1n∑i=1nsiβ2,i+1x2q2F12q,2;i+1;1−ϑ2x,Λ8=1n∑i=1nβ2,1x2q2F12q,2;3;1−ϑ2x−1n∑i=1nsiβi+2,1x2q2F12q,i+2;i+3;1−ϑ2x.Proof.
Using Lemma37, power mean inequality, n–polynomial harmonically s–type convexity of ψ′q and properties of modulus, we have
(58)ψx-ϑ1ϑ2ϑ2-ϑ1∫ϑ1ϑ2ψudu≤ϑ1ϑ2ϑ2-ϑ1·x-ϑ12∫01κκϑ1+1-κx2ψ′ϑ1xκϑ1+1-κxdκ+ϑ2-x2∫01κκϑ2+1-κx2ψ′ϑ2xκϑ2+1-κxdκ≤ϑ1ϑ2x-ϑ12ϑ2-ϑ1∫01κdκ1-1/q·∫01κκϑ1+1-κx2qψ′ϑ1xκϑ1+1-txqdκ1/q+ϑ1ϑ2ϑ2-x2ϑ2-ϑ1∫01κdκ1-1/q·∫01κκϑ2+1-κx2qψ′ϑ2xκϑ2+1-κxqdκ1/q≤ϑ1ϑ2x-ϑ12ϑ2-ϑ1121-1/q·∫01κ1/n∑i=1n1-s1-κiψ′xq+1/n∑i=1n1-sκiψ′ϑ1qκϑ1+1-κx2qdκ1/q+ϑ1ϑ2ϑ2-x2ϑ2-ϑ1121-1/q·∫01κ1/n∑i=1n1-s1-κiψ′xq+1/n∑i=1n1-sκiψ′ϑ2qκϑ2+1-κx2qdκ1/q≤ϑ1ϑ2ϑ2-ϑ1121-1/q×1n∑i=1n∫01κ1-s1-κiκϑ1+1-κx2qdκψ′xqx-ϑ12·1n∑i=1n∫01κ1-s1-κiκϑ1+1-κx2qdκψ′xq+1n∑i=1n∫01κ1-sκiκϑ1+1-κx2qdκψ′ϑ1q1/q+ϑ2-x2·1n∑i=1n∫01κ1-s1-κiκϑ2+1-κx2qdκψ′xq+1n∑i=1n∫01κ1-sκiκϑ2+1-κx2qdκψ′ϑ2q1/q≤ϑ1ϑ2ϑ2-ϑ1121-1/q×x-ϑ12Λ5ψ′xq+Λ6ψ′ϑ1q1/q+ϑ2-x2Λ7ψ′xq+Λ8ψ′ϑ2q1/q,which completes the proof.Theorem 40.
Letψ:ϑ1,ϑ2⊆0,+∞→ℝ be a differentiable function on Io, where ϑ1,ϑ2∈I with ϑ1<ϑ2 and ψ′∈Lϑ1,ϑ2. If the function ψ′q is an n–polynomial harmonically s–type convex, for q≥1 and s∈0,1, then for all x∈ϑ1,ϑ2, one has
(59)∣ψx−ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψuu2du∣≤ϑ1ϑ2ϑ2−ϑ1x−ϑ11ϑ1−lnx−lnϑ1x−ϑ11−1/qΛ9ψ′xq+Λ10ψ′ϑ1q1/q+ϑ2−xlnx−lnϑ1x−ϑ1−1ϑ21−1/qΛ11ψ′xq+Λ12ψ′ϑ2q1/q,where
(60)Λ9=1n∑i=1nβ1,1x2q2F12q,1;2;1−ϑ1x−1n∑i=1nsiβ1,i+2x2q2F12q,1;i+2;1−ϑ1x,Λ10=1n∑i=1nβ1,1x2q2F12q,1;2;1−ϑ1x−1n∑i=1nsiβi+1,1x2q2F12q,i+1;i+2;1−ϑ1x,Λ11=1n∑i=1nβ1,1x2q2F12q,2;3;1−ϑ2x−1n∑i=1nsiβ1,i+2x2q2F12q,1;i+2;1−ϑ2x,Λ12=1n∑i=1nβ1,1x2q2F12q,2;3;1−ϑ2x−1n∑i=1nsiβi+1,1x2q2F12q,i+1;i+2;1−ϑ2x.Proof.
Using Lemma37, power mean inequality, n–polynomial harmonically s–type convexity of ψ′q and properties of modulus, we have
(61)ψx-ϑ1ϑ2ϑ2-ϑ1∫ϑ1ϑ2ψudu≤ϑ1ϑ2ϑ2-ϑ1·x-ϑ12∫01κκϑ1+1-κx2ψ′ϑ1xκϑ1+1-κxdκ+ϑ2-x2∫01κκϑ2+1-κx2ψ′ϑ2xκϑ2+1-κxdκ≤ϑ1ϑ2x-ϑ12ϑ2-ϑ1∫01κκϑ1+1-κx2dκ1-1/q·∫011κϑ1+1-κx2qψ′ϑ1xκϑ1+1-κxqdκ1/q+ϑ1ϑ2ϑ2-x2ϑ2-ϑ1∫01κκϑ2+1-κx2dκ1-1/q·∫011κϑ2+1-κx2qψ′ϑ2xκϑ2+1-κxqdκ1/q≤ϑ1ϑ2x-ϑ12ϑ2-ϑ11x-ϑ11ϑ1-lnx-lnϑ1x-ϑ11-1/q×∫011/n∑i=1n1-s1-κiψ′xq+1/n∑i=1n1-sκiψ′ϑ1qκϑ1+1-κx2qdκ1/q+ϑ1ϑ2ϑ2-x2ϑ2-ϑ11ϑ2-xlnx-lnϑ1x-ϑ1-1ϑ21-1/q×∫011/n∑i=1n1-s1-κiψ′xq+1/n∑i=1n1-sκiψ′ϑ2qκϑ2+1-κx2qdκ1/q≤ϑ1ϑ2x-ϑ12ϑ2-ϑ11x-ϑ11ϑ1-lnx-lnϑ1x-ϑ11-1/q×1n∑i=1n∫011-s1-κiκϑ1+1-κx2qdκψ′xq+1n∑i=1n∫011-sκiκϑ1+1-κx2qdκψ′ϑ1q1/q+ϑ1ϑ2ϑ2-x2ϑ2-ϑ11ϑ2-xlnx-lnϑ1x-ϑ1-1ϑ21-1/q×1n∑i=1n∫011-s1-κiκϑ2+1-κx2qdκψ′xq+1n∑i=1n∫011-sκiκϑ2+1-κx2qdκψ′ϑ2q1/q≤ϑ1ϑ2ϑ2-ϑ1·x-ϑ11ϑ1-lnx-lnϑ1x-ϑ11-1/qΛ9ψ′xq+Λ10ψ′ϑ1q1/q+ϑ2-xlnx-lnϑ1x-ϑ1-1ϑ21-1/qΛ11ψ′xq+Λ12ψ′ϑ2q1/q,which completes the proof.Theorem 41.
Letψ:ϑ1,ϑ2⊆0,+∞→ℝ be a differentiable function on Io, where ϑ1,ϑ2∈I with ϑ1<ϑ2 and ψ′∈Lϑ1,ϑ2. If the function ψ′q is an n–polynomial harmonically s–type convex, for q>1, 1/p+1/q=1 and s∈0,1, then
(62)∣ψx−ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψuu2du∣≤ϑ1ϑ2ϑ2−ϑ11p+11/p×x−ϑ12Λ9ψ′xq+Λ10ψ′ϑ1q1/q+ϑ2−x2Λ11ψ′xq+Λ12ψ′ϑ2q1/q,where Λ9,Λ10,Λ11, and Λ12 are defined in Theorem 40.Proof.
Using Lemma37, Hölder’s inequality, n–polynomial harmonically s–type convexity of ψ′q, and properties of modulus, we have
(63)∣ψx−ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψudu∣≤ϑ1ϑ2ϑ2−ϑ1×x−ϑ12∫01κκϑ1+1−κx2∣ψ′ϑ1xκϑ1+1−κx∣dκ+ϑ2−x2∫01κκϑ2+1−κx2∣ψ′ϑ2xκϑ2+1−κx∣dκ≤ϑ1ϑ2x−ϑ12ϑ2−ϑ1∫01κdκ1/p∫011κϑ1+1−κx2qψ′ϑ1xκϑ1+1−κxqdκ1/q+ϑ1ϑ2ϑ2−x2ϑ2−ϑ1∫01κdκ1/p∫011κϑ2+1−κx2qψ′ϑ2xκϑ2+1−κxqdκ1/q≤ϑ1ϑ2x−ϑ12ϑ2−ϑ11p+11/p∫011/n∑i=1n1−s1−κiψ′xq+1/n∑i=1n1−sκiψ′ϑ1qκϑ1+1−κx2qdκ1/q+ϑ1ϑ2ϑ2−x2ϑ2−ϑ11p+11/p∫011/n∑i=1n1−s1−κiψ′xq+1/n∑i=1n1−sκiψ′ϑ2qκϑ2+1−κx2qdκ1/q≤ϑ1ϑ2ϑ2−ϑ11p+11/p×x−ϑ12Λ9ψ′xq+Λ10ψ′ϑ1q1/q+ϑ2−x2Λ11ψ′xq+Λ12ψ′ϑ2q1/q,which completes the proof.Theorem 42.
Letψ:ϑ1,ϑ2⊆0,+∞→ℝ be a differentiable function on Io, where ϑ1,ϑ2∈I with ϑ1<ϑ2 and ψ′∈Lϑ1,ϑ2. If the function ψ′q is an n–polynomial harmonically s–type convex, for q>1, 1/p+1/q=1 and s∈0,1, then
(64)∣ψx−ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψuu2du∣≤ϑ1ϑ2ϑ2−ϑ11n∑i=1ni+1−sii+11/q×x−ϑ12βp+1,1x2p2F12p,p+1,p+2,1−ϑ1xψ′xq+ψ′ϑ1q1/q+ϑ2−x2β1,1+pϑ22p2F12p,1;p+2,1−xϑ2ψ′xq+ψ′ϑ2q1/q.Proof.
Using Lemma37, Hölder’s inequality, n–polynomial harmonically s–type convexity of ψ′q, and properties of modulus, we have
(65)∣ψx−ϑ1ϑ2ϑ2−ϑ1∫ϑ1ϑ2ψudu∣≤ϑ1ϑ2ϑ2−ϑ1x−ϑ12∫01κκϑ1+1−κx2∣ψ′ϑ1xκϑ1+1−κx∣dκ+ϑ2−x2∫01κκϑ2+1−κx2∣ψ′ϑ2xκϑ2+1−κx∣dκ≤ϑ1ϑ2x−ϑ12ϑ2−ϑ1∫01κpκϑ1+1−κx2dκ1/p∫01ψ′ϑ1xκa+1−κxqdk1/q+ϑ1ϑ2ϑ2−x2ϑ2−ϑ1∫01κpκϑ2+1−κx2dκ1/p∫01ψ′ϑ2xκϑ2+1−κxqdκ1/q≤ϑ1ϑ2x−ϑ12ϑ2−ϑ1βp+1,1x2p2F12p,p+1,p+2,1−ϑ1x×∫011n∑i=1n1−s1−κiψ′xq+1n∑i=1n1−sκiψ′ϑ1qdk1/q+ϑ1ϑ2ϑ2−x2ϑ2−ϑ1β1,1+pϑ22p2F12p,1;p+2,1−xϑ2×∫011n∑i=1n1−s1−κiψ′xq+1n∑i=1n1−sκiψ′ϑ2qdk1/q≤ϑ1ϑ2x−ϑ12ϑ2−ϑ1βp+1,1x2p2F12p,p+1,p+2,1−ϑ1x×∑i=1ni+1−sii+1ψ′xq+ψ′ϑ1qn1/q+ϑ1ϑ2ϑ2−x2ϑ2−ϑ1β1,1+pϑ22p2F12p,1;p+2,1−xϑ2×∑i=1ni+1−sii+1ψ′xq+ψ′ϑ2qn1/q≤ϑ1ϑ2ϑ2−ϑ11n∑i=1ni+1−sii+11/q×x−ϑ12βp+1,1x2p2F12p,p+1,p+2,1−ϑ1xψ′xq+ψ′ϑ1q1/q+ϑ2−x2β1,1+pϑ22p2F12p,1;p+2,1−xϑ2ψ′xq+ψ′ϑ2q1/q,which completes the proof.


## 7. Applications

In this section, we recall the following special means for two positive real numbersϑ1,ϑ2 where ϑ1<ϑ2:
(1)
The arithmetic mean(66)A=Aϑ1,ϑ2=ϑ1+ϑ22(2)
The geometric mean(67)G=Gϑ1,ϑ2=ϑ1ϑ2(3)
The harmonic mean(68)H=Hϑ1,ϑ2=2ϑ1ϑ2ϑ1+ϑ2(4)
The logarithmic mean(69)L=Lϑ1,ϑ2=ϑ2−ϑ1lnϑ2−lnϑ1(5)
Thep-logarithmic mean(70)Lp=Lpϑ1,ϑ2=ϑ2p+1−ϑ1p+1p+1ϑ2−ϑ11/p,p∈ℝ\0(6)
The identric mean(71)I=Iϑ1,ϑ2=1eϑ2ϑ2ϑ1ϑ11/ϑ2−ϑ1These means have a lot of applications in areas and different type of numerical approximations. However, the following simple relationship are known in the literature.(72)Hϑ1,ϑ2≤Gϑ1,ϑ2≤Lϑ1,ϑ2≤Iϑ1,ϑ2≤Aϑ1,ϑ2.Proposition 43.
Let0<ϑ1<ϑ2. Then, we get the following inequality
(73)n2∑i=1n1−s/2iHϑ1,ϑ2≤G2ϑ1,ϑ2Lϑ1,ϑ2≤Aϑ1,ϑ22n∑i=1ni+1−sii+1.Proof.
Takingψx=x for x>0 in Theorem 18, then inequality (73) is easily captured.Proposition 44.
Let0<ϑ1<ϑ2. Then we get the following inequality
(74)n2∑i=1n1−s/2iH2ϑ1,ϑ2≤G2ϑ1,ϑ2≤Aϑ12,ϑ222n∑i=1ni+1−sii+1.Proof.
Takingψx=x2 for x>0 in Theorem 18, then inequality (74) is easily captured.Proposition 45.
Let0<ϑ1<ϑ2. Then, we get the following inequality
(75)n2∑i=1n1−s/2iHϑ1,ϑ2≤Iϑ1,ϑ2≤Gϑ1,ϑ22n∑i=1ni+1−sii+1.Proof.
Takingψx=lnx for x>0 in Theorem 18, then inequality (75) is easily captured.Proposition 46.
Let0<ϑ1<ϑ2. Then, we get the following inequality
(76)n2∑i=1n1−s/2iH2ϑ1,ϑ2lnHϑ1,ϑ2≤G2ϑ1,ϑ2lnIϑ1,ϑ2≤Aϑ12lnϑ1,ϑ22lnϑ22n∑i=1ni+1−sii+1.Proof.
Takingψx=x2lnx for x>0 in Theorem 18, then inequality (76) is easily captured.Proposition 47.
Let0<ϑ1<ϑ2. Then, we get the following inequality
(77)n2∑i=1n1−s/2iHp+2ϑ1,ϑ2≤G2ϑ1,ϑ2Lppϑ1,ϑ2≤Aϑ1p+2,ϑ2p+22n∑i=1ni+1−sii+1.Proof.
Takingψx=xp+2 for x>0 in Theorem 18, then inequality (77) is easily captured.


## 8. Conclusion

We have introduced and investigated some algebraic properties of a new class of functions namelyn–polynomial harmonically s–type convex. We showed that this class of functions had some nice properties, which other convex functions had as well. We proved that our new class of n–polynomial harmonically s–type convex function is very larger with respect to the known class of functions, like n–polynomial convex and n–polynomial harmonically convex. A new version of Hermite–Hadamard type inequality and an integral identity for the differentiable functions are obtained. In recent years, many researchers put the effort into the theory of inequalities to bring a new dimension to mathematical analysis and applied mathematics with different features in the literature. Due to widespread views and applications, the theory of inequalities has become an attractive, interesting, and absorbing field for the researchers. It is high time to find the applications of these inequalities along with efficient numerical methods. We believe that our newly introduced idea and concept will have very deep research in this captivating field of analysis and inequalities. The amazing techniques and wonderful ideas of this article can be extended and generalized on the coordinates along with fractional integral calculus. Our aim in the future is that we will approach and continue our research work in this direction furthermore.



---

*Source: 6615948-2021-01-30.xml*