Open AccessArticle

Trapezium-like Inequalities Involving k-th Order Differentiable $R_{γ}$ -Convex Functions and Applications

Miguel Vivas-Cortez

Muhammad Uzair Awan

^2,*

Sadia Talib

²,

Muhammad Aslam Noor

and

Khalida Inayat Noor

Escuela de Ciencias Físicas y Matemáticas, Facultad de Ciencias Exactas y Naturales, Pontificia Universidad Católica del Ecuador, Av. 12 de Octubre 1076, Apartado, Quito 17-01-2184, Ecuador

Department of Mathematics, Government College University, Faisalabad 38000, Pakistan

Department of Mathematics, COMSATS University Islamabad, Islamabad 44000, Pakistan

Author to whom correspondence should be addressed.

Symmetry 2022, 14(3), 448; https://doi.org/10.3390/sym14030448

Submission received: 27 January 2022 / Revised: 3 February 2022 / Accepted: 11 February 2022 / Published: 23 February 2022

(This article belongs to the Special Issue Mathematical Inequalities, Special Functions and Symmetry)

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Abstract

We introduce the class of

R_{γ}

-convex functions and discuss that it relates to some other classes of convexity. We study the class of

R_{γ}

-convex functions in the perspective of trapezium-like inequalities, for which we also derive a new integral identity involving a

k

-th order differentiable function. In order to show the significance of our results, we also discuss several special cases and offer some interesting applications.

Keywords:

convex;

R_{γ}

-convex functions; k-times differentiable; trapezium inequalities; means

MSC:

26A51; 26D15

1. Introduction and Preliminaries

The theory of convexity has great significance in modern analysis and also plays an important role in different branches of pure and applied sciences through its numerous applications. For example, convexity plays a significant role in optimization, mathematical economics, and operations research. The concepts of convexity and symmetry also have a close relation. There are many important properties of symmetric convex sets. A significance of the relation between between convexity and symmetry is that we work on one and apply it to the other. In recent decades the classical concepts of convex sets and convex functions have been extended and generalized in different directions according to the need of the problems. In 1978, Brckner [1] introduced the notion of

s_{1}

-convex functions and noticed that we can recapture the classical convexity from

s_{1}

-convexity by taking

s = 1

. In 1995, Dragomir et al. [2] introduced and studied the class of P-convex functions. Godunova and Levin [3] introduced the notion of Godunova–Levin type convex functions. Varosanec [4] introduced and studied the class of h-convex functions. It is worth mentioning here that the class of h-convex functions unifies all of these discussed classes and also enjoys some favorable properties of classical convex functions. In 2015, Dragomir [5] introduced the notion of

s_{2}

-Godunova–Levin type convex functions and observed that this class is also contained in the class of h-convex functions. In 2013, Dragomir [6] introduced another generalization of classical convexity that is called

φ

-convex functions. Taking inspiration from this, Awan et al. [7] introduced another significant generalization of classical convexity that is called

γ

-preinvex functions [8]. Very recently, Cortez et al. [9] introduced the class of

R

-convex sets as:

Definition 1

([9]). Let

ρ, λ > 0

and

σ = (σ (0), \dots, σ (k), \dots)

be a bounded sequence of positive real numbers. A non-empty set

L

is said to be generalized convex, if:

\begin{matrix} υ_{1} + τ R_{ρ, λ, σ} (υ_{2} - υ_{1}) \in L, \forall υ_{1}, υ_{2} \in L, τ \in [0, 1] . \end{matrix}

Here

R_{ρ, λ, σ} (\cdot)

is Raina’s function and is defined as follows:

\begin{matrix} R_{ρ, λ, σ} (z) = R_{ρ, λ}^{σ (0), σ (1), \dots} (z) = \sum_{k = 0}^{\infty} \frac{σ (k)}{Γ (ρ k + λ)} z^{k}, \end{matrix}

where

ρ, λ > 0

∣ z ∣ < R

and

σ = {σ (0), σ (1), \dots, σ (k), \dots}

is a bounded sequence of positive real numbers. For details, see [10].

Using

R

-convex sets as a domain, Cortez et al. [9] also defined the class of

R

-convex functions as:

Definition 2

([9]). Let

ρ, λ > 0

and

σ = (σ (0), \dots, σ (k), \dots)

be a bounded sequence of positive real numbers. A function

Ψ : L \to R

is said to be generalized convex, if:

\begin{matrix} Ψ (υ_{1} + τ R_{ρ, λ, σ} (υ_{2} - υ_{1})) \leq (1 - τ) Ψ (υ_{1}) + τ Ψ (υ_{2}), \forall υ_{1}, υ_{2} \in L, τ \in [0, 1] . \end{matrix}

For some more details regarding generalizations, extensions and applications of classical convexity, see [4,7,11,12,13,14,15,16].

As a matter of fact, the theory of convex functions has a close relationship with the theory of inequalities. We can easily obtain a huge number of inequalities by just using the convexity property of the functions and their generalizations. Hermite–Hadamard’s inequality is one of the most important results that can be viewed as an equivalent property of the convex functions. This inequality is also known as a trapezium inequality. In recent years, this result has been extended and generalized in different directions using novel and innovative ideas.

Cortez et al. [9] derived a new version of Hermite–Hadamard’s inequality using the class of generalized convex functions. This result reads as follows:

Theorem 1.

Let

Ψ : L = [c_{1}, c_{1} + R_{ρ, λ, σ} (c_{2} - c_{1})] \to R

be a generalized convex function; then:

\begin{matrix} Ψ (\frac{2 c_{1} + R_{ρ, λ, σ} (c_{2} - c_{1})}{2}) \leq \frac{1}{R_{ρ, λ, σ} (c_{2} - c_{1})} \int_{c_{1}}^{c_{1} + R_{ρ, λ, σ} (c_{2} - c_{1})} Ψ (x) d x \leq \frac{Ψ (c_{1}) + Ψ (c_{2})}{2} . \end{matrix}

Note that if we take

R_{ρ, λ, σ} (c_{2} - c_{1}) = c_{2} - c_{1}

, then we can recapture the classical Hermite–Hadamard’s inequality from the above inequality, which reads as:

Theorem 2.

Let

Ψ : I = [c_{1}, c_{2}] \to R

be a convex function; then:

\begin{matrix} Ψ (\frac{c_{1} + c_{2}}{2}) \leq \frac{1}{c_{2} - c_{1}} \int_{c_{1}}^{c_{2}} Ψ (x) d x \leq \frac{Ψ (c_{1}) + Ψ (c_{2})}{2} . \end{matrix}

For more details regarding Hermite–Hadamard’s inequality and its applications, see [14,17,18].

The main objective of this paper is to generalize the notion of

R

-convex functions and introduce the class of

R_{γ}

-convex functions. We show that the class of

R_{γ}

-convex functions includes some other classes of classical convexity. We study the class of

R_{γ}

-convex functions in the perspective of trapezium-like inequalities. To obtain our main results, we derive a new integral identity involving a

k

-th order differentiable function. Using this identity as an auxiliary result, we then derive associated trapezium-like inequalities essentially using the class of

k

-th order

R_{γ}

-convex functions. We also discuss several special cases that can be deduced from the main results of our paper. In order to show the significance of our results, we also discuss several special cases and offer some interesting applications. We hope that the ideas and techniques of this paper will inspire interested readers working in this field.

2. Main Results

In this section, we discuss our main results.

2.1. $R_{γ}$ -Convex Functions

We now define the class of

R_{γ}

-convex functions.

Definition 3.

Let

γ : (0, 1) \to (0, \infty)

be a real function. Let

ρ, λ > 0

and

σ = (σ (0), \dots, σ (k), \dots)

be a bounded sequence of positive real numbers. A function

Ψ : L \to R

is said to be

R_{γ}

-convex, if:

\begin{matrix} Ψ (υ_{1} + μ R_{ρ, λ, σ} (υ_{2} - υ_{1})) \leq (1 - μ) γ (1 - μ) Ψ (υ_{1}) + μ γ (μ) Ψ (υ_{2}), \forall υ_{1}, υ_{2} \in K, μ \in [0, 1] . \end{matrix}

We now discuss some special cases of Definition 3.

I. If we take

γ (μ) = 1

in Definition 3, then we have the class of generalized convex functions [9].

II. If we take

γ (μ) = μ^{- 1}

in Definition 3, then we have the definition of

R_{P}

-convex functions.

Definition 4.

Let

ρ, λ > 0

and

σ = (σ (0), \dots, σ (k), \dots)

be a bounded sequence of positive real numbers. A function

Ψ : L \to R

is said to be

R_{P}

-convex, if:

\begin{matrix} Ψ (υ_{1} + μ R_{ρ, λ, σ} (υ_{2} - υ_{1})) \leq Ψ (υ_{1}) + Ψ (υ_{2}), \forall υ_{1}, υ_{2} \in K, μ \in [0, 1] . \end{matrix}

III. If we take

γ (μ) = μ^{s - 1}

in Definition 3, where

s \in (0, 1)

, then we have the class of

R_{s_{1}}

-convex functions of Breckner type.

Definition 5.

Let

s \in (0, 1]

. Let

ρ, λ > 0

and

σ = (σ (0), \dots, σ (k), \dots)

be a bounded sequence of positive real numbers. A function

Ψ : L \to R

is said to be

R_{s_{1}}

-convex, if:

\begin{matrix} Ψ (υ_{1} + μ R_{ρ, λ, σ} (υ_{2} - υ_{1})) \leq {(1 - μ)}^{s} Ψ (υ_{1}) + μ^{s} Ψ (υ_{2}), \forall υ_{1}, υ_{2} \in K, μ \in [0, 1] . \end{matrix}

IV. If we take

γ (μ) = μ^{- s - 1}

, then Definition 3 reduces to the definition of

R_{s_{2}}

-Godunova–Levin convex function.

Definition 6.

Let

s \in [0, 1]

. Let

ρ, λ > 0

and

σ = (σ (0), \dots, σ (k), \dots)

be a bounded sequence of positive real numbers. A function

Ψ : L \to R

is said to be

R_{s_{2}}

-convex, if:

\begin{matrix} Ψ (υ_{1} + μ R_{ρ, λ, σ} (υ_{2} - υ_{1})) \leq {(1 - μ)}^{- s} Ψ (υ_{1}) + μ^{- s} Ψ (υ_{2}), \forall υ_{1}, υ_{2} \in K, μ \in (0, 1) . \end{matrix}

V. If we take

γ (μ) = 1 - μ

in Definition 3, then we have the definition of

R_{t g s}

-convex function.

Definition 7.

Let

ρ, λ > 0

and

σ = (σ (0), \dots, σ (k), \dots)

be a bounded sequence of positive real numbers. A function

Ψ : L \to R

is said to be

R_{t g s}

-convex, if:

\begin{matrix} Ψ (υ_{1} + μ R_{ρ, λ, σ} (υ_{2} - υ_{1})) \leq μ (1 - μ) [Ψ (υ_{1}) + Ψ (υ_{2})], \forall υ_{1}, υ_{2} \in K, μ \in [0, 1] . \end{matrix}

VI. If we take

R_{ρ, λ, σ} (υ_{2} - υ_{1}) = υ_{2} - υ_{1}

in Definition 3, then we have the definition of

γ

-convex function, see [6].

Our next result depends upon condition M, which was introduced by Noor and Noor [19].

Condition M. Assume that the function

R (\cdot)

satisfies the following condition:

R (θ R (v - u)) = θ R (v - u), θ \in [0, 1] .

We now give a new version of Hermite–Hadamard’s inequality essentially using the class of

R_{γ}

-convex functions.

Theorem 3.

Let

Ψ : L = [c_{1}, Δ] \to R

be a

R_{γ}

-convex function and

R_{ρ, λ, σ} (\cdot)

satisfy condition M; then for

R_{ρ, λ, σ} (c_{2} - c_{1}) > 0

and

γ (\frac{1}{2}) \neq 0

, we have:

\begin{matrix} \frac{1}{γ (\frac{1}{2})} Ψ (\frac{2 c_{1} + R_{ρ, λ, σ} (c_{2} - c_{1})}{2}) \leq \frac{1}{R_{ρ, λ, σ} (c_{2} - c_{1})} \int_{c_{1}}^{c_{1} + R_{ρ, λ, σ} (c_{2} - c_{1})} Ψ (x) d x \leq (Ψ (c_{1}) + Ψ (c_{2})) \int_{0}^{1} μ γ (μ) d μ . \end{matrix}

Proof.

It is given that

R_{ρ, λ, σ} (\cdot)

satisfies condition M and since

Ψ

is a

R_{γ}

-convex function, by taking

x = c_{1} + μ R_{ρ, λ, σ} (c_{2} - c_{1})

and

y = c_{1} + (1 - μ) R_{ρ, λ, σ} (c_{2} - c_{1})

, we have:

\begin{matrix} Ψ (\frac{2 c_{1} + R_{ρ, λ, σ} (c_{2} - c_{1})}{2}) & = Ψ (\frac{2 x + R_{ρ, λ, σ} (y - x)}{2}) \\ \leq γ (\frac{1}{2}) \frac{1}{R_{ρ, λ, σ} (c_{2} - c_{1})} \int_{c_{1}}^{c_{1} + R_{ρ, λ, σ} (c_{2} - c_{1})} Ψ (x) d x . \end{matrix}

This implies:

\begin{matrix} \frac{1}{γ (\frac{1}{2})} Ψ (\frac{2 c_{1} + R_{ρ, λ, σ} (c_{2} - c_{1})}{2}) \leq \frac{1}{R_{ρ, λ, σ} (c_{2} - c_{1})} \int_{c_{1}}^{c_{1} + R_{ρ, λ, σ} (c_{2} - c_{1})} Ψ (x) d x . \end{matrix}

(1)

Additionally, using the fact that

Ψ

is a

R_{γ}

-convex function, we have:

\begin{matrix} \frac{1}{R_{ρ, λ, σ} (c_{2} - c_{1})} \int_{c_{1}}^{c_{1} + R_{ρ, λ, σ} (c_{2} - c_{1})} Ψ (x) d x \leq (Ψ (c_{1}) + Ψ (c_{2})) \int_{0}^{1} μ γ (μ) d μ . \end{matrix}

(2)

Combining (1) and (2) completes the proof. □

Remark 1.

Note that for different suitable choices of the function

γ (\cdot)

in Theorem 3, we obtain other versions of Hermite–Hadamard’s inequality. For instance, if we take

γ (μ) = 1, μ^{- 1}, μ^{s - 1}, μ^{- s - 1}

and

γ (μ) = 1 - μ

, then we obtain Hermite–Hadamard’s inequality for generalized convex functions,

R_{P}

-convex functions,

R_{s_{1}}

-convex functions,

R_{s_{2}}

-convex functions and for

R_{t g s}

-convex functions, respectively.

2.2. Auxiliary Result

In this section, we derive a new integral identity involving

k

-times differentiable functions. For the sake of simplicity, we now consider

Ω_{μ} : = c_{1} + μ R_{ρ, λ, σ} (c_{2} - c_{1})

Ω_{1 - μ} : = c_{1} + (1 - μ) R_{ρ, λ, σ} (c_{2} - c_{1})

Δ : = c_{1} + R_{ρ, λ, σ} (c_{2} - c_{1})

and

Λ (c_{1}, μ) : = {c_{1}}_{k} μ^{k} + \dots + {c_{1}}_{1} μ + {c_{1}}_{0}

Lemma 1.

Let

Ψ : L \to R

be a

k

-times differentiable function on

L^{\circ}

with

c_{1}, c_{2} \in L

and

R_{ρ, λ, σ} (c_{2} - c_{1}) > 0

where

k

is an even number. If

Ψ^{k} \in L [c_{1}, c_{1} + R_{ρ, λ, σ} (c_{2} - c_{1})]

, then

\begin{matrix} M (c_{1}, c_{2}; k) = \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! {c_{1}}_{k}} \int_{0}^{1} (Λ (c_{1}, μ)) [Ψ^{k} (Ω_{μ}) + Ψ^{k} (Ω_{1 - μ})] d μ . \end{matrix}

where:

\begin{matrix} M (c_{1}, c_{2}; k) & = \frac{1}{R_{ρ, λ, σ} (c_{2} - c_{1})} \int_{c_{1}}^{Δ} Ψ (x) d x - \frac{Ψ (c_{1}) + Ψ (Δ)}{2} + \dots \\ - \frac{(k (k - 1) (k - 2) {c_{1}}_{k} + \dots + 4.3.2 {c_{1}}_{4}) ϕ^{k - 4} (c_{2}, c_{1})}{2 k! {c_{1}}_{k}} [Ψ^{k - 4} (Δ) + Ψ^{k - 4} (c_{1})] \\ + \frac{(k (k - 1) {c_{1}}_{k} + \dots + 3.2 {c_{1}}_{3}) ϕ^{k - 3} (c_{2}, c_{1})}{2 k! {c_{1}}_{k}} [Ψ^{k - 3} (Δ) - Ψ^{k - 3} (c_{1})] \\ - \frac{(k {c_{1}}_{k} + \dots + 2 {c_{1}}_{2}) ϕ^{k - 2} (c_{2}, c_{1})}{2 k! {c_{1}}_{k}} [Ψ^{k - 2} (Δ) + Ψ^{k - 2} (c_{1})] \\ + \frac{({c_{1}}_{k} + \dots + {c_{1}}_{1} + 2 {c_{1}}_{0}) ϕ^{k - 1} (c_{2}, c_{1})}{2 k! {c_{1}}_{k}} [Ψ^{k - 1} (Δ) - Ψ^{k - 1} (c_{1})] . \end{matrix}

Proof.

Consider the following integral and integrating by parts repeatedly, we have:

\begin{matrix} K_{1} & = \int_{0}^{1} (Λ (c_{1}, μ)) Ψ^{k} (Ω_{1 - μ}) d μ \\ = - (Λ (c_{1}, μ)) \frac{Ψ^{k - 1} (Ω_{1 - μ})}{R_{ρ, λ, σ} (c_{2} - c_{1})} ∣_{0}^{1} \\ + \frac{1}{R_{ρ, λ, σ} (c_{2} - c_{1})} \int_{0}^{1} (k {c_{1}}_{k} μ^{k - 1} + \dots + {c_{1}}_{1}) Ψ^{k - 1} (Ω_{1 - μ}) d μ \\ = \frac{1}{R_{ρ, λ, σ} (c_{2} - c_{1})} [{c_{1}}_{0} Ψ^{k - 1} (Δ) - ({c_{1}}_{k} + \dots + {c_{1}}_{1} + {c_{1}}_{0}) Ψ^{k - 1} (c_{1})] \\ + \frac{1}{R_{ρ, λ, σ} (c_{2} - c_{1})} \int_{0}^{1} (k {c_{1}}_{k} μ^{k - 1} + \dots + {c_{1}}_{1}) Ψ^{k - 1} (Ω_{1 - μ}) d μ \\ = \frac{1}{R_{ρ, λ, σ} (c_{2} - c_{1})} [{c_{1}}_{0} Ψ^{k - 1} (Δ) - ({c_{1}}_{k} + \dots + {c_{1}}_{1} + {c_{1}}_{0}) Ψ^{k - 1} (c_{1})] \\ - (k {c_{1}}_{k} μ^{k - 1} + \dots + 2 {c_{1}}_{2} μ + {c_{1}}_{1}) \frac{Ψ^{k - 2} (Ω_{1 - μ})}{R_{ρ, λ, σ}^{2} (c_{2} - c_{1})} ∣_{0}^{1} \\ - \frac{1}{R_{ρ, λ, σ}^{2} (c_{2} - c_{1})} \int_{0}^{1} (k (k - 1) {c_{1}}_{k} μ^{k - 2} + \dots + 3.2 {c_{1}}_{3} μ + 2 {c_{1}}_{2}) Ψ^{k - 2} (Ω_{1 - μ}) d μ \end{matrix}

\begin{matrix} = \frac{1}{R_{ρ, λ, σ} (c_{2} - c_{1})} [{c_{1}}_{0} Ψ^{k - 1} (Δ) - ({c_{1}}_{k} + \dots + {c_{1}}_{1} + {c_{1}}_{0}) Ψ^{k - 1} (c_{1})] \\ - \frac{1}{R_{ρ, λ, σ}^{2} (c_{2} - c_{1})} [(k {c_{1}}_{k} + \dots + 2 {c_{1}}_{2} + {c_{1}}_{1}) Ψ^{k - 2} (c_{1}) - {c_{1}}_{1} Ψ^{k - 2} (Δ)] \\ - \frac{1}{R_{ρ, λ, σ}^{2} (c_{2} - c_{1})} \int_{0}^{1} (k (k - 1) {c_{1}}_{k} μ^{k - 2} + \dots + 3.2 {c_{1}}_{3} μ + 2 {c_{1}}_{2}) Ψ^{k - 2} (Ω_{1 - μ}) d μ \\ = \frac{1}{R_{ρ, λ, σ} (c_{2} - c_{1})} [{c_{1}}_{0} Ψ^{k - 1} (Δ) - ({c_{1}}_{k} + \dots + {c_{1}}_{1} + {c_{1}}_{0}) Ψ^{k - 1} (c_{1})] \\ - \frac{1}{R_{ρ, λ, σ}^{2} (c_{2} - c_{1})} [(k {c_{1}}_{k} + \dots + 2 {c_{1}}_{2} + {c_{1}}_{1}) Ψ^{k - 2} (c_{1}) - {c_{1}}_{1} Ψ^{k - 2} (Δ)] \\ + \frac{1}{R_{ρ, λ, σ}^{3} (c_{2} - c_{1})} [2 {c_{1}}_{2} Ψ^{k - 3} (Δ) - (k (k - 1) {c_{1}}_{k} + \dots + 3.2 {c_{1}}_{3} + 2 {c_{1}}_{2}) Ψ^{k - 3} (c_{1})] \\ - \frac{1}{R_{ρ, λ, σ}^{4} (c_{2} - c_{1})} [(k (k - 1) (k - 2) {c_{1}}_{k} + \dots + 4.3.2 {c_{1}}_{4} + 3.2 {c_{1}}_{3}) Ψ^{k - 4} (c_{1}) \\ - 3.2 {c_{1}}_{3} Ψ^{k - 4} (Δ)] \\ + \dots - \frac{1}{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})} [(k! {c_{1}}_{k} + (k - 1)! {c_{1}}_{k - 1}) Ψ (c_{1}) - (k - 1)! {c_{1}}_{k - 1} Ψ (Δ)] \\ + \frac{k! {c_{1}}_{k}}{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})} \int_{0}^{1} Ψ (Ω_{1 - μ}) d μ . \end{matrix}

(3)

In addition,

\begin{matrix} K_{2} & = \int_{0}^{1} (Λ (c_{1}, μ)) Ψ^{k} (Ω_{μ}) d μ \\ = (Λ (c_{1}, μ)) \frac{Ψ^{k - 1} (Ω_{μ})}{R_{ρ, λ, σ} (c_{2} - c_{1})} ∣_{0}^{1} \\ - \frac{1}{R_{ρ, λ, σ} (c_{2} - c_{1})} \int_{0}^{1} (k {c_{1}}_{k} μ^{k - 1} + \dots + {c_{1}}_{1}) Ψ^{k - 1} (Ω_{μ}) d μ \\ = \frac{1}{R_{ρ, λ, σ} (c_{2} - c_{1})} [({c_{1}}_{k} + \dots + {c_{1}}_{1} + {c_{1}}_{0}) Ψ^{k - 1} (Δ) - {c_{1}}_{0} Ψ^{k - 1} (c_{1})] \\ - \frac{1}{R_{ρ, λ, σ} (c_{2} - c_{1})} \int_{0}^{1} (k {c_{1}}_{k} μ^{k - 1} + \dots + {c_{1}}_{1}) Ψ^{k - 1} (Ω_{μ}) d μ \\ = \frac{1}{R_{ρ, λ, σ} (c_{2} - c_{1})} [({c_{1}}_{k} + \dots + {c_{1}}_{1} + {c_{1}}_{0}) Ψ^{k - 1} (Δ) - {c_{1}}_{0} Ψ^{k - 1} (c_{1})] \\ - (k {c_{1}}_{k} μ^{k - 1} + \dots + 2 {c_{1}}_{2} μ + {c_{1}}_{1}) \frac{Ψ^{k - 2} (Ω_{μ})}{R_{ρ, λ, σ}^{2} (c_{2} - c_{1})} ∣_{0}^{1} \\ + \frac{1}{R_{ρ, λ, σ}^{2} (c_{2} - c_{1})} \int_{0}^{1} (k (k - 1) {c_{1}}_{k} μ^{k - 2} + \dots + 3.2 {c_{1}}_{3} μ + 2 {c_{1}}_{2}) Ψ^{k - 2} (Ω_{μ}) d μ \end{matrix}

\begin{matrix} = \frac{1}{R_{ρ, λ, σ} (c_{2} - c_{1})} [({c_{1}}_{k} + \dots + {c_{1}}_{1} + {c_{1}}_{0}) Ψ^{k - 1} (Δ) - {c_{1}}_{0} Ψ^{k - 1} (c_{1})] \\ - \frac{1}{R_{ρ, λ, σ}^{2} (c_{2} - c_{1})} [(k {c_{1}}_{k} + \dots + 2 {c_{1}}_{2} + {c_{1}}_{1}) Ψ^{k - 2} (Δ) - {c_{1}}_{1} Ψ^{k - 2} (c_{1})] \\ + \frac{1}{R_{ρ, λ, σ}^{2} (c_{2} - c_{1})} \int_{0}^{1} (k (k - 1) {c_{1}}_{k} μ^{k - 2} + \dots + 3.2 {c_{1}}_{3} μ + 2 {c_{1}}_{2}) Ψ^{k - 2} (Ω_{μ}) d μ \\ = \frac{1}{R_{ρ, λ, σ} (c_{2} - c_{1})} [({c_{1}}_{k} + \dots + {c_{1}}_{1} + {c_{1}}_{0}) Ψ^{k - 1} (Δ) - {c_{1}}_{0} Ψ^{k - 1} (c_{1})] \\ - \frac{1}{R_{ρ, λ, σ}^{2} (c_{2} - c_{1})} [(k {c_{1}}_{k} + \dots + 2 {c_{1}}_{2} + {c_{1}}_{1}) Ψ^{k - 2} (Δ) - {c_{1}}_{1} Ψ^{k - 2} (c_{1})] \\ + \frac{1}{R_{ρ, λ, σ}^{3} (c_{2} - c_{1})} [(k (k - 1) {c_{1}}_{k} + \dots + 3.2 {c_{1}}_{3} + 2 {c_{1}}_{2}) Ψ^{k - 3} (Δ) - 2 {c_{1}}_{2} Ψ^{k - 3} (c_{1})] \\ - \frac{1}{R_{ρ, λ, σ}^{4} (c_{2} - c_{1})} [(k (k - 1) (k - 2) {c_{1}}_{k} + \dots + 4.3.2 {c_{1}}_{4} + 3.2 {c_{1}}_{3}) Ψ^{k - 4} (Δ) - 3.2 {c_{1}}_{3} Ψ^{k - 4} (c_{1})] \\ + \dots - \frac{1}{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})} [(k! {c_{1}}_{k} + (k - 1)! {c_{1}}_{k - 1}) Ψ (Δ) - (k - 1)! {c_{1}}_{k - 1} Ψ (c_{1})] \\ + \frac{k! {c_{1}}_{k}}{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})} \int_{0}^{1} Ψ (Ω_{μ}) d μ . \end{matrix}

(4)

Summing up Equations (3) and (4), we have:

\begin{matrix} K_{1} + K_{2} & = \int_{0}^{1} (Λ (c_{1}, μ)) [Ψ^{k} (Ω_{μ}) + Ψ^{k} (Ω_{1 - μ})] d μ \\ = \frac{{c_{1}}_{0}}{R_{ρ, λ, σ} (c_{2} - c_{1})} [Ψ^{k - 1} (Δ) - Ψ^{k - 1} (c_{1})] \\ + \frac{({c_{1}}_{k} + \dots + {c_{1}}_{1} + {c_{1}}_{0})}{R_{ρ, λ, σ} (c_{2} - c_{1})} [Ψ^{k - 1} (Δ) - Ψ^{k - 1} (c_{1})] \\ + \frac{{c_{1}}_{1}}{R_{ρ, λ, σ}^{2} (c_{2} - c_{1})} [Ψ^{k - 2} (Δ) + Ψ^{k - 2} (c_{1})] \\ - \frac{(k {c_{1}}_{k} + \dots + 2 {c_{1}}_{2} + {c_{1}}_{1})}{R_{ρ, λ, σ}^{2} (c_{2} - c_{1})} [Ψ^{k - 2} (Δ) + Ψ^{k - 2} (c_{1})] \\ + \frac{2 {c_{1}}_{2}}{R_{ρ, λ, σ}^{3} (c_{2} - c_{1})} [Ψ^{k - 3} (Δ) - Ψ^{k - 3} (c_{1})] \\ + \frac{(k (k - 1) {c_{1}}_{k} + \dots + 3.2 {c_{1}}_{3} + 2 {c_{1}}_{2})}{R_{ρ, λ, σ}^{3} (c_{2} - c_{1})} [Ψ^{k - 3} (Δ) - Ψ^{k - 3} (c_{1})] \\ + \frac{3.2 {c_{1}}_{3}}{R_{ρ, λ, σ}^{4} (c_{2} - c_{1})} [Ψ^{k - 4} (Δ) + Ψ^{k - 4} (c_{1})] \end{matrix}

\begin{matrix} - \frac{(k (k - 1) (k - 2) {c_{1}}_{k} + \dots + 4.3.2 {c_{1}}_{4} + 3.2 {c_{1}}_{3})}{R_{ρ, λ, σ}^{4} (c_{2} - c_{1})} [Ψ^{k - 4} (Δ) - Ψ^{k - 4} (c_{1})] \\ + \dots - \frac{(k! {c_{1}}_{k}}{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})} [Ψ (c_{1}) + Ψ (Δ)] + \frac{2 k! {c_{1}}_{k}}{R_{ρ, λ, σ}^{k + 1} (c_{2} - c_{1})} \int_{c_{1}}^{Δ} Ψ (x) d x \\ = \frac{({c_{1}}_{k} + \dots + {c_{1}}_{1} + 2 {c_{1}}_{0})}{R_{ρ, λ, σ} (c_{2} - c_{1})} [Ψ^{k - 1} (Δ) - Ψ^{k - 1} (c_{1})] \\ - \frac{(k {c_{1}}_{k} + \dots + 2 {c_{1}}_{2})}{R_{ρ, λ, σ}^{2} (c_{2} - c_{1})} [Ψ^{k - 2} (Δ) + Ψ^{k - 2} (c_{1})] \\ + \frac{(k (k - 1) {c_{1}}_{k} + \dots + 3.2 {c_{1}}_{3})}{R_{ρ, λ, σ}^{3} (c_{2} - c_{1})} [Ψ^{k - 3} (Δ) - Ψ^{k - 3} (c_{1})] \\ - \frac{(k (k - 1) (k - 2) {c_{1}}_{k} + \dots + 4.3.2 {c_{1}}_{4})}{R_{ρ, λ, σ}^{4} (c_{2} - c_{1})} [Ψ^{k - 4} (Δ) - Ψ^{k - 4} (c_{1})] \\ + \dots - \frac{k! {c_{1}}_{k}}{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})} [Ψ (c_{1}) + Ψ (Δ)] + \frac{2 k! {c_{1}}_{k}}{R_{ρ, λ, σ}^{k + 1} (c_{2} - c_{1})} \int_{c_{1}}^{Δ} Ψ (x) d x . \end{matrix}

Multiplying the above equality by

\frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! {c_{1}}_{k}}

, we obtain:

\begin{matrix} \frac{1}{R_{ρ, λ, σ} (c_{2} - c_{1})} \int_{c_{1}}^{Δ} Ψ (x) d x - \frac{Ψ (c_{1}) + Ψ (Δ)}{2} + \dots \\ - \frac{(k (k - 1) (k - 2) {c_{1}}_{k} + \dots + 4.3.2 {c_{1}}_{4}) ϕ^{k - 4} (c_{2}, c_{1})}{2 k! {c_{1}}_{k}} [Ψ^{k - 4} (Δ) + Ψ^{k - 4} (c_{1})] \\ + \frac{(k (k - 1) {c_{1}}_{k} + \dots + 3.2 {c_{1}}_{3}) ϕ^{k - 3} (c_{2}, c_{1})}{2 k! {c_{1}}_{k}} [Ψ^{k - 3} (Δ) - Ψ^{k - 3} (c_{1})] \\ - \frac{(k {c_{1}}_{k} + \dots + 2 {c_{1}}_{2}) ϕ^{k - 2} (c_{2}, c_{1})}{2 k! {c_{1}}_{k}} [Ψ^{k - 2} (Δ) + Ψ^{k - 2} (c_{1})] \\ + \frac{({c_{1}}_{k} + \dots + {c_{1}}_{1} + 2 {c_{1}}_{0}) ϕ^{k - 1} (c_{2}, c_{1})}{2 k! {c_{1}}_{k}} [Ψ^{k - 1} (Δ) - Ψ^{k - 1} (c_{1})] \\ = \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! {c_{1}}_{k}} \int_{0}^{1} (Λ (c_{1}, μ)) [Ψ^{k} (Ω_{μ}) + Ψ^{k} (Ω_{1 - μ})] d μ . \end{matrix}

This completes the proof. □

2.3. Trapezium-like Inequalities

Now using Lemma 1, we derive new trapezium-like inequalities.

Theorem 4.

Let

Ψ : L \to R

be a function such that

Ψ^{(k)}

exists on

L^{\circ}

and

Ψ^{(k)}

is integrable on

L

, where

c_{1}, c_{2} \in L

with

R_{ρ, λ, σ} (c_{2} - c_{1}) > 0

and

k

is an even number. If

∣ Ψ^{(k)} ∣

is the

R_{γ}

-convex function on

L

, then:

\begin{matrix} ∣ M (c_{1}, c_{2}; k) ∣ & \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} [∣ Ψ^{k} (c_{1}) ∣ + ∣ Ψ^{k} (c_{2}) ∣] \\ \times \int_{0}^{1} [∣ {c_{1}}_{k} μ^{k} ∣ + \dots + ∣ {c_{1}}_{1} μ ∣ + ∣ {c_{1}}_{0} ∣] [μ γ (μ) + (1 - μ) γ (1 - μ)] d μ . \end{matrix}

Proof.

Using Lemma 1 and the

γ

-preinvexity of

∣ Ψ^{(k)} ∣

, we have:

\begin{matrix} ∣ M (c_{1}, c_{2}; k) ∣ \\ \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} \int_{0}^{1} ∣ Λ (c_{1}, μ) ∣ ∣ Ψ^{k} (Ω_{μ}) + Ψ^{k} (Ω_{1 - μ}) ∣ d μ \\ \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} \int_{0}^{1} ∣ Λ (c_{1}, μ) ∣ [∣ Ψ^{k} (Ω_{μ}) + Ψ^{k} (Ω_{1 - μ}) ∣] d μ \\ \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} \int_{0}^{1} ∣ Λ (c_{1}, μ) ∣ [∣ Ψ^{k} (Ω_{μ}) ∣ + ∣ Ψ^{k} (Ω_{1 - μ}) ∣] d μ \\ \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} \int_{0}^{1} ∣ Λ (c_{1}, μ) ∣ \\ \times [(1 - μ) γ (1 - μ) ∣ Ψ^{k} (c_{1}) ∣ + μ γ (μ) ∣ Ψ^{k} (c_{2}) ∣ + μ γ (μ) ∣ Ψ^{k} (c_{1}) ∣ + (1 - μ) γ (1 - μ) ∣ Ψ^{k} (c_{2}) ∣] d μ \\ = \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} [∣ Ψ^{k} (c_{1}) ∣ + ∣ Ψ^{k} (c_{2}) ∣] \\ \times \int_{0}^{1} ∣ Λ (c_{1}, μ) [μ γ (μ) + (1 - μ) γ (1 - μ)] d μ \\ \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} [∣ Ψ^{k} (c_{1}) ∣ + ∣ Ψ^{k} (c_{2}) ∣] \\ \times \int_{0}^{1} [∣ {c_{1}}_{k} μ^{k} ∣ + \dots + ∣ {c_{1}}_{1} μ ∣ + ∣ {c_{1}}_{0} ∣] [μ γ (μ) + (1 - μ) γ (1 - μ)] d μ . \end{matrix}

This completes the proof. □

Now we will discuss some special cases of Theorem 4.

I. If

γ (μ) = 1

, then Theorem 4 reduces to the following result in the class of generalized convex functions.

Corollary 1.

Under the assumptions of Theorem 4 if

∣ Ψ^{(k)} ∣

is generalized convex function on

L

, then

\begin{matrix} ∣ M (c_{1}, c_{2}; k) ∣ \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} [∣ Ψ^{k} (c_{1}) ∣ + ∣ Ψ^{k} (c_{2}) ∣] \sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{j + 1} . \end{matrix}

II. If

γ (μ) = μ^{- 1}

, then Theorem 4 reduces to the following result in the class of

R_{P}

-convex function.

Corollary 2.

Under the assumptions of Theorem 4 if

∣ Ψ^{(k)} ∣

is an

R_{P}

-convex function on

L

, then:

\begin{matrix} ∣ M (c_{1}, c_{2}; k) ∣ \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{k! ∣ {c_{1}}_{k} ∣} [∣ Ψ^{k} (c_{1}) ∣ + ∣ Ψ^{k} (c_{2}) ∣] \sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{j + 1} . \end{matrix}

III. If

γ (μ) = μ^{s - 1}

, then Theorem 4 reduces to the following result in the class of

R_{s_{1}}

-convex functions.

Corollary 3.

Under the assumptions of Theorem 4 if

∣ Ψ^{(k)} ∣

is an

R_{s_{1}}

-convex function on

L

, then

\begin{matrix} ∣ M (c_{1}, c_{2}; k) ∣ \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} [∣ Ψ^{k} (c_{1}) ∣ + ∣ Ψ^{k} (c_{2}) ∣] \sum_{j = 0}^{k} ∣ {c_{1}}_{j} ∣ (\frac{1}{j + s + 1} - B (j + 1, s + 1)), \end{matrix}

where

B (\cdot, \cdot)

is the beta function and is defined as:

B (x, y) = \int_{0}^{1} ν^{x - 1} {(1 - ν)}^{y - 1} d ν = \frac{Γ (x) Γ (y)}{Γ (x + y)} .

IV. If

γ (μ) = μ^{- s - 1}

, then Theorem 4 reduces to the following result in the class of

R_{s_{2}}

-convex functions.

Corollary 4.

Under the assumptions of Theorem 4, if

∣ Ψ^{(k)}

is an

R_{s_{2}}

-convex function on

L

, then:

\begin{matrix} ∣ M (c_{1}, c_{2}; k) ∣ \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} [∣ Ψ^{k} (c_{1}) ∣ + ∣ Ψ^{k} (c_{2}) ∣] \sum_{j = 0}^{k} ∣ {c_{1}}_{j} ∣ (\frac{1}{j - s + 1} - B (j + 1, 1 - s)) . \end{matrix}

V. If

γ (μ) = 1 - μ

, then Theorem 4 reduces to the following result in the class of

R_{t g s}

-convex functions.

Corollary 5.

Under the assumptions of Theorem 4, if

∣ Ψ^{(k)} ∣

is an

R_{t g s}

-convex function on

L

, then:

\begin{matrix} ∣ M (c_{1}, c_{2}; k) ∣ \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} [∣ Ψ^{k} (c_{1}) ∣ + ∣ Ψ^{k} (c_{2}) ∣] \sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{(j + 2) (j + 3)} . \end{matrix}

Theorem 5.

Let

Ψ : L \to R

be a function such that

Ψ^{(k)}

exists on

L^{\circ}

and

Ψ^{(k)}

is integrable on

L

, where

c_{1}, c_{2} \in L

with

R_{ρ, λ, σ} (c_{2} - c_{1}) > 0

and

k

is an even number. If

∣ Ψ^{(k)} ∣^{q}

is the

R_{γ}

-convex function on

L

, then for

p^{- 1} + q^{- 1} = 1

, we have:

\begin{matrix} ∣ M (c_{1}, c_{2}; k) ∣ & \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} (\sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{{(j p + 1)}^{\frac{1}{p}}}) [{(∣ Ψ^{k} (c_{1}) ∣^{q} \int_{0}^{1} (1 - μ) γ (1 - μ) d μ + {∣ Ψ^{k} (c_{2}) ∣}^{q} \int_{0}^{1} μ γ (μ) d μ)}^{\frac{1}{q}} \\ + {(∣ Ψ^{k} (c_{1}) ∣^{q} \int_{0}^{1} μ γ (μ) d μ + {∣ Ψ^{k} (c_{2}) ∣}^{q} (1 - μ) γ (1 - μ) d μ)}^{\frac{1}{q}}] . \end{matrix}

Proof.

Using Lemma 1, Hölder’s inequality, Minkowski’s integral inequality, and the

γ

-preinvexity of

∣ Ψ^{(k)} ∣^{q}

, we have:

\begin{matrix} ∣ M (c_{1}, c_{2}; k) ∣ & \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} \int_{0}^{1} ∣ Λ (c_{1}, μ) ∣ ∣ Ψ^{k} (Ω_{μ}) + Ψ^{k} (Ω_{1 - μ}) ∣ d μ \\ \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} \int_{0}^{1} ∣ Λ (c_{1}, μ) ∣ [∣ Ψ^{k} (Ω_{μ}) + Ψ^{k} (Ω_{1 - μ}) ∣] d μ \\ \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} \int_{0}^{1} ∣ Λ (c_{1}, μ) ∣ [∣ Ψ^{k} (Ω_{μ}) ∣ + ∣ Ψ^{k} (Ω_{1 - μ}) ∣] d μ \end{matrix}

\begin{matrix} \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} {(\int_{0}^{1} {∣ Λ (c_{1}, μ) ∣}^{p} d μ)}^{\frac{1}{p}} {(\int_{0}^{1} {∣ Ψ^{k} (Ω_{μ}) ∣}^{q} d μ)}^{\frac{1}{q}} \\ + \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} {(\int_{0}^{1} {∣ Λ (c_{1}, μ) ∣}^{p} d μ)}^{\frac{1}{p}} {(\int_{0}^{1} {∣ Ψ^{k} (Ω_{1 - μ}) ∣}^{q} d μ)}^{\frac{1}{q}} \\ \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} {(\int_{0}^{1} {∣ Λ (c_{1}, μ) ∣}^{p} d μ)}^{\frac{1}{p}} \\ \times {(\int_{0}^{1} [(1 - μ) γ (1 - μ) ∣ Ψ^{k} (c_{1}) ∣ + μ γ (μ) {∣^{q} Ψ^{k} (c_{2}) ∣}^{q}] d μ)}^{\frac{1}{q}} \\ + \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} {(\int_{0}^{1} {∣ Λ (c_{1}, μ) ∣}^{p} d μ)}^{\frac{1}{p}} \\ \times {(\int_{0}^{1} [μ γ (μ) ∣ Ψ^{k} (c_{1}) ∣^{q} + (1 - μ) γ (1 - μ) {∣ Ψ^{k} (c_{2}) ∣}^{q}] d μ)}^{\frac{1}{q}} \\ \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} (\sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{{(j p + 1)}^{\frac{1}{p}}}) [{(\int_{0}^{1} [(1 - μ) γ (1 - μ) ∣ Ψ^{k} (c_{1}) ∣ + μ γ (μ) {∣^{q} Ψ^{k} (c_{2}) ∣}^{q}] d μ)}^{\frac{1}{q}} \\ + {(\int_{0}^{1} [μ γ (μ) ∣ Ψ^{k} (c_{1}) ∣^{q} + (1 - μ) γ (1 - μ) {∣ Ψ^{k} (c_{2}) ∣}^{q}] d μ)}^{\frac{1}{q}}] \\ = \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} (\sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{{(j p + 1)}^{\frac{1}{p}}}) [{(∣ Ψ^{k} (c_{1}) ∣^{q} \int_{0}^{1} (1 - μ) γ (1 - μ) d μ + {∣ Ψ^{k} (c_{2}) ∣}^{q} \int_{0}^{1} μ γ (μ) d μ)}^{\frac{1}{q}} \\ + {(∣ Ψ^{k} (c_{1}) ∣^{q} \int_{0}^{1} μ γ (μ) d μ + {∣ Ψ^{k} (c_{2}) ∣}^{q} (1 - μ) γ (1 - μ) d μ)}^{\frac{1}{q}}] . \end{matrix}

This completes the proof. □

Now we will discuss some special cases of Theorem 5.

I. If

γ (μ) = 1

, then Theorem 5 reduces to the following result in the class of generalized convex functions.

Corollary 6.

Under the assumptions of Theorem 5, if

∣ Ψ^{(k)} ∣^{q}

is a generalized convex function on

L

, then:

\begin{matrix} ∣ M (c_{1}, c_{2}; k) ∣ & \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{k! ∣ {c_{1}}_{k} ∣} (\sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{{(j p + 1)}^{\frac{1}{p}}}) {(\frac{∣ Ψ^{k} (c_{1}) ∣^{q} + {∣ Ψ^{k} (c_{2}) ∣}^{q}}{2})}^{\frac{1}{q}} . \end{matrix}

II. If

γ (μ) = μ^{- 1}

, then Theorem 5 reduces to the following result in the class of

R_{P}

-convex functions.

Corollary 7.

Under the assumptions of Theorem 5, if

∣ Ψ^{(k)} ∣^{q}

is an

R_{P}

-convex function on

L

, then:

\begin{matrix} ∣ M (c_{1}, c_{2}; k) ∣ & \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{k! ∣ {c_{1}}_{k} ∣} (\sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{{(j p + 1)}^{\frac{1}{p}}}) {(∣ Ψ^{k} (c_{1}) ∣^{q} + {∣ Ψ^{k} (c_{2}) ∣}^{q})}^{\frac{1}{q}} . \end{matrix}

III. If

γ (μ) = μ^{s - 1}

, then Theorem 5 reduces to the following result in the class of

R_{s_{1}}

-convex functions.

Corollary 8.

Under the assumptions of Theorem 5, if

∣ Ψ^{(k)} ∣^{q}

is an

R_{s_{1}}

-convex function on

L

, then:

\begin{matrix} ∣ M (c_{1}, c_{2}; k) ∣ & \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{k! ∣ {c_{1}}_{k} ∣} (\sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{{(j p + 1)}^{\frac{1}{p}}}) {(\frac{∣ Ψ^{k} (c_{1}) ∣^{q} + {∣ Ψ^{k} (c_{2}) ∣}^{q}}{1 + s})}^{\frac{1}{q}} . \end{matrix}

IV. If

γ (μ) = μ^{- s - 1}

, then Theorem 5 reduces to the following result in the class of

R_{s_{2}}

-convex functions.

Corollary 9.

Under the assumptions of Theorem 5, if

∣ Ψ^{(k)} ∣^{q}

is an

R_{s_{2}}

-convex function on

L

, then:

\begin{matrix} ∣ M (c_{1}, c_{2}; k) ∣ & \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{k! ∣ {c_{1}}_{k} ∣} (\sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{{(j p + 1)}^{\frac{1}{p}}}) {(\frac{∣ Ψ^{k} (c_{1}) ∣^{q} + {∣ Ψ^{k} (c_{2}) ∣}^{q}}{1 - s})}^{\frac{1}{q}} . \end{matrix}

V. If

γ (μ) = 1 - μ

, then Theorem 5 reduces to the following result in the class of

R_{t g s}

-convex functions.

Corollary 10.

Under the assumptions of Theorem 5, if

∣ Ψ^{(k)} ∣^{q}

is an

R_{t g s}

-convex function on

L

, then:

\begin{matrix} ∣ M (c_{1}, c_{2}; k) ∣ & \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{k! ∣ {c_{1}}_{k} ∣} (\sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{{(j p + 1)}^{\frac{1}{p}}}) {(\frac{∣ Ψ^{k} (c_{1}) ∣^{q} + {∣ Ψ^{k} (c_{2}) ∣}^{q}}{6})}^{\frac{1}{q}} . \end{matrix}

Theorem 6.

Let

Ψ : L \to R

be a function such that

Ψ^{(k)}

exists on

L^{\circ}

and

Ψ^{(k)}

is integrable on

L

, where

c_{1}, c_{2} \in L

with

R_{ρ, λ, σ} (c_{2} - c_{1}) > 0 d

k

is an even number. If

∣ Ψ^{(k)} ∣^{q}

is the

R_{γ}

-convex function on

L

for

q \geq 1

, then:

\begin{matrix} ∣ M (c_{1}, c_{2}; k) ∣ & \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} {(\sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{(j + 1)})}^{1 - \frac{1}{q}} \\ \times [{(∣ Ψ^{k} (c_{1}) ∣^{q} \int_{0}^{1} (1 - μ) γ (1 - μ) ∣ Λ (c_{1}, μ) ∣ d μ + ∣ Ψ^{k} (c_{2}) ∣^{q} \int_{0}^{1} μ γ (μ) ∣ Λ (c_{1}, μ) ∣ d μ)}^{\frac{1}{q}} \\ + {(∣ Ψ^{k} (c_{1}) ∣^{q} \int_{0}^{1} μ γ (μ) ∣ Λ (c_{1}, μ) ∣ d μ + ∣ Ψ^{k} (c_{2}) ∣^{q} \int_{0}^{1} (1 - μ) γ (1 - μ) ∣ Λ (c_{1}, μ) ∣ d μ)}^{\frac{1}{q}}] . \end{matrix}

Proof.

Using Lemma 1, the power mean integral inequality, and the

γ

-preinvexity of

∣ Ψ^{(k)} ∣^{q}

, we have:

\begin{matrix} ∣ M (c_{1}, c_{2}; k) ∣ \\ \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} \int_{0}^{1} ∣ Λ (c_{1}, μ) ∣ ∣ Ψ^{k} (Ω_{μ}) + Ψ^{k} (Ω_{1 - μ}) ∣ d μ \\ \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} \int_{0}^{1} ∣ Λ (c_{1}, μ) ∣ [∣ Ψ^{k} (Ω_{μ}) + Ψ^{k} (Ω_{1 - μ}) ∣] d μ \\ \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} \int_{0}^{1} ∣ Λ (c_{1}, μ) ∣ [∣ Ψ^{k} (Ω_{μ}) ∣ + ∣ Ψ^{k} (Ω_{1 - μ}) ∣] d μ \\ \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} {(\int_{0}^{1} ∣ Λ (c_{1}, μ) ∣ d μ)}^{1 - \frac{1}{q}} {(\int_{0}^{1} ∣ Λ (c_{1}, μ) ∣ ∣ Ψ^{k} (Ω_{μ}) ∣^{q} d μ)}^{\frac{1}{q}} \\ + \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} {(\int_{0}^{1} ∣ Λ (c_{1}, μ) ∣ d μ)}^{1 - \frac{1}{q}} {(\int_{0}^{1} ∣ Λ (c_{1}, μ) ∣ ∣ Ψ^{k} (Ω_{1 - μ}) ∣^{q} d μ)}^{\frac{1}{q}} \\ \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} {(\sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{j + 1})}^{1 - \frac{1}{q}} \\ \times {(\int_{0}^{1} ∣ Λ (c_{1}, μ) ∣ [(1 - μ) γ (1 - μ) ∣ Ψ^{k} (c_{1}) ∣ + μ γ (μ) {∣^{q} Ψ^{k} (c_{2}) ∣}^{q}] d μ)}^{\frac{1}{q}} \\ + \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} {(\sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{j + 1})}^{1 - \frac{1}{q}} \\ \times {(\int_{0}^{1} ∣ Λ (c_{1}, μ) ∣ [μ γ (μ) ∣ Ψ^{k} (c_{1}) ∣^{q} + (1 - μ) γ (1 - μ) {∣ Ψ^{k} (c_{2}) ∣}^{q}] d μ)}^{\frac{1}{q}} \\ \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! {c_{1}}_{k}} {(\sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{(j + 1)})}^{1 - \frac{1}{q}} \\ \times [{(∣ Ψ^{k} (c_{1}) ∣^{q} \int_{0}^{1} (1 - μ) γ (1 - μ) ∣ Λ (c_{1}, μ) ∣ d μ + ∣ Ψ^{k} (c_{2}) ∣^{q} \int_{0}^{1} μ γ (μ) ∣ Λ (c_{1}, μ) ∣ d μ)}^{\frac{1}{q}} \\ + {(∣ Ψ^{k} (c_{1}) ∣^{q} \int_{0}^{1} μ γ (μ) ∣ Λ (c_{1}, μ) ∣ d μ + ∣ Ψ^{k} (c_{2}) ∣^{q} \int_{0}^{1} (1 - μ) γ (1 - μ) ∣ Λ (c_{1}, μ) ∣ d μ)}^{\frac{1}{q}}] . \end{matrix}

This completes the proof. □

Now we will discuss some special cases of Theorem 6.

I. If

γ (μ) = 1

, then Theorem 6 reduces to the following result in the class of generalized convex functions.

Corollary 11.

Under the assumptions of Theorem 6, if

∣ Ψ^{(k)} ∣^{q}

is a generalized convex function on

L

, then:

\begin{matrix} ∣ M (c_{1}, c_{2}; k) ∣ & \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} {(\sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{(j + 1)})}^{1 - \frac{1}{q}} \\ \times [{(∣ Ψ^{k} (c_{1}) ∣^{q} \sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{(j + 1) (j + 2)} + {∣ Ψ^{k} (c_{2}) ∣}^{q} \sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{j + 2})}^{\frac{1}{q}} \\ + {(∣ Ψ^{k} (c_{1}) ∣^{q} \sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{j + 2} + {∣ Ψ^{k} (c_{2}) ∣}^{q} \sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{(j + 1) (j + 2)})}^{\frac{1}{q}}] . \end{matrix}

II. If

γ (μ) = μ^{- 1}

, then Theorem 6 reduces to the following result in the class of

R_{P}

-convex functions.

Corollary 12.

Under the assumptions of Theorem 6, if

∣ Ψ^{(k)} ∣^{q}

is an

R_{P}

-convex function on

L

, then:

\begin{matrix} ∣ M (c_{1}, c_{2}; k) ∣ & \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{k! ∣ {c_{1}}_{k} ∣} (\sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{(j + 1)}) {(∣ Ψ^{k} (c_{1}) ∣^{q} + {∣ Ψ^{k} (c_{2}) ∣}^{q})}^{\frac{1}{q}} . \end{matrix}

III. If

γ (μ) = μ^{s - 1}

, then Theorem 6 reduces to the following result in the class of

R_{s_{1}}

-convex functions.

Corollary 13.

Under the assumptions of Theorem 6, if

∣ Ψ^{(k)} ∣^{q}

is an

R_{s_{1}}

-convex function on

L

, then:

\begin{matrix} ∣ M (c_{1}, c_{2}; k) ∣ & \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} {(\sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{(j + 1)})}^{1 - \frac{1}{q}} \\ \times [{(∣ Ψ^{k} (c_{1}) ∣^{q} \sum_{j = 0}^{k} B (j + 1, s + 1) + {∣ Ψ^{k} (c_{2}) ∣}^{q} \sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{(j + s + 1)})}^{\frac{1}{q}} \\ + {(∣ Ψ^{k} (c_{1}) ∣^{q} \sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{(j + s + 1)} + {∣ Ψ^{k} (c_{2}) ∣}^{q} \sum_{j = 0}^{k} B (j + 1, s + 1))}^{\frac{1}{q}}] . \end{matrix}

IV. If

γ (μ) = μ^{- s - 1}

, then Theorem 6 reduces to the following result in the class of

R_{s_{2}}

-convex functions.

Corollary 14.

Under the assumptions of Theorem 6, if

∣ Ψ^{(k)} ∣^{q}

is an

R_{s_{2}}

-convex function on

L

, then:

\begin{matrix} ∣ M (c_{1}, c_{2}; k) ∣ & \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 k! ∣ {c_{1}}_{k} ∣} {(\sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{(j + 1)})}^{1 - \frac{1}{q}} \\ \times [{(∣ Ψ^{k} (c_{1}) ∣^{q} \sum_{j = 0}^{k} B (j + 1, 1 - s) + {∣ Ψ^{k} (c_{2}) ∣}^{q} \sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{(j - s + 1)})}^{\frac{1}{q}} \\ + {(∣ Ψ^{k} (c_{1}) ∣^{q} \sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{(j - s + 1)} + {∣ Ψ^{k} (c_{2}) ∣}^{q} \sum_{j = 0}^{k} B (j + 1, 1 - s))}^{\frac{1}{q}}] . \end{matrix}

V. If

γ (μ) = 1 - μ

, then Theorem 6 reduces to the following result in the class of

R_{t g s}

-convex functions.

Corollary 15.

Under the assumptions of Theorem 6, if

∣ Ψ^{(k)} ∣^{q}

is an

R_{t g s}

-convex function on

L

, then:

\begin{matrix} ∣ M (c_{1}, c_{2}; k) ∣ & \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{k! ∣ {c_{1}}_{k} ∣} {(\sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{(j + 1)})}^{1 - \frac{1}{q}} {(\sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{(j + 2) (j + 3)})}^{\frac{1}{q}} {(∣ Ψ^{k} (c_{1}) ∣^{q} + {∣ Ψ^{k} (c_{2}) ∣}^{q})}^{\frac{1}{q}} . \end{matrix}

2.4. Applications

We now consider the following special means for different positive real numbers

c_{1}

and

c_{2},

where

c_{1} < c_{2}

Arithmetic mean: $A (c_{1}, c_{2}) = \frac{c_{1} + c_{2}}{2}$ ;
$k$ –logarithmic mean: $L_{k} (c_{1}, c_{2}) = {(\frac{{c_{2}}^{k + 1} - {c_{1}}^{k + 1}}{(k + 1) (c_{2} - c_{1})})}^{\frac{1}{k}}$ , $k \in R \ {0, - 1}$ .

Proposition 1.

Let

c_{1}, c_{2} \in R, 0 < c_{1} < c_{2}

, where

k

is an even number; then:

\begin{matrix} ∣ L_{k + 1}^{k + 1} (c_{1}, c_{2}) - A ({c_{1}}^{k}, {c_{2}}^{k}) + \dots \\ - \frac{(k (k - 1) (k - 2) {c_{1}}_{k} + \dots + 4.3.2 {c_{1}}_{4}) {(c_{2} - c_{1})}^{k - 4}}{2.4! {c_{1}}_{k}} [{c_{2}}^{4} + {c_{1}}^{4}] \\ + \frac{(k (k - 1) {c_{1}}_{k} + \dots + 3.2 {c_{1}}_{3}) {(c_{2} - c_{1})}^{k - 3}}{2.3! {c_{1}}_{k}} [{c_{2}}^{3} - {c_{1}}^{3}] \\ - \frac{(k {c_{1}}_{k} + \dots + 2 {c_{1}}_{2}) {(c_{2} - c_{1})}^{k - 2}}{2.2! {c_{1}}_{k}} [{c_{2}}^{2} + {c_{1}}^{2}] \\ + \frac{({c_{1}}_{k} + \dots + {c_{1}}_{1} + 2 {c_{1}}_{0}) {(c_{2} - c_{1})}^{k - 1}}{2 {c_{1}}_{k}} (c_{2} - c_{1}) ∣ \\ \leq \frac{{(c_{2} - c_{1})}^{k}}{2 ∣ {c_{1}}_{k} ∣} [∣ {c_{1}}^{k} ∣ + ∣ {c_{2}}^{k} ∣] \sum_{j = 0}^{k} ∣ {c_{1}}_{j} ∣ (\frac{1}{j + s + 1} - B (j + 1, s + 1)) . \end{matrix}

Proof.

The proof directly follows from Corollary 3 by taking

Ψ : [0, 1] \to [0, 1], Ψ (x) = x^{k}

and

R_{ρ, λ, σ} (c_{2} - c_{1}) = c_{2} - c_{1} .

□

Proposition 2.

Let

c_{1}, c_{2} \in R, 0 < c_{1} < c_{2}

, where

k

is an even number; then:

\begin{matrix} ∣ L_{k + 1}^{k + 1} (c_{1}, c_{2}) - A ({c_{1}}^{k}, {c_{2}}^{k}) + \dots \\ - \frac{(k (k - 1) (k - 2) {c_{1}}_{k} + \dots + 4.3.2 {c_{1}}_{4}) {(c_{2} - c_{1})}^{k - 4}}{2.4! {c_{1}}_{k}} [{c_{2}}^{4} + {c_{1}}^{4}] \\ + \frac{(k (k - 1) {c_{1}}_{k} + \dots + 3.2 {c_{1}}_{3}) {(c_{2} - c_{1})}^{k - 3}}{2.3! {c_{1}}_{k}} [{c_{2}}^{3} - {c_{1}}^{3}] \\ - \frac{(k {c_{1}}_{k} + \dots + 2 {c_{1}}_{2}) {(c_{2} - c_{1})}^{k - 2}}{2.2! {c_{1}}_{k}} [{c_{2}}^{2} + {c_{1}}^{2}] \\ + \frac{({c_{1}}_{k} + \dots + {c_{1}}_{1} + 2 {c_{1}}_{0}) {(c_{2} - c_{1})}^{k - 1}}{2 {c_{1}}_{k}} (c_{2} - c_{1}) ∣ \\ \leq \frac{{(c_{2} - c_{1})}^{k}}{{(k!)}^{1 - \frac{1}{q}} ∣ {c_{1}}_{k} ∣} (\sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{{(j p + 1)}^{\frac{1}{p}}}) {(\frac{∣ {c_{1}}^{k} ∣^{q} + {∣ {c_{2}}^{k} ∣}^{q}}{1 + s})}^{\frac{1}{q}} . \end{matrix}

Proof.

The proof directly follows from Corollary 8 by taking

Ψ : [0, 1] \to [0, 1], Ψ (x) = x^{k}

and

R_{ρ, λ, σ} (c_{2} - c_{1}) = c_{2} - c_{1} .

□

Proposition 3.

Let

c_{1}, c_{2} \in R, 0 < c_{1} < c_{2}

, where

k

is an even number; then:

\begin{matrix} ∣ L_{k + 1}^{k + 1} (c_{1}, c_{2}) - A ({c_{1}}^{k}, {c_{2}}^{k}) + \dots \\ - \frac{(k (k - 1) (k - 2) {c_{1}}_{k} + \dots + 4.3.2 {c_{1}}_{4}) {(c_{2} - c_{1})}^{k - 4}}{2.4! {c_{1}}_{k}} [{c_{2}}^{4} + {c_{1}}^{4}] \\ + \frac{(k (k - 1) {c_{1}}_{k} + \dots + 3.2 {c_{1}}_{3}) {(c_{2} - c_{1})}^{k - 3}}{2.3! {c_{1}}_{k}} [{c_{2}}^{3} - {c_{1}}^{3}] \\ - \frac{(k {c_{1}}_{k} + \dots + 2 {c_{1}}_{2}) {(c_{2} - c_{1})}^{k - 2}}{2.2! {c_{1}}_{k}} [{c_{2}}^{2} + {c_{1}}^{2}] \\ + \frac{({c_{1}}_{k} + \dots + {c_{1}}_{1} + 2 {c_{1}}_{0}) {(c_{2} - c_{1})}^{k - 1}}{2 {c_{1}}_{k}} (c_{2} - c_{1}) ∣ \\ \leq \frac{R_{ρ, λ, σ}^{k} (c_{2} - c_{1})}{2 ∣ {c_{1}}_{k} ∣} {(\frac{1}{k!} \sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{(j + 1)})}^{1 - \frac{1}{q}} \\ \times [{(∣ {c_{1}}^{k} ∣^{q} \sum_{j = 0}^{k} B (j + 1, s + 1) + {∣ {c_{2}}^{k} ∣}^{q} \sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{(j + s + 1)})}^{\frac{1}{q}} \\ + {(∣ {c_{1}}^{k} ∣^{q} \sum_{j = 0}^{k} \frac{∣ {c_{1}}_{j} ∣}{(j + s + 1)} + {∣ {c_{2}}^{k} ∣}^{q} \sum_{j = 0}^{k} B (j + 1, s + 1))}^{\frac{1}{q}}] . \end{matrix}

Proof.

The proof directly follows from Corollary 13 by taking

Ψ : [0, 1] \to [0, 1], Ψ (x) = x^{k}

and

R_{ρ, λ, σ} (c_{2} - c_{1}) = c_{2} - c_{1} .

□

3. Conclusions

We have introduced the class of

R_{γ}

-convex functions involving Raina’s function. We have shown that by making suitable choices of the real function

γ (\cdot)

, we can recapture some other new classes of the classical convexity. This shows that our new class relates to several other unrelated classes of the convex functions. We then studied this class from the perspective of integral inequalities of the trapezium type. In order to establish our main results, we have derived a new integral identity involving

k

-th order differentiable functions. Finally, we have presented some applications to means that show the significance of our obtained results. We hope that the ideas and techniques of this paper will inspire interested readers working in this field. We would like to point out here that the results of this paper can be extended by using the class of higher-order

R_{γ}

-convex functions. This will be an interesting problem for future research.

Author Contributions

Writing to Original draft, M.V.-C., M.U.A., S.T., M.A.N. and K.I.N. All authors contributed equally to the writing of this paper. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Dirección de Investigación from Pontificia Universidad Católica del Ecuador in the research project entitled, “Some integrals inequalities and generalized convexity” (Algunas desigualdades integrales para funciones con algún tipo de convexidad generalizada y aplicaciones).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

No data were used to support this study.

Acknowledgments

The authors are thankful to the editor and the anonymous reviewers for their valuable comments and suggestions.

Conflicts of Interest

The authors declare that they have no competing interests.

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Vivas-Cortez, M.; Awan, M.U.; Talib, S.; Noor, M.A.; Noor, K.I. Trapezium-like Inequalities Involving k-th Order Differentiable $R_{γ}$ -Convex Functions and Applications. Symmetry 2022, 14, 448. https://doi.org/10.3390/sym14030448

AMA Style

Vivas-Cortez M, Awan MU, Talib S, Noor MA, Noor KI. Trapezium-like Inequalities Involving k-th Order Differentiable $R_{γ}$ -Convex Functions and Applications. Symmetry. 2022; 14(3):448. https://doi.org/10.3390/sym14030448

Chicago/Turabian Style

Vivas-Cortez, Miguel, Muhammad Uzair Awan, Sadia Talib, Muhammad Aslam Noor, and Khalida Inayat Noor. 2022. "Trapezium-like Inequalities Involving k-th Order Differentiable $R_{γ}$ -Convex Functions and Applications" Symmetry 14, no. 3: 448. https://doi.org/10.3390/sym14030448

APA Style

Vivas-Cortez, M., Awan, M. U., Talib, S., Noor, M. A., & Noor, K. I. (2022). Trapezium-like Inequalities Involving k-th Order Differentiable $R_{γ}$ -Convex Functions and Applications. Symmetry, 14(3), 448. https://doi.org/10.3390/sym14030448

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Trapezium-like Inequalities Involving k-th Order Differentiable $R_{γ}$ -Convex Functions and Applications

Abstract

1. Introduction and Preliminaries

2. Main Results

2.1. $R_{γ}$ -Convex Functions

2.2. Auxiliary Result

2.3. Trapezium-like Inequalities

2.4. Applications

3. Conclusions

Author Contributions

Funding

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Trapezium-like Inequalities Involving k-th Order Differentiable Rγ-Convex Functions and Applications

Abstract

1. Introduction and Preliminaries

2. Main Results

2.1. R γ -Convex Functions

2.2. Auxiliary Result

2.3. Trapezium-like Inequalities

2.4. Applications

3. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

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Acknowledgments

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Trapezium-like Inequalities Involving k-th Order Differentiable $R_{γ}$ -Convex Functions and Applications

2.1. $R_{γ}$ -Convex Functions