Mathematical Methods in the Applied Sciences
Volume 48, Issue 12 pp. 12567-12576
RESEARCH ARTICLE
Open Access

Refinements of the Jensen Inequality and Estimates of the Jensen Gap Based on Interval-Valued Functions

İzzettin Demir, 

Corresponding Author

İzzettin Demir

Department of Mathematics, Faculty of Science and Arts, Duzce University, Düzce, Türkİye

Correspondence:

İzzettin Demir ([email protected])

Contribution: Conceptualization, ​Investigation, Writing - original draft, Methodology, Validation, Visualization, Writing - review & editing, Formal analysis, Supervision, Resources

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First published: 12 May 2025
https://doi.org/10.1002/mma.11046

Funding: The author received no specific funding for this work.

ABSTRACT

The significance of the Jensen inequality stems from its impactful and compelling outcomes. As a generalization of classical convexity, it plays a key role in deriving other well-known inequalities such as Hermite–Hadamard, Hölder, Minkowski, arithmetic-geometric, and Young's inequalities. So, this inequality has become an influential concept in a wide range of scientific fields. Besides, interval analysis provides methods for managing uncertainty in data, making it possible to build mathematical and computer models of various deterministic real-world phenomena. In this paper, taking into account all of these, we first present several refinements of the Jensen inequality for the left and right convex interval-valued functions. We also provide examples with corresponding graphs to demonstrate these refinements more clearly. Next, we adopt a novel approach to derive several bounds for the Jensen gap in integral form using the gH-differentiable interval valued functions as well as various related notions. Moreover, we obtain the proposed bounds by utilizing the renowned Ostrowski inequality. The fundamental benefit of the newly discovered inequalities is that they extend to many known inequalities in the literature, as discussed in this work.

1 Introduction

The theory of convex functions, a key area of mathematics, has broad applications in fields such as optimization theory, control theory, energy systems, information theory, and physics. Moreover, convexity theory and its practical uses have provided solutions to a wide range of mathematical problems. In particular, the discovery of many basic inequalities from convex functions has accelerated the growth of this field. Besides, the application of integral inequalities along with the idea of convexity has been extensively used to achieve many new results in the theory of inequalities. The first essential outcome for a convex function is known as the Hermite–Hadamard inequality, which was investigated by C. Hermite and J. Hadamard [1, 2] stated as the follows:
f a + b 2 ≤ 1 b − a ∫ a b f ( x ) d x ≤ f ( a ) + f ( b ) 2 $$ f\left(\frac{a+b}{2}\right)\le \frac{1}{b-a}\int_a^bf(x) dx\le \frac{f(a)+f(b)}{2} $$ (1)
where f : I → ℝ $$ f:I\to \mathbb{R} $$ is a convex function on the interval I $$ I $$ of real numbers and a , b ∈ I $$ a,b\in I $$ with a < b $$ a&lt;b $$ . If f $$ f $$  is concave, then both inequalities in the statement hold in the reverse direction.

The inequality, which provides bounds for the average value of a convex function over a compact interval, is widely used in integral calculus, probability theory, statistics, optimization, and number theory, and forms the basis for a wide range of other inequalities. Additionally, it is used to solve real-world problems in physics, engineering, economics, and other disciplines. With the emergence of new problems, the use of this inequality continues to expand, establishing it as a vital tool for resolving not only complex mathematical problems but also issues in various other disciplines. On the other hand, the Hermite–Hadamard inequality is identified by the trapezoid inequality on its right part and the midpoint inequality on its left part. Trapezoid-type inequalities for the case of the convex functions were first established by Dragomir and Agarwal in [3], whereas midpoint-type inequalities for the case of the convex functions were first proved by Kırmacı in [4]. Since these inequalities have been emerged, a lot of research has been done in this area [5-8].

A multivariable extension of the convexity property is the renowned Jensen inequality, which states that if f : J ⊆ ℝ → ℝ $$ f:J\subseteq \mathbb{R}\to \mathbb{R} $$ is a convex function and a 1 , a 2 , … , a n ∈ J $$ {a}_1,{a}_2,\dots, {a}_n\in J $$ , then the following inequality holds:
f ∑ i = 1 n λ i a i ≤ ∑ i = 1 n λ i f ( a i ) , $$ f\left(\sum \limits_{i&#x0003D;1}&#x0005E;n{\lambda}_i{a}_i\right)\le \sum \limits_{i&#x0003D;1}&#x0005E;n{\lambda}_if\left({a}_i\right), $$
where λ 1 , λ 2 , … , λ n > 0 $$ {\lambda}_1,{\lambda}_2,\dots, {\lambda}_n&gt;0 $$ with ∑ i = 1 n λ i = 1 $$ \sum \limits_{i&#x0003D;1}&#x0005E;n{\lambda}_i&#x0003D;1 $$ .

Jensen inequality, well-known in classical analysis, has been influential in fields like probability theory, information theory, mathematical economics, and control theory. The problem of generalizing Jensen inequalities is a significant topic in today's mathematical research. This inequality is also a fundamental result in the study of convex functions and has various refinements and extensions. These refinements have offered not only a deeper understanding of convexity but also opened up new applications in various fields. For instance, Abramovich et al. [9] defined refined versions of Jensen inequality that provide more precise bounds compared to the conventional approach. In [10], Horvath established refined bounds for the integral form of Jensen inequality. Then, Khan et al. [11] offered bounds for the identities regarding the generalization of Jensen inequality with the aid of Cebysev functionals and also obtained Ostrowski types inequalities for these functionals. Next, Sarıkaya [12] proposed a new refinement of the Jensen inequality for convex functions, leading to the derivation of several inequalities, with special attention to Bullen's inequality.

In [13] Bullen proved an alternative Hermite–Hadamard type inequality, known as Bullen's inequality: If f : I → ℝ $$ f:I\to \mathbb{R} $$ is a convex function on the interval I $$ I $$ of real numbers and a , b ∈ I $$ a,b\in I $$ with a < b $$ a&lt;b $$ , then
1 b − a ∫ a b f ( x ) d x ≤ 1 2 f a + b 2 + f ( a ) + f ( b ) 2 $$ \frac{1}{b-a}\int_a&#x0005E;bf(x) dx\le \frac{1}{2}\left[f\left(\frac{a&#x0002B;b}{2}\right)&#x0002B;\frac{f(a)&#x0002B;f(b)}{2}\right] $$ (2)

The most important objective of these kinds of inequalities is that they provide more accurate and stronger information about error estimates of the widely examined quadrature and cubature rules. The inequality (2) also gives the error bounds for the remainder of Bullen quadrature schemes. So, researchers have focused their studies on these types of inequalities. For instance, Çakmak [14] obtained some new inequalities for differentiable functions based on h $$ h $$ -convex functions including Bullen-type inequalities. In [15], İşcan et al. established a new general identity for differentiable functions and therefore obtained some new general inequalities containing all of the Hermite–Hadamard and Bullen type for functions whose derivatives in absolute value at certain power are convex. Tseng et al. [16]  studied some Hadamard-type and Bullen-type inequalities with help of Lipschitz functions and established some applications by utilizing the special means. Afterwards, Cortez et al. [17] established a new set of Bullen-type inequalities concerning the Jensen–Mecer inequality.

On the other hand, interval analysis, also referred to as interval arithmetic or interval computation, is a valuable numerical technique for reliably solving uncertain or nonlinear problems. As a particular case of set-valued analysis [18-20], interval analysis deals with the study of intervals in mathematical analysis. While its roots go back to Archimedes' work on the circumference of a circle, the formal exploration of this method started in the 1920s. Based on our knowledge, significant progress in this domain was not made until the 1950s. In 1966, the first book about interval analysis was presented by Moore [21], and this analysis was initially employed to determine the error bounds in the numerical solutions of a finite state machine. Moreover, this approach was developed to tackle interval uncertainty that occurs in various mathematical and computer models of absolute events in the real world. Following the release of his book, many researchers started investigating both the theoretical and application of interval computation. Nowadays, as a powerful tool for handling uncertain data, this method finds applications in a range of fields, including computer graphics, computational physics, error analysis, robotics and a variety of other well-known scientific and technology domains [22-24].

Following that, many authors have uncovered a strong connection between the inequalities and the interval-valued functions by utilizing inclusion relations through various integral operators. In [25, 26], Chalco-Cano et al. established Ostrowski type inequalities for interval-valued functions via generalized Hukuhara derivative which was given by Stefanini and Bede [27]. Then, Zhao et al. [28]  offered new Jensen and type inequalities for interval-valued functions by means of the h $$ h $$ -convexity. Also, Roman Flores et al. [29] established Minkowski and Beckenbach's inequalities for interval-valued functions. Next, Zhang et al. [30] developed a new version of Jensen inequalities for set-valued and fuzzy set-valued functions using a pseudo order relation. They demonstrated that these generalized Jensen inequalities extend a form of Costa's Jensen inequalities shown in [31]. Khan et al. [32] proposed new forms of Hermite–Hadamard type inequalities for fuzzy interval-valued functions, proving their effectiveness with the aid of non-trivial examples. For more information, see [33, 34] and the references therein.

Inspired by these works we initially put forward some refinements for the Jensen inequality along with the left and right convex interval-valued functions. To better demonstrate the newly discovered refinements, we give several examples accompanied by their related graphs. Then, we adopt an innovative approach to present various bounds for the Jensen gap, incorporating gH-differentiable interval-valued functions and some related notions. Moreover, we achieve the proposed bounds by employing the well-known Ostrowski inequality. Finally, the inequalities given in this study expand various known inequalities in the literature, which is the main advantage of the newly found inequalities.

2 Preliminaries

In this section, we give an overview of interval-valued functions, the theory of convexity, and interval-valued integration, all of which will be used in this article. Let ℝ 𝒞 be the collection of all closed intervals of ℝ $$ \mathbb{R} $$ , that is, ℝ 𝒞 = { α ∗ , α ∗ : α ∗ , α ∗ ∈ ℝ and α ∗ ≤ α ∗ } $$ {\alpha}_{\ast}\le {\alpha}&#x0005E;{\ast}\Big\} $$ .  If   α ∗ > 0 $$ {\alpha}_{\ast }&gt;0 $$ , then α ∗ , α ∗ $$ \left[{\alpha}_{\ast },{\alpha}&#x0005E;{\ast}\right] $$ is called a positive closed interval. Consider the set of all positive closed intervals of ℝ $$ \mathbb{R} $$ as ℝ 𝒞 + .

Firstly, we offer essential arithmetic operations concerning interval analysis. For α = [ α ∗ , α ∗ ] , β = [ β ∗ , β ∗ ] ∈ ℝ 𝒞 and w ∈ ℝ $$ w\in \mathbb{R} $$ , arithmetic operations are defined by
α + β = α ∗ + β ∗ , α ∗ + β ∗ , w . α = w α ∗ , w α ∗ if w > 0 { 0 } , if w = 0 w α ∗ , w α ∗ , if w < 0 α − β = α ∗ − β ∗ , α ∗ − β ∗ $$ {\displaystyle \begin{array}{cc}\hfill \alpha &#x0002B;\beta &amp; &#x0003D;\left[{\alpha}_{\ast }&#x0002B;{\beta}_{\ast },{\alpha}&#x0005E;{\ast }&#x0002B;{\beta}&#x0005E;{\ast}\right],\hfill \\ {}\hfill w.\alpha &amp; &#x0003D;\left\{\begin{array}{cc}\left[w{\alpha}_{\ast },w{\alpha}&#x0005E;{\ast}\right]&amp; \mathrm{if}\kern0.3em w&gt;0\\ {}\left\{0\right\},&amp; \mathrm{if}\kern0.3em w&#x0003D;0\\ {}\left[w{\alpha}&#x0005E;{\ast },w{\alpha}_{\ast}\right],&amp; \mathrm{if}\kern0.3em w&lt;0\end{array}\right.\hfill \\ {}\hfill \alpha -\beta &amp; &#x0003D;\left[{\alpha}_{\ast }-{\beta}&#x0005E;{\ast },{\alpha}&#x0005E;{\ast }-{\beta}_{\ast}\right]\hfill \end{array}} $$
and
α . β = min α ∗ β ∗ , α ∗ β ∗ , α ∗ β ∗ , α ∗ β ∗ , max α ∗ β ∗ , α ∗ β ∗ , α ∗ β ∗ , α ∗ β ∗ . $$ {\displaystyle \begin{array}{cc}\hfill \alpha .\beta &amp; &#x0003D;\left[\min \left\{{\alpha}_{\ast }{\beta}_{\ast },{\alpha}_{\ast }{\beta}&#x0005E;{\ast },{\alpha}&#x0005E;{\ast }{\beta}_{\ast },{\alpha}&#x0005E;{\ast }{\beta}&#x0005E;{\ast}\right\},\right.\hfill \\ {}\hfill &amp; \max \left.\left\{{\alpha}_{\ast }{\beta}_{\ast },{\alpha}_{\ast }{\beta}&#x0005E;{\ast },{\alpha}&#x0005E;{\ast }{\beta}_{\ast },{\alpha}&#x0005E;{\ast }{\beta}&#x0005E;{\ast}\right\}\right].\hfill \end{array}} $$
Moreover, the Pompeiu–Hausdorff metric or distance between the intervals α $$ \alpha $$ and β $$ \beta $$ is defined by
H α , β = max α ∗ − β ∗ , α ∗ − β ∗ . $$ H\left(\alpha, \beta \right)&#x0003D;\max \left\{\left&#x0007C;{\alpha}_{\ast }-{\beta}_{\ast}\right&#x0007C;,\left&#x0007C;{\alpha}&#x0005E;{\ast }-{\beta}&#x0005E;{\ast}\right&#x0007C;\right\}. $$
It can be readily observed that ℝ 𝒞 , H is a complete metric space [35].

Let F : [ a , b ] → ℝ 𝒞 be an interval-valued function such that F ( t ) = F ∗ ( t ) , F ∗ ( t ) $$ F(t)&#x0003D;\left[{F}_{\ast }(t),{F}&#x0005E;{\ast }(t)\right] $$ for all t ∈ [ a , b ] $$ t\in \left[a,b\right] $$ . The functions F ∗ $$ {F}_{\ast } $$ and F ∗ $$ {F}&#x0005E;{\ast } $$ are called the lower and the upper (endpoint) functions of F $$ F $$ , respectively.

For the interval-valued functions, it is obvious that F : [ a , b ] → ℝ 𝒞 is continuous at t 0 ∈ [ a , b ] $$ {t}_0\in \left[a,b\right] $$ if lim t → t 0 F ( t ) = F ( t 0 ) $$ {\lim}_{t\to {t}_0}F(t)&#x0003D;F\left({t}_0\right) $$ where the limit is taken in the metric space ℝ 𝒞 , H . So, F $$ F $$ is continuous at t 0 ∈ [ a , b ] $$ {t}_0\in \left[a,b\right] $$ if and only if its endpoint functions F ∗ $$ {F}_{\ast } $$ and F ∗ $$ {F}&#x0005E;{\ast } $$ are continuous at t 0 ∈ [ a , b ] $$ {t}_0\in \left[a,b\right] $$ . The family of all continuous interval-valued functions is denoted by C ( [ a , b ] , ℝ 𝒞 ) . For the quasilinear space C ( [ a , b ] , ℝ 𝒞 ) , a quasinorm . ∞ $$ {\left&#x0007C;\left&#x0007C;.\right&#x0007C;\right&#x0007C;}_{\infty } $$ is given by F ∞ = sup t ∈ [ a , b ] H ( F ( t ) , { 0 } ) $$ {\left&#x0007C;\left&#x0007C;F\right&#x0007C;\right&#x0007C;}_{\infty }&#x0003D;\underset{t\in \left[a,b\right]}{\sup }H\left(F(t),\left\{0\right\}\right) $$ . For further insights on quasilinear spaces and quasinorms, see [36].

Remark 1. ([[30]])

  1. For all α , β ∈ ℝ 𝒞 , the relation “ ≤ p $$ {\le}_p $$ ” on ℝ 𝒞 is defined by

    α ≤ p β ⇔ α ∗ ≤ β ∗ and α ∗ ≤ β ∗ , $$ \alpha {\le}_p\beta \iff {\alpha}_{\ast}\le {\beta}_{\ast}\kern0.3em \mathrm{and}\kern0.3em {\alpha}&#x0005E;{\ast}\le {\beta}&#x0005E;{\ast }, $$
    and it is a pseudo order relation. In the interval analysis case, the pseudo order relation ( ≤ p $$ {\le}_p $$ ) coincidents to partial order relation ( ≤ $$ \le $$ ).

  2. It is evident that ( ≤ p $$ {\le}_p $$ ) looks similar to “left and right” on the real line ℝ $$ \mathbb{R} $$ and therefore, we refer to “ ≤ p $$ {\le}_p $$ ”  as the “left and right”  or “LR” order for short.

A partition of [ a , b ] $$ \left[a,b\right] $$ is any finite ordered subset P $$ P $$ having the form
P : a = κ 0 < κ 1 < . . . < κ n = b . $$ P:a&#x0003D;{\kappa}_0&lt;{\kappa}_1&lt;\dots &lt;{\kappa}_n&#x0003D;b. $$
The mesh of a partition P $$ P $$ is defined by
m e s h ( P ) = max { κ i − κ i − 1 : i = 1 , 2 , . . . , n } . $$ mesh(P)&#x0003D;\max \left\{{\kappa}_i-{\kappa}_{i-1}:i&#x0003D;1,2,\dots, n\right\}. $$
We denote by 𝒫 ( [ a , b ] ) the set of all partitions of [ a , b ] $$ \left[a,b\right] $$ . Let 𝒫 ( δ , [ a , b ] ) be the set of all P ∈ 𝒫 ( [ a , b ] ) such that m e s h ( P ) < δ $$ mesh(P)&lt;\delta $$ . Choosing an arbitrary point ξ i $$ {\xi}_i $$ in [ κ i − 1 , κ i ] $$ \left[{\kappa}_{i-1},{\kappa}_i\right] $$ for all i = 1 , 2 , … , n $$ i&#x0003D;1,2,\dots, n $$ , we define the sum
S ( F , P , δ ) = ∑ i = 1 n F ( ξ i ) ( κ i − κ i − 1 ) , $$ S\left(F,P,\delta \right)&#x0003D;\sum \limits_{i&#x0003D;1}&#x0005E;nF\left({\xi}_i\right)\left({\kappa}_i-{\kappa}_{i-1}\right), $$
where F : [ a , b ] → ℝ 𝒞 . We call S ( F , P , δ ) $$ S\left(F,P,\delta \right) $$ a Riemann sum of F $$ F $$ corresponding to P ∈ 𝒫 ( δ , [ a , b ] ) . [37]

Definition 1. ([[38, 39]])A function F : [ a , b ] → ℝ 𝒞 is called interval Riemann integrable on [ a , b ] $$ \left[a,b\right] $$ if there exists an A ∈ ℝ 𝒞 such that, for each ϵ > 0 $$ \epsilon &gt;0 $$ , there exists a δ > 0 $$ \delta &gt;0 $$ , where

H ( S ( F , P , δ ) , A ) < ϵ $$ H\left(S\left(F,P,\delta \right),A\right)&lt;\epsilon $$
for every Riemann sum S $$ S $$ of F $$ F $$ corresponding to each P ∈ 𝒫 ( δ , [ a , b ] ) and independent of choice ξ i ∈ [ κ i − 1 , κ i ] $$ {\xi}_i\in \left[{\kappa}_{i-1},{\kappa}_i\right] $$ for 1 ≤ i ≤ n $$ 1\le i\le n $$ . In this case, A $$ A $$ is called the interval Riemann integral of F $$ F $$ on [ a , b ] $$ \left[a,b\right] $$ and is denoted by
A = ∫ a b F ( t ) d t $$ A&#x0003D;\int_a&#x0005E;bF(t) dt $$

The following theorem gives a relation between ℛ a , b $$ {\mathcal{R}}_{\left[a,b\right]} $$ and ℐ ℛ [ a , b ] $$ \mathcal{I}{\mathcal{R}}_{\left[a,b\right]} $$ , where ℛ a , b $$ {\mathcal{R}}_{\left[a,b\right]} $$ and ℐ ℛ [ a , b ] $$ \mathcal{I}{\mathcal{R}}_{\left[a,b\right]} $$ are the collections of all Riemann integrable real valued functions and all Riemann integrable interval-valued functions on [ a , b ] $$ \left[a,b\right] $$ , respectively ([40], p. 131):

Theorem 1. F ∈ ℐ ℛ [ a , b ] $$ F\in \mathcal{I}{\mathcal{R}}_{\left[a,b\right]} $$ if and only if F ∗ $$ {F}_{\ast } $$ , F ∗ ∈ ℛ [ a , b ] $$ {F}&#x0005E;{\ast}\in {\mathcal{R}}_{\left[a,b\right]} $$ and

∫ a b F ( t ) d t = ∫ a b F ∗ ( t ) d t , ∫ a b F ∗ ( t ) d t . $$ \int_a&#x0005E;bF(t) dt&#x0003D;\left[\int_a&#x0005E;b{F}_{\ast }(t) dt,\int_a&#x0005E;b{F}&#x0005E;{\ast }(t) dt\right]. $$

It is easily seen that if F ( t ) ≤ p G ( t ) $$ F(t){\le}_pG(t) $$ for all t ∈ [ a , b ] $$ t\in \left[a,b\right] $$ , then ∫ a b F ( t ) d t ≤ p ∫ a b G ( t ) d t $$ \int_a&#x0005E;bF(t) dt{\le}_p\int_a&#x0005E;bG(t) dt $$ .

A fundamental step in establishing a workable definition of the derivative for interval-valued functions is to determine a suitable subtraction operation between intervals. So, Stefanini and Bede [27] offered the following definition:

Definition 2.Let α , β ∈ ℝ 𝒞 . The generalized Hukuhara difference (gH-difference, for short) between α $$ \alpha $$ and β $$ \beta $$ is defined by

α ⊖ g h β = γ ⇔ α = β + γ or β = α + ( − 1 ) γ . $$ \alpha {\ominus}_{gh}\beta &#x0003D;\gamma \iff \left\{\begin{array}{l}\alpha &#x0003D;\beta &#x0002B;\gamma \\ {}\mathrm{or}\\ {}\beta &#x0003D;\alpha &#x0002B;\left(-1\right)\gamma .\end{array}\right. $$

Next, Stefanini and Bede [27] obtained the concept of a generalized Hukuhara differentiable interval-valued function:

Definition 3.The generalized Hukuhara derivative of an interval-valued function F : a , b → ℝ 𝒞 at t 0 ∈ [ a , b ] $$ {t}_0\in \left[a,b\right] $$ is defined as

F ′ ( t 0 ) = lim h → 0 F ( t 0 + h ) ⊖ g h F ( t 0 ) h $$ {F}&#x0005E;{\prime}\left({t}_0\right)&#x0003D;\underset{h\to 0}{\lim}\frac{F\left({t}_0&#x0002B;h\right){\ominus}_{gh}F\left({t}_0\right)}{h} $$ (3)

If F ′ ( t 0 ) ∈ $$ {F}&#x0005E;{\prime}\left({t}_0\right)\in $$ ℝ 𝒞 satisfying (3) exists, we say that F $$ F $$ is a generalized Hukuhara differentiable (gH-differentiable) at t 0 ∈ [ a , b ] $$ {t}_0\in \left[a,b\right] $$ .

Afterwards, Chalco-Cano et al. [25] provided the interval version of the mean value theorem using the gH-derivative, which will be crucial for deriving our main results.

Theorem 2.Let F : [ a , b ] → ℝ 𝒞 be a continuously gH-differentiable interval-valued function on [ a , b ] $$ \left[a,b\right] $$ with a finite number of switching points at precisely a = c 0 < c 1 < c 2 < . . . < c n < c n + 1 = b $$ a&#x0003D;{c}_0&lt;{c}_1&lt;{c}_2&lt;\dots &lt;{c}_n&lt;{c}_{n&#x0002B;1}&#x0003D;b $$ . Then, we have

H ( F ( b ) , F ( a ) ) ≤ F ′ ∞ ( b − a ) . $$ H\left(F(b),F(a)\right)\le {\left&#x0007C;\left&#x0007C;{F}&#x0005E;{\prime}\right&#x0007C;\right&#x0007C;}_{\infty}\left(b-a\right). $$

In [30], Zhang et al. introduced a kind of convex interval-valued function as follows:

Definition 4.The interval valued function F : [ a , b ] → ℝ 𝒞 + is named as L R $$ LR $$ -convex interval-valued function  [ a , b ] $$ \left[a,b\right] $$ if for all x , y ∈ [ a , b ] $$ x,y\in \left[a,b\right] $$ and t ∈ [ 0 , 1 ] $$ t\in \left[0,1\right] $$ , the inequality

F ( t x + ( 1 − t ) y ) ≤ p t F ( x ) + ( 1 − t ) F ( y ) $$ F\left( tx&#x0002B;\left(1-t\right)y\right){\le}_p tF(x)&#x0002B;\left(1-t\right)F(y) $$ (4)
is valid. If the inequality (4) is reversed, then F $$ F $$ is named as L R $$ LR $$ -concave interval-valued function on [ a , b ] $$ \left[a,b\right] $$ .

Theorem 3. ([[30]]) F : [ a , b ] → ℝ 𝒞 + be an interval-valued function such that F ( t ) = [ F ∗ ( t ) , F ∗ ( t ) ] $$ F(t)&#x0003D;\left[{F}_{\ast }(t),{F}&#x0005E;{\ast }(t)\right] $$ for all t ∈ [ a , b ] $$ t\in \left[a,b\right] $$ . Then, F $$ F $$ is an L R $$ LR $$ -convex interval-valued function on [ a , b ] $$ \left[a,b\right] $$ if and only if both the functions F ∗ $$ {F}_{\ast } $$ and F ∗ $$ {F}&#x0005E;{\ast } $$ are convex functions.

Remark 2. ([[30]])According to 3, it is apparent that if F ∗ ( t ) = F ∗ ( t ) $$ {F}_{\ast }(t)&#x0003D;{F}&#x0005E;{\ast }(t) $$ , then the L R $$ LR $$ -convex interval-valued function turns into classical convex function.

3 Refinements of the Jensen Inequality

In this section, we bring out some refinements of Jensen inequality by using L R $$ LR $$ -convex interval-valued functions. Also, we present many examples accompanied by the graphs to offer a clearer demonstration of these refinements.

Theorem 4.Let F : [ a , b ] → ℝ 𝒞 be an LR-convex interval-valued function on [ a , b ] $$ \left[a,b\right] $$ and g : [ a , b ] → ℝ $$ g:\left[a,b\right]\to \mathbb{R} $$ be a continuous function on [ a , b ] $$ \left[a,b\right] $$ with a < b $$ a&lt;b $$ . Then, the following inequalities hold:

F 1 b − a ∫ a b g ( t ) d t ≤ p x − a b − a F 1 x − a ∫ a x g ( t ) d t + b − x b − a F 1 b − x ∫ x b g ( t ) d t ≤ p 1 b − a ∫ a b F g ( t ) d t ≤ p F ( x ) − x F ( b ) − F ( a ) b − a + F ( x ) − F ( a ) ( x − a ) ( b − a ) ∫ a x g ( t ) d t + F ( b ) − F ( x ) ( b − x ) ( b − a ) ∫ x b g ( t ) d t , $$ {\displaystyle \begin{array}{cc}\hfill &amp; F\left(\frac{1}{b-a}\int_a&#x0005E;bg(t) dt\right)\hfill \\ {}\hfill {\le}_p&amp; \kern0.2em \frac{x-a}{b-a}F\left(\frac{1}{x-a}\int_a&#x0005E;xg(t) dt\right)&#x0002B;\frac{b-x}{b-a}F\left(\frac{1}{b-x}\int_x&#x0005E;bg(t) dt\right)\hfill \\ {}\hfill {\le}_p&amp; \kern0.2em \frac{1}{b-a}\int_a&#x0005E;bF\left(g(t)\right) dt\hfill \\ {}\hfill {\le}_p&amp; \kern0.2em F(x)-x\frac{F(b)-F(a)}{b-a}&#x0002B;\frac{F(x)-F(a)}{\left(x-a\right)\left(b-a\right)}\int_a&#x0005E;xg(t) dt\hfill \\ {}\hfill &amp; &#x0002B;\frac{F(b)-F(x)}{\left(b-x\right)\left(b-a\right)}\int_x&#x0005E;bg(t) dt,\hfill \end{array}} $$ (5)
where x ∈ ( a , b ) $$ x\in \left(a,b\right) $$ .

Proof.By Theorem 3, since F $$ F $$ is an LR-convex interval-valued function on [ a , b ] $$ \left[a,b\right] $$ , we obtain that both F ∗ $$ {F}_{\ast } $$ and F ∗ $$ {F}&#x0005E;{\ast } $$ are convex functions. Considering the convexity of F ∗ $$ {F}_{\ast } $$ , this implies that

F ∗ 1 b − a ∫ a b g ( t ) d t = F ∗ x − a b − a 1 x − a ∫ a x g ( t ) d t + b − x b − a 1 b − x ∫ x b g ( t ) d t ≤ x − a b − a F ∗ 1 x − a ∫ a x g ( t ) d t + b − x b − a F ∗ 1 b − x ∫ x b g ( t ) d t ≤ x − a b − a 1 x − a ∫ a x F ∗ g ( t ) d t + b − x b − a 1 b − x ∫ x b F ∗ g ( t ) d t = 1 b − a ∫ a b F ∗ g ( t ) d t . $$ {\displaystyle \begin{array}{cc}\hfill &amp; {F}_{\ast}\left(\frac{1}{b-a}\int_a&#x0005E;bg(t) dt\right)\hfill \\ {}\hfill &#x0003D;&amp; \kern0.2em {F}_{\ast}\left(\frac{x-a}{b-a}\frac{1}{x-a}\int_a&#x0005E;xg(t) dt&#x0002B;\frac{b-x}{b-a}\frac{1}{b-x}\int_x&#x0005E;bg(t) dt\right)\hfill \\ {}\hfill \le &amp; \kern0.2em \frac{x-a}{b-a}{F}_{\ast}\left(\frac{1}{x-a}\int_a&#x0005E;xg(t) dt\right)&#x0002B;\frac{b-x}{b-a}{F}_{\ast}\left(\frac{1}{b-x}\int_x&#x0005E;bg(t) dt\right)\hfill \\ {}\hfill \le &amp; \kern0.2em \frac{x-a}{b-a}\frac{1}{x-a}\int_a&#x0005E;x{F}_{\ast}\left(g(t)\right) dt&#x0002B;\frac{b-x}{b-a}\frac{1}{b-x}\int_x&#x0005E;b{F}_{\ast}\left(g(t)\right) dt\hfill \\ {}\hfill &#x0003D;&amp; \kern0.2em \frac{1}{b-a}\int_a&#x0005E;b{F}_{\ast}\left(g(t)\right) dt.\hfill \end{array}} $$ (6)

Also, the fact that F ∗ $$ {F}_{\ast } $$ is convex results in

1 b − a ∫ a b F ∗ g ( t ) d t = 1 b − a ∫ a x F ∗ x − g ( t ) x − a a + g ( t ) − a x − a x d t + 1 b − a ∫ x b F ∗ b − g ( t ) b − x x + g ( t ) − x b − x b d t $$ {\displaystyle \begin{array}{cc}\hfill &amp; \frac{1}{b-a}\int_a&#x0005E;b{F}_{\ast}\left(g(t)\right) dt\hfill \\ {}\hfill &#x0003D;&amp; \kern0.2em \frac{1}{b-a}\int_a&#x0005E;x{F}_{\ast}\left(\frac{x-g(t)}{x-a}a&#x0002B;\frac{g(t)-a}{x-a}x\right) dt\hfill \\ {}\hfill &amp; &#x0002B;\frac{1}{b-a}\int_x&#x0005E;b{F}_{\ast}\left(\frac{b-g(t)}{b-x}x&#x0002B;\frac{g(t)-x}{b-x}b\right) dt\hfill \end{array}} $$
≤ 1 b − a ∫ a x x − g ( t ) x − a F ∗ ( a ) + g ( t ) − a x − a F ∗ ( x ) d t + 1 b − a ∫ x b b − g ( t ) b − x F ∗ ( x ) + g ( t ) − x b − x F ∗ ( b ) d t = x F ∗ ( a ) − a F ∗ ( x ) b − a + F ∗ ( x ) − F ∗ ( a ) ( x − a ) ( b − a ) ∫ a x g ( t ) d t + b F ∗ ( x ) − x F ∗ ( b ) b − a + F ∗ ( b ) − F ∗ ( x ) ( b − x ) ( b − a ) ∫ x b g ( t ) d t = F ∗ ( x ) − x F ∗ ( b ) − F ∗ ( a ) b − a + F ∗ ( x ) − F ∗ ( a ) ( x − a ) ( b − a ) ∫ a x g ( t ) d t + F ∗ ( b ) − F ∗ ( x ) ( b − x ) ( b − a ) ∫ x b g ( t ) d t . $$ {\displaystyle \begin{array}{cc}\hfill \le &amp; \kern0.2em \frac{1}{b-a}\int_a&#x0005E;x\left[\frac{x-g(t)}{x-a}{F}_{\ast }(a)&#x0002B;\frac{g(t)-a}{x-a}{F}_{\ast }(x)\right] dt\hfill \\ {}\hfill &amp; &#x0002B;\frac{1}{b-a}\int_x&#x0005E;b\left[\frac{b-g(t)}{b-x}{F}_{\ast }(x)&#x0002B;\frac{g(t)-x}{b-x}{F}_{\ast }(b)\right] dt\hfill \\ {}\hfill &#x0003D;&amp; \kern0.2em \frac{x{F}_{\ast }(a)-a{F}_{\ast }(x)}{b-a}&#x0002B;\frac{F_{\ast }(x)-{F}_{\ast }(a)}{\left(x-a\right)\left(b-a\right)}\int_a&#x0005E;xg(t) dt\hfill \\ {}\hfill &amp; &#x0002B;\frac{b{F}_{\ast }(x)-x{F}_{\ast }(b)}{b-a}&#x0002B;\frac{F_{\ast }(b)-{F}_{\ast }(x)}{\left(b-x\right)\left(b-a\right)}\int_x&#x0005E;bg(t) dt\hfill \\ {}\hfill &#x0003D;&amp; \kern0.2em {F}_{\ast }(x)-x\frac{F_{\ast }(b)-{F}_{\ast }(a)}{b-a}&#x0002B;\frac{F_{\ast }(x)-{F}_{\ast }(a)}{\left(x-a\right)\left(b-a\right)}\int_a&#x0005E;xg(t) dt\hfill \\ {}\hfill &amp; &#x0002B;\frac{F_{\ast }(b)-{F}_{\ast }(x)}{\left(b-x\right)\left(b-a\right)}\int_x&#x0005E;bg(t) dt.\hfill \end{array}} $$ (7)

If we analyze the convexity of F ∗ $$ {F}&#x0005E;{\ast } $$ in a similar manner as above, we again arrive at

F ∗ 1 b − a ∫ a b g ( t ) d t ≤ x − a b − a F ∗ 1 x − a ∫ a x g ( t ) d t + b − x b − a F ∗ 1 b − x ∫ x b g ( t ) d t = 1 b − a ∫ a b F ∗ g ( t ) d t ≤ F ∗ ( x ) − x F ∗ ( b ) − F ∗ ( a ) b − a + F ∗ ( x ) − F ∗ ( a ) ( x − a ) ( b − a ) ∫ a x g ( t ) d t + F ∗ ( b ) − F ∗ ( x ) ( b − x ) ( b − a ) ∫ x b g ( t ) d t . $$ {\displaystyle \begin{array}{cc}\hfill &amp; {F}&#x0005E;{\ast}\left(\frac{1}{b-a}\int_a&#x0005E;bg(t) dt\right)\hfill \\ {}\hfill \le &amp; \kern0.2em \frac{x-a}{b-a}{F}&#x0005E;{\ast}\left(\frac{1}{x-a}\int_a&#x0005E;xg(t) dt\right)&#x0002B;\frac{b-x}{b-a}{F}&#x0005E;{\ast}\left(\frac{1}{b-x}\int_x&#x0005E;bg(t) dt\right)\hfill \\ {}\hfill &#x0003D;&amp; \kern0.2em \frac{1}{b-a}\int_a&#x0005E;b{F}&#x0005E;{\ast}\left(g(t)\right) dt\hfill \\ {}\hfill \le &amp; \kern0.2em {F}&#x0005E;{\ast }(x)-x\frac{F&#x0005E;{\ast }(b)-{F}&#x0005E;{\ast }(a)}{b-a}&#x0002B;\frac{F&#x0005E;{\ast }(x)-{F}&#x0005E;{\ast }(a)}{\left(x-a\right)\left(b-a\right)}\int_a&#x0005E;xg(t) dt\hfill \\ {}\hfill &amp; &#x0002B;\frac{F&#x0005E;{\ast }(b)-{F}&#x0005E;{\ast }(x)}{\left(b-x\right)\left(b-a\right)}\int_x&#x0005E;bg(t) dt.\hfill \end{array}} $$ (8)

Thus, by employing property of L R $$ LR $$ -order along with the inequalities (6), (7), and (8), we derive the intended inequality (5).

Remark 3.If F ∗ = F ∗ $$ {F}&#x0005E;{\ast }&#x0003D;{F}_{\ast } $$ , then the above theorem reduces to Theorem 1 in [12].

Corollary 1.Under the assumptions of Theorem 4, we have

F a + b 2 ≤ p x − a b − a F x + a 2 + b − x b − a F x + b 2 ≤ p 1 b − a ∫ a b F ( t ) d t ≤ p F ( x ) 2 + ( x − a ) F ( a ) + ( b − x ) F ( b ) 2 ( b − a ) . $$ {\displaystyle \begin{array}{cc}\hfill &amp; F\left(\frac{a&#x0002B;b}{2}\right)\hfill \\ {}\hfill {\le}_p&amp; \kern0.2em \frac{x-a}{b-a}F\left(\frac{x&#x0002B;a}{2}\right)&#x0002B;\frac{b-x}{b-a}F\left(\frac{x&#x0002B;b}{2}\right)\hfill \\ {}\hfill {\le}_p&amp; \kern0.2em \frac{1}{b-a}\int_a&#x0005E;bF(t) dt\hfill \\ {}\hfill {\le}_p&amp; \kern0.2em \frac{F(x)}{2}&#x0002B;\frac{\left(x-a\right)F(a)&#x0002B;\left(b-x\right)F(b)}{2\left(b-a\right)}.\hfill \end{array}} $$ (9)

Proof.By choosing g ( t ) = t $$ g(t)&#x0003D;t $$ for all t ∈ [ a , b ] $$ t\in \left[a,b\right] $$ in the inequality (5), we achieve the required inequality (9).

Remark 4.If we take F ∗ ( t ) = F ∗ ( t ) $$ {F}&#x0005E;{\ast }(t)&#x0003D;{F}_{\ast }(t) $$ and x = a + b 2 $$ x&#x0003D;\frac{a&#x0002B;b}{2} $$ in (9), then we obtain the classical Bullen's integral inequality (2).

Example 1.Let us take the interval-valued function F : [ a , b ] = 1 , 3 → ℝ 𝒞 + given by F ( t ) = [ t 2 , 2 t 2 ] $$ F(t)&#x0003D;\left[{t}&#x0005E;2,2{t}&#x0005E;2\right] $$ . Since end point functions F ∗ ( t ) = t 2 $$ {F}_{\ast }(t)&#x0003D;{t}&#x0005E;2 $$ and F ∗ ( t ) = 2 t 2 $$ {F}&#x0005E;{\ast }(t)&#x0003D;2{t}&#x0005E;2 $$ are convex functions, F $$ F $$ is an L R $$ LR $$ -convex interval-valued function. Besides, taking into account the continuous function g : [ 1 , 3 ] → ℝ $$ g:\left[1,3\right]\to \mathbb{R} $$ defined by g ( t ) = t $$ g(t)&#x0003D;t $$ , we see that

F ∗ 1 b − a ∫ a b g ( t ) d t = 1 2 ∫ 1 3 t d t 2 = 4 = Ψ ∗ , F ∗ 1 b − a ∫ a b g ( t ) d t = 2 1 2 ∫ 1 3 t d t 2 = 8 = Ψ ∗ , $$ {\displaystyle \begin{array}{cc}\hfill {F}_{\ast}\left(\frac{1}{b-a}\int_a&#x0005E;bg(t) dt\right)&amp; &#x0003D;{\left(\frac{1}{2}\int_1&#x0005E;3 tdt\right)}&#x0005E;2&#x0003D;4={\Psi}_{\ast },\hfill \\ {}\hfill {F}&#x0005E;{\ast}\left(\frac{1}{b-a}\int_a&#x0005E;bg(t) dt\right)&amp; &#x0003D;2{\left(\frac{1}{2}\int_1&#x0005E;3 tdt\right)}&#x0005E;2&#x0003D;8={\Psi}&#x0005E;{\ast },\hfill \end{array}} $$
x − a b − a F ∗ 1 x − a ∫ a x g ( t ) d t + b − x b − a F ∗ 1 b − x ∫ x b g ( t ) d t = − 2 x 2 + 8 x + 26 8 = Ω ∗ , x − a b − a F ∗ 1 x − a ∫ a x g ( t ) d t + b − x b − a F ∗ 1 b − x ∫ x b g ( t ) d t = − 2 x 2 + 8 x + 26 4 = Ω ∗ , $$ {\displaystyle \begin{array}{cc}\hfill &amp; \frac{x-a}{b-a}{F}_{\ast}\left(\frac{1}{x-a}\int_a&#x0005E;xg(t) dt\right)\hfill \\ {}\hfill &amp; &#x0002B;\frac{b-x}{b-a}{F}_{\ast}\left(\frac{1}{b-x}\int_x&#x0005E;bg(t) dt\right)&#x0003D;\frac{-2{x}&#x0005E;2&#x0002B;8x&#x0002B;26}{8}&#x0003D;{\Omega}_{\ast },\hfill \\ {}\hfill &amp; \frac{x-a}{b-a}{F}&#x0005E;{\ast}\left(\frac{1}{x-a}\int_a&#x0005E;xg(t) dt\right)\hfill \\ {}\hfill &amp; &#x0002B;\frac{b-x}{b-a}{F}&#x0005E;{\ast}\left(\frac{1}{b-x}\int_x&#x0005E;bg(t) dt\right)&#x0003D;\frac{-2{x}&#x0005E;2&#x0002B;8x&#x0002B;26}{4}&#x0003D;{\Omega}&#x0005E;{\ast },\hfill \end{array}} $$
1 b − a ∫ a b F ∗ g ( t ) d t = 13 3 = Φ ∗ , 1 b − a ∫ a b F ∗ g ( t ) d t = 26 3 = Φ ∗ , $$ {\displaystyle \begin{array}{cc}\hfill &amp; \frac{1}{b-a}\int_a&#x0005E;b{F}_{\ast}\left(g(t)\right) dt&#x0003D;\frac{13}{3}&#x0003D;{\Phi}_{\ast },\hfill \\ {}\hfill &amp; \frac{1}{b-a}\int_a&#x0005E;b{F}&#x0005E;{\ast}\left(g(t)\right) dt&#x0003D;\frac{26}{3}&#x0003D;{\Phi}&#x0005E;{\ast },\hfill \end{array}} $$
and
F ∗ ( x ) − x F ∗ ( b ) − F ∗ ( a ) b − a + F ∗ ( x ) − F ∗ ( a ) ( x − a ) ( b − a ) ∫ a b g ( t ) d t + F ∗ ( b ) − F ∗ ( x ) ( b − x ) ( b − a ) ∫ x b g ( t ) d t = 2 x 2 − 8 x + 26 4 = Λ ∗ , F ∗ ( x ) − x F ∗ ( b ) − F ∗ ( a ) b − a + F ∗ ( x ) − F ∗ ( a ) ( x − a ) ( b − a ) ∫ a b g ( t ) d t + F ∗ ( b ) − F ∗ ( x ) ( b − x ) ( b − a ) ∫ x b g ( t ) d t = 2 x 2 − 8 x + 26 2 = Λ ∗ , $$ {\displaystyle \begin{array}{cc}\hfill {F}_{\ast }(x)&amp; -x\frac{F_{\ast }(b)-{F}_{\ast }(a)}{b-a}&#x0002B;\frac{F_{\ast }(x)-{F}_{\ast }(a)}{\left(x-a\right)\left(b-a\right)}\int_a&#x0005E;bg(t) dt\hfill \\ {}\hfill &amp; &#x0002B;\frac{F_{\ast }(b)-{F}_{\ast }(x)}{\left(b-x\right)\left(b-a\right)}\int_x&#x0005E;bg(t) dt&#x0003D;\frac{2{x}&#x0005E;2-8x&#x0002B;26}{4}&#x0003D;{\Lambda}_{\ast },\hfill \\ {}\hfill {F}&#x0005E;{\ast }(x)&amp; -x\frac{F&#x0005E;{\ast }(b)-{F}&#x0005E;{\ast }(a)}{b-a}&#x0002B;\frac{F&#x0005E;{\ast }(x)-{F}&#x0005E;{\ast }(a)}{\left(x-a\right)\left(b-a\right)}\int_a&#x0005E;bg(t) dt\hfill \\ {}\hfill &amp; &#x0002B;\frac{F&#x0005E;{\ast }(b)-{F}&#x0005E;{\ast }(x)}{\left(b-x\right)\left(b-a\right)}\int_x&#x0005E;bg(t) dt&#x0003D;\frac{2{x}&#x0005E;2-8x&#x0002B;26}{2}&#x0003D;{\Lambda}&#x0005E;{\ast },\hfill \end{array}} $$
where Ψ = [ Ψ ∗ , Ψ ∗ ] , Ω = [ Ω ∗ , Ω ∗ ] , Φ = [ Φ ∗ , Φ ∗ ] $$ \Psi &#x0003D;\left[{\Psi}_{\ast },{\Psi}&#x0005E;{\ast}\right],\kern0.3em \Omega &#x0003D;\left[{\Omega}_{\ast },{\Omega}&#x0005E;{\ast}\right],\kern0.3em \Phi &#x0003D;\left[{\Phi}_{\ast },{\Phi}&#x0005E;{\ast}\right] $$ and Λ = [ Λ ∗ , Λ ∗ ] $$ \Lambda &#x0003D;\left[{\Lambda}_{\ast },{\Lambda}&#x0005E;{\ast}\right] $$ . Therefore, we get
[ Ψ ∗ , Ψ ∗ ] ≤ p [ Ω ∗ , Ω ∗ ] ≤ p [ Φ ∗ , Φ ∗ ] ≤ p [ Λ ∗ , Λ ∗ ] . $$ \left[{\Psi}_{\ast },{\Psi}&#x0005E;{\ast}\right]{\le}_p\left[{\Omega}_{\ast },{\Omega}&#x0005E;{\ast}\right]{\le}_p\left[{\Phi}_{\ast },{\Phi}&#x0005E;{\ast}\right]{\le}_p\left[{\Lambda}_{\ast },{\Lambda}&#x0005E;{\ast}\right]. $$ (10)

Thus, it is apparent from Figure 1 that for every x $$ x $$ in the interval ( 1 , 3 ) $$ \left(1,3\right) $$ , the inequality (10) is satisfied.

Details are in the caption following the image
FIGURE 1
Open in figure viewerPowerPoint
The graph of four parts of the inequality (5) according to Example 1, depending on x ∈ ( 1 , 3 ) $$ x\in \left(1,3\right) $$ . [Colour figure can be viewed at wileyonlinelibrary.com]

Definition 5.An interval-valued function F : [ a , b ] → ℝ 𝒞 is said to be L R $$ LR $$ -increasing if for all t 1 , t 2 ∈ [ a , b ] $$ {t}_1,{t}_2\in \left[a,b\right] $$ ,

t 1 ≤ t 2 implies F ( t 1 ) ≤ p F ( t 2 ) . $$ {t}_1\le {t}_2\kern0.3em \mathrm{implies}\kern0.3em F\left({t}_1\right){\le}_pF\left({t}_2\right). $$

Theorem 5.Let F : [ a , b ] → ℝ 𝒞 be an interval-valued function such that F ( t ) = [ F ∗ ( t ) , F ∗ ( t ) ] $$ F(t)&#x0003D;\left[{F}_{\ast }(t),{F}&#x0005E;{\ast }(t)\right] $$ for all t ∈ [ a , b ] $$ t\in \left[a,b\right] $$ . Then, F $$ F $$ is an L R $$ LR $$ -increasing interval-valued function on [ a , b ] $$ \left[a,b\right] $$ if and only if both the functions F ∗ $$ {F}_{\ast } $$ and F ∗ $$ {F}&#x0005E;{\ast } $$ are increasing functions.

Proof.It follows easily from the concepts of L R $$ LR $$ -order and the increasing function in the classical sense.

Theorem 6.Let F : [ a , b ] → ℝ $$ F:\left[a,b\right]\to \mathbb{R} $$ be an LR-increasing convex interval-valued function on [ a , b ] $$ \left[a,b\right] $$ and g : [ a , b ] → ℝ $$ g:\left[a,b\right]\to \mathbb{R} $$ be a convex function on [ a , b ] $$ \left[a,b\right] $$ with a < b $$ a&lt;b $$ . Then, the following inequalities hold:

F 1 b − a ∫ a b g ( t ) d t ≤ p x − a b − a F 1 x − a ∫ a x g ( t ) d t + b − x b − a F 1 b − x ∫ x b g ( t ) d t ≤ p 1 b − a ∫ a b F g ( t ) d t ≤ p 1 b − a ∫ a x F x − t x − a g ( a ) + t − a x − a g ( x ) d t + 1 b − a ∫ x b F b − t b − x g ( x ) + t − x b − x g ( b ) d t ≤ p 1 2 F g ( x ) + ( x − a ) F ( g ( a ) ) + ( b − x ) F g ( b ) 2 ( b − a ) . $$ {\displaystyle \begin{array}{cc}\hfill &amp; F\left(\frac{1}{b-a}\int_a&#x0005E;bg(t) dt\right)\hfill \\ {}\hfill &amp; {\le}_p\frac{x-a}{b-a}F\left(\frac{1}{x-a}\int_a&#x0005E;xg(t) dt\right)&#x0002B;\frac{b-x}{b-a}F\left(\frac{1}{b-x}\int_x&#x0005E;bg(t) dt\right)\hfill \\ {}\hfill &amp; {\le}_p\frac{1}{b-a}\int_a&#x0005E;bF\left(g(t)\right) dt\hfill \\ {}\hfill &amp; {\le}_p\frac{1}{b-a}\int_a&#x0005E;xF\left(\frac{x-t}{x-a}g(a)&#x0002B;\frac{t-a}{x-a}g(x)\right) dt\hfill \\ {}\hfill &amp; &#x0002B;\frac{1}{b-a}\int_x&#x0005E;bF\left(\frac{b-t}{b-x}g(x)&#x0002B;\frac{t-x}{b-x}g(b)\right) dt\hfill \\ {}\hfill &amp; {\le}_p\frac{1}{2}F\left(g(x)\right)&#x0002B;\frac{\left(x-a\right)F\left(g(a)\right)&#x0002B;\left(b-x\right)F\left(g(b)\right)}{2\left(b-a\right)}.\hfill \end{array}} $$ (11)

Proof.Based on Theorems 3 and 5, as F $$ F $$ is an L R $$ LR $$ -increasing convex interval-valued function, it follows that both F ∗ $$ {F}_{\ast } $$ and F ∗ $$ {F}&#x0005E;{\ast } $$ are increasing convex functions. So, by utilizing the convexity property of g $$ g $$ , we obtain

1 b − a ∫ a b F ∗ g ( t ) d t = 1 b − a ∫ a x F ∗ g x − t x − a a + t − a x − a x + 1 b − a ∫ x b F ∗ g b − t b − x x + t − x b − x b d t ≤ 1 b − a ∫ a x F ∗ x − t x − a g ( a ) + t − a x − a g ( x ) d t + 1 b − a ∫ x b F ∗ b − t b − x g ( x ) + t − x b − x g ( b ) d t ≤ 1 b − a ∫ a x x − t x − a F ∗ g ( a ) d t + 1 b − a ∫ a x t − a x − a F ∗ g ( x ) d t + 1 b − a ∫ x b b − t b − x F ∗ g ( x ) d t + 1 b − a ∫ x b t − x b − x F ∗ g ( b ) d t = 1 2 F ∗ g ( x ) + ( x − a ) F ∗ g ( a ) + ( b − x ) F ∗ g ( b ) 2 ( b − a ) . $$ {\displaystyle \begin{array}{cc}\hfill &amp; \frac{1}{b-a}\int_a&#x0005E;b{F}_{\ast}\left(g(t)\right) dt\hfill \\ {}\hfill &#x0003D;&amp; \kern0.2em \frac{1}{b-a}\int_a&#x0005E;x{F}_{\ast}\left(g\left(\frac{x-t}{x-a}a&#x0002B;\frac{t-a}{x-a}x\right)\right)\hfill \\ {}\hfill &amp; &#x0002B;\frac{1}{b-a}\int_x&#x0005E;b{F}_{\ast}\left(g\left(\frac{b-t}{b-x}x&#x0002B;\frac{t-x}{b-x}b\right)\right) dt\hfill \\ {}\hfill \le &amp; \kern0.2em \frac{1}{b-a}\int_a&#x0005E;x{F}_{\ast}\left(\frac{x-t}{x-a}g(a)&#x0002B;\frac{t-a}{x-a}g(x)\right) dt\hfill \\ {}\hfill &amp; &#x0002B;\frac{1}{b-a}\int_x&#x0005E;b{F}_{\ast}\left(\frac{b-t}{b-x}g(x)&#x0002B;\frac{t-x}{b-x}g(b)\right) dt\hfill \\ {}\hfill \le &amp; \kern0.2em \frac{1}{b-a}\int_a&#x0005E;x\frac{x-t}{x-a}{F}_{\ast}\left(g(a)\right) dt&#x0002B;\frac{1}{b-a}\int_a&#x0005E;x\frac{t-a}{x-a}{F}_{\ast}\left(g(x)\right) dt\hfill \\ {}\hfill &amp; &#x0002B;\frac{1}{b-a}\int_x&#x0005E;b\frac{b-t}{b-x}{F}_{\ast}\left(g(x)\right) dt&#x0002B;\frac{1}{b-a}\int_x&#x0005E;b\frac{t-x}{b-x}{F}_{\ast}\left(g(b)\right) dt\hfill \\ {}\hfill &#x0003D;&amp; \kern0.2em \frac{1}{2}{F}_{\ast}\left(g(x)\right)&#x0002B;\frac{\left(x-a\right){F}_{\ast}\left(g(a)\right)&#x0002B;\left(b-x\right){F}_{\ast}\left(g(b)\right)}{2\left(b-a\right)}.\hfill \end{array}} $$ (12)

If we analyze the convexity of F ∗ $$ {F}&#x0005E;{\ast } $$ in a manner similar to the previous analysis, we reach the following outcome:

1 b − a ∫ a b F ∗ g ( t ) d t = 1 b − a ∫ a x F ∗ g x − t x − a a + t − a x − a x + 1 b − a ∫ x b F ∗ g b − t b − x x + t − x b − x b d t $$ {\displaystyle \begin{array}{cc}\hfill \frac{1}{b-a}\int_a&#x0005E;b{F}&#x0005E;{\ast}\left(g(t)\right) dt&#x0003D;&amp; \kern0.2em \frac{1}{b-a}\int_a&#x0005E;x{F}&#x0005E;{\ast}\left(g\left(\frac{x-t}{x-a}a&#x0002B;\frac{t-a}{x-a}x\right)\right)\hfill \\ {}\hfill &amp; &#x0002B;\frac{1}{b-a}\int_x&#x0005E;b{F}&#x0005E;{\ast}\left(g\left(\frac{b-t}{b-x}x&#x0002B;\frac{t-x}{b-x}b\right)\right) dt\hfill \end{array}} $$
≤ 1 b − a ∫ a x F ∗ x − t x − a g ( a ) + t − a x − a g ( x ) d t + 1 b − a ∫ x b F ∗ b − t b − x g ( x ) + t − x b − x g ( b ) d t ≤ 1 b − a ∫ a x x − t x − a F ∗ g ( a ) d t + 1 b − a ∫ a x t − a x − a F ∗ g ( x ) d t + 1 b − a ∫ x b b − t b − x F ∗ g ( x ) d t + 1 b − a ∫ x b t − x b − x F ∗ g ( b ) d t = 1 2 F ∗ g ( x ) + ( x − a ) F ∗ g ( a ) + ( b − x ) F ∗ g ( b ) 2 ( b − a ) . $$ {\displaystyle \begin{array}{cc}\hfill \le &amp; \kern0.2em \frac{1}{b-a}\int_a&#x0005E;x{F}&#x0005E;{\ast}\left(\frac{x-t}{x-a}g(a)&#x0002B;\frac{t-a}{x-a}g(x)\right) dt\hfill \\ {}\hfill &amp; &#x0002B;\frac{1}{b-a}\int_x&#x0005E;b{F}&#x0005E;{\ast}\left(\frac{b-t}{b-x}g(x)&#x0002B;\frac{t-x}{b-x}g(b)\right) dt\hfill \\ {}\hfill \le &amp; \kern0.2em \frac{1}{b-a}\int_a&#x0005E;x\frac{x-t}{x-a}{F}&#x0005E;{\ast}\left(g(a)\right) dt&#x0002B;\frac{1}{b-a}\int_a&#x0005E;x\frac{t-a}{x-a}{F}&#x0005E;{\ast}\left(g(x)\right) dt\hfill \\ {}\hfill &amp; &#x0002B;\frac{1}{b-a}\int_x&#x0005E;b\frac{b-t}{b-x}{F}&#x0005E;{\ast}\left(g(x)\right) dt&#x0002B;\frac{1}{b-a}\int_x&#x0005E;b\frac{t-x}{b-x}{F}&#x0005E;{\ast}\left(g(b)\right) dt\hfill \\ {}\hfill &#x0003D;&amp; \kern0.2em \frac{1}{2}{F}&#x0005E;{\ast}\left(g(x)\right)&#x0002B;\frac{\left(x-a\right){F}&#x0005E;{\ast}\left(g(a)\right)&#x0002B;\left(b-x\right){F}&#x0005E;{\ast}\left(g(b)\right)}{2\left(b-a\right)}.\hfill \end{array}} $$ (13)

Thus, through the use of the L R $$ LR $$ -order property and the inequalities (12) and (13), we obtain third and fourth parts of the inequality (11). On the other hand, since every convex function is continuous, the first and second parts of the inequality (11) are obtained from Theorem 4, completing the proof

Remark 5.If F ∗ = F ∗ $$ {F}&#x0005E;{\ast }&#x0003D;{F}_{\ast } $$ , then the above theorem reduces to Theorem 2 in [12].

Example 2.Consider the interval valued function F : [ a , b ] = 0 , 2 → ℝ 𝒞 + defined by F ( t ) = [ t 2 , e t ] $$ F(t)&#x0003D;\left[{t}&#x0005E;2,{e}&#x0005E;t\right] $$ . Because end point functions F ∗ ( t ) = t 2 $$ {F}_{\ast }(t)&#x0003D;{t}&#x0005E;2 $$ and F ∗ ( t ) = e t $$ {F}&#x0005E;{\ast }(t)&#x0003D;{e}&#x0005E;t $$ are increasing convex functions on [ 0 , 2 ] , F $$ \left[0,2\right],\kern0.3em F $$ is an L R $$ LR $$ -increasing convex interval valued function. In addition, upon choosing the continuous function g : [ 0 , 2 ] → ℝ $$ g:\left[0,2\right]\to \mathbb{R} $$ as g ( t ) = t 2 $$ g(t)&#x0003D;\frac{t}{2} $$ , we find that

F ∗ 1 b − a ∫ a b g ( t ) d t = 1 4 ∫ 0 2 t d t 2 = 1 4 = Ψ ∗ , F ∗ 1 b − a ∫ a b g ( t ) d t = e 1 4 ∫ 0 2 t d t = e 1 2 = Ψ ∗ , x − a b − a F ∗ 1 x − a ∫ a x g ( t ) d t + b − x b − a F ∗ 1 b − x ∫ x b g ( t ) d t = − x 2 + 2 x + 4 16 = Ω ∗ , x − a b − a F ∗ 1 x − a ∫ a x g ( t ) d t + b − x b − a F ∗ 1 b − x ∫ x b g ( t ) d t = e x 4 x + 2 e − x e 2 = Ω ∗ , 1 b − a ∫ a b F ∗ g ( t ) d t = 1 3 = Φ ∗ , 1 b − a ∫ a b F ∗ g ( t ) d t = e − 1 = Φ ∗ , $$ {\displaystyle \begin{array}{cc}\hfill &amp; {F}_{\ast}\left(\frac{1}{b-a}\int_a&#x0005E;bg(t) dt\right)&#x0003D;{\left(\frac{1}{4}\int_0&#x0005E;2 tdt\right)}&#x0005E;2&#x0003D;\frac{1}{4}&#x0003D;{\Psi}_{\ast },\kern0.90em \hfill \\ {}\hfill &amp; {F}&#x0005E;{\ast}\left(\frac{1}{b-a}\int_a&#x0005E;bg(t) dt\right)&#x0003D;{e}&#x0005E;{\left(\frac{1}{4}\int_0&#x0005E;2 tdt\right)}&#x0003D;{e}&#x0005E;{\frac{1}{2}}&#x0003D;{\Psi}&#x0005E;{\ast },\hfill \\ {}\hfill &amp; \frac{x-a}{b-a}{F}_{\ast}\left(\frac{1}{x-a}\int_a&#x0005E;xg(t) dt\right)\hfill \\ {}\hfill &amp; &#x0002B;\frac{b-x}{b-a}{F}_{\ast}\left(\frac{1}{b-x}\int_x&#x0005E;bg(t) dt\right)&#x0003D;\frac{-{x}&#x0005E;2&#x0002B;2x&#x0002B;4}{16}&#x0003D;{\Omega}_{\ast },\hfill \\ {}\hfill &amp; \frac{x-a}{b-a}{F}&#x0005E;{\ast}\left(\frac{1}{x-a}\int_a&#x0005E;xg(t) dt\right)\hfill \\ {}\hfill &amp; &#x0002B;\frac{b-x}{b-a}{F}&#x0005E;{\ast}\left(\frac{1}{b-x}\int_x&#x0005E;bg(t) dt\right)&#x0003D;\frac{e&#x0005E;{\frac{x}{4}}\left(x&#x0002B;2\sqrt{e}-x\sqrt{e}\right)}{2}&#x0003D;{\Omega}&#x0005E;{\ast },\hfill \\ {}\hfill &amp; \frac{1}{b-a}\int_a&#x0005E;b{F}_{\ast}\left(g(t)\right) dt&#x0003D;\frac{1}{3}&#x0003D;{\Phi}_{\ast },\hfill \\ {}\hfill &amp; \frac{1}{b-a}\int_a&#x0005E;b{F}&#x0005E;{\ast}\left(g(t)\right) dt&#x0003D;e-1&#x0003D;{\Phi}&#x0005E;{\ast },\hfill \end{array}} $$
1 b − a ∫ a x F ∗ x − t x − a g ( a ) + t − a x − a g ( x ) d t + 1 b − a ∫ x b F ∗ b − t b − x g ( x ) + t − x b − x g ( b ) d t = 1 3 = Λ ∗ , 1 b − a ∫ a x F ∗ x − t x − a g ( a ) + t − a x − a g ( x ) d t + 1 b − a ∫ x b F ∗ b − t b − x g ( x ) + t − x b − x g ( b ) d t = e − 1 = Λ ∗ , $$ {\displaystyle \begin{array}{ccc}\hfill &amp; \frac{1}{b-a}\int_a&#x0005E;x{F}_{\ast}\left(\frac{x-t}{x-a}g(a)&#x0002B;\frac{t-a}{x-a}g(x)\right) dt\hfill &amp; \hfill \\ {}\hfill &amp; &#x0002B;\frac{1}{b-a}\int_x&#x0005E;b{F}_{\ast}\left(\frac{b-t}{b-x}g(x)&#x0002B;\frac{t-x}{b-x}g(b)\right) dt\hfill &amp; \hfill &#x0003D;\frac{1}{3}&#x0003D;{\Lambda}_{\ast },\\ {}\hfill &amp; \frac{1}{b-a}\int_a&#x0005E;x{F}&#x0005E;{\ast}\left(\frac{x-t}{x-a}g(a)&#x0002B;\frac{t-a}{x-a}g(x)\right) dt\hfill &amp; \hfill \\ {}\hfill &amp; &#x0002B;\frac{1}{b-a}\int_x&#x0005E;b{F}&#x0005E;{\ast}\left(\frac{b-t}{b-x}g(x)&#x0002B;\frac{t-x}{b-x}g(b)\right) dt\hfill &amp; \hfill &#x0003D;e-1&#x0003D;{\Lambda}&#x0005E;{\ast },\end{array}} $$
and
1 2 F ∗ ( g ( x ) ) + ( x − a ) F ∗ ( g ( a ) ) + ( b − x ) F ∗ ( g ( b ) ) 2 ( b − a ) = x 2 − 2 x + 4 8 = Γ ∗ , 1 2 F ∗ ( g ( x ) ) + ( x − a ) F ∗ ( g ( a ) ) + ( b − x ) F ∗ ( g ( b ) ) 2 ( b − a ) = 2 e x 2 + x + 2 e − x e 4 = Γ ∗ , $$ {\displaystyle \begin{array}{cc}\hfill \frac{1}{2}{F}_{\ast}\left(g(x)\right)&amp; &#x0002B;\frac{\left(x-a\right){F}_{\ast}\left(g(a)\right)&#x0002B;\left(b-x\right){F}_{\ast}\left(g(b)\right)}{2\left(b-a\right)}\hfill \\ {}\hfill &#x0003D;&amp; \kern0.2em \frac{x&#x0005E;2-2x&#x0002B;4}{8}&#x0003D;{\Gamma}_{\ast },\hfill \\ {}\hfill \frac{1}{2}{F}&#x0005E;{\ast}\left(g(x)\right)&amp; &#x0002B;\frac{\left(x-a\right){F}&#x0005E;{\ast}\left(g(a)\right)&#x0002B;\left(b-x\right){F}&#x0005E;{\ast}\left(g(b)\right)}{2\left(b-a\right)}\hfill \\ {}\hfill &#x0003D;&amp; \kern0.2em \frac{2{e}&#x0005E;{\frac{x}{2}}&#x0002B;x+2e- xe}{4}&#x0003D;{\Gamma}&#x0005E;{\ast },\hfill \end{array}} $$
where Ψ = [ Ψ ∗ , Ψ ∗ ] , Ω = [ Ω ∗ , Ω ∗ ] , Φ = [ Φ ∗ , Φ ∗ ] , Λ = [ Λ ∗ , Λ ∗ ] $$ \Psi &#x0003D;\left[{\Psi}_{\ast },{\Psi}&#x0005E;{\ast}\right],\kern0.3em \Omega &#x0003D;\left[{\Omega}_{\ast },{\Omega}&#x0005E;{\ast}\right],\kern0.3em \Phi &#x0003D;\left[{\Phi}_{\ast },{\Phi}&#x0005E;{\ast}\right]\kern0.3em ,\kern0.3em \Lambda &#x0003D;\left[{\Lambda}_{\ast },{\Lambda}&#x0005E;{\ast}\right] $$ and Γ = [ Γ ∗ , Γ ∗ ] $$ \Gamma &#x0003D;\left[{\Gamma}_{\ast },{\Gamma}&#x0005E;{\ast}\right] $$ . So, we have
[ Ψ ∗ , Ψ ∗ ] ≤ p [ Ω ∗ , Ω ∗ ] ≤ p [ Φ ∗ , Φ ∗ ] ≤ p [ Λ ∗ , Λ ∗ ] ≤ p [ Γ ∗ , Γ ∗ ] . $$ \left[{\Psi}_{\ast },{\Psi}&#x0005E;{\ast}\right]{\le}_p\left[{\Omega}_{\ast },{\Omega}&#x0005E;{\ast}\right]{\le}_p\left[{\Phi}_{\ast },{\Phi}&#x0005E;{\ast}\right]{\le}_p\left[{\Lambda}_{\ast },{\Lambda}&#x0005E;{\ast}\right]{\le}_p\left[{\Gamma}_{\ast },{\Gamma}&#x0005E;{\ast}\right]. $$ (14)

Hence, from Figure 2, it is evident that the inequality (14) is valid for every x $$ x $$ in the interval ( 0 , 2 ) $$ \left(0,2\right) $$ .

Details are in the caption following the image
FIGURE 2
Open in figure viewerPowerPoint
The graph of five parts of the inequality (11) according to Example 2, depending on x ∈ ( 0 , 2 ) $$ x\in \left(0,2\right) $$ . [Colour figure can be viewed at wileyonlinelibrary.com]

4 Estimates of the Jensen Gap

The aim of this section is to establish several bounds for the Jensen gap, involving gH-differentiable interval-valued functions and associated concepts. We also obtain these bounds by utilizing the renowned Ostrowski inequality.

Let g : [ a , b ] → ℝ $$ g:\left[a,b\right]\to \mathbb{R} $$ be a differentiable mapping on ( a , b ) $$ \left(a,b\right) $$ . If g ′ ( t ) ≤ M $$ \left&#x0007C;{g}&#x0005E;{\prime }(t)\right&#x0007C;\le M $$ for all t ∈ ( a , b ) $$ t\in \left(a,b\right) $$ , then the following inequality holds:
g ( t ) − 1 b − a ∫ a b g ( s ) d s ≤ M ( b − a ) 1 4 + t − a + b 2 2 ( b − a ) 2 $$ \left&#x0007C;g(t)-\frac{1}{b-a}\int_a&#x0005E;bg(s) ds\right&#x0007C;\le M\left(b-a\right)\left[\frac{1}{4}&#x0002B;\frac{{\left(t-\frac{a&#x0002B;b}{2}\right)}&#x0005E;2}{{\left(b-a\right)}&#x0005E;2}\right] $$ (15)

This result is referred to as the Ostrowski inequality in the literature and is a significant and practical integral inequality [41]. This inequality results in an upper bound for the approximation of the integral average 1 b − a ∫ a b g ( s ) d s $$ \frac{1}{b-a}\int_a&#x0005E;bg(s) ds $$ by the value of g ( s ) $$ g(s) $$ at the point t ∈ [ a , b ] $$ t\in \left[a,b\right] $$ .

Theorem 7.Let F : [ a , b ] → ℝ 𝒞  be a continuously gH-differentiable interval-valued function on [ a , b ] $$ \left[a,b\right] $$ with a finite number of switching points at precisely a = c 0 < c 1 < c 2 < . . . < c n < c n + 1 = b $$ a&#x0003D;{c}_0&lt;{c}_1&lt;{c}_2&lt;\dots &lt;{c}_n&lt;{c}_{n&#x0002B;1}&#x0003D;b $$ . Let g : [ a , b ] → ℝ $$ g:\left[a,b\right]\to \mathbb{R} $$ be a differentiable mapping on ( a , b ) $$ \left(a,b\right) $$ . If g ′ ( t ) ≤ M $$ \left&#x0007C;{g}&#x0005E;{\prime }(t)\right&#x0007C;\le M $$ , then the following inequality holds:

H 1 b − a ∫ a b F g ( t ) d t , F 1 b − a ∫ a b g ( s ) d s ≤ F ′ ∞ ( b − a ) 3 M $$ H\left(\frac{1}{b-a}\int_a&#x0005E;bF\left(g(t)\right) dt,F\left(\frac{1}{b-a}\int_a&#x0005E;bg(s) ds\right)\right)\le {\left&#x0007C;\left&#x0007C;{F}&#x0005E;{\prime}\right&#x0007C;\right&#x0007C;}_{\infty}\frac{\left(b-a\right)}{3}M $$ (16)

Proof.Considering the properties of the Pompeiu–Hausdorff metric and from Theorem 2, it follows that

H 1 b − a ∫ a b F g ( t ) d t , F 1 b − a ∫ a b g ( s ) d s = H 1 b − a ∫ a b F g ( t ) d t , 1 b − a ∫ a b F 1 b − a ∫ a b g ( s ) d s d t ≤ 1 b − a ∫ a b H F g ( t ) , F 1 b − a ∫ a b g ( s ) d s d t ≤ F ′ ∞ b − a ∫ a b g ( t ) − 1 b − a ∫ a b g ( s ) d s d t . $$ {\displaystyle \begin{array}{cc}\hfill &amp; H\left(\frac{1}{b-a}\int_a&#x0005E;bF\left(g(t)\right) dt,F\left(\frac{1}{b-a}\int_a&#x0005E;bg(s) ds\right)\right)\hfill \\ {}\hfill &#x0003D;&amp; \kern0.2em H\left(\frac{1}{b-a}\int_a&#x0005E;bF\left(g(t)\right) dt,\frac{1}{b-a}\int_a&#x0005E;bF\left(\frac{1}{b-a}\int_a&#x0005E;bg(s) ds\right) dt\right)\hfill \\ {}\hfill \le &amp; \kern0.2em \frac{1}{b-a}\int_a&#x0005E;bH\left(F\left(g(t)\right),F\left(\frac{1}{b-a}\int_a&#x0005E;bg(s) ds\right)\right) dt\hfill \\ {}\hfill \le &amp; \kern0.2em \frac{{\left&#x0007C;\left&#x0007C;{F}&#x0005E;{\prime}\right&#x0007C;\right&#x0007C;}_{\infty }}{b-a}\int_a&#x0005E;b\left&#x0007C;g(t)-\frac{1}{b-a}\int_a&#x0005E;bg(s) ds\right&#x0007C; dt.\hfill \end{array}} $$
Due to g ′ ( t ) ≤ M $$ \left&#x0007C;{g}&#x0005E;{\prime }(t)\right&#x0007C;\le M $$ on ( a , b ) $$ \left(a,b\right) $$ , with the application of the inequality (15), we derive the following:
H 1 b − a ∫ a b F g ( t ) d t , F 1 b − a ∫ a b g ( s ) d s ≤ F ′ ∞ ( b − a ) 3 M . $$ H\left(\frac{1}{b-a}\int_a&#x0005E;bF\left(g(t)\right) dt,F\left(\frac{1}{b-a}\int_a&#x0005E;bg(s) ds\right)\right)\le {\left&#x0007C;\left&#x0007C;{F}&#x0005E;{\prime}\right&#x0007C;\right&#x0007C;}_{\infty}\frac{\left(b-a\right)}{3}M. $$

Now, we proceed by presenting a more general result than Theorem 7.

Theorem 8.Under the assumptions of Theorem 7, we have

H 1 b − a ∫ a b F g ( t ) d t , ( 1 − α ) F 1 b − a ∫ a b g ( s ) d s + F ( a ) + F ( b ) 2 α ≤ F ′ ∞ ( 1 − α ) ( b − a ) 3 M + α 2 ( b − a ) ∫ a b g ( t ) − a + g ( t ) − b d t . $$ {\displaystyle \begin{array}{cc}\hfill &amp; H\left(\frac{1}{b-a}\int_a&#x0005E;bF\left(g(t)\right) dt,\left(1-\alpha \right)F\left(\frac{1}{b-a}\int_a&#x0005E;bg(s) ds\right)\right.\hfill \\ {}\hfill &amp; &#x0002B;\left.\frac{F(a)&#x0002B;F(b)}{2}\alpha \right)\hfill \\ {}\hfill \le &amp; \kern0.2em {\left&#x0007C;\left&#x0007C;{F}&#x0005E;{\prime}\right&#x0007C;\right&#x0007C;}_{\infty}\left[\left(1-\alpha \right)\frac{\left(b-a\right)}{3}M\right.\hfill \\ {}\hfill &amp; &#x0002B;\left.\frac{\alpha }{2\left(b-a\right)}\int_a&#x0005E;b\left&#x0007C;g(t)-a\right&#x0007C;&#x0002B;\left&#x0007C;g(t)-b\right&#x0007C; dt\right].\hfill \end{array}} $$ (17)

Proof.In light of the properties of the Pompeiu–Hausdorff metric and using Theorem 2, we obtain that

H 1 b − a ∫ a b F g ( t ) d t , ( 1 − α ) F 1 b − a ∫ a b g ( s ) d s + F ( a ) + F ( b ) 2 α = H 1 b − a ∫ a b F g ( t ) ( 1 − α ) + F g ( t ) α d t , 1 b − a ∫ a b ( 1 − α ) F 1 b − a ∫ a b g ( s ) d s + F ( a ) + F ( b ) 2 α d t ≤ 1 b − a ∫ a b H F g ( t ) ( 1 − α ) + F g ( t ) α , ( 1 − α ) F 1 b − a ∫ a b g ( s ) d s + F ( a ) + F ( b ) 2 α d t $$ {\displaystyle \begin{array}{cc}\hfill &amp; H\left(\frac{1}{b-a}\int_a&#x0005E;bF\left(g(t)\right) dt,\left(1-\alpha \right)F\left(\frac{1}{b-a}\int_a&#x0005E;bg(s) ds\right)\right.\hfill \\ {}\hfill &amp; &#x0002B;\left.\frac{F(a)&#x0002B;F(b)}{2}\alpha \right)\hfill \\ {}\hfill &#x0003D;&amp; \kern0.2em H\left(\frac{1}{b-a}\int_a&#x0005E;bF\left(g(t)\right)\left(1-\alpha \right)&#x0002B;F\left(g(t)\right)\alpha dt,\right.\hfill \\ {}\hfill &amp; \left.\frac{1}{b-a}\int_a&#x0005E;b\left[\left(1-\alpha \right)F\left(\frac{1}{b-a}\int_a&#x0005E;bg(s) ds\right)&#x0002B;\frac{F(a)&#x0002B;F(b)}{2}\alpha \right] dt\right)\hfill \\ {}\hfill \le &amp; \kern0.2em \frac{1}{b-a}\int_a&#x0005E;bH\left(F\left(g(t)\right)\left(1-\alpha \right)\right.\hfill \\ {}\hfill &amp; &#x0002B;\left.F\left(g(t)\right)\alpha, \left(1-\alpha \right)F\left(\frac{1}{b-a}\int_a&#x0005E;bg(s) ds\right)&#x0002B;\frac{F(a)&#x0002B;F(b)}{2}\alpha \right) dt\hfill \end{array}} $$
≤ 1 b − a ∫ a b ( 1 − α ) H F g ( t ) , F 1 b − a ∫ a b g ( s ) d s + α H F g ( t ) , F ( a ) + F ( b ) 2 d t ≤ F ′ ∞ ( 1 − α ) ( b − a ) 3 M + α b − a ∫ a b H F g ( t ) , F ( a ) + F ( b ) 2 d t . $$ {\displaystyle \begin{array}{cc}\hfill \le &amp; \kern0.2em \frac{1}{b-a}\int_a&#x0005E;b\left[\left(1-\alpha \right)H\left(F\left(g(t)\right),F\left(\frac{1}{b-a}\int_a&#x0005E;bg(s) ds\right)\right)\right.\hfill \\ {}\hfill &amp; &#x0002B;\left.\alpha H\left(F\left(g(t)\right),\frac{F(a)&#x0002B;F(b)}{2}\right)\right] dt\hfill \\ {}\hfill \le &amp; \kern0.2em {\left&#x0007C;\left&#x0007C;{F}&#x0005E;{\prime}\right&#x0007C;\right&#x0007C;}_{\infty}\left(1-\alpha \right)\frac{\left(b-a\right)}{3}M\hfill \\ {}\hfill &amp; &#x0002B;\frac{\alpha }{b-a}\int_a&#x0005E;bH\left(F\left(g(t)\right),\frac{F(a)&#x0002B;F(b)}{2}\right) dt.\hfill \end{array}} $$
From the fact that
∫ a b H F g ( t ) , F ( a ) + F ( b ) 2 d t = ∫ a b H F g ( t ) 2 + F g ( t ) 2 , F ( a ) 2 + F ( b ) 2 d t ≤ ∫ a b H F g ( t ) 2 , F ( a ) 2 + H F g ( t ) 2 , F ( b ) 2 d t ≤ F ′ ∞ 2 ∫ a b g ( t ) − a + g ( t ) − b d t $$ {\displaystyle \begin{array}{cc}\hfill &amp; \int_a&#x0005E;bH\left(F\left(g(t)\right),\frac{F(a)&#x0002B;F(b)}{2}\right) dt\hfill \\ {}\hfill &#x0003D;&amp; \kern0.2em \int_a&#x0005E;bH\left(\frac{F\left(g(t)\right)}{2}&#x0002B;\frac{F\left(g(t)\right)}{2},\frac{F(a)}{2}&#x0002B;\frac{F(b)}{2}\right) dt\hfill \\ {}\hfill \le &amp; \kern0.2em \int_a&#x0005E;bH\left(\frac{F\left(g(t)\right)}{2},\frac{F(a)}{2}\right)&#x0002B;H\left(\frac{F\left(g(t)\right)}{2},\frac{F(b)}{2}\right) dt\hfill \\ {}\hfill \le &amp; \kern0.2em \frac{{\left&#x0007C;\left&#x0007C;{F}&#x0005E;{\prime}\right&#x0007C;\right&#x0007C;}_{\infty }}{2}\int_a&#x0005E;b\left&#x0007C;g(t)-a\right&#x0007C;&#x0002B;\left&#x0007C;g(t)-b\right&#x0007C; dt\hfill \end{array}} $$
it follows that the desired inequality (17) is achieved.

Remark 6.By choosing α = 0 $$ \alpha &#x0003D;0 $$ in Theorem 8, the inequality (16) is obtained.

On the other hand, if we use the interval-valued function F ( t ) = { f ( t ) } $$ F(t)&#x0003D;\left\{f(t)\right\} $$ with α = 0 $$ \alpha &#x0003D;0 $$ , then we arrive at the following inequality:

1 b − a ∫ a b f g ( t ) d t − f 1 b − a ∫ a b g ( s ) d s ≤ f ′ ∞ ( b − a ) 3 M , $$ \left&#x0007C;\frac{1}{b-a}\int_a&#x0005E;bf\left(g(t)\right) dt-f\left(\frac{1}{b-a}\int_a&#x0005E;bg(s) ds\right)\right&#x0007C;\le {\left&#x0007C;\left&#x0007C;{f}&#x0005E;{\prime}\right&#x0007C;\right&#x0007C;}_{\infty}\frac{\left(b-a\right)}{3}M, $$
which was proved by Sarıkaya [12].

5 Conclusion

In the present paper, we first introduce some refinements of the Jensen inequality for the left and right convex interval-valued functions. Then, we prove the correctness of our main findings using specific examples enriched by graphical representations. Next, we utilize a new method to establish many bounds for the Jensen gap in integral form through the use of the gH-differentiable interval-valued functions with related concepts. Additionally, we establish the proposed bounds by applying the famous Ostrowski inequality. The primary advantage of these newly established inequalities is their ability to generalize numerous inequalities previously published in the literature.

In future studies, we believe that readers will be motivated to look into more general inequalities and their applications via our approaches and findings into the realm of interval-valued functions and inequalities. Therefore, interested readers can generalize our work to other models using fractional calculus such as (fuzzy) Riemann–Liouville fractional operators and Atangana–Baleanu fractional operators for (fuzzy) interval-valued functions. Moreover, the researchers can develop new Jensen-type inequalities by employing different types of convexity by means of the interval-valued functions.

Author Contributions

İzzettin Demir: conceptualization; investigation; writing – original draft; methodology; validation; visualization; writing – review and editing; formal analysis; supervision; resources.

Conflicts of Interest

The author declares no conflicts of interest.

Data Availability Statement

Data sharing is not applicable to this paper as no data sets were generated or analyzed during the current study.

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