Hermite-Hadamard type inequalities for interval-valued exponential type pre-invex functions via Riemann-Liouville fractional integrals

1.   Introduction

The classical Hermite-Hadamard inequality is one of the most well-established inequalities in the theory of convex functions with geometrical interpretation and it has many applications. This inequality may be regarded as a refinement of the concept of convexity. Hermite-Hadamard inequality for convex functions has received renewed attention in recent years and a remarkable refinements and generalizations have been studied ^[1,2].

The importance of the study of set-valued analysis from a theoretical point of view as well as from their applications is well known. Many advances in set-valued analysis have been motivated by control theory and dynamical games. Optimal control theory and mathematical programming were an engine driving these domains since the dawn of the sixties. Interval analysis is a particular case and it was introduced as an attempt to handle interval uncertainty that appears in many mathematical or computer models of some deterministic real-world phenomena.

Furthermore, a few significant inequalities like Hermite-Hadamard and Ostrowski type inequalities have been established for interval valued functions in recent years. In ^[3,4], Chalco-Cano et al. established Ostrowski type inequalities for interval valued functions by using Hukuhara derivatives for interval valued functions. In ^[5], Román-Flores et al. established Minkowski and Beckenbach's inequalities for interval valued functions. For other related results we refer to the readers ^[6].

In this paper, we establish Hermite-Hadamard type inequalities and He's inequality for interval-valued exponential type pre-invex functions in the Riemann-Liouville interval-valued fractional operator settings.

2.   Preliminaries

We begin with recalling some basic concepts and notions in the convex analysis.

Let the space of all intervals of $\Re$ is $\Re_{c}$ and $\varLambda\in\Re_{c}$ given by

Various binary operations are given as follows ^[7]:

Scalar multiplication: $\tau\in\Re,$

Difference, addition, product and reciprocal for $\varLambda_{1}, \varLambda_{2}\in\Re_{c}$ are respectively given by

Let $\Re_{\varLambda}, \Re_{\varLambda}^{+}\; and\; \Re_{\varLambda}^{-}$ denote the collection of all closed intervals of $\Re$, the collection of all positive intervals of $\Re$ and the collection of all negative intervals of $\Re$ respectively. In this paper, we examine a few algebraic properties of interval arithmetic.

Definition 2.1. ^[7] A mapping $\varOmega$ is called an interval-valued function of $\upsilon$ on $[a_{1}, b_{1}]$ if it assigns a nonempty interval to every $v\in[a_{1}, b_{1}]$, that is

where $\overleftrightarrow{\varOmega}(\upsilon)\; and\; \underleftrightarrow{\varOmega}(\upsilon)$ are both real valued functions.

Consider any finite ordered subset $\complement$ be the partition of $[a_{1}, b_{1}]$, that is

The mesh of $\complement$ is

The Riemann sum of $\varOmega:[a_{1}, b_{1}]\rightarrow \Re_{\varLambda}$ can be defined by

where $mesh(\complement) < c$.

Definition 2.2. ^[8] A mapping $\varOmega:[a_{1}, b_{1}]\rightarrow \Re_{\varLambda}$ is called an interval-Riemann integrable on $[a_{1}, b_{1}] \; if\; \exists\; \varLambda\in\Re_{\varLambda}$ such that for every $\delta > 0$ satisfying

we have

Lemma 2.1. ^[9] Let $\varOmega:[a_{1}, b_{1}]\rightarrow \Re_{\varLambda}$ be an interval-valued function as in (2.1), then it is interval-Riemann integrable if and only if

In simple words, $\varOmega$ is interval-Riemann integrable if and only if $\overleftrightarrow{\varOmega}(v)\; and\; \underleftrightarrow{\varOmega}(v)$ are both Riemann integrable functions.

Definition 2.3. ^[10] Let $\varOmega\in L_{1}[a_{1}, b_{1}]$, then the Riemann-Liouville fractional integrals of order $m > 0$ with $0\leq a_{1}$ are defined by

Definition 2.4. ^[11,12] Let $\varOmega:[a_{1}, b_{1}]\rightarrow \Re_{\varLambda}$ be an interval-valued, interval-Riemann integrable function as in (2.1), then the interval Riemann-Liouville fractional integrals of order $m > 0$ with $0\leq a_{1}$ are defined by

Corollary 2.1. ^[12] Let $\varOmega:[a_{1}, b_{1}]\rightarrow \Re_{\varLambda}$ be an interval-valued function as in (2.1) such that $\overleftrightarrow{\varOmega}(v)\; and\; \underleftrightarrow{\varOmega}(v)$ are Riemann integrable functions, then

Definition 2.5. ^[13] A set $\varLambda\subset\Re^{n}$ with respect to a vector function $\eta:\Re^{n}\times\Re^{n}\rightarrow \Re^{n}$ is called an invex set if

Definition 2.6. ^[13] A function $\varOmega$ on the invex set $\varLambda$ with respect to a vector function $\eta:\varLambda\times \varLambda\rightarrow \Re^{n}$ is called pre-invex function if

Lemma 2.2. ^[14,15] If $\varLambda$ is open and $\eta:\varLambda\times \varLambda\rightarrow \Re$, then $\forall a_{1}, b_{1}\in \varLambda, \; \tau, \tau_{1}, \tau_{2}\in[0, 1]$, we have

In ^[16], Noor presented Hermite-Hadamard-inequality for pre-invex function, as follows:

Definition 2.7. ^[15] Let us consider an interval-valued function $\varOmega$ on the set $\varLambda$, then $\varOmega$ is pre-invex interval valued function with respect to $\eta$ on an invex set $\varLambda\subset\Re^{n}$ with respect to a vector function $\eta:\varLambda\times \varLambda\rightarrow \Re^{n}$ if

Taking motivation from the exponential type convexity proposed in ^[17], we introduce the following notion:

Definition 2.8. A function $\varOmega$ on the invex set $\varLambda$ is called exponential-type pre-invex function with respect to a vector function $\eta:\varLambda\times \varLambda\rightarrow \Re^{n}$ if

It is important to note that a pre-invex function need not to be convex function. For example, the function $f(x) = -|x|$ is not a convex function but it is a pre-invex function with respect to $\eta$, where

Theorem 2.1. Let $\varOmega:[a_{1}, b_{1}]\rightarrow \Re$ be an exponential-type pre-invex function with respect to a vector function $\eta:\varLambda\times \varLambda\rightarrow \Re^{n}$. If $a_{1} < b_{1}$ and $\varOmega\in L[a_{1}, b_{1}]$, then we have

Proof. At first, from exponential-type-pre-invexity of $\varOmega$, we have

Integrating the above inequality with respect to $\tau_{1}\in[0, 1]$ yields

Now, taking $v = b_{1}+\tau_{1}\eta(a_{1}, b_{1})$ gives

This completes the proof.

By merging the concepts of pre-invexity and exponential type pre-invexity, we propose the following notion:

Definition 2.9. Let $\varLambda\subset\Re^{n}$ be an invex set with respect to a vector function $\eta:\varLambda\times \varLambda\rightarrow \Re^{n}$. The interval valued function $\varOmega$ on the set $\varLambda$ is exponential-type pre-invex interval valued function with respect to $\eta$ if

Remark 2.1. In Definition 2.9, by taking $h(\tau_{1}) = e^{\tau_{1}}-1$, where $h:[0, 1]\subset[a_{1}, b_{1}]\rightarrow \Re$ and $h\neq0$, then we get $h$-pre-invex interval valued function with respect to $\eta$, that is

Remark 2.2. Let $\varLambda\subset\Re^{n}$ be an invex set with respect to a vector function $\eta:\Re^{n}\times\Re^{n}\rightarrow \Re^{n}$. The interval valued function $\varOmega$ on the set $\varLambda$ is exponential-type-pre-invex function with respect to $\eta$ if and only if $\overleftrightarrow{\varOmega}, \underleftrightarrow{\varOmega}$ are exponential-type pre-invex functions with respect to $\eta$, that is

Remark 2.3. If $\overleftrightarrow{\varOmega}(v) = \underleftrightarrow{\varOmega}(v)$, then we get (2.12).

Remark 2.4. Since $\tau_{1}\leq e^{\tau_{1}}-1$ and $1-\tau_{1}\leq e^{1-\tau_{1}}-1$ for all $\; \tau_{1}\in[0, 1]$, so every nonnegative pre-invex interval valued function with respect to $\eta$ is also exponential-type pre-invex interval valued function with respect to $\eta$.

3.   Interval fractional Hermite-Hadamard type inequalities

In this section, we establish fractional Hermite-Hadamard type inequality for interval-valued exponential type pre-invex. The family of Lebesgue measurable interval-valued functions is denoted by $L([v_{1}, v_{2}], \Re_{0})$.

Theorem 3.1. Let $\varLambda\subset\Re$ be an open invex set with respect to $\eta:\varLambda\times \varLambda\rightarrow \Re$ and $a_{1}, b_{1}\in \varLambda$ with $a_{1} < a_{1}+\eta(b_{1}, a_{1})$. If $\varOmega:[a_{1}, a_{1}+\eta(b_{1}, a_{1})]\rightarrow \Re$ is an exponential type pre-invex interval-valued function such that $\varOmega\in L[a_{1}, a_{1}+\eta(b_{1}, a_{1})]$ and $m > 0$, then we have (considering Lemma 2.2 holds)

where

Proof. Since $\varOmega$ is an exponential type pre-invex interval-valued function, so

Taking $a_{1} = c_{1}+(1-\tau_{1})\eta(d_{1}, c_{1})$ and $b_{1} = c_{1}+(\tau_{1})\eta(d_{1}, c_{1})$ gives

implies

By multiplying by $\tau_{1}^{m-1}$ on both sides and integrating over $[0, 1]$ with respect to $\tau_{1}$, we get

Similarly

From (3.3)–(3.5), we get

Now, from the interval valued exponential type pre-invexity of $\varOmega$, we have

Similarly

Thus, by adding (3.7) and (3.8), we get

By multiplying by $\tau_{1}^{m-1}$ on both sides and integrating over $[0, 1]$ with respect to $\tau_{1}$, we get

Now, from (3.2) we get

Also from (3.4), (3.5) and (3.9), we get

Combining (3.6) and (3.10), we get

Corollary 3.1. If $\overleftrightarrow{\varOmega}(v) = \underleftrightarrow{\varOmega}(v)$, then (3.1) leads to the following fractional inequality for exponential type pre-invex function:

Theorem 3.2. Let $\varLambda\subset\Re$ be an open invex set with respect to $\eta:\varLambda\times \varLambda\rightarrow \Re$ and $a_{1}, b_{1}\in \varLambda$ with $a_{1} < a_{1}+\eta(b_{1}, a_{1})$. If $\varOmega, \varOmega_{1}:[a_{1}, a_{1}+\eta(b_{1}, a_{1})]\rightarrow \Re$ are exponential type pre-invex interval-valued functions such that $\varOmega, \varOmega_{1}\in L[a_{1}, a_{1}+\eta(b_{1}, a_{1})]$ and $m > 0$, then we have (considering Lemma 2.2 holds)

where

and

Proof. Since $\varOmega\; and\; \varOmega_{1}$ are exponential type pre-invex interval-valued functions, so we have

and

Since $\varOmega, \varOmega_{1}\in\Re_{\varLambda}^{+}$, so

Similarly, we have

Adding (3.16) and (3.17) yields

From (3.14) and (3.15), we have

Multiplying by $\tau_{1}^{m-1}$ on both sides and integrating over $[0, 1]$ with respect to $\tau_{1}$ gives

So

and

From (3.12) and (3.13), we get

and

Thus,

Corollary 3.2. If $\overleftrightarrow{\varOmega}(v) = \underleftrightarrow{\varOmega}(v)$, then (3.11) leads to the following fractional inequality for exponential type pre-invex function:

Theorem 3.3. Let $\varLambda\subset\Re$ be an open invex set with respect to $\eta:\varLambda\times \varLambda\rightarrow \Re$ and $a_{1}, b_{1}\in \varLambda$ with $a_{1} < a_{1}+\eta(b_{1}, a_{1})$. If $\varOmega, \varOmega_{1}:[a_{1}, a_{1}+\eta(b_{1}, a_{1})]\rightarrow \Re$ are exponential type pre-invex interval-valued functions such that $\varOmega, \varOmega_{1}\in L[a_{1}, a_{1}+\eta(b_{1}, a_{1})]$ and $m > 0$, then from (3.12)–(3.15), we have (considering Lemma 2.2 holds)

Proof. Since $\varOmega$ is an exponential type pre-invex interval-valued function, so we have

Taking $a_{1} = c_{1}+(1-\tau_{1})\eta(d_{1}, c_{1})$ and $b_{1} = c_{1}+(\tau_{1})\eta(d_{1}, c_{1})$ gives

implies

Similarly

Multiplying (3.19) and (3.20) gives

Since $\varOmega, \varOmega_{1}\in\Re_{\varLambda}^{+}, \;$ are exponential type pre-invex interval-valued functions for $\tau_{1}\in[0, 1]$, so we have

Similarly

Adding (3.22) and (3.23) yields

Now from (3.21), we can write

Multiplying by $\tau_{1}^{m-1}$ on both sides and integrating over $[0, 1]$ with respect to $\tau_{1}$ yields

Thus from (3.12)–(3.15), we get

Corollary 3.3. If $\overleftrightarrow{\varOmega}(v) = \underleftrightarrow{\varOmega}(v)$, then (3.18) leads to the following fractional inequality for exponential type pre-invex function:

4.   Interval fractional Hermite-Hadamard type Inequality via He's fractional derivative

In this section, we establish Hermite-hadamard type inequality in the setting of the He's fractional derivatives introduced in ^[18].

Definition 4.1. Let $\varOmega$ be an $L_{1}$ function defined on an interval $[0, n_{1}]$. Then the $k_{1}$-th He's fractional derivative of $\varOmega(n_{1})$ is defined by

The interval He's fractional derivative based on left and right end point functions can be defined by

where

and

Theorem 4.1. Let $\varOmega:[n_{1}, n_{2}]\rightarrow \Re$ be an exponential type pre-invex interval-valued function defined on $[n_{1}, n_{2}]\subset \varLambda$, where $\varLambda$ is an open invex set with respect to $\eta:\varLambda\times \varLambda\rightarrow \Re$ and $\varOmega:[n_{1}, n_{2}]\subset\Re\rightarrow \Re_{c}^{+}$ is given by $\varOmega(n) = [\underleftrightarrow{\varOmega}(n), \overleftrightarrow{\varOmega}(n)]$ for all $n\in[n_{1}, n_{2}]$. If $\varOmega\in L_{1}([n_{1}, n_{2}], \Re)$, then

Proof. Let $\varOmega:[n_{1}, n_{2}]\rightarrow \Re$ be an exponential type pre-invex interval-valued function defined on $[n_{1}, n_{2}]$, then

and

Taking $n_{2} = 0\; ,\; 0\leq n_{1}$ and multiplying by $\dfrac{(\tau_{1}-n)^{i-k_{1}-1}}{\Gamma(i-k_{1})}$, we get

Integrating with respect to $\tau_{1}$ over $[0, n_{1}]$ gives

implies

Getting $i$-th derivative on both sides and using (4.1), we get

Similarly

Thus, we can write

So,

Corollary 4.1. If $\overleftrightarrow{\varOmega}(v) = \underleftrightarrow{\varOmega}(v)$, then (4.3) leads to the following fractional inequality for exponential type pre-invex function:

5.   Conclusions

In this paper we studied the interval-valued exponential type pre-invex functions. We established He's and Hermite-Hadamard type inequalities for interval-valued exponential type pre-invex functions in the setting of Riemann-Liouville interval-valued fractional operator.

Acknowledgments

This work was sponsored in part by Henan Science and Technology Project of China (No:182102110292).

Conflict of interest

The author declares no conflict of interest.