Some inequalities via fractional conformable integral operators

In this paper, we adopt conformable fractional integral to develop integral inequalities such as Minkowski and Hermite–Hadamard inequalities. Our results are the generalization of the inequalities obtained by Dahmani and Bougoffa cited in the literature.

The theory of fractional integral inequalities plays a vital role in the field of mathematical sciences. There is one of the most famous inequalities for convex functions known as Hermite–Hadamard inequality. Many researchers studied this inequality and published various generalizations and extensions by using fractional integral. We begin with the Hermite–Hadamard inequality, which is defined as follows: Let \(f:I\subseteq \mathbb{R}\rightarrow \mathbb{R}\) be a convex function and \(a, b\in I\) with \(a< b\), then

Further generalizations and extensions can be found in, e.g., [3, 8, 10, 17]. In [15], the Riemann–Liouville fractional integrals \(\mathfrak{I}_{a^{+}}^{\alpha }\) and \(\mathfrak{I}_{b^{-}}^{\alpha }\) of order \(\alpha >0\) are defined respectively by

and

where Γ is the gamma function (see [18]). In [7], the left- and right-sided fractional conformable integral operators are respectively defined by

and

where \(\beta \in \mathbb{C}\) and \(\Re (\beta )>0\). Obviously, if we consider \(a=0\), \(b=0\), and \(\alpha =1\) in (4) and (5), then we get the Riemann–Liouville fractional integrals (2) and (3) respectively. In [16], Set et al. defined the following one-sided conformable fractional integral operator:

Recently Rahman et al. [13, 14] established some new inequalities of the Grüss type and certain Chebyshev-type inequalities for conformable fractional integrals. In [5, 9, 11, 12], various researchers established generalized k-fractional conformable integral inequalities, Minkowski and Chebyshev type integral inequalities involving generalized k-fractional conformable integrals. The Hermite–Hadamard type inequalities for k-fractional conformable integrals are found in [6]. A significant contribution by Guessab and Schmeisser [4] is an investigation of sharp integral inequalities of the Hermite–Hadamard type.

The paper is arranged as follows: In Sect. 2, the main results, which are reverse Minkowski and related Hermite–Hadamard type integral inequalities, are established by employing fractional conformable integral operators. The concluding remarks are given in Sect. 3.

In this section, we use fractional conformable integral operator to develop reverse Minkowski and Hermite–Hadamard integral inequalities. The reverse Minkowski fractional integral inequality is presented in the following theorems.

Theorem 2.1

Let \(\beta ,\alpha >0\), \(\sigma \geq 1\), and let Φ, Ψ be two positive functions on \([0,\infty )\) such that, for all \(x>0\), \({}^{\beta }\mathfrak{I}^{\alpha }\varPhi ^{\sigma }(x)<\infty \), \({}^{\beta }\mathfrak{I}^{\alpha }\varPsi ^{\sigma }(x)<\infty \). If \(0< m\leq \frac{\varPhi (t)}{\varPsi (t)}\leq M\), \(t\in [0,x]\), then the following inequality holds:

Proof

Using the condition \(\frac{\varPhi (t)}{\varPsi (t)}< M\), \(t\in [0,x]\), \(x>0\), we have

Multiplying both sides of (8) by \(\frac{1}{\varGamma (\beta )} (\frac{x^{\alpha }-t^{\alpha }}{\alpha } )^{\beta -1}t^{\alpha -1}\) and integrating the resultant inequality with respect to t from 0 to x, we have

which can be written as

Hence, it follows that

Now, using the condition \(m\varPsi (t)\leq \varPhi (t)\), we have

which yields

Multiplying both sides of (10) by \(\frac{1}{\varGamma (\beta )} (\frac{x^{\alpha }-t^{\alpha }}{\alpha } )^{\beta -1}t^{\alpha -1}\) and integrating the resultant inequality with respect to t from 0 to x, we get

Thus, adding inequalities (9) and (11) yields the desired inequality. □

Theorem 2.2

Let \(\beta , \alpha >0\), \(\beta \in \mathbb{C}\), \(\sigma \geq 1\), and let Φ, Ψ be two positive functions on \([0,\infty )\) such that, for all \(x>0\), \({}^{\beta }\mathfrak{I}^{\alpha }\varPhi ^{\sigma }(x)< \infty \), \({}^{\beta }\mathfrak{I}^{\alpha }\varPsi ^{\sigma }(x)<\infty \). If \(0< m\leq \frac{\varPhi (t)}{\varPsi (t)}\leq M\), \(t\in [0,x]\), then the following inequality holds:

Proof

From the multiplication of inequalities (9) and (11), we have

Now, applying the Minkowski inequality to the right-hand side of (13), we obtain

Thus, from inequalities (13) and (14), we get the desired inequality (12). □

Lemma 2.3

([2])

Let f be a concave function on \([a,b]\), then the following inequalities hold:

Theorem 2.4

Let \(\beta , \alpha >0\), \(\beta \in \mathbb{C}\), \(r, s>1\), and let Φ and Ψ be two positive functions on \([0,\infty )\). If \(\varPhi ^{r}\) and \(\varPsi ^{s}\) are two concave functions on \([0,\infty )\), then the following inequality holds:

Proof

Since the functions \(\varPhi ^{r}\) and \(\varPsi ^{s}\) are concave on \([0,\infty )\), therefore for any \(x>0\), \(\alpha >0\) and by Lemma 2.3, we have

and

Multiplying both sides of (17) and (18) by \(\frac{1}{\varGamma (\beta )} (\frac{x^{\alpha }-t^{\alpha }}{\alpha } )^{\beta -1}t^{\alpha \beta -1}\), \(t\in (0,x)\), and integrating the resultant inequalities from 0 to x, we get

and

Taking \(x^{\alpha }-t^{\alpha }=y^{\alpha }\), we have

and

Thus the use of (19) and (21) yields

Similarly, the use of (20) and (22) yields

From inequalities (23) and (24), it follows that

Since Φ and Ψ are positive functions, therefore for any \(x>0\), \(\alpha >0\), \(r\geq 1\), and \(s\geq 1\), we have

and

Hence, it follows that

and

From inequalities (28) and (29), we obtain

Thus, by combining (25) and (30), we get the desired result. □

Theorem 2.5

Let \(\beta , \mu , \alpha >0\), \(\beta , \mu \in \mathbb{C}\), \(r>1\), \(s>1\), and let Φ, Ψ be two positive functions on \([0,\infty )\). If \(\varPhi ^{r}\) and \(\varPsi ^{s}\) are two concave functions on \([0,\infty )\), then we have the following inequality:

Proof

Multiplying both sides of inequalities (17) and (18) by \(\frac{1}{\varGamma (\beta )} (\frac{x^{\alpha }-t ^{\alpha }}{\alpha } )^{\beta -1}t^{\mu \alpha -1}\), \(t\in (0,x)\) and then integrating the resultant inequalities with respect to t from 0 to x, we have

and

Now, using \(x^{\alpha }-t^{\alpha }=y^{\alpha }\), we have

and

Thus, from (32) and (34), we can write

Similarly, from inequalities (33) and (35), we obtain

From (36) and (37), it follows that

Since Φ and Ψ are positive functions, therefore for any \(x>0\), \(\alpha >0\), \(r\geq 1\), \(s\geq 1\), we have

and

Thus from (39) and (40) it follows that

Combining inequalities (38) and (41), we get the desired proof. □

Remark 1

Letting \(\beta =\mu \) in Theorem 2.5, we obtain Theorem 2.4.

In this paper, we established Minkowski and Hermite–Hadamard inequalities for conformable fractional integral operator. If we consider \(\alpha =1\) throughout the paper, then the obtained results will reduce to the said inequalities obtained by Dahmani [2]. Similarly, if we consider \(\alpha =\beta =1\), then all the results will lead to the classical inequalities obtained by [1].

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Affiliations

1. Department of Mathematics, College of Arts and Sciences, Prince Sattam Bin Abdulaziz University, Wadi Aldawasir, Saudi Arabia

   • Kottakkaran Sooppy Nisar

2. College of Computer and Information Sciences, Majmaah University, Al Majmaah 11952, Saudi Arabia

   • Asifa Tassaddiq

3. Department of Mathematics, Shaheed Benazir Bhutto University, Sheringal, Upper Dir, Pakistan

   • Gauhar Rahman
   •  & Aftab Khan

Contributions

All authors contributed equally. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Asifa Tassaddiq.

Competing interests

The authors declare that they have no competing interests.

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