Abstract
Recent research has gained more attention on conformable integrals and derivatives to derive the various type of inequalities. One of the recent advancements in the field of fractional calculus is the generalized nonlocal proportional fractional integrals and derivatives lately introduced by Jarad et al. (Eur. Phys. J. Special Topics 226:3457–3471, 2017) comprising the exponential functions in the kernels. The principal aim of this paper is to establish reverse Minkowski inequalities and some other fractional integral inequalities by utilizing generalized proportional fractional integrals. Also, two new theorems connected with this inequality as well as other inequalities associated with the generalized proportional fractional integrals are established.
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1 Introduction
Fractional calculus is a study of integrals and derivatives of arbitrary order which was a natural outgrowth of conventional definitions of calculus integral and derivative. Fractional integral has been comprehensively studied in the literature. The idea has been defined by numerous mathematicians with a slightly different formula, for example, Riemann–Liouville, Weyl, Erdélyi–Kober, Hadamard integral, Liouville and Katugampola fractional integral (see [18, 22, 23, 26, 34]). In the last few years, Khalil et al. [24] and Abdeljawad [1] established a new class of fractional derivatives and integrals called fractional conformable derivatives and integrals. Jarad et al. [21] introduced the fractional conformable integral operators. On the basis of that idea, one can obtain the generalizations of the inequalities: Hadamard, Hermite–Hadamard, Opial, Grüss, Ostrowski, Chebyshev, among others [19, 35, 37,38,39]).
Later on in [6], Anderson and Ulness improved the idea of the fractional conformable derivative by introducing the idea of local derivatives. In [2, 3, 7, 9, 27] researchers introduced new fractional derivative operators by using exponential and Mittag-Leffler functions in their kernels. In [20], Jarad et al. proposed the left and right generalized nonlocal proportional fractional integral and derivative operators. Such generalizations motivate future research to present more innovative ideas to unify the fractional operators and obtain the inequalities involving such fractional operators. The integral inequalities and their applications play an essential role in the theory of differential equations and applied mathematics. A variety of various types of some classical integral inequalities and their generalizations have been established by utilizing the classical fractional integral, fractional derivative operators (see, e.g., [4, 12, 14,15,16,17, 25, 28,29,30, 32, 33, 36, 41, 42, 46, 47]).
The reverse Minkowski fractional integral inequalities are perceived in [13]. Anber et al. [5] have gained some fractional integral inequalities by using Riemann–Liouville fractional integral. In [11], the authors established Minkowski inequalities and some other inequalities by employing Katugampola fractional integral operators. In [10, 45], the authors established the reverse Minkowski inequality for Hadamard fractional integral operators. In [31], Mubeen et al. recently established the reverse Minkowski inequalities and some related inequalities for generalized k-fractional conformable integrals.
This paper is organized as follows: In the second section, we present some known results and basic definitions. In the third section, the reverse Minkowski inequalities are presented. In the fourth section, some other related inequalities involving generalized nonlocal proportional fractional integrals are presented.
2 Preliminaries
This section is devoted to some known definitions and results associated with the classical Riemann–Liouville fractional integrals and their generalization involving the Riemann–Liouville fractional integrals. Set et al. [40] presented Hermite–Hadamard and reverse Minkowski inequalities for Riemann–Liouville fractional integrals. In [8], Bougoffa also presented Hardy’s and reverse Minkowski inequalities. The following theorems involving the reverse Minkowski inequalities are the motivation of work performed so far, involving the classical Riemann integrals.
Theorem 2.1
([40])
Let \(r\geq 1\) and let g, h be two positive functions on \([0,\infty )\). If \(0< m\leq \frac{g(\rho )}{h(\rho )} \leq M\), \(\vartheta \in [a,b]\), then the following inequality holds:
Theorem 2.2
([40])
Let \(r\geq 1\) and let g, h be two positive functions on \([0,\infty )\). If \(0< m\leq \frac{g(\rho )}{h(\rho )} \leq M\), \(\vartheta \in [a,b]\), then the following inequality holds:
Definition 2.1
The left and right R-L fractional integrals of order λ are respectively defined by
and
where \(\lambda \in \mathbb{C}\) and \(\Re (\lambda )>0\).
In [13], Dahmani introduced the following reverse Minkowski inequalities involving the R-L fractional integral operators.
Theorem 2.3
([13])
Let \(\lambda \in \mathbb{C}\), \(\Re (\lambda )>0\), \(r\geq 1\), and let g, h be two positive functions on \([0,\infty )\) such that, for all \(\vartheta >0\), \(\mathfrak{I}^{\lambda }g^{r}( \vartheta )<\infty \), \(\mathfrak{I}^{\lambda }h^{r}(\vartheta )< \infty \). If \(0< m\leq \frac{g(\rho )}{h(\rho )}\leq M\), \(\rho \in [a, \vartheta ]\), then the following inequality holds:
Theorem 2.4
([13])
Let \(\lambda \in \mathbb{C}\), \(\Re (\lambda )>0\), \(r\geq 1\), and let g, h be two positive functions on \([0,\infty )\) such that, for all \(\vartheta >0\), \(\mathfrak{I}^{\lambda }g^{r}( \vartheta )<\infty \), \(\mathfrak{I}^{\lambda }h^{r}(\vartheta )< \infty \). If \(0< m\leq \frac{g(\rho )}{h(\rho )}\leq M\), \(\rho \in [a, \vartheta ]\), then the following inequality holds:
Definition 2.2
([20])
The left and right generalized nonlocal proportional integral operators are respectively defined by
and
where \(\eta \in (0,1]\) and \(\lambda \in \mathbb{C}\) and \(\Re (\lambda )>0\).
Remark 2.1
If we consider \(\eta =1\) in (7) and (8), then we get the left and right Riemann–Liouville (3) and (4) respectively.
3 Reverse Minkowski inequalities via generalized proportional fractional integral operator
In this section, we use generalized nonlocal proportional fractional integral operator to develop reverse Minkowski integral inequalities. The reverse Minkowski fractional integral inequality is presented in the following theorem.
Theorem 3.1
Let \(\eta \in (0,1]\), \(\lambda \in \mathbb{C}\), \(\Re (\lambda )>0\), \(r\geq 1\), and let g, h be two positive functions on \([0,\infty )\) such that, for all \(\vartheta >0\), \({}_{a}\mathfrak{I}^{\lambda , \eta }g^{r}(\vartheta )<\infty \), \({}_{a}\mathfrak{I}^{\lambda , \eta }h^{r}(\vartheta )<\infty \). If \(0< m\leq \frac{g(\rho )}{h( \rho )}\leq M\), \(\rho \in [a,\vartheta ]\), then the following inequality holds:
Proof
Under the condition stated in Theorem 3.1, \(\frac{g(\rho )}{h( \rho )}\leq M\), \(\rho \in [0,\vartheta ]\), \(\vartheta >0\), we have
Consider a function
We observe that the function \(\mathfrak{F}(\vartheta ,\rho )\) remains positive for all \(\rho \in (a,\vartheta )\), \(a<\vartheta \leq b\), since each term of the above function is positive in view of conditions stated in Theorem 3.1.
Multiplying both sides of (10) by \(\mathfrak{F}(\vartheta , \rho )\) and integrating the resultant inequality with respect to ρ from a to ϑ, we have
which can be written as
Hence, it follows that
Now, using the condition \(mg(\rho )\leq h(\rho )\), we have
it follows that
Multiplying both sides of (13) by \(\mathfrak{F}(\vartheta , \rho )\) and integrating the resultant inequality with respect to ρ from a to ϑ, we have
Thus adding inequalities (12) and (14) yields the desired inequality. □
Theorem 3.2
Let \(\eta \in (0,1]\), \(\lambda \in \mathbb{C}\), \(\Re (\lambda )>0\), \(r\geq 1\), and let g, h be two positive functions on \([0,\infty )\) such that, for all \(\vartheta >0\), \({}_{a}\mathfrak{I}^{\lambda , \eta }g^{r}(\vartheta )<\infty \), \({}_{a}\mathfrak{I}^{\lambda , \eta }h^{r}(\vartheta )<\infty \). If \(0< m\leq \frac{g(\rho )}{h( \rho )}\leq M\), \(\rho \in [a,\vartheta ]\), then the following inequality holds:
Proof
The multiplication of inequalities (12) and (14) yields
Now, applying the Minkowski inequality to the right-hand side of (16), we obtain
Thus, from inequalities (16) and (17), we get the desired inequality (15). □
4 Certain related inequalities via generalized proportional fractional integral operator
This section is devoted to deriving certain related inequalities involving a generalized proportional fractional integral operator.
Theorem 4.1
Let \(\eta \in (0,1]\), \(\lambda \in \mathbb{C}\), \(\Re (\lambda )>0\), \(r>1\), \(1/r+1/s =1\), and let g, h be two positive functions on \([0,\infty )\) such that \({}_{a}\mathfrak{I}^{\lambda ,\eta }[g(\vartheta )]<\infty \), \({}_{a}\mathfrak{I}^{\lambda ,\eta }[h(\vartheta )]< \infty \). If \(0< m\leq \frac{g(\rho )}{h(\rho )}\leq M<\infty \), \(\rho \in [a,\vartheta ]\), \(\vartheta >a\), we have
Proof
Since \(\frac{g(\rho )}{h(\rho )}\leq M<\infty \), \(\rho \in [a,\vartheta ]\), \(\vartheta >a\), therefore we have
It follows that
Multiplying both sides of (20) by \(\mathfrak{F}(\vartheta , \rho )\) where \(\mathfrak{F}(\vartheta ,\rho )\) is defined by (11) and integrating the resultant inequality with respect to ρ from a to ϑ, we have
It follows that
Consequently, we have
On the other hand, \(m g(\rho )\leq h(\rho )\), \(\rho \in [a,\vartheta ]\), \(\vartheta >a\), therefore we have
It follows that
Again, multiplying both sides of (25) by \(\mathfrak{F}( \vartheta ,\rho )\) where \(\mathfrak{F}(\vartheta ,\rho )\) is defined by (11) and integrating the resultant inequality with respect to ρ from a to ϑ, we have
Hence, we can write
Multiplying (23) and (27), we get the desired inequality. □
Theorem 4.2
Let \(\eta \in (0,1]\), \(\lambda \in \mathbb{C}\), \(\Re (\lambda )>0\), \(r>1\), \(\frac{1}{r}+{1/s}=1\), and let g, h be two positive functions on \([0,\infty )\) such that \({}_{a}\mathfrak{I}^{\lambda ,\eta }[g^{r}( \vartheta )]<\infty \), \({}_{a}\mathfrak{I}^{\lambda ,\eta }[h^{s}( \vartheta )]<\infty \). If \(0< m\leq \frac{g(\rho )^{r}}{h(\rho )^{s}} \leq M<\infty \), \(\rho \in [a,\vartheta ]\), \(\vartheta >a\), we have
Proof
Replacing \(g(\vartheta )\) and \(h(\vartheta )\) by \(g^{r}(\vartheta )\) and \(h^{r}(\vartheta )\), \(a<\vartheta \leq b\) in Theorem 4.1, we get the desired inequality (28). □
Theorem 4.3
Let \(\eta \in (0,1]\), \(\lambda \in \mathbb{C}\), \(\Re (\lambda )>0\), \(r>1\), \(\frac{1}{r}+{1/s}=1\), and let g, h be two positive functions on \([0,\infty )\) such that \({}_{a}\mathfrak{I}^{\lambda ,\eta }[g^{r}( \vartheta )]<\infty \), \({}_{a}\mathfrak{I}^{\lambda ,\eta }[h^{s}( \vartheta )]<\infty \). If \(0< m\leq \frac{g^{r}(\rho )}{h^{s}(\rho )} \leq M<\infty \) where \(m, M\in \mathbb{R}\), \(\rho \in [a,\vartheta ]\), \(\vartheta >a\), then the following inequality for left generalized proportional fractional integral holds:
Proof
By the given hypothesis \(\frac{g(\rho )}{h(\rho )}\leq M\), we have
Multiplying both sides of inequality (30) by \(\mathfrak{F}( \vartheta ,\rho )\) where \(\mathfrak{F}(\vartheta ,\rho )\) is defined by (11) and integrating the resultant identity with respect to ρ over \((a,\vartheta )\), we get
It follows that
On the other hand, using \(m\leq \frac{g(\rho )}{h(\rho )}\), \(a< t<\vartheta \), we have
Again, multiplying both sides of inequality (33) by \(\mathfrak{F}(\vartheta ,\rho )\) where \(\mathfrak{F}(\vartheta , \rho )\) is defined by (11) and integrating the resultant identity with respect to ρ over \((a,\vartheta )\), we get
Now, using Young’s inequality, we have
Multiplying both sides of inequality (33) by \(\mathfrak{F}( \vartheta ,\rho )\) where \(\mathfrak{F}(\vartheta ,\rho )\) is defined by (11) and integrating the resultant identity with respect to ρ over \((a,\vartheta )\), we get
With the aid of (32) and (34), (36) can be written as
Now, using the inequality \((\rho +\omega )^{r}\leq 2^{s-1}(\rho ^{r}+ \omega ^{r})\), \(r>1\), \(\rho , \omega >0\), one can obtain
and
Hence the proof of (29) can follow from (37), (38), and (39). □
Theorem 4.4
Let \(\eta \in (0,1]\), \(\lambda \in \mathbb{C}\), \(\Re (\lambda )>0\), \(r\geq 1\), and let g, h be two positive functions on \([0,\infty )\) such that \({}_{a}\mathfrak{I}^{\lambda ,\eta }[g^{r}(\vartheta )]< \infty \), \({}_{a}\mathfrak{I}^{\lambda ,\eta }[h^{r}(\vartheta )]< \infty \). If \(0< k< m\leq \frac{g(\rho )}{h(\rho )}\leq M<\infty \), where \(m, M\in \mathbb{R}\), \(\rho \in [a,\vartheta ]\), \(\vartheta >a\), then the following inequality for left generalized proportional fractional integral holds:
Proof
Under the given hypothesis \(0< k< m\leq \frac{g^{r}(\rho )}{h^{s}( \rho )}\leq M<\infty \), we have
It can be written as
Also, we have
It follows that
Also, we have
It follows that
Multiplying both sides of inequality (41) by \(\mathfrak{F}( \vartheta ,\rho )\) where \(\mathfrak{F}(\vartheta ,\rho )\) is defined by (11) and integrating the resultant identity with respect to ρ over \((a,\vartheta )\), we get
It follows that
Again, multiplying both sides of inequality (42) by \(\mathfrak{F}(\vartheta ,\rho )\) where \(\mathfrak{F}(\vartheta , \rho )\) is defined by (11) and integrating the resultant identity with respect to ρ over \((a,\vartheta )\), we get
Hence, by adding inequalities (43) and (44), we get the desired inequality (40). □
Theorem 4.5
Let \(\eta \in (0,1]\), \(\lambda \in \mathbb{C}\), \(\Re (\lambda )>0\), \(r\geq 1\), and let g, h be two positive functions on \([0,\infty )\) such that \({}_{a}\mathfrak{I}^{\lambda ,\eta }[g^{r}(\vartheta )]< \infty \), \({}_{a}\mathfrak{I}^{\lambda ,\eta }[h^{r}(\vartheta )]< \infty \). If \(0\leq \alpha \leq g(\rho )\leq \mathcal{A}\) and \(0\leq \sigma \leq h(\rho )\leq \mathcal{B}\) for all \(\rho \in [a, \vartheta ]\), \(\vartheta >a\), then the following inequality for left generalized proportional fractional integral holds:
Proof
Under the given hypothesis, we have
The product of inequality (46) with \(0\leq \alpha \leq g( \rho )\leq \mathcal{A}\) yields
From (47), we obtain
and
Now, multiplying both sides of inequalities (48) and (49) respectively by \(\mathfrak{F}(\vartheta ,\rho )\) where \(\mathfrak{F}(\vartheta ,\rho )\) is defined by (11) and integrating the resultant identity with respect to ρ over \((a,\vartheta )\), we obtain
and
Hence, by adding (50) and (51), we get the desired proof. □
Theorem 4.6
Let \(\eta \in (0,1]\), \(\lambda \in \mathbb{C}\), \(\Re (\lambda )>0\), \(r\geq 1\), and let g, h be two positive functions on \([0,\infty )\) such that \({}_{a}\mathfrak{I}^{\lambda ,\eta }[g(\vartheta )]<\infty \), \({}_{a}\mathfrak{I}^{\lambda ,\eta }[h(\vartheta )]<\infty \). If \(0< m\leq \frac{g(\rho )}{h(\rho )}\leq M\) where \(m, M\in \mathbb{R}\) for all \(\rho \in [a,\vartheta ]\), \(\vartheta >a\), then the following inequality for the left generalized proportional fractional integral holds:
Proof
Under the given hypothesis, \(0< m\leq \frac{g(\rho )}{h(\rho )}\leq M\), we have
Also, we have \(\frac{1}{M}\leq \frac{h(\rho )}{g(\rho )}\leq \frac{1}{m}\), which gives
The multiplication of (53) and (54) yields
Now, multiplying both sides of inequality (55) by \(\mathfrak{F}(\vartheta ,\rho )\) where \(\mathfrak{F}(\vartheta ,\rho )\) is defined by (11) and integrating the resultant identity with respect to ρ over \((a,\vartheta )\), we have
It follows that
which completes the desired proof. □
Theorem 4.7
Let \(\eta \in (0,1]\), \(\lambda \in \mathbb{C}\), \(\Re (\lambda )>0\), \(r\geq 1\), and let g, h be two positive functions on \([0,\infty )\) such that \({}_{a}\mathfrak{I}^{\lambda ,\eta }[g(\vartheta )]<\infty \), \({}_{a}\mathfrak{I}^{\lambda ,\eta }[h(\vartheta )]<\infty \). If \(0< m\leq \frac{g(\rho )}{h(\rho )}\leq M\), where \(m, M\in \mathbb{R}\) for all \(\rho \in [a,\vartheta ]\), \(\vartheta >a\), then the following inequality for the left generalized proportional fractional integral holds:
where \(h (g(\vartheta ),h(\vartheta ) )=\max \{M [ (\frac{M}{m}+1 )g(\rho )-Mh(\rho ) ],\frac{(m+M)h( \rho )-g(\rho )}{m} \}\).
Proof
Under the given hypothesis \(0< m\leq \frac{g(\rho )}{h(\rho )}\leq M\), where \(\rho \in [a,\vartheta ]\), \(\vartheta >a\), we have
and
where \(h (g(\vartheta ),h(\vartheta ) )=\max \{M [ (\frac{M}{m}+1 )g(\rho )-Mh(\rho ) ],\frac{(m+M)h( \rho )-g(\rho )}{m} \}\). Also, from the given hypothesis \(0<\frac{1}{M}\leq \frac{h(\rho )}{g(\rho )}\leq \frac{1}{m}\), we have
and
It follows that
From (60) and (64), we can write
and
Now, multiplying both sides of inequalities (65) and (62) respectively by \(\mathfrak{F}(\vartheta ,\rho )\) where \(\mathfrak{F}(\vartheta ,\rho )\) is defined by (11) and integrating the resultant identity with respect to ρ over \((a,\vartheta )\), we get
It follows that
Similarly, from (62), we obtain
5 Concluding remarks
In this paper, we presented the Minkowski inequalities and some other related inequalities via generalized nonlocal proportional fractional integral operators. The results exhibited in Sect. 3 generalized the work earlier done by Dahmani [13] for Riemann–Liouville fractional integral operator. Also, the special cases of the results presented in Sect. 3 are found in [40]. The inequalities established in Sect. 4 generalized the inequalities earlier obtained by Suliman [44]. Also, our result will reduce to some classical results which are found in the work of Sroysang [43].
References
Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015). https://doi.org/10.1016/j.cam.2014.10.016
Abdeljawad, T., Baleanu, D.: Monotonicity results for fractional difference operators with discrete exponential kernels. Adv. Differ. Equ. 2017, 78 (2017). https://doi.org/10.1186/s13662-017-1126-1
Abdeljawad, T., Baleanu, D.: On fractional derivatives with exponential kernel and their discrete versions. Rep. Math. Phys. 80, 11–27 (2017). https://doi.org/10.1016/S0034-4877(17)30059-9
Alzabut, J., Abdeljawad, T., Jarad, F., Sudsutad, W.: A Gronwall inequality via the generalized proportional fractional derivative with applications. J. Inequal. Appl. 2019, 101 (2019)
Anber, A., Dahmani, Z., Bendoukha, B.: New integral inequalities of Feng Qi type via Riemann-Liouville fractional integration. Facta Univ., Ser. Math. Inform. 27(2), 13–22 (2012)
Anderson, D.R., Ulness, D.J.: Newly defined conformable derivatives. Adv. Dyn. Syst. Appl. 10(2), 109–137 (2015)
Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20, 763–769 (2016). https://doi.org/10.2298/TSCI160111018A
Bougoffa, L.: On Minkowski and Hardy integral inequalities. J. Inequal. Pure Appl. Math. 7(2), Article ID 60 (2006)
Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 73–85 (2015)
Chinchane, V.L., Pachpatte, D.B.: New fractional inequalities via Hadamard fractional integral. Int. J. Funct. Anal. Oper. Theory Appl. 5, 165–176 (2013)
da Vanterler, J., Sousa, C., Capelas de Oliveira, E.: The Minkowski’s inequality by means of a generalized fractional integral. AIMS Ser. Appl. Math. 3, 131–147 (2018). https://doi.org/10.3934/Math.2018.1.131
da Vanterler, J., Sousa, C., Oliveira, D.S., Capelas de Oliveira, E.: Grüss-type inequalities by means of generalized fractional integrals. Bull. Braz. Math. Soc. (2019). https://doi.org/10.1007/s00574-019-00138-z
Dahmani, Z.: On Minkowski and Hermite-Hadamard integral inequalities via fractional integral. Ann. Funct. Anal. 1, 51–58 (2010)
Dahmani, Z.: New inequalities in fractional integrals. Int. J. Nonlinear Sci. 9(4), 493–497 (2010)
Dahmani, Z., Tabharit, L.: On weighted Gruss type inequalities via fractional integration. J. Adv. Res. Pure Math. 2, 31–38 (2010)
Dragomir, S.S.: A generalization of Gruss’s inequality in inner product spaces and applications. J. Math. Anal. Appl. 237(1), 74–82 (1999)
Dragomir, S.S.: Some integral inequalities of Gruss type. Indian J. Pure Appl. Math. 31(4), 397–415 (2002)
Herrmann, R.: Fractional Calculus: An Introduction for Physicists. World Scientific, Singapore (2011)
Huang, C.J., Rahman, G., Nisar, K.S., Ghaffar, A., Qi, F.: Some inequalities of Hermite-Hadamard type for k-fractional conformable integrals. Aust. J. Math. Anal. Appl. 16(1), 1–9 (2019)
Jarad, F., Abdeljawad, T., Alzabut, J.: Generalized fractional derivatives generated by a class of local proportional derivatives. Eur. Phys. J. Spec. Top. 226, 3457–3471 (2017). https://doi.org/10.1140/epjst/e2018-00021-7
Jarad, F., Ugrlu, E., Abdeljawad, T., Baleanu, D.: On a new class of fractional operators. Adv. Differ. Equ. 2017(1), 247 (2017). https://doi.org/10.1186/s13662-017-1306-z
Katugampola, U.N.: A new approach to generalized fractional derivatives. Bull. Math. Anal. Appl. 6, 1–15 (2014)
Katugampola, U.N.: New fractional integral unifying six existing fractional integrals (2016) arXiv:1612.08596
Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264(65), 65–70 (2014)
Khan, H., Abdeljawad, T., Tunç, C., Alkhazzan, A., Khan, A.: Minkowski’s inequality for the AB-fractional integral operator. J. Inequal. Appl. 2019, 96 (2019)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 207. Elsevier, Amsterdam (2006)
Losada, J., Nieto, J.J.: Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 87–92 (2015)
McD Mercer, A.: An improvement of the Gruss inequality. JIPAM. J. Inequal. Pure Appl. Math. 10(4), Article ID 93 (2005)
McD Mercer, A., Mercer, P.: New proofs of the Gruss inequality. Aust. J. Math. Anal. Appl. 1(2), Article ID 12 (2004)
Mitrinovic, D.S., Pecaric, J.E., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer Academic Publishers, Dordrecht (1993)
Mubeen, S., Habib, S., Naeem, M.N.: The Minkowski inequality involving generalized k-fractional conformable integral, Mubeen et al. J. Inequal. Appl. 2019, 81 (2019). https://doi.org/10.1186/s13660-019-2040-8
Nisar, K.S., Qi, F., Rahman, G., Mubeen, S., Arshad, M.: Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function. J. Inequal. Appl. 2018, 135 (2018)
Nisar, K.S., Rahman, G., Choi, J., Mubeen, S., Arshad, M.: Certain Gronwall type inequalities associated with Riemann-Liouville k- and Hadamard k-fractional derivatives and their applications. East Asian Math. J. 34(3), 249–263 (2018)
Podlubny, I.: Fractional Differential Equation. Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1999)
Qi, F., Rahman, G., Hussain, S.M., Du, W.S., Nisar, K.S.: Some inequalities of Čebyšev type for conformable k-fractional integral operators. Symmetry 10, 614 (2018). https://doi.org/10.3390/sym10110614
Rahman, G., Nisar, K.S., Mubeen, S., Choi, J.: Certain inequalities involving the \((k,\rho )\)-fractional integral operator. Far East J. Math. Sci.: FJMS 103(11), 1879–1888 (2018)
Rahman, G., Nisar, K.S., Qi, F.: Some new inequalities of the Gruss type for conformable fractional integrals. AIMS Ser. Appl. Math. 3(4), 575–583 (2018)
Rahman, G., Ullah, Z., Khan, A., Set, E., Nisar, K.S.: Certain Chebyshev type inequalities involving fractional conformable integral operators. Mathematics 7, 364 (2019). https://doi.org/10.3390/math7040364
Set, E., Mumcu, İ., Demirbaş, S.: Conformable fractional integral inequalities of Chebyshev type. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 2253–2259 (2019). https://doi.org/10.1007/s13398-018-0614-9
Set, E., Özdemir, M., Dragomir, S.: On the Hermite-Hadamard inequality and other integral inequalities involving two functions. J. Inequal. Appl. 2010, 148102 (2010)
Set, E., Tomar, M., Sarikaya, M.Z.: On generalized Grüss type inequalities for k-fractional integrals. Appl. Math. Comput. 269, 29–34 (2015)
Sousa, J., Capelas de Oliveira, E.: The Minkowski’s inequality by means of a generalized fractional integral. AIMS Ser. Appl. Math. 3(1), 131–147 (2018)
Sroysang, B.: More on reverses of Minkowski’s integral inequality. Math. Æterna 3, 597–600 (2013)
Sulaiman, W.T.: Reverses of Minkowski’s, Hölder’s, and Hardy’s integral inequalities. Int. J. Mod. Math. Sci. 1, 14–24 (2012)
Taf, S., Brahim, K.: Some new results using Hadamard fractional integral. Int. J. Nonlinear Anal. Appl. 7, 103–109 (2015)
Usta, F., Budak, H., Ertuǧral, F., Sarıkaya, M.Z.: The Minkowski’s inequalities utilizing newly defined generalized fractional integral operators. Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 68(1), 686–701 (2019)
Vanterlerda, J., Sousa, C., Capelas de Oliveira, E.: On the Ψ-fractional integral and applications. Comput. Appl. Math. 38, 4 (2019). https://doi.org/10.1007/s40314-019-0774-z
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Rahman, G., Khan, A., Abdeljawad, T. et al. The Minkowski inequalities via generalized proportional fractional integral operators. Adv Differ Equ 2019, 287 (2019). https://doi.org/10.1186/s13662-019-2229-7
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DOI: https://doi.org/10.1186/s13662-019-2229-7