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Chebyshev type inequalities via generalized fractional conformable integrals
Journal of Inequalities and Applications volume 2019, Article number: 245 (2019)
Abstract
Our aim in this present paper is to establish several Chebyshev type inequalities involving generalized fractional conformable integral operator recently introduced by T.U. Khan and M.A. Khan (J. Comput. Appl. Math. 346:378–389, 2019). Also, we present Chebyshev type inequalities involving Riemann–Liouville type fractional conformable integral operators as a particular result of our main result.
1 Introduction
Fractional calculus is an extremely useful tool to carry out differentiation and integration of real or complex number orders. This subject has received great consideration from researchers and mathematicians throughout the last few decades. Beginning with the classical Riemann–Liouville fractional integral and derivative operators, a large number of fractional integral and derivative operators and their generalizations have been presented (see, e.g., [2,3,4]). Among a large number of the fractional integral operators developed, due to applications in many fields of sciences, the Riemann–Liouville fractional integral operator has been extensively investigated. Integrations with weight functions are utilized in numerous mathematical problems such as approximation theory, spectral analysis, statistical analysis, and the theory of statistical distributions. Recently Khan et al. [5] established certain inequalities of the Hermite–Hadamard type with applications. In [6], the authors established Ostrowski type inequalities involving conformable fractional integrals. In [7,8,9,10], the authors introduced Hermite–Hadamard–Fejer inequalities for conformable fractional integrals via preinvex functions, generalized inequalities via GG-convexity and GA-convexity, Hermite–Hadamard type inequalities via the Montgomery identity, and Hermite–Hadamard type inequalities pertaining conformable fractional integrals and their applications. Rahman et al. [11] established certain Chebyshev type inequalities involving fractional conformable integral operators. In [12], the authors introduced the Minkowski inequalities via generalized proportional fractional integral operators. Some new inequalities involving fractional conformable integrals are found in the work of Nisar et al. [13]. Also, many researchers (see, e.g., [14,15,16,17,18,19,20,21,22,23]) have established a variety of inequalities by employing Riemann–Liouville fractional integral and derivative operators and their generalizations.
Recall the Chebyshev inequality [24] for two integrable and synchronous functions Φ and Ψ defined on \([a,b]\)
The two functions Φ and Ψ are said to be synchronous on \([a,b]\) if
In [25,26,27,28], various researchers studied and introduced various generalizations of inequality (1). In [29], Belarbi and Dahmani established the following results related to the Chebyshev inequalities via Riemann–Liouville fractional integral operators.
Theorem 1.1
Suppose that Φ and Ψ are two synchronous functions defined on \([0,\infty )\). Then the following inequality holds for all \(\theta > 0\), \(\xi > 0\):
Theorem 1.2
Suppose that Φ and Ψ are two synchronous functions defined on \([0,\infty )\). Then the following inequality holds for all \(\theta > 0\), \(\xi > 0\), \(\lambda > 0\):
Theorem 1.3
Suppose that \((\varPhi _{j})_{j=1,\ldots,n}\) are n positive increasing functions defined on \([0,\infty )\). Then the following inequality holds for any \(\theta > 0\), \(\xi > 0\):
Theorem 1.4
Suppose that Φ and Ψ are two functions defined on \([0,\infty )\) such that Φ is increasing, Ψ is differentiable, and there exists a real number \(m := \inf_{\theta \geq 0} g^{\prime }(\theta )\). Then the following inequality is valid for all \(\theta > 0\), \(\xi > 0\):
The left and right generalized fractional conformable integral operators are presented respectively in [1] as follows:
and
where \(\lambda \in \mathbb{C}\), \(\Re (\lambda )>0\), \(\xi \in (0,1] \), \(\eta \in \mathbb{R}\), \(\xi +\eta \neq 0\), and Γ is the well-known gamma function [30].
Remark 1
(i) If \(\eta =0\) in (7) and (8), then we have the following Riemann–Liouville type fractional conformable integral operators:
and
where \(\lambda \in \mathbb{C}\), \(\Re (\lambda )>0\), \(\xi \in (0,1] \).
(ii) If \(\xi =1\) in (9) and (10), then we obtain the following Riemann–Liouville fractional integral operators:
and
where \(\lambda \in \mathbb{C}\), \(\Re (\lambda )>0\).
In this paper, we consider the following one-sided fractional conformable integral operator for conformable integrable function Φ:
where \(\lambda \in \mathbb{C}\), \(\Re (\lambda )>0\), \(\xi \in (0,1] \), \(\eta \in \mathbb{R}\), and \(\xi +\eta \neq 0\).
Remark 2
(i) When we let \(\eta =0\), then (13) will lead to the following Riemann–Liouville type fractional conformable integral operator:
(ii) When we let \(\eta =0\) and \(\xi =1\), then (13) will lead to the following Riemann–Liouville fractional integral operator:
Our aim is to establish Chebyshev type inequalities with two synchronous functions for a new generalized integral.
2 Main results
In this section, we present Chebyshev type inequalities involving generalized fractional conformable integral operator (13).
Theorem 2.1
Let Φ and Ψ be two integrable functions which are synchronous on \([0,\infty )\). Then
where \(\lambda \in \mathbb{C}\), \(\Re (\lambda )>0\), \(\xi \in (0,1]\), \(\eta \in \mathbb{R}\), \(\xi +\eta \neq 0\), and Γ is the gamma function.
Proof
Since Φ and Ψ are synchronous on \([0,\infty )\), we have
or equivalently
Multiplying both sides of (18) by \(\frac{1}{\varGamma (\lambda ) \zeta ^{1-\xi -\eta }} (\frac{\theta ^{\xi +\eta } - \zeta ^{\xi +\eta }}{\xi +\eta } )^{\lambda -1}\) and integrating the resultant inequality with respect to ζ over \((0,\theta )\), we get
It follows that
Thus, we obtain
where
Multiplying both sides of (19) by \(\frac{1}{ \varGamma (\lambda ) \varrho ^{1-\xi -\eta }} (\frac{\theta ^{\xi +\eta } - \varrho ^{\xi +\eta }}{\xi +\eta } )^{\lambda -1}\) and integrating the resultant identity with respect to ϱ over \((0,\theta )\), we get
It follows that
which completes the desired proof. □
Corollary 1
Let Φ and Ψ be two integrable functions which are synchronous on \([0,\infty )\). Then
where \(\xi \in (0,1] \), \(\lambda \in \mathbb{C}\), and \(\Re (\lambda )>0\).
Proof
If we take \(\eta =0\) in Theorem 2.1, then we get the desired inequality involving Riemann–Liouville type conformable fractional integral. □
Remark 3
Applying Theorem 2.1 for \(\eta =0\) and \(\xi =1\) will give Theorem 1.1.
Theorem 2.2
Let Φ and Ψ be two integrable functions which are synchronous on \([0,\infty )\). Then
where \(\lambda , \tau \in \mathbb{C}\), \(\Re (\lambda )>0\), \(\Re (\tau )>0\), \(\xi \in (0,1]\), \(\eta \in \mathbb{R}\), and \(\xi +\eta \neq 0\).
Proof
Multiplying both sides of (19) by \(\frac{1}{\varGamma (\tau ) \varrho ^{1-\xi -\eta }} (\frac{\theta ^{\xi +\eta } - \varrho ^{\xi +\eta }}{\xi +\eta } )^{\tau -1}\) and integrating the resultant inequality with respect to ϱ over \((0,\theta )\), we get
Therefore, we have
which completes the desired proof of Theorem 2.2. □
Remark 4
If we consider \(\tau = \lambda \) in Theorem 2.2, then we obtain Theorem 2.1.
Corollary 2
Suppose that Φ and Ψ are two integrable functions which are synchronous on \([0,\infty )\). Then
where \(\lambda , \tau \in \mathbb{C}\), \(\Re (\lambda )>0\), \(\Re (\tau )>0\), \(\xi \in (0,1]\).
Proof
If we take \(\eta = 0\) in Theorem 2.2, then we get the desired inequality involving Riemann–Liouville type fractional conformable integral operator. □
Remark 5
If we consider \(\eta =0\) and \(\xi =1\) 2.2, then we get Theorem 1.2.
Remark 6
The inequalities in Theorems 2.1 and 2.2 will be reversed if the functions are asynchronous on \([0,\infty )\).
Theorem 2.3
Let \((\varPhi _{j})_{j=1,2,\ldots,n}\) be n positive increasing functions on \([0,\infty )\). Then, for \(\theta >0\), \(\xi \in (0,1]\), \(\eta \in \mathbb{R}\), \(\lambda \in \mathbb{C}\), we have
Proof
To prove this theorem, we apply induction on n. Obviously, for \(n = 1\), we have
holds. For \(n = 2\), since \(\varPhi _{1}\) and \(\varPhi _{2}\) are positive and increasing functions, therefore we have
Hence, by applying Theorem 2.1, we obtain
Now, assume that by induction hypothesis
Since \(\varPhi _{j}\); \(j=1,2,\ldots ,n\), are positive increasing functions on \(\mathbb{R^{+}}\), therefore \(g:=\prod_{j=1}^{n-1} \varPhi _{j}\) is increasing on \(\mathbb{R^{+}}\). Let \(h:=\varPhi _{n}\). Applying Theorem 2.1 to the functions Φ and Ψ, we have
By using (24), we obtain
which completes the desired proof. □
Corollary 3
Suppose that \((\varPhi _{j})_{j=1,2,\ldots,n}\) are n positive increasing functions on \([0,\infty )\). Then, for \(\theta >0\), \(\xi \in (0,1]\), \(\lambda \in \mathbb{C}\), and \(\Re (\lambda )>0\), we have
Proof
If we let \(\eta =0\) in Theorem 2.3, then we get the desired corollary involving Riemann–Liouville type fractional conformable integral operator. □
Remark 7
If we let \(\eta =0\) and \(\xi =1\) in Theorem 2.3, we get Theorem 1.3.
Theorem 2.4
Let \(\xi \in (0,1]\), \(\eta \in \mathbb{R}\), \(\lambda \in \mathbb{C}\), \(\Re (\lambda )>0\), and \(\xi +\eta \neq 0\). Also, let two functions \(\varPhi , \varPsi : \mathbb{R}_{0}^{+} \rightarrow \mathbb{R}\) such that Φ is increasing and Ψ is differentiable with \(\varPsi ^{ \prime }\) bounded below, and let \(m = \inf_{\theta \in \mathbb{R}_{0} ^{+}} \varPsi ^{\prime }(\theta )\). Then
where \(i(\theta )\) is the identity function.
Proof
Let \(\varPsi (\theta ) = \varPsi (\theta ) - m\theta ^{\xi +\eta }\). We find that Ψ is differentiable and increasing on \(\mathbb{R}_{0}^{+}\). As in the process of Theorem 2.3, for clarity, let \(p(\theta ) := m\theta ^{\xi +\eta }\), we obtain
We have
and
Finally using (29) and (30) in (28), we obtain the desired result. □
Corollary 4
Let \(\xi \in (0,1]\), \(\lambda \in \mathbb{C}\), and \(\Re (\lambda )>0\). Also, let two functions \(\varPhi , \varPsi : \mathbb{R}_{0}^{+} \rightarrow \mathbb{R}\) such that Φ is increasing and Ψ is differentiable with \(\varPsi ^{\prime }\) bounded below, and let \(m = \inf_{\theta \in \mathbb{R}_{0}^{+}} \varPsi ^{\prime }(\theta )\). Then
where \(i(\theta )\) is the identity function.
Proof
If we take \(\eta = 0\) in Theorem 2.4, then we get the desired Corollary 4, which involves Riemann–Liouville type fractional conformable integral operator. □
Remark 8
If we let \(\xi =1\) and \(\eta =0\) in Theorem 2.4, then we obtain Theorem 1.4.
3 Concluding remarks
Several Chebyshev type inequalities involving generalized conformable fractional integral operators are introduced in this paper. Also, we presented some particular results which involve Riemann–Liouville type conformable fractional integral operator. We observed that if we let \(\xi =1\) and \(\eta =0\), then the inequalities obtained in this paper will reduce to the inequalities obtained earlier by Belarbi and Dahmani [29].
References
Khan, T.U., Khan, M.A.: Generalized conformable fractional integral operators. J. Comput. Appl. Math. 346, 378–389 (2019). https://doi.org/10.1016/j.cam.2018.07.018
Kilbas, A.A., Srivastava, H.M., Truhillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Podlubny, I.: Fractional Differential Equations. Academic Press, London (1999)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Reading (1993)
Khan, M.A., Khurshid, Y., Dragomir, S.S., Ullah, R.: Inequalities of the Hermite–Hadamard type with applications. Punjab Univ. J. Math. 50(3), 1–12 (2018)
Khan, M.A., Begum, S., Khurshid, Y., Chu, Y.M.: Ostrowski type inequalities involving conformable fractional integrals. J. Inequal. Appl. 2018, 70, 1–14 (2018)
Khurshid, Y., Adil Khan, M., Chu, Y.M.: Hermite–Hadamard–Fejer inequalities for conformable fractional integrals via preinvex functions. J. Funct. Spaces 2019, 1–9 Article ID 3146210 (2019)
Khurshid, Y., Khan, M.A., Chu, Y.M.: Generalized inequalities via GG-convexity and GA-convexity. J. Funct. Spaces 2019, Article ID 6926107, 1–8 (2019)
Khan, M.A., Khurshid, Y., Chu, Y.M.: Hermite–Hadamard type inequalities via the Montgomery identity. Commun. Math. Appl. 10(1), 85–97 (2019)
Iqbal, A., Khan, M.A., Ullah, S., Kashuri, A., Chu, Y.M.: Hermite–Hadamard type inequalities pertaining conformable fractional integrals and their applications. AIP Adv. 8, 075101, 1–18 (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
Rahman, G., Khan, A., Abdeljawad, T., Nisar, K.S.: The Minkowski inequalities via generalized proportional fractional integral operators. Adv. Differ. Equ. 2019, 287 (2019). https://doi.org/10.1186/s13662-019-2229-7
Nisar, K.S., Tassaddiq, A., Rahman, G., Khan, A.: Some inequalities via fractional conformable integral operators. J. Inequal. Appl. 2019, 217 (2019)
Dahmani, Z., Tabharit, L.: On weighted Gruss type inequalities via fractional integration. J. Adv. Res. Pure Math. 2, 31–38 (2010)
Dahmani, Z.: New inequalities in fractional integrals. Int. J. Nonlinear Sci. 9(4), 493–497 (2010)
Kiryakova, V.: Generalized Fractional Calculus and Applications. Pitman Res. Notes Math. Ser., vol. 301. Longman, New York (1994)
Katrakhov, V.V., Sitnik, S.M.: The transmutation method and boundary-value problems for singular elliptic equations. Contemp. Math Fundam. Dir. 64, 211–426 (2018)
Ntouyas, K.S., Agarwal, P., Tariboon, J.: On Polya–Szego and Chebyshev types inequalities involving the Riemann–Liouville fractional integral operators. J. Math. Inequal. 10(2), 491–504 (2016)
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)
Sarikaya, M.Z., Dahmani, Z., Kiris, M.E., Ahmad, F.: \((k, s)\)-Riemann–Liouville fractional integral and applications. Hacet. J. Math. Stat. 45(1), 77–89 (2016)
Set, E., Tomar, M., Sarikaya, M.Z.: On generalized Grüss type inequalities for k-fractional integrals. Appl. Math. Comput. 269, 29–34 (2015)
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)
Chebyshev, P.L.: Sur les expressions approximatives des integrales definies par les autres prises entre les mmes limites. Proc. Math. Soc. Charkov 2, 93–98 (1882)
Rahman, G., Nisar, K.S., Qi, F.: Some new inequalities of the Gruss type for conformable fractional integrals. AIMS Math. 3(4), 575–583 (2018)
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
Özdemir, M.E., Set, E., Akdemir, A.O., Sarkaya, M.Z.: Some new Chebyshev type inequalities for functions whose derivatives belong to \(L_{p}\) spaces. Afr. Math. 26, 1609–1619 (2015)
Set, E., Dahmani, Z., Mumcu, I.: New extensions of Chebyshev type inequalities using generalized Katugampola integrals via Polya–Szeg inequality. Int. J. Optim. Control Theor. Appl. 8(2), 137–144 (2018)
Belarbi, S., Dahmani, Z.: On some new fractional integral inequalities. J. Inequal. Pure Appl. Math. 10(3), Article 86, 5 pp. (2009)
Srivastava, H.M., Choi, J.: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam (2012)
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This project was supported by the Deanship of Scientific Research at Prince Sattam Bin Abdulaziz University under the research project # 2019/01/10384.
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Nisar, K.S., Rahman, G. & Mehrez, K. Chebyshev type inequalities via generalized fractional conformable integrals. J Inequal Appl 2019, 245 (2019). https://doi.org/10.1186/s13660-019-2197-1
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DOI: https://doi.org/10.1186/s13660-019-2197-1