Abstract
We study a generalized left sided tempered fractional (GTF)-integral concerning another function Ψ in the kernel. Then we investigate several kinds of inequalities such as Grüss-type and certain other related inequalities by utilizing the GTF-integral. Additionally, we present various special cases of the main result. By utilizing the connection between GTF-integral and Riemann–Liouville integral concerning another function Ψ in the kernel, certain distinct particular cases of the main result are also presented. Furthermore, certain other inequalities can be formed by applying various kinds of conditions on the function Ψ.
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1 Introduction
The field of fractional calculus deals with the integrals and differentiation of an arbitrary non-integer order. In the last three centuries, this field has been considered as the most power tool in describing the anomalous kinetics and its wide applications in diverse domains. Numerous mathematical, statistical, engineering, physical, chemical, and biological phenomena can be modeled by utilizing ordinary differential equations involving fractional derivatives. Many a mathematician and physicist has contributed to the development of the theories of fractional calculus. The interesting reader is referred to [1–3] and the references therein. In practical applications, a numerous types of fractional integrals and derivatives operators, such as the Riemann–Liouville, Caputo, Riesz, Hilfer, Hadamard, Erdélyi–Kolber, Saigo, and Marichev–Saigo–Maeda operators, were extensively studied by various researchers. We refer the reader to [2–4].
Later, mathematicians introduced the notion of fractional conformable integrals and derivatives which are cited therein. Khalil et al. [5] introduced fractional conformable derivatives operators with some shortcoming. Abdeljawad [6] investigated the properties of the fractional conformable derivative operators. Jarad et al. [7] defined generalized fractional conformable integral and derivative operators. Anderson and Unless [8] presented the idea of a conformable derivative by employing local proportional derivatives. In [9], Abdeljawad and Baleanu gave certain monotonicity results for fractional difference operators with discrete exponential kernels. In [10], Abdeljawad and Baleanu have defined a fractional derivative operator with exponential kernel and the discrete version. In [11], Atangana and Baleanu defined a new fractional derivative operator with the non-local and non-singular kernel. A fractional derivative without a singular kernel can be found in the work of Caputo and Fabrizio [12]. Certain properties of fractional derivatives without a singular kernel can be found in the work of Losada and Nieto [13]. In [14–16], the authors studied stability analysis and a numerical scheme for fractional Klein–Gordon equations, existence results in Banach space for a nonlinear impulsive system and results for mild solutions of fractional coupled hybrid boundary value problems.
A variety of such types of distinguished operators led researchers to establish new ideas and fractional integral inequalities by utilizing these new operators. In [17, 18], Hasib et al. established various inequalities by using AB-fractional and Saigo fractional integral operators. Recently Alzabut et al. and Rahman et al. [19, 20] studied generalized proportional derivatives and integral operators and established a certain Gronwall inequality and the Minkowski inequalities involving the said operators. Rahman et al. [21, 22] presented fractional integral inequalities for a family of positive continuous and decreasing functions and inequalities for convex functions by employing proportional Hadamard fractional integrals. Recently, researchers presented several various remarkable inequalities with properties and applications for the fractional conformable integrals and proportional integrals. The interested reader may consult [23–33].
2 Preliminaries
In this section, we consider some well-known definitions and mathematical preliminaries.
Definition 2.1
([34])
Suppose that the functions \(\mathcal{U},\mathcal{V}: [a_{1},b_{1}]\rightarrow \mathbb{R}\) are positive with \(\mathcal{A}\leq \mathcal{U(\vartheta )}\leq \mathcal{B}\) and \(\mathcal{C}\leq \mathcal{V(\vartheta )}\leq \mathcal{D}\), for all \(\vartheta \in [a_{1},b_{1}]\), then the following inequality holds:
where the constants \(\mathcal{B}\), \(\mathcal{A}\), \(\mathcal{C}\), \(\mathcal{D}\in \mathbb{R}\) and \(\frac{1}{4}\) is the sharp value of inequality (2.1).
Definition 2.2
The function \(\mathcal{U(\varrho )}\) will be in the space \(L_{p,r}{[0,\infty [}\) if
If we consider \(r=0\), then (2.2) gives
Definition 2.3
([37])
Suppose that we have the function \(\mathcal{U}\in L_{1}{[0,\infty [}\) and assume that the function Ψ is positive, monotone and an increasing on \({[0,\infty [}\) and let \(\varPsi ^{\prime }\) be continuous on \({[0,\infty [}\) with \(\varPsi (0)=0\). Then the Lebesgue real-valued measurable function \(\mathcal{U}\) defined on \({[0,\infty [}\) is said to be in the space \(\chi _{\varPsi }^{p}(0,\infty )\), \((1\leq p<\infty )\) for which
When \(p=\infty \), then
Note that the space \(\chi _{\varPsi }^{p}(0,\infty )\) coincides with the space \(L_{p}{[0,\infty [}\) if \(\varPsi (\varrho )=\varrho \) for \(1\leq p<\infty \) and similarly with the space \(L_{p,r}[1,\infty [\) if \(\varPsi (\varrho )=\ln \varrho \) for \(1\leq p<\infty \).
The tempered fractional integral was first studied by Buschman [38], but Li et al. [39] and Meerschaert et al. [40] have described the associated tempered fractional calculus more explicitly. Fernandez and Ustaoǧlu [41] investigated several analytic properties of tempered fractional integrals.
Definition 2.4
Suppose that \([a,b]\) is a real interval and \(\kappa ,\xi \in \mathbb{C}\) with \(\Re (\kappa )>0\) and \(\Re (\xi )\geq 0\), then the left sided tempered fractional integral is defined by
Remark 2.1
Setting \(\xi =0\) in (2.3) yields the following Riemann–Liouville fractional integral:
The tempered fractional integral (2.3) satisfies the following semigroup property:
In [42], Fahad et al. defined the following general form of the generalized tempered fractional integral concerning another function.
Definition 2.5
Let \(\mathcal{U}\) be an integrable function in the space \(\chi _{\varPsi }^{p}(0,\infty )\) and assume that the function Ψ is positive, monotone and increasing on \({[0,\infty [}\) and let \(\varPsi ^{\prime }\) be continuous on \({[0,\infty [}\) with \(\varPsi (0)=0\). Then the left (GTF)-integral of a function \(\mathcal{U}\) concerning another function Ψ in the kernel is defined by
where \(\xi >0\), \(\kappa \in \mathbb{C}\) with \(\Re (\kappa )>0\) and \(\varGamma (\cdot)\) is the well-known gamma function.
Remark 2.2
The left GTF-integral (2.5) will reduce the following fractional integrals:
- i.
setting \(\varPsi (\rho )=\rho \), then the left tempered fractional integral (2.3) will be obtained,
- ii.
setting \(\xi =0\), then the left generalized RL-fractional integral operator defined by [1] will be obtained,
- iii.
setting \(\varPsi (\rho )=\ln \rho \), the left Hadamard GTF-integral defined by [42] will be obtained,
$$\begin{aligned} & \bigl({}_{a}^{\varPsi }\mathcal{J}^{\kappa ,\xi } \mathcal{U} \bigr) ( \rho ) \\ &\quad =\frac{1}{\varGamma (\kappa )} \int _{a}^{\rho }\exp \bigl[-\xi (\ln \rho -\ln \varrho )\bigr] (\ln \rho - \ln \varrho )^{ \kappa -1}\frac{\mathcal{U(\varrho )}}{\varrho }\,d\varrho ,\quad a< \rho , \end{aligned}$$(2.6) - iv.
setting \(\varPsi (\rho )=\frac{\rho ^{\kappa }}{\kappa }\), \(\kappa >0\) and \(\xi =0\), the left Katugampola [36] fractional integral operator will be obtained,
- v.
setting \(\varPsi (\rho )=\rho \) and \(\xi =0\), the left Riemann–Liouville fractional integral (2.4) will be obtained,
- vi.
setting \(\varPsi (\rho )=\frac{\rho ^{\alpha +s}}{\alpha +s}\) and \(\xi =0\) (where \(\alpha \in (0,1]\), \(s\in \mathbb{R}\) and \(\mu +s\neq 0\)), then the generalized fractional conformable integral given in [43] will be obtained.
In this manuscript, we will consider the following one sided GTF-integral.
Definition 2.6
Let \(\mathcal{U}\) be an integrable function in the space \(\chi _{\varPsi }^{p}(0,\infty )\) and assume that the function Ψ is positive, monotone and increasing on \({[0,\infty [}\) and let \(\varPsi ^{\prime }\) is continuous on \({[0,\infty [}\) with \(\varPsi (0)=0\). Then the one sided (GTF) integral of a function \(\mathcal{U}\) concerning another function Ψ in the kernel is defined by
One can easily derive the following results.
Theorem 2.1
Let\(\mathcal{U}:[0,\rho ]\subseteq {[0,\infty [}\rightarrow \mathbb{R}\)be an GTF-integral operator concerning another functionΨ, then we have
where\(\kappa , \mu >0\).
Theorem 2.2
The GTF-integral operator\({}^{\varPsi }\mathcal{J}^{\kappa ,\xi }: L_{1}[0,\rho ]\rightarrow L_{1}[0, \rho ]\)will satisfy the following linearity property:
where\(\mathcal{U}_{1}, \mathcal{U}_{2}\in L_{1}[0,\rho ]\)and\(\alpha , \beta \in \mathbb{R}\).
The main goal of this manuscript is to establish certain inequalities such as Grüss-type and several other inequalities by utilizing the GTF-integral (2.1). Also, certain special and particular cases of the main result are presented.
3 Main results
We present generalizations of certain inequalities by utilizing the GTF-integral operator (2.7) containing another function Ψ in its kernel in this section.
Theorem 3.1
Suppose that we have the function\(\mathcal{U}\in \chi _{\varPsi }^{p}(0,\infty )\)and assume that the functionΨis positive, monotone and increasing on\({[0,\infty [}\)and its derivative\(\varPsi ^{\prime }\)is continuous on\({[0,\infty [}\)with\(\varPsi (0)=0\). Moreover, let\(\phi _{1}\)and\(\phi _{2}\)be two integrable functions defined on\({[0,\infty [}\)such that
Then, for\(\varrho >0\), \(\kappa , \lambda >0\), we have
Proof
Applying (3.1) for all \(\rho \geq 0\) and \(\zeta \geq 0\), we have
It follows that
Multiplying both sides of (3.3) with \(\frac{1}{\varGamma (\kappa )}e^{-\xi (\varPsi ( \varrho )-\varPsi (\rho ) )} (\varPsi (\varrho )-\varPsi (\rho ) )^{\kappa -1}\varPsi ^{\prime }(\rho )\) and integrating the resulting estimate with respect to ρ from 0 to ϱ, we get
which with the aid of (2.7) becomes
Again, multiplying both sides of (3.4) with \(\frac{1}{\varGamma (\lambda )}e^{-\xi (\varPsi ( \varrho )-\varPsi (\zeta ) )} (\varPsi (\varrho )-\varPsi (\zeta ) )^{\lambda -1}\varPsi ^{\prime }(\zeta )\) and integrating the resulting estimate with respect to ζ from 0 to ϱ, we obtain
which proves the desired assertion (3.2). □
Theorem 3.2
Suppose that the two positive functions\(\mathcal{U}\)and\(\mathcal{V}\)are defined on\({[0,\infty [}\)and assume that the functionΨis positive, monotone and increasing on\({[0,\infty [}\)and its derivative\(\varPsi ^{\prime }\)is continuous on\({[0,\infty [}\)with\(\varPsi (0)=0\). Assume that (3.1) holds and let\(\psi _{1}\), \(\psi _{2}\)be two integrable functions defined on\({[0,\infty [}\)such that
Then, for\(\varrho >0\)and\(\kappa ,\lambda >0\), the following four inequalities hold:
Proof
To prove inequality (3.7), using (3.1) and (3.6) for \(\rho , \zeta \in {[0,\infty [}\) yields
It follows that
Multiplying both sides of inequality (3.11) with \(\frac{1}{\varGamma (\kappa )}e^{-\xi (\varPsi ( \varrho )-\varPsi (\rho ) )} (\varPsi (\varrho )-\varPsi (\rho ) )^{\kappa -1}\varPsi ^{\prime }(\rho )\) and integrating the resulting estimate with respect to ρ from 0 to ϱ, we get
which with the aid of (2.7) becomes
Again, multiplying both sides of (3.12) with \(\frac{1}{\varGamma (\lambda )}e^{-\xi (\varPsi ( \varrho )-\varPsi (\zeta ) )} (\varPsi (\varrho )-\varPsi (\zeta ) )^{\lambda -1}\varPsi ^{\prime }(\zeta )\), integrating the resulting estimate with respect to ζ from 0 to ϱ and using (2.7), we obtain
which completes the desired assertion (3.7). Inequalities (3.8)–(3.10) can be proved by using the following identities:
and
respectively. □
4 Certain other inequalities via GTF-integral concerning another function
Certain other types of inequalities which involving generalized tempered fractional (GTF) integral (2.7) are presented in this section.
Theorem 4.1
Suppose that the two positive functions\(\mathcal{U}\)and\(\mathcal{V}\)are defined on\({[0,\infty [}\)and assume that the functionΨis positive, monotone and increasing on\({[0,\infty [}\)and its derivative\(\varPsi ^{\prime }\)is continuous on\({[0,\infty [}\)with\(\varPsi (0)=0\). If\(p_{1}, q_{1}>1\)are such that\(\frac{1}{p_{1}}+\frac{1}{q_{1}}=1\), then, for\(\varrho >0\), the following inequalities hold:
and
Proof
Consider the well-known Young’s inequality [44]:
Applying (4.5) for \(\mathfrak{a}=\mathcal{U(\rho )}\mathcal{V(\zeta )}\) and \(\mathfrak{b}=\mathcal{U(\zeta )}\mathcal{V(\rho )}\), \(\rho , \zeta >0\), we have
Multiplying both sides of inequality (4.6) with \(\frac{1}{\varGamma (\kappa )}e^{-\xi (\varPsi ( \varrho )-\varPsi (\rho ) )} (\varPsi (\varrho )-\varPsi (\rho ) )^{\kappa -1}\varPsi ^{\prime }(\rho )\) and integrating the resulting estimate with respect to ρ from 0 to ϱ, we get
which in view of (2.7) becomes
Now, multiplying both sides of (4.7) with \(\frac{1}{\varGamma (\lambda )}e^{-\xi (\varPsi ( \varrho )-\varPsi (\zeta ) )} (\varPsi (\varrho )-\varPsi (\zeta ) )^{\lambda -1}\varPsi ^{\prime }(\zeta )\), integrating the resulting estimate with respect to ζ from 0 to ϱ and using (2.7), we obtain
which is the desired assertion (4.1). The inequalities (4.2), (4.3) and (4.4) can be obtained by employing the similar procedure and setting the following parameters in (4.5), respectively:
and
□
Theorem 4.2
Suppose that the two positive functions\(\mathcal{U}\)and\(\mathcal{V}\)are defined on\({[0,\infty [}\)and assume that the functionΨis positive, monotone and increasing on\({[0,\infty [}\)and its derivative\(\varPsi ^{\prime }\)is continuous on\({[0,\infty [}\)with\(\varPsi (0)=0\). If\(p_{1}, q_{1}>1\)are such that\(\frac{1}{p_{1}}+\frac{1}{q_{1}}=1\), then, for\(\varrho >0\), the following inequalities hold:
and
Proof
Consider the following well-known \(\mathcal{AM-GM}\) inequality:
Applying (4.15) for \(\mathfrak{a}=\mathcal{U(\rho )}\mathcal{V(\zeta )}\) and \(\mathfrak{b}=\mathcal{U(\zeta )}\mathcal{V(\rho )}\), \(\rho ,\zeta >0\), we have
Multiplying both sides of inequality (4.16) with \(\frac{1}{\varGamma (\kappa )}e^{-\xi (\varPsi ( \varrho )-\varPsi (\rho ) )} (\varPsi (\varrho )-\varPsi (\rho ) )^{\kappa -1}\varPsi ^{\prime }(\rho )\) and integrating the resulting estimate with respect to ρ from 0 to ϱ, we get
which in view of (2.7) becomes
Again, multiplying both sides of (4.17) with \(\frac{1}{\varGamma (\lambda )}e^{-\xi (\varPsi ( \varrho )-\varPsi (\zeta ) )} (\varPsi (\varrho )-\varPsi (\zeta ) )^{\lambda -1}\varPsi ^{\prime }(\zeta )\), integrating the resulting estimate with respect to ζ from 0 to ϱ and using (2.7), we obtain the desired assertion (4.11),
The inequalities (4.12), (4.13) and (4.14) can be easily obtained by following the same procedure and using the following parameters in (4.15), respectively. We have
and
□
Theorem 4.3
Suppose that the two positive functions\(\mathcal{U}\)and\(\mathcal{V}\)are defined on\({[0,\infty [}\)and assume that the functionΨis positive, monotone and increasing on\({[0,\infty [}\)and its derivative\(\varPsi ^{\prime }\)is continuous on\({[0,\infty [}\)with\(\varPsi (0)=0\). Let\(p_{1}, q_{1}>1\)be such that\(\frac{1}{p_{1}}+\frac{1}{q_{1}}=1\). Suppose
Then, for\(\varrho >0\), the following inequalities hold:
and
Proof
From (4.21), we have
It follows that
Multiplying both sides of inequality (4.25) with \(\frac{1}{\varGamma (\kappa )}e^{-\xi (\varPsi ( \varrho )-\varPsi (\rho ) )} (\varPsi (\varrho )-\varPsi (\rho ) )^{\kappa -1}\varPsi ^{\prime }(\rho )\) and integrating the resulting estimate with respect to ρ from 0 to ϱ and using (2.7), we obtain
Now, since \(\mathcal{K}\mathcal{H}>0\) and
it follows that
Hence, by using (4.26) and (4.27), we have
which gives the desired assertion (4.22).
Now, from (4.28), we have
Subtraction of \(({}^{\varPsi }\mathcal{J}^{\kappa ,\xi }\mathcal{U}( \varrho )\mathcal{V}(\varrho ) )\) from (4.29) yields the desired assertion (4.23). Similarly, we can easily prove the assertion (4.24) from (4.22). □
5 Special cases
This section is devoted to certain special cases of the main result obtained in Sects. 3 and 4.
(I) Applying Theorem 3.1 for \(\varPsi (\varrho )=\varrho \), we attain the following result for a one sided tempered fractional integral.
Corollary 5.1
Suppose that we have the function\(\mathcal{U}\in L_{1}{[0,\infty [}\)and let\(\phi _{1}\)and\(\phi _{2}\)be two integrable functions defined on\({[0,\infty [}\)such that
Then, for\(\varrho >0\), \(\kappa , \lambda >0\), we have
(II) Applying Theorem 3.1 for \(\xi =1\), we attain the following result for the one sided generalized Riemann–Liouville fractional integral proved earlier by Kacar et al. [37].
Corollary 5.2
Suppose that the function\(\mathcal{U}\in \chi _{\varPsi }^{p}(0,\infty )\)and assume that the functionΨis positive, monotone and increasing on\({[0,\infty [}\)and its derivative\(\varPsi ^{\prime }\)is continuous on\({[0,\infty [}\)with\(\varPsi (0)=0\). Moreover, let\(\phi _{1}\)and\(\phi _{2}\)be two integrable functions defined on\({[0,\infty [}\)such that
Then, for\(\varrho >0\), \(\kappa , \lambda >0\), we have
(III) Applying Theorem 3.1 for \(\varPsi (\varrho )=\varrho \) and \(\xi =1\), we attain the following result for one sided Riemann–Liouville fractional integral proved earlier by Tariboon et al. [45].
Corollary 5.3
Suppose that the function\(\mathcal{U}\in L_{1}{[0,\infty [}\)and let\(\phi _{1}\)and\(\phi _{2}\)be two integrable functions defined on\({[0,\infty [}\)such that
Then, for\(\varrho >0\), \(\kappa , \lambda >0\), we have
The special cases of Theorem 3.2 are presented by the following corollaries.
(I) Setting \(\varPsi (\varrho )=\varrho \), we attain the following result for one sided tempered fractional integral.
Corollary 5.4
Suppose that the two positive functions\(\mathcal{U}\)and\(\mathcal{V}\)are defined on\({[0,\infty [}\). Assume that (3.1) holds and\(\psi _{1}\), \(\psi _{2}\)are two integrable functions on\({[0,\infty [}\)such that
Then, for\(\varrho >0\)and\(\kappa ,\lambda >0\), the following four inequalities hold:
(II) Applying Theorem 3.2 for \(\xi =1\), we attain the following result for the one sided generalized Riemann–Liouville fractional integral proved earlier by Kacar et al. [37].
Corollary 5.5
Suppose that the two positive functions\(\mathcal{U}\)and\(\mathcal{V}\)are defined on\({[0,\infty [}\)and assume that the functionΨis positive, monotone and increasing on\({[0,\infty [}\)and its derivative\(\varPsi ^{\prime }\)is continuous on\({[0,\infty [}\)with\(\varPsi (0)=0\). Assume that (3.1) holds and\(\psi _{1}\), \(\psi _{2}\)are two integrable functions defined on\({[0,\infty [}\)such that
Then, for\(\varrho >0\)and\(\kappa ,\lambda >0\), the following four inequalities hold:
(III) Applying Theorem 3.2 for \(\varPsi (\varrho )=\varrho \) and \(\xi =1\), we attain the following result for the one sided Riemann–Liouville fractional integral proved earlier by Tariboon et al. [45].
Corollary 5.6
Suppose that the functions\(\mathcal{U}\)and\(\mathcal{V}\)are two positive functions defined on\({[0,\infty [}\). Assume that (3.1) holds and\(\psi _{1}\), \(\psi _{2}\)are two integrable functions defined on\({[0,\infty [}\)such that
Then, for\(\varrho >0\)and\(\kappa ,\lambda >0\), then the following four inequalities hold:
The following corollaries represent special cases of Theorem 4.1.
(I) Setting \(\varPsi (\varrho )=\varrho \), we attain the following result for a one sided tempered fractional integral.
Corollary 5.7
Suppose that the functions\(\mathcal{U}\)and\(\mathcal{V}\)are two positive functions defined on\({[0,\infty [}\)and let\(p_{1}, q_{1}>1\)be such that\(\frac{1}{p_{1}}+\frac{1}{q_{1}}=1\). Then, for\(\varrho >0\), the following inequalities hold:
and
(II) Applying Theorem 4.1 for \(\xi =1\), we attain the following new result for a one sided generalized Riemann–Liouville fractional integral.
Corollary 5.8
Suppose that the functions\(\mathcal{U}\)and\(\mathcal{V}\)are two positive functions defined on\({[0,\infty [}\)and assume that the functionΨis positive, monotone and increasing on\({[0,\infty [}\)and its derivative\(\varPsi ^{\prime }\)is continuous on\({[0,\infty [}\)with\(\varPsi (0)=0\). If\(p_{1}, q_{1}>1\)are such that\(\frac{1}{p_{1}}+\frac{1}{q_{1}}=1\), then, for\(\varrho >0\), the following inequalities hold:
and
In a similar way, we can obtain the special cases of Theorems 4.2 and 4.3 by applying similar procedures.
6 Particular cases
Here, we present certain new particular cases of our main result by employing the connection of GTF-integral (2.7) with the classical Riemann–Liouville expression containing another function in the kernel.
Li et al. [39] defined the connection of a tempered fractional integral (2.3) with the Riemann–Liouville fractional integral by
Here, we propose the following connection of the GTF-integral (2.7) with the generalized Riemann–Liouville fractional integral as
where \({}^{\varPsi }\mathcal{J}^{\kappa } [e^{\xi \varPsi (\varrho )} \mathcal{ U(\varrho )} ]\) is the generalized Riemann–Liouville fractional integral concerning another function.
Applying the above connection (6.1) to Theorem 3.1, one can get the following new result in terms of the generalized Riemann–Liouville fractional integral in the sense of another function.
Theorem 6.1
Suppose that the function\(\mathcal{U}\in \chi _{\varPsi }^{p}(0,\infty )\)and assume that the functionΨis positive, monotone and increasing on\({[0,\infty [}\)and its derivative\(\varPsi ^{\prime }\)is continuous on\({[0,\infty [}\)with\(\varPsi (0)=0\). Moreover, let\(\phi _{1}\)and\(\phi _{2}\)be two integrable functions defined on\({[0,\infty [}\)such that
Then, for\(\varrho >0\), \(\kappa , \lambda >0\), we have
Applying (6.1) to Theorem 3.2, one can obtain easily the following new results.
Theorem 6.2
Suppose that the two positive functions\(\mathcal{U}\)and\(\mathcal{V}\)are defined on\({[0,\infty [}\)and assume that the functionΨis positive, monotone and increasing on\({[0,\infty [}\)and its derivative\(\varPsi ^{\prime }\)is continuous on\({[0,\infty [}\)with\(\varPsi (0)=0\). Assume that (6.2) holds and\(\psi _{1}\), \(\psi _{2}\)are two integrable functions defined on\({[0,\infty [}\)such that
Then, for\(\varrho >0\)and\(\kappa ,\lambda >0\), the following four inequalities hold:
and
One can obtain the following new result of Theorem 4.1 in terms of the generalized Riemann–Liouville fractional integral in the sense of another function by utilizing (6.1).
Theorem 6.3
Suppose that the two positive functions\(\mathcal{U}\)and\(\mathcal{V}\)are defined on\({[0,\infty [}\)and assume that the functionΨis positive, monotone and increasing on\({[0,\infty [}\)and its derivative\(\varPsi ^{\prime }\)is continuous on\({[0,\infty [}\)with\(\varPsi (0)=0\). If\(p_{1}, q_{1}>1\)is such that\(\frac{1}{p_{1}}+\frac{1}{q_{1}}=1\), then, for\(\varrho >0\), the following inequalities hold:
and
Similarly, we can obtain particular new results of Theorems 4.2 and 4.3 in terms of the generalized Riemann–Liouville fractional integral in the sense of another function by utilizing (6.1). Also, one can easily obtain certain new results of Theorems presented in Sect. 6 by utilizing the special cases discussed in Remark 2.2.
7 Conclusion
In this paper, we presented various types of inequalities such as Grüss-type inequalities and certain other inequalities by employing a generalized tempered fractional (GTF)-integral in the sense of another function Ψ. Furthermore, we have discussed several special cases by using Remark 2.2. Also, we proposed a connection between the GTF-integral with the classical Riemann–Liouville fractional integral and derived certain new results in terms of the Riemann–Liouville fractional integral concerning another function. One can easily obtain several other types of inequalities, such as Hadamard fractional integral inequalities and generalized fractional conformable inequalities by utilizing Remark 2.2. Moreover, certain new inequalities can be derived by utilizing the inequalities discussed in Sect. 6.
References
Kilbas, A.A., Sarivastava, H.M., Trujillo, J.J.: Theory and Application of Fractional Differential Equation. North-Holland Mathematics Studies. Elsevier, Amsterdam (2006)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives, Theory and Applications. Gordon & Breach, Yverdon (1993) Edited and with a foreword by S.M. Nikol’skǐ, Translated from the 1987 Russian original, Revised by the authors
Podlubny, I.: Fractional Differential Equations. Academic Press, London (1999)
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264(65), 65–70 (2014)
Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015). https://doi.org/10.1016/j.cam.2014.10.016
Jarad, F., Ugurlu, E., Abdeljawad, T., Baleanu, D.: On a new class of fractional operators. Adv. Differ. Equ. 2017, 247 (2017)
Anderson, D.R., Ulness, D.J.: Newly defined conformable derivatives. Adv. Dyn. Syst. Appl. 10(2), 109–137 (2015)
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
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
Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 73–85 (2015)
Losada, J., Nieto, J.J.: Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 87–92 (2015)
Khan, H., Khan, A., Chen, W., Shah, K.: Stability analysis and a numerical scheme for fractional Klein–Gordon equations. Math. Methods Appl. Sci. 42(2), 723–732 (2019)
Khan, H., Khan, A., Abdeljawad, T., Alkhazzan, A.: Existence results in Banach space for a nonlinear impulsive system. Adv. Differ. Equ. 2019, 18 (2019)
Baleanu, D., Jafari, H., Khan, H., Johnston, S.J.: Results for mild solution of fractional coupled hybrid boundary value problems. Open Math. 13, 601–608 (2015)
Khan, H., Abdeljawad, T., Tunç, C., et al.: Minkowski’s inequality for the AB-fractional integral operator. J. Inequal. Appl. 2019, 96 (2019). https://doi.org/10.1186/s13660-019-2045-3
Khan, H., Tunç, C., Baleanu, D., Khan, A., Alkhazzan, A.: Inequalities for n-class of functions using the Saigo fractional integral operator. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113, 2407–2420 (2019)
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)
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
Rahman, G., Abdeljawad, T., Khan, A., Nisar, K.S.: Some fractional proportional integral inequalities. J. Inequal. Appl. 2019, 244 (2019). https://doi.org/10.1186/s13660-019-2199-z
Rahman, G., Abdeljawad, T., Jarad, F., Khan, A., Nisar, K.S.: Certain inequalities via generalized proportional Hadamard fractional integral operators. Adv. Differ. Equ. 2019, 454 (2019)
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)
Mubeen, S., Habib, S., Naeem, M.N.: The Minkowski inequality involving generalized k-fractional conformable integral. J. Inequal. Appl. 2019, 81 (2019). https://doi.org/10.1186/s13660-019-2040-8
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
Niasr, K.S., Tassadiq, A., Rahman, G., Khan, A.: Some inequalities via fractional conformable integral operators. J. Inequal. Appl. 2019, 217 (2019). https://doi.org/10.1186/s13660-019-2170-z
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., Qi, F.: Some new inequalities of the Gruss type for conformable fractional integrals. AIMS Math. 3(4), 575–583 (2018)
Rahman, G., Nisar, K.S., Ghaffar, A., Qi, F.: Some inequalities of the Grüss type for conformable k-fractional integral operators. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 114, 9 (2020). https://doi.org/10.1007/s13398-019-00731-3.
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
Rahmnan, G., Abdeljawad, T., Jarad, F., Nisar, K.S.: Bounds of generalized proportional fractional integrals in general form via convex functions and their applications. Mathematics 8, 113 (2020). https://doi.org/10.3390/math8010113
Rashid, S., Jarad, F., Noor, M.A., Kalsoom, H., Chu, Y.-M.: Inequalities by means of generalized proportional fractional integral operators with respect to another function. Mathematics 7, 1225 (2019). https://doi.org/10.3390/math7121225
Mohammed, P.O., Hamasalh, F.K.: New conformable fractional integral inequalities of Hermite–Hadamard type for convex functions. Symmetry 11, 263 (2019). https://doi.org/10.3390/sym11020263
Grüss, G.: Uber, das Maximum des absoluten Betrages von \(\frac{1}{b-a}\int _{a}^{b}\mathcal{U(\varrho )} \mathcal{V(\varrho )}\,d\varrho -\frac{1}{(b-a)^{2}}\int _{a}^{b} \mathcal{U(\varrho )}\,d\varrho \int _{a}^{b}\mathcal{V(\varrho )}\,d \varrho \). Math. Z. 39, 215–226 (1935)
Yildirim, H., Kirtay, Z.: Ostrowski inequality for generalized fractional integral and related inequalities. Malaya J. Mat. 2, 322–329 (2014)
Katugampola, U.N.: Approach to a generalized fractional integral. Appl. Math. Comput. 218, 860–865 (2011)
Kacar, E., Kacar, Z., Yildirim, H.: Integral inequalities for Riemann–Liouville fractional integrals of a function with respect to another function. Iran. J. Math. Sci. Inform. 13, 1–13 (2018)
Buschman, R.G.: Decomposition of an integral operator by use of Mikusenski calculus. SIAM J. Math. Anal. 3(1), 83–85 (1972)
Li, C., Deng, W., Zhao, L.: Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations. Discrete Contin. Dyn. Syst., Ser. B 24(4), 1989–2015 (2019)
Meerschaert, M.M., Sabzikar, F., Chen, J.: Tempered fractional calculus. J. Comput. Phys. 293, 14–28 (2015)
Fernandez, A., Ustaǧlu, C.: On some analytic properties of tempered fractional calculus. J. Comput. Appl. Math. (2019). https://doi.org/10.1016/j.cam.2019.112400
Fahad, H.M., Fernandez, A., Rehman, M.U., Siddiqi, M.: Tempered and Hadamard-type fractional calculus with respect to functions (11 Dec 2019) arXiv:1907.04551v2 [math.CA]
Khan, T.U., Khan, M.A.: Generalized conformable fractional integral operators. J. Comput. Appl. Math. (2018). https://doi.org/10.1016/j.cam.2018.07.018
Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, New York (1989)
Tariboon, J., Ntouyas, S.K., Sudsutad, W.: Some new Riemann–Liouville fractional integral inequalities. Int. J. Math. Sci. 2014, 869434 (2014)
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The fourth author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.
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Rahman, G., Nisar, K.S., Rashid, S. et al. Certain Grüss-type inequalities via tempered fractional integrals concerning another function. J Inequal Appl 2020, 147 (2020). https://doi.org/10.1186/s13660-020-02420-x
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DOI: https://doi.org/10.1186/s13660-020-02420-x