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New estimates considering the generalized proportional Hadamard fractional integral operators
Advances in Difference Equations volume 2020, Article number: 275 (2020)
Abstract
In the article, we describe the Grüss type inequality, provide some related inequalities by use of suitable fractional integral operators, address several variants by utilizing the generalized proportional Hadamard fractional (GPHF) integral operator. It is pointed out that our introduced new integral operators with nonlocal kernel have diversified applications. Our obtained results show the computed outcomes for an exceptional choice to the GPHF integral operator with parameter and the proportionality index. Additionally, we illustrate two examples that can numerically approximate these operators.
1 Introduction
A revolution inside the discipline of differentiation and integration has been witnessed: classical differentiation has become extended by the use of nonlocal operators. The classical derivative was combined with a strength regulation sort of kernel and ultimately this provided the upward thrust to new calculus referred to as the fractional calculus. The newly proposed calculus permits one to depict progressively complex problems with various properties, for example, in thermal conduction where the heat is streaming inside a medium with two distinct properties and a new mathematical model of heat conduction, one considered isotropic generalized thermoelasticity, with a three-phase lag, this model being considered in terms of the methodology of fractional calculus. Several significant results have been obtained [1–10].
Fractional calculus in continuous and discrete operators has also been comprehensively utilized in numerous fields [11–30]. But the concept has been propagated and implemented in applied mathematics, physics and porous media as a mathematical model. Various recognized generalized fractional operators comprise the Hadamard operator, the Erdélyi–Kober operator, the Saigo operator, the Gaussian hypergeometric operator, the Marichev–Saigo–Maeda fractional integral operator and so on. Out of these operators, the Riemann–Liouville fractional integral operator has been broadly applied by scientists in research just from an application viewpoint. For more details concerning fractional calculus operators, one may also refer to the expositions by Miller and Ross [31], Mathai [32], Kiryakova [33] and Baleanu et al. [34]. A captivating feature of this study is that there are numerous fractional operators that have fertile utilities in numerous areas of pure and applied mathematics. There are ideas with various qualities, and this grants the users an opportunity to select the suitable operator for exhibiting the issue under consideration. Moreover, as a consequence of its smoothness in the real world, experts have provided much contemplation to currently resolve fractional operators without singular kernels [35–40], and presently, a while later, various articles dealing with these sorts of fractional operators have been published.
Recently, the generalized proportional fractional integral operator contemplated by Jarad et al. [41] has potential application in statistical theory and also there have been enthralling presentations in the theory of fractional Schrödinger equations [42, 43]. These kinds of speculations elevate future studies to establish novel ideas to modify the fractional operators and help us to attain integral inequalities via such generalized fractional operators. Integral inequalities and their utilities perform- a crucial job in the theory of differential and difference equations. An assortment of distinct kinds of classical variants and their modifications have been built up by employing the classical fractional integral, derivative operators and their speculative ideas in the matter [44–46]. Recently, the Gronwall and the Minkowski inequalities concerning to the generalized proportional fractional derivative and fractional integral were explored by Alzabut et al. [3] and Rahman et al. [43]. Approving this predilection, we present an adjusted form for many recognized Grüss type inequalities [47] in the sense of the generalized proportional Hadamard fractional integral that could be more proficient and much more pertinent than the current ones. The well-known Grüss inequality can be stated as follows.
Let \(\mathcal{F},\mathcal{G}:[\nu _{1},\nu _{2}]\rightarrow (0, \infty )\) be two positive real-valued functions such that \(m\leq \mathcal{F}(l)\leq \mathcal{M}\) and \(n\leq \mathcal{G}(l)\leq \mathcal{N}\) for all \(l\in [\nu _{1},\nu _{2}]\). Then the inequality
holds with the best possible constant \(1/4\).
Inequality (1.1) is a marvelous instrument for exploring various systematic areas of science comprising chaos, porous media, biotechnology, heat transfer, time scale analysis and so on. There has been a continuous growth of eagerness for such a field of research to address the problems of various usages of these generalizations. Such discoveries had been analyzed by various investigators who in this way used assorted techniques for exploring and proposing these variations [48–53].
In this paper a new concept of integration that takes into account the fractional operator and also several generalizations will be introduced. Another important problem that could be handled by the new operators is the Grüss type and several other generalizations. We present, in general, three numerical schemes (Young, weighted arithmetic and geometric mean) that can be used to find solutions via the generalized proportional Hadamard fractional integral. Interestingly, several existing results recaptured by the results presented are Hadamard fractional integral inequalities. Therefore, the concept is rather novel and appears to make it possible to explore new directions of research as regards distinct scientific areas in pure and applied mathematics. We observe that the GPHF integral is able to show some kind of self-similarities.
2 Prelude
Now, we demonstrate concisely some essential preliminaries on fractional calculus for the convenience of the reader. The basic information is given in the monograph [37].
The left and right sides generalized proportional integral operators were introduced by Jarad et al. [41], they are defined by
and
where the proportionality index \(\rho \in (0,1]\), \(\gamma \in \mathbf{C}\) with \(\mathfrak{R}(\gamma )>0\), and \(\varGamma (l)=\int _{0}^{\infty }\vartheta ^{l-1}e^{-\vartheta }\,d \vartheta \) is the Euler gamma function [54–56].
Remark 2.1
Letting \(\rho =1\). Then (2.1) and (2.2) reduce to the following left and right side Riemann–Liouville fractional integral operators:
and
Next, we recall the concept of GPHF integral operator, which was introduced by Rahman et al. in [49]
Definition 2.2
([49])
Let \(\gamma >0\) and the proportionality index \(\rho \in (0,1]\). Then the left and right side GPHF integrals are defined by
and
Definition 2.3
Let \(\gamma >0\) and the proportionality index \(\rho \in (0,1]\). Then the one-sided GPHF integral is defined as
Remark 2.4
Letting \(\rho =1\). Then (2.5)–(2.7) become the Hadamard fractional integrals
and
For convenience, we give the semigroup property
which implies the commutative property,
Remark 2.5
If we choose \(\rho =1\), then (2.11) becomes the result of [32].
3 Main results
In the section, we will provide the refinements for some classical variants by utilizing the GPHF integral operator defined in (2.7).
Theorem 3.1
Let\(\rho \in (0,1]\), \(\gamma >0\), and\(\mathcal{F}\)be a positive real-valued function defined on\([1,\infty )\). Assume that
\((I)\)There exist two integrable functions\(\varphi _{1}\)and\(\varphi _{2}\)defined on\([1,\infty )\)such that
for all\(l\in [1,\infty )\).
Then one has
for all\(l>1\), \(\gamma >0\)and\(\varrho \in (0, 1]\).
Proof
Let \(\theta \geq 1\) and \(\varsigma \geq 1\). Then from \((I)\) we have
Therefore,
Multiplying both sides of (3.4) by \(\frac{1}{\rho ^{\gamma }\varGamma (\gamma )} \frac{\exp [\frac{\rho -1}{\rho }\ln (\frac{l}{\theta })] (\ln (\frac{l}{\theta }) )^{\gamma -1}}{\theta }\) and integrating the obtained inequality from 1 to l, we get
that is,
Multiplying both sides of (3.6) by \(\frac{1}{\rho ^{\varrho }\varGamma (\varrho )} \frac{\exp [\frac{\rho -1}{\rho }\ln (\frac{l}{\varsigma })] (\ln (\frac{l}{\varsigma }) )^{\varrho -1}}{\varsigma }\) and integrating the obtained results from 1 to l, we have
From (3.7) we immediately get the desired inequality (3.2). □
Corollary 3.2 is a special case of Theorem 3.1.
Corollary 3.2
Letting\(\rho =1\). Then Theorem 3.1leads to the Hadamard fractional integrals inequality
which was proved by Sudsutad et al. in [48].
Theorem 3.3
Let\(\rho \in (0,1]\), \(\gamma,\varrho >0\), and\(\mathcal{F}\)and\(\mathcal{G}\)be two positive real-valued functions defined on\([1,\infty )\)such that condition\((I)\)given in Theorem 3.1and condition\((\mathit{II})\)hold.
\((\mathit{II})\)There exist two integrable functions\(\omega _{1}\)and\(\omega _{2}\)defined on\([1,\infty )\)such that
for all\(l\in [1,\infty )\).
Then, for all\(x,\gamma,\varrho >0\), we have the following inequalities:
Proof
We first prove \((N_{1})\). Let \(l\in [1,\infty )\). Then it follows from \((I)\) and \((\mathit{II})\) that
that is,
Multiplying both sides of (3.11) by \(\frac{1}{\rho ^{\gamma }\varGamma (\gamma )} \frac{\exp [\frac{\rho -1}{\rho }\ln (\frac{l}{\theta })] (\ln (\frac{l}{\theta }) )^{\gamma -1}}{\theta }\) and integrating the obtained inequality from 1 to l lead to
Inequality (3.12) can be rewritten as
Multiplying both sides of (3.13) by \(\frac{1}{\rho ^{\varrho }\varGamma (\varrho )} \frac{\exp [\frac{\rho -1}{\rho }\ln (\frac{l}{\varsigma })] (\ln (\frac{l}{\varsigma }) )^{\varrho -1}}{\varsigma }\) and integrating the obtained inequality from 1 to l we get
which leads to the desired inequality \((N_{1})\)
Inequalities \((N_{2})\)–\((N_{4})\) can be proved by using the similar arguments as in the proof of inequality \((N_{1})\) and the facts that
□
As a special case of Theorem 3.3, we have Corollary 3.4.
Corollary 3.4
If\(\rho =1\), then Theorem 3.3leads to the Hadamard fractional integrals inequalities [48]
Theorem 3.5
Let\(\rho \in (0,1]\), \(\gamma,\varrho >0\), \(p, q>0\)with\(1/p+1/q=1\), and\(\mathcal{F}\)and\(\mathcal{G}\)be two positive real-valued functions defined on\([1,\infty )\). Then the following inequalities hold for\(l>1\):
Proof
It follows from the Young inequality [38] that
for all \(\mu, \nu \geq 0\).
Let \(\theta,\varsigma >1\), \(\mu =\mathcal{F}(\theta )\mathcal{G}(\varsigma )\) and \(\nu =\mathcal{F}(\varsigma )\mathcal{G}(\theta )\). Then inequality (3.15) becomes
Multiplying both sides of inequality (3.16) by \(\frac{\exp [\frac{\rho -1}{\rho }(\ln (\frac{l}{\theta })] (\ln (\frac{l}{\theta }) )^{\gamma -1}}{\theta \rho ^{\gamma }\varGamma (\gamma )}\) and integrating the obtained inequality from 1 to l give
Inequality (3.17) can be rewritten as
Multiplying both sides of inequality (3.18) by \(\frac{\exp [\frac{\rho -1}{\rho }(\ln (\frac{l}{\varsigma })] (\ln (\frac{l}{\varsigma }) )^{\varrho -1}}{\varsigma \rho ^{\varrho }\varGamma (\varrho )}\) and integrating the obtained result from 1 to l, one has
Inequality (3.19) leads to the conclusion that
which completes the proof of inequality \((N_{9})\).
Let
and
in the Young inequality, respectively. Then inequalities \((N_{10})\)–\((N_{12})\) can be proved by use of similar arguments to the proof of inequality \((N_{9})\). □
Corollary 3.6
Let\(\rho =1\). Then Theorem 3.5leads to the Hadamard fractional integrals inequalities
Theorem 3.7
Let\(\rho \in (0,1]\), \(\gamma,\varrho >0\), \(p, q>1\)with\(1/p+1/q=1\), and\(\mathcal{F}\)and\(\mathcal{G}\)be two integrable functions defined on\([1,\infty )\). Then the inequalities
hold for\(l>1\).
Proof
It follows from the well-known weighted arithmetic–geometric mean inequality that
for \(\mu, \nu \geq 0\).
Let \(\varsigma,\theta >1\), \(\mu =\mathcal{F}(\theta )\mathcal{G}(\varsigma )\) and \(\nu =\mathcal{F}(\varsigma )\mathcal{G}(\theta )\). Then inequality (3.22) leads to
Multiplying both sides of inequality (3.23) by
and integrating the obtained inequality from 1 to l, we have
Inequality (3.24) can be rewritten as
which completes the proof of inequality \((N_{17})\).
Let
in the arithmetic–geometric mean inequality, respectively. Then inequalities \((N_{18})\)–\((N_{20})\) can be proved by using the similar arguments as in the proof of inequality \((N_{17})\). □
Corollary 3.8
Let\(\rho =1\). Then Theorem 3.7leads to the Hadamard fractional integrals inequalities
Example 3.9
Let \(l>1\), \(\gamma,\varrho >0\), \(\mathcal{F}\) and \(\mathcal{G}\) be two integrable functions defined on \([1,\infty )\), and
Then we have the following three inequalities:
Proof
It follows from (3.26) that
Inequality (3.27) can be rewritten as
Multiplying both sides of inequality (3.28) by
and integrating the obtained inequality from 1 to l lead to
Inequality (3.29) implies that
Alternately, it follows from \(\hbar \mathcal{H}>0\) and
that
Therefore inequality \((1)\) follows easily from inequalities (3.30) and (3.32). Similarly, we also can prove inequalities \((2)\) and \((3)\). □
Example 3.10
Let \(l>1\), \(\gamma,\varrho >0\), \(p, q>1\) with \(1/p+1/q=1\), \(\mathcal{F}\) be an integrable function defined on \([1,\infty )\), and \(\mathfrak{J}_{1^{-},l}^{\gamma,\rho }\mathcal{F}\) be the generalized proportional Hadamard fractional integral operator. Then we have
where
and
is the incomplete gamma function [57–60].
Proof
It follows from Definition 2.3 and the modulus property that
for \(\vartheta >1\).
Making use of the well-known Hölder inequality, we obtain
Let \(v=\ln (\frac{l}{\vartheta })\). Then elaborated computations lead to
□
4 Conclusion
In this paper, we have derived numerous inequalities in the framework of a novel proposed GPHF integral operator with proportionality index ρ. Our obtained results are refinements of the Grüss inequality. In the special case of \(\rho =1\), it is worth mentioning that this allows for recapturing some existing operators from the GPHF integral operator, therefore, the GPHF integral operator is superior to many existing operators. In addition, our new approach recaptures the Grüss type inequalities and their variants proposed by Sudsutad et al. [48]. Our ideas may lead to a lot of follow-up research.
References
Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)
Abdeljawad, T., Baleanu, D.: Monotonicity results for fractional difference operators with discrete exponential kernels. Adv. Differ. Equ. 2017, Article ID 78 (2017)
Alzabut, J., Abdeljawad, T., Jarad, F., Sudsutad, W.: A Gronwall inequality via the generalized proportional fractional derivative with applications. J. Inequal. Appl. 2019, Article ID 101 (2019)
Doungmo, G., Emile, F., Kumar, S., Mugisha, S.B.: Similarities in a fifth-order evolution equation with and with no singular kernel. Chaos Solitons Fractals 130, Article ID 109467 (2020)
Ghanbari, B., Kumar, S., Kumar, R.: A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative. Chaos Solitons Fractals 133, Article ID 109619 (2020)
Jleli, M., Kumar, S., Kumar, R., Samet, B.: Analytical approach for time fractional wave equations in the sense of Yang–Abdel–Aty–Cattani via the homotopy perturbation transform method. Alex. Eng. J. https://doi.org/10.1016/j.aej.2019.12.022
Kumar, S., Kumar, A., Abbas, S., Al Qurashi, M., Baleanu, D.: A modified analytical approach with existence and uniqueness for fractional Cauchy reaction-diffusion equations. Adv. Differ. Equ. 2020, Article ID 28 (2020)
Kumar, S., Kumar, R., Agarwal, R.P., Samet, B.: A study of fractional Lotka-Volterra population model by using Haar wavelet and Adams–Bashforth–Moulton methods. Math. Methods Appl. Sci. 2020, 43 (2020). https://doi.org/10.1002/mma.6297
Kumar, S., Kumar, R., Singh, J., Nisar, K.S., Kumar, D.: An efficient numerical scheme for fractional model of HIV-1 infection of \(CD4^{+}\) T-cells with the effect of antiviral drug therapy. Alex. Eng. J. https://doi.org/10.1016/j.aej.2019.12.046
Kumar, S., Nisar, K.S., Kumar, R., Cattani, C., Samet, B.: A new Rabotnov fractiona-exponential functio-based fractional derivative for diffusion equation under external force. Math. Methods Appl. Sci. 43(7), 4460–4471 (2020)
Wu, J., Liu, Y.-C.: Fixed point theorems for monotone operators and applications to nonlinear elliptic problems. Fixed Point Theory Appl. 2013, Article ID 134 (2013)
Wu, J.: Some fixed-point theorems for mixed monotone operators in partially ordered probabilistic metric spaces. Fixed Point Theory Appl. 2014, Article ID 49 (2014)
Huang, C.-X., Guo, S., Liu, L.-Z.: Boundedness on Morrey space for Toeplitz type operator associated to singular integral operator with variable Calderón–Zygmund kernel. J. Math. Inequal. 8(3), 453–464 (2014)
Zhou, X.-S.: Weighted sharp function estimate and boundedness for commutator associated with singular integral operator satisfying a variant of Hörmander’s condition. J. Math. Inequal. 9(2), 587–596 (2015)
Huang, C.-X., Liu, L.-Z.: Boundedness of multilinear singular integral operator with a non-smooth kernel and mean oscillation. Quaest. Math. 40(3), 295–312 (2017)
Tan, Y.-X., Liu, L.-Z.: Weighted boundedness of multilinear operator associated to singular integral operator with variable Calderón–Zygmund kernel. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 111(4), 931–946 (2017)
Hu, H.-J., Liu, L.-Z.: Weighted inequalities for a general commutator associated to a singular integral operator satisfying a variant of Hörmander’s condition. Math. Notes 101(5–6), 830–840 (2017)
Rashid, S., Jarad, F., Noor, M.A., Kalsoom, H., Chu, Y.-M.: Inequalities by means of generalized proportional fractional integral operators with respect another function. Mathematics 7(12), Article ID 1225 (2019)
Rashid, S., Jarad, F., Chu, Y.-M.: A note on reverse Minkowski inequality via generalized proportional fractional integral operator with respect to another function. Math. Probl. Eng. 2020, Article ID 7630260 (2020)
Huang, C.-X., Liu, L.-Z.: Sharp function inequalities and boundness for Toeplitz type operator related to general fractional singular integral operator. Publ. Inst. Math. 92(106), 165–167 (2012)
Wu, J., Liu, Y.-C.: Uniqueness results and convergence of successive approximations for fractional differential equations. Hacet. J. Math. Stat. 42(2), 149–158 (2013)
Zhou, X.-S., Huang, C.-X., Hu, H.-J., Liu, L.: Inequality estimates for the boundedness of multilinear singular and fractional integral operators. J. Inequal. Appl. 2013, Article ID 303 (2013)
Liu, F.-W., Feng, L.-B., Anh, V., Li, J.: Unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time-space fractional Bloch–Torrey equations on irregular convex domains. Comput. Math. Appl. 78(5), 1637–1650 (2019)
Jiang, Y.-J., Xu, X.-J.: A monotone finite volume method for time fractional Fokker–Planck equations. Sci. China Math. 62(4), 783–794 (2019)
Zhou, S.-H., Jiang, Y.-J.: Finite volume methods for N-dimensional time fractional Fokker–Planck equations. Bull. Malays. Math. Sci. Soc. 42, 3167–3186 (2019)
Pratap, A., Raja, R., Cao, J.-D., Alzabut, J., Huang, C.-X.: Finite-time synchronization criterion of graph theory perspective fractional-order coupled discontinuous neural networks. Adv. Differ. Equ. 2020, Article ID 97 (2020)
Iqbal, A., Adil Khan, M., Ullah, S., Chu, Y.-M.: Some new Hermite–Hadamard-type inequalities associated with conformable fractional integrals and their applications. J. Funct. Spaces 2020, Article ID 9845407 (2020)
Rashid, S., Jarad, F., Kalsoom, H., Chu, Y.-M.: On Pólya–Szegö and Ćebyšev type inequalities via generalized k-fractional integrals. Adv. Differ. Equ. 2020, Article ID 125 (2020)
Awan, M.U., Talib, S., Chu, Y.-M., Noor, M.A., Noor, K.I.: Some new refinements of Hermite–Hadamard-type inequalities involving \(\varPsi _{k}\)-Riemann–Liouville fractional integrals and applications. Math. Probl. Eng. 2020, Article ID 3051920 (2020)
Chu, Y.-M., Adil Khan, M., Ali, T., Dragomir, S.S.: Inequalities for α-fractional differentiable functions. J. Inequal. Appl. 2017, Article ID 93 (2017)
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
Mathai, A.M.: A pathway to matrix-variate gamma and normal densities. Linear Algebra Appl. 396, 317–328 (2005)
Kiryakova, V.: Generalized Fractional Calculus and Applications. Longman, Harlow (1994)
Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus: Models and Numerical Methods. World Scientific, Hackensack (2012)
Jarad, F., Uğurlu, E., Abdeljawad, T., Baleanu, D.: On a new class of fractional operators. Adv. Differ. Equ. 2017, Article ID 247 (2017)
Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, New York (1989)
Sharma, B., Kumar, S., Cattani, C., Baleanu, D.: Nonlinear dynamics of Cattaneo–Christov heat flux model for third-grade power-law fluid. J. Comput. Nonlinear Dyn. 15(1), Article ID CND-19-1131 (2020)
Sharma, B., Kumar, S., Paswan, M.K.: Analytical solution for mixed convection and MHD flow of electrically conducting non-Newtonian nanofluid with different nanoparticles: a comparative study. Int. J. Heat Mass Transf. 36(3), 987–996 (2018)
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)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon (1993)
Rahman, G., Khan, A., Abdeljawad, T., Nisar, K.S.: The Minkowski inequalities via generalized proportional fractional integral operators. Adv. Differ. Equ. 2019, Article ID 287 (2019)
Dahmani, Z.: New inequalities in fractional integrals. Int. J. Nonlinear Sci. 9(4), 493–497 (2010)
Dahmani, Z., Tabharit, L., Taf, S.: New generalisations of Gruss inequality using Riemann–Liouville fractional integrals. Bull. Math. Anal. Appl. 2(3), 93–99 (2010)
Denton, Z., Vatsala, A.S.: Fractional integral inequalities and applications. Comput. Math. Appl. 59(3), 1087–1094 (2010)
Grüss, G.: Über das Maximum des absoluten Betrages von \(\frac{1}{b-a}\int _{a}^{b}f(x)g(x)\,dx-\frac{1}{(b-a)^{2}}\int _{a}^{b}f(x)\,dx \int _{a}^{b}g(x) \,dx\). Math. Z. 39(1), 215–226 (1935)
Sudsutad, W., Ntouyas, S.K., Tariboon, J.: Fractional integral inequalities via Hadamard’s fractional integral. Abstr. Appl. Anal. 2014, Article ID 563096 (2014)
Rahman, G., Abdeljawad, T., Jarad, F., Khan, A., Nisar, K.S.: Certain inequalities via generalized proportional Hadamard fractional integral operators. Adv. Differ. Equ. 2019, Article ID 454 (2019)
Rafeeq, S., Kalsoom, S., Hussain, S., Rashid, S., Chu, Y.-M.: Delay dynamic double integral inequalities on time scales with applications. Adv. Differ. Equ. 2020, Article ID 40 (2020)
Rashid, S., Ashraf, R., Noor, M.A., Noor, K.I., Chu, Y.-M.: New weighted generalizations for differentiable exponentially convex mapping with application. AIMS Math. 5(4), 3525–3546 (2020)
Rashid, S., Noor, M.A., Noor, K.I., Safdar, F., Chu, Y.-M.: Hermite–Hadamard type inequalities for the class of convex functions on time scale. Mathematics 7(10), Article ID 956 (2019)
Rashid, S., Noor, M.A., Noor, K.I., Chu, Y.-M.: Ostrowski type inequalities in the sense of generalized \(\mathcal{K}\)-fractional integral operator for exponentially convex functions. AIMS Math. 5(3), 2629–2645 (2020)
Zhao, T.-H., Chu, Y.-M., Wang, H.: Logarithmically complete monotonicity properties relating to the gamma function. Abstr. Appl. Anal. 2011, Article ID 896483 (2011)
Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On rational bounds for the gamma function. J. Inequal. Appl. 2017, Article ID 210 (2017)
Zhao, T.-H., Shi, L., Chu, Y.-M.: Convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 114(2), Article ID 96 (2020). https://doi.org/10.1007/s13398-020-00825-3
Zaheer Ullah, S., Adil Khan, M., Chu, Y.-M.: A note on generalized convex functions. J. Inequal. Appl. 2019, Article ID 291 (2019)
Yang, Z.-H., Qian, W.-M., Zhang, W., Chu, Y.-M.: Notes on the complete elliptic integral of the first kind. Math. Inequal. Appl. 23(1), 77–93 (2020)
Wang, M.-K., Hong, M.-Y., Xu, Y.-F., Shen, Z.-H., Chu, Y.-M.: Inequalities for generalized trigonometric and hyperbolic functions with one parameter. J. Math. Inequal. 14(1), 1–21 (2020)
Wang, M.-K., Chu, H.-H., Li, Y.-M., Chu, Y.-M.: Answers to three conjectures on convexity of three functions involving complete elliptic integrals of the first kind. Appl. Anal. Discrete Math. 14, 255–271 (2020)
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The work was supported by the Natural Science Foundation of China (Grant Nos. 11971142, 61673169, 11871202, 11701176, 11626101, 11601485).
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Zhou, SS., Rashid, S., Jarad, F. et al. New estimates considering the generalized proportional Hadamard fractional integral operators. Adv Differ Equ 2020, 275 (2020). https://doi.org/10.1186/s13662-020-02730-w
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DOI: https://doi.org/10.1186/s13662-020-02730-w
MSC
- 26D15
- 26D10
- 90C23
- 26E60
Keywords
- Grüss inequality
- Fractional calculus
- Generalized proportional Hadamard fractional integral operator
- Riemann–Liouville fractional integral operator