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Preface: Special issue on approximation by linear and nonlinear operators with applications. Part I
Iyengar-Hilfer fractional inequalities
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA |
Here we present Iyengar type integral inequalities. At the univariate level they involve $ \psi $-Hilfer left and right fractional derivatives. At the multivariate level they involve Hilfer left and right fractional derivatives, and they deal with radial and non-radial functions on the ball and spherical shell. All estimates are with respect to norms $ \left \Vert \cdot \right \Vert _{p} $, $ 1\leq p\leq \infty $. At the end we provide an application.
References:
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R. Almeida,
A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 460-481.
doi: 10.1016/j.cnsns.2016.09.006. |
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S. S. Dragomir,
Inequalities for the Riemann-Stieljes integral of $\left( p,q\right) -H-$dominated integrators with applications, Appl. Math. E-Notes, 15 (2015), 243-260.
|
| [3] |
K. S. K. Iyengar,
Note on an inequality, Math. Stud., 6 (1938), 75-76.
|
| [4] |
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differentiation Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006. |
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W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., New York-Toronto, Ont. -London, 1966. |
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S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. |
| [7] |
D. W. Stroock, A Concise Introduction to the Theory of Integration, Birkhaüser Boston, Inc, Boston, MA, 1999. |
| [8] |
Ž. Tomovski, R. Hilfer and H. M. Srivastava,
Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Integral Transforms Spec Funct., 21 (2010), 797-814.
doi: 10.1080/10652461003675737. |
| [9] |
J. Vanterler da C. Sousa and E. Capelas de Oliveira,
On the $\psi $-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72-91.
doi: 10.1016/j.cnsns.2018.01.005. |
show all references
References:
| [1] |
R. Almeida,
A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 460-481.
doi: 10.1016/j.cnsns.2016.09.006. |
| [2] |
S. S. Dragomir,
Inequalities for the Riemann-Stieljes integral of $\left( p,q\right) -H-$dominated integrators with applications, Appl. Math. E-Notes, 15 (2015), 243-260.
|
| [3] |
K. S. K. Iyengar,
Note on an inequality, Math. Stud., 6 (1938), 75-76.
|
| [4] |
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differentiation Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006. |
| [5] |
W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., New York-Toronto, Ont. -London, 1966. |
| [6] |
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. |
| [7] |
D. W. Stroock, A Concise Introduction to the Theory of Integration, Birkhaüser Boston, Inc, Boston, MA, 1999. |
| [8] |
Ž. Tomovski, R. Hilfer and H. M. Srivastava,
Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Integral Transforms Spec Funct., 21 (2010), 797-814.
doi: 10.1080/10652461003675737. |
| [9] |
J. Vanterler da C. Sousa and E. Capelas de Oliveira,
On the $\psi $-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72-91.
doi: 10.1016/j.cnsns.2018.01.005. |
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