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doi: 10.3934/mfc.2021004

Iyengar-Hilfer fractional inequalities

George A. Anastassiou

Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA

Received  February 2021 Published  March 2021

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.

Keywords: Fractional calculus, fractional derivatives, integral inequalities, fractional Iyengar inequality, right and left $ \psi $-Hilfer and Hilfer fractional derivatives.
Mathematics Subject Classification: 26A33, 26D10, 26D15.
Citation: George A. Anastassiou. Iyengar-Hilfer fractional inequalities. Mathematical Foundations of Computing, doi: 10.3934/mfc.2021004
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.  Google Scholar

[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.   Google Scholar

[3]

K. S. K. Iyengar, Note on an inequality, Math. Stud., 6 (1938), 75-76.   Google Scholar

[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.  Google Scholar

[5]

W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., New York-Toronto, Ont. -London, 1966.  Google Scholar

[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.  Google Scholar

[7]

D. W. Stroock, A Concise Introduction to the Theory of Integration, Birkhaüser Boston, Inc, Boston, MA, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.   Google Scholar

[3]

K. S. K. Iyengar, Note on an inequality, Math. Stud., 6 (1938), 75-76.   Google Scholar

[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.  Google Scholar

[5]

W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., New York-Toronto, Ont. -London, 1966.  Google Scholar

[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.  Google Scholar

[7]

D. W. Stroock, A Concise Introduction to the Theory of Integration, Birkhaüser Boston, Inc, Boston, MA, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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