On Pólya-Szegö integral inequalities using k-Hilfer fractional derivative

Asha B. Nale; Vaijanath L. Chinchane; Ould Melha Khellaf; Christophe Chesneau

doi:10.2478/aupcsm-2025-0008

Abstract

The main aim of this paper is to use the k-Hilfer fractional derivative to derive certain Pólya-Szegö fractional integral inequalities. Further fractional integral inequalities are obtained. The results presented here extend and generalize various existing inequalities associated with the Riemann-Liouville, Caputo, Saigo, and Hilfer fractional operators. The findings contribute to the growing theory of fractional calculus and offer potential tools for the analysis of fractional differential and integral equations.

References

Ahmad, Hijaz, et al. “Refinements of Ostrowski type integral inequalities involving Atangana-Baleanu fractional integral operator.” Symmetry 13, no. 11 (2021): Article 2059. Cited on 78.
Search in Google Scholar Back to article
Anastassiou, George A. Fractional Differentiation Inequalities. New York: Springer, 2009. Cited on 78.
Search in Google Scholar Back to article
Anastassiou, George A., et al. “Montgomery identities for fractional integrals and related fractional inequalities.” J. Inequal. Pure Appl. Math. 10, no. 4 (2009): Article 97. Cited on 78.
Search in Google Scholar Back to article
Baleanu, D., J.A.T. Machado, and C.J. Luo. Fractional Dynamics and Control. New York: Springer, 2012. Cited on 78.
Search in Google Scholar Back to article
Belarbi, S., and Z. Dahmani. “On some new fractional integral inequality.” J. Inequal. Pure Appl. Math. 10, no. 3 (2009): Article 86. Cited on 78.
Search in Google Scholar Back to article
Chebyshev, P.L. “Sur les expressions approximatives des intégrales définies par les autres prises entre les mêmes limites.” Proc. Sci. Charkov 2 (1882): 93-98. Cited on 78.
Search in Google Scholar Back to article
Chinchane, V.L., et al. “On some fractional integral inequalities involving Caputo-Fabrizio integral operator.” Axioms 10, no. 4 (2021): Article 255. Cited on 78.
Search in Google Scholar Back to article
Chinchane, V.L., A.B. Nale, and S.K. Panchal. “On the Minkowski fractional integral inequality using k-Hilfer fractional derivative.” Novi Sad J. Math. 54, no. 2 (2024): 161-171. Cited on 78.
Search in Google Scholar Back to article
Chinchane, V.L., and D.B. Pachpatte. “A note on some integral inequalities via Hadamard integral.” J. Fractional Calculus Appl. 4, no. 11 (2013): 1-5. Cited on 78.
Search in Google Scholar Back to article
Chinchane, V.L., and D.B. Pachpatte. “A note on fractional integral inequalities involving convex functions using Saigo fractional integral.” Indian J. Math. 61, no. 1 (2019): 27-39. Cited on 78.
Search in Google Scholar Back to article
Chinchane, V.L. “New approach to Minkowski fractional inequalities using generalized k-fractional integral operator.” J. Indian Math. Soc. 85, no. 1-2 (2018): 32-41. Cited on 78.
Search in Google Scholar Back to article
Chinchane, V.L. “On Chebyshev type inequalities using generalized k-fractional integral operator.” Progr. Fract. Differ. Appl. 3, no. 3 (2017): 1-8. Cited on 78.
Search in Google Scholar Back to article
Dorrego, G.A., and R.A. Cerutti. “The k-fractional Hilfer derivative.” Int. J. Math. Anal. 7, no. 9-12 (2013): 543-550. Cited on 80.
Search in Google Scholar Back to article
Diaz, R., and E. Pariguan. “On hypergeometric functions and the Pochhammer k-symbol.” Divulg. Mat. 15, no. 2 (2007): 179-192. Cited on 79.
Search in Google Scholar Back to article
Deniz, E., A.O. Akdemir, and E. Yüksel. “New extensions of Chebyshev-Pólya-Szegö type inequalities via conformable integrals.” AIMS Math. 5, no. 2 (2020): 956-965. Cited on 78.
Search in Google Scholar Back to article
Dahmani, Z. “New inequalities in fractional integrals.” Int. J. Nonlinear Sci. 4, no. 9 (2010): 493-497. Cited on 78.
Search in Google Scholar Back to article
Dahmani, Z. “Some results associated with fractional integrals involving the extended Chebyshev.” Acta Univ. Apulensis Math. Inform. 27 (2011): 217-224. Cited on 78.
Search in Google Scholar Back to article
Dragomir, S.S., and C.E. Pearce. Selected Topics in Hermite-Hadamard Inequality. Victoria: Victoria Univ. Monographs, 2000. Cited on 78.
Search in Google Scholar Back to article
[19] Grüss, G. “Über das Maximum des absoluten Betrages von 1b−a∫abf(x)g(x)dx−1(b−a)2∫abf(x)dx∫abg(x)dx \frac{1}{b-a} \int_a^b f(x) g(x) d x- \frac{1}{(b-a)^2} \int_a^b f(x) d x \int_a^b g(x) d x .” 39 (1935): 215-226. Cited on 78.
Back to article
Hilfer, R. Applications of Fractional Calculus in Physics. Singapore: World Scientific Publishing, 2000. Cited on 80.
Search in Google Scholar Back to article
Iqbal, S., et al. “Some new Grüss inequalities associated with generalized fractional derivative.” AIMS Math. 8, no. 1 (2022): 213-227. Cited on 78.
Search in Google Scholar Back to article
Iqbal, S., et al. “On some new Grüss inequalities concerning the Caputo k-fractional derivative.” Punjab Univ. J. Math. 52 (2020): 17-29. Cited on 80.
Search in Google Scholar Back to article
Jain, S., et al. “Certain recent fractional inequalities associated with hypergeo-metric operators.” J. King Saud Univ. Sci. 28 (2016): 82-86. Cited on 78.
Search in Google Scholar Back to article
Katugampola, U.N. “A new approach to generalized fractional derivatives.” Bull. Math. Anal. Appl. 6, no. 4 (2014): 1-15. Cited on 78.
Search in Google Scholar Back to article
Kilbas, A.A., H.M. Srivastava, and J.J. Trujillo. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier, 2006. Cited on 78 and 80.
Search in Google Scholar Back to article
Kiryakova, V. “On two Saigo’s fractional integral operators in the class of univalent functions.” Fract. Calc. Appl. Anal. 9, no. 2 (2006): 159-176. Cited on 78.
Search in Google Scholar Back to article
Miller, K.S., and B. Ross. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley, 1993. Cited on 78.
Search in Google Scholar Back to article
Nchama, G.A.M., A.L. Mecias, and M.R. Richard. “The Caputo-Fabrizio fractional integral to generate some new inequalities.” Inf. Sci. Lett. 2, no. 8 (2019): 73-80. Cited on 78.
Search in Google Scholar Back to article
Nale, A.B., S.K. Panchal, and V.L. Chinchane. “Minkowski-type inequalities using generalized proportional Hadamard fractional integral operators.” Filomat 35, no. 9 (2021): 2973-2984. Cited on 78.
Search in Google Scholar Back to article
Nale, A.B., et al. “Fractional integral inequalities using Marichev-Saigo-Maeda fractional integral operator.” Progr. Fract. Differ. Appl. 7, no. 4 (2021): 1-7. Cited on 78.
Search in Google Scholar Back to article
Nale, A.B., et al. “Pólya-Szegö integral inequalities using the Caputo-Fabrizio approach.” Axioms 11, no. 2 (2022): Article 79. Cited on 78.
Search in Google Scholar Back to article
Nale, A.B., D.B. Pachpatte, and V.L. Chinchane. “On Pólya-Szegö fractional inequality via Hadamard fractional integral.” Eurasian Bull. Math. 4, no. 3 (2021): 211-219. Cited on 78.
Search in Google Scholar Back to article
Oliveira, D.S., and E. Capelas de Oliveira. “Hilfer-Katugampola fractional derivative.” Comput. Appl. Math. 37 (2018): 3672-3690. Cited on 78.
Search in Google Scholar Back to article
Pachpatte, B.G. “A note on Chebyshev-Grüss type inequalities for differential function.” Tamsui Oxf. J. Math. Sci. 22, no. 10 (2006): 29-36. Cited on 77.
Search in Google Scholar Back to article
Podlubny, I. Fractional Differential Equations. San Diego: Academic Press, 1999. Cited on 78.
Search in Google Scholar Back to article
Rahman, G., et al. “Some new tempered fractional Pólya-Szegö and Chebyshev-type inequalities with respect to another function.” J. Math. 2020 (2020): Article 9858671. Cited on 78.
Search in Google Scholar Back to article
Rashid, S., et al. “On Pólya-Szegö and Čebyšev type inequalities via generalized k-fractional integrals.” Adv. Differ. Equ. 2020 (2020): Article 125. Cited on 78.
Search in Google Scholar Back to article
Sahoo, S.K., et al. “Some novel fractional integral inequalities over a new class of generalized convex function.” Fractal Fract. 6 (2022): Article 42. Cited on 78.
Search in Google Scholar Back to article
Sahoo, S.K., et al. “Hermite-Hadamard type inequalities involving k-fractional operator for (¯h, m)-convex functions.” Symmetry 13 (2021): Article 1686. Cited on 78.
Search in Google Scholar Back to article
Somko, S.G., A.A. Kilbas, and O.I. Marichev. Fractional Integral and Derivative Theory and Application. Switzerland: Gordon and Breach, 1993. Cited on 78 and 80.
Search in Google Scholar Back to article

On Pólya-Szegö integral inequalities using k-Hilfer fractional derivative

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