Research Article
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The Minkowski type inequalities for weighted fractional operators

Year 2022, Volume: 71 Issue: 3, 884 - 897, 30.09.2022
https://doi.org/10.31801/cfsuasmas.1054069

Abstract

In this article, inequalities of reverse Minkowski type involving weighted fractional operators are investigated. In addition, new fractional integral inequalities related to Minkowski type are also established.

References

  • Abdeljawad, T., On conformable fractional calculus, J. Comput. Appl. Math, 279 (2015), 57–66. https://doi.org/10.1016/j.cam.2014.10.016
  • Akdemir, A.O., Butt, S.I., Nadeem, M., Ragusa, M.A., New general variants of Chebyshev type inequalities via generalized fractional integral operators, Mathematics, 9(2) (2021), 122. https://doi.org/10.3390/math9020122
  • Bayraktar, B., Some integral inequalities of Hermite-Hadamard type for differentiable (s,m)-convex functions via fractional integrals, TWMS Jour. App. Engin. Math., 10(3) (2020), 625-637.
  • Belarbi, S., Dahmani, Z., On some new fractional integral inequalities, J. Ineq. Pure and Appl. Math., 10(3) (2009), art. 86.
  • Bougoffa, L., On Minkowski and Hardy integral inequalities, J. Ineq. Pure & Appl. Math., 7(2) (2006), art. 60.
  • Butt, S.I., Nadeem, M., Farid, G., On Caputo fractional derivatives via exponential s-convex functions, Turkish Jour. Sci., 5(2) (2020), 140-146.
  • Butt, S.I., Umar, M., Rashid, S., Akdemir, A.O., Chu, Y.M., New Hermite-Jensen-Mercer-type inequalities via k-fractional integrals, Adv. Diff. Equat., 1 (2020), 1-24. https://doi.org/10.1186/s13662-020-03093-y
  • Chinchane, V.L., Pachpatte, D.B., New fractional inequalities via Hadamard fractional integral, Int. J. Funct. Anal., 5 (2013), 165-176.
  • Dahmani, Z., On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Ann. Funct. Anal., 1(1) (2010), 51–58. https://doi.org/10.15352/afa/1399900993
  • Ekinci, A., Özdemir, M.E., Some new integral inequalities via Riemann-Liouville integral operators, App. Comp. Math., 3(18) (2019), 288–295.
  • Ekinci, A., Özdemir, M.E., Set, E., New integral inequalities of Ostrowski type for quasi−convex functions with applications, Turkish Jour. Sci., 5(3) (2020), 290-304.
  • Hardy, G.H., Littewood, J.E., P`olya, G., Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, 1988, pp.30-32, 123, 146-150.
  • Jarad, F., Abdeljawad, T., Shah, K., On the weighted fractional operators of a function with respect to another function, Fractals, 28(08) (2020), 2040011. https://doi.org/10.1142/S0218348X20400113
  • Jarad, F., Ugurlu, U., Abdeljawad, T., Baleanu, D., On a new class of fractional operators, Adv. Differ. Equ, 2017(247) (2017), 1-16. https://doi.org/10.1186/s13662-017-1306-z
  • Katugampola, U.N., New approach to generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865. https://doi.org/10.3390/math10040573
  • Katugampola, U.N., A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1-15.
  • Khalil, R., Al Horani, M., Yousef, A., Sababheh, M., A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002
  • Khan, T.U., Khan, M.A., Generalized conformable fractional integral operators, J. Comput. Appl. Math., 346 (2019), 378–389. https://doi.org/10.1016/j.cam.2018.07.018
  • Kızıl, S¸., Ardı¸c, M.A., Inequalities for strongly convex functions via Atangana-Baleanu integral operators, Turkish Jour. Sci., 6(2) (2021), 96-109.
  • Liouville, J., Memoire sur quelques questions de geometrie et de mecanique, et sur un nouveau genre de calcul pour resoudre ces questions, Journal de l’ Ecole Polytechnique, 13 (1832), 1–69.
  • Miller, K.S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
  • Mitrinovic, D.S., Peˇcari´c, J.E., Fink, A.M., Classical and New Inequalities in Analysis, Kluwer Academic Publishers, London, 1993.
  • Mohammed, P.O., Abdeljawad, T., Alqudah, M.A., Jarad, F., New discrete inequalities of Hermite–Hadamard type for convex functions, Adv. Diff. Equat., 1 (2021), 1-10. https://doi.org/10.1186/s13662-021-03290-3
  • Mohammed, P.O., Abdeljawad, T., Kashuri, A., Fractional Hermite-Hadamard-Fejer inequalities for a convex function with respect to an increasing function involving a positive weighted symmetric function, Symmetry, 12(9) (2020), 1503. https://doi.org/10.3390/sym12091503
  • Mohammed, P.O., Abdeljawad, T., Zeng, S., Kashuri, A., Fractional Hermite-Hadamard integral inequalities for a new class of convex functions, Symmetry, 12(9) (2020), 1485. https://doi.org/10.3390/sym12091485
  • Özdemir, M.E., Dragomir, S.S., Yıldız, Ç., The Hadamard inequality for convex function via fractional integrals, Acta Math. Sci., 33B(5) (2013), 1293-1299. https://doi.org/10.1016/S0252-9602(13)60081-8
  • Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, 1999.
  • Sarıkaya, M.Z., Yaldız, H., On generalization integral inequalities for fractional inegrals, Nihonkai Math. J., 25 (2014), 93-104.
  • Set, E., Akdemir, A.O., Gözpınar, A., Jarad, F., Ostrowski type inequalities via new fractional conformable integrals, AIMS Mathematics, 4(6) (2019), 1684–1697. https://doi.org/10.3934/math.2019.6.1684
  • Set, E., Özdemir, M.E., Dragomir, S.S., On the Hermite–Hadamard inequality and other integral inequalities involving two functions, J. Ineq. Appl., 2010(148102) (2010), 1-9. https://doi.org/10.1155/2010/148102
  • Sousa, J.V.D.C., de Oliveira, E.C., The Minkowski’s inequality by means of a generalized fractional integral, AIMS Mathematics, 3(1) (2018), 131-142. https://doi.org/10.3934/Math.2018.1.131
  • Yavuz, M., Özdemir, N., Comparing the new fractional derivative operators involving exponential and Mittag-Leffler kernel, Discrete Contin. Dyn. Syst., Ser., 13(3) (2020), 995. https://doi.org/10.3934/dcdss.2020058
Year 2022, Volume: 71 Issue: 3, 884 - 897, 30.09.2022
https://doi.org/10.31801/cfsuasmas.1054069

Abstract

References

  • Abdeljawad, T., On conformable fractional calculus, J. Comput. Appl. Math, 279 (2015), 57–66. https://doi.org/10.1016/j.cam.2014.10.016
  • Akdemir, A.O., Butt, S.I., Nadeem, M., Ragusa, M.A., New general variants of Chebyshev type inequalities via generalized fractional integral operators, Mathematics, 9(2) (2021), 122. https://doi.org/10.3390/math9020122
  • Bayraktar, B., Some integral inequalities of Hermite-Hadamard type for differentiable (s,m)-convex functions via fractional integrals, TWMS Jour. App. Engin. Math., 10(3) (2020), 625-637.
  • Belarbi, S., Dahmani, Z., On some new fractional integral inequalities, J. Ineq. Pure and Appl. Math., 10(3) (2009), art. 86.
  • Bougoffa, L., On Minkowski and Hardy integral inequalities, J. Ineq. Pure & Appl. Math., 7(2) (2006), art. 60.
  • Butt, S.I., Nadeem, M., Farid, G., On Caputo fractional derivatives via exponential s-convex functions, Turkish Jour. Sci., 5(2) (2020), 140-146.
  • Butt, S.I., Umar, M., Rashid, S., Akdemir, A.O., Chu, Y.M., New Hermite-Jensen-Mercer-type inequalities via k-fractional integrals, Adv. Diff. Equat., 1 (2020), 1-24. https://doi.org/10.1186/s13662-020-03093-y
  • Chinchane, V.L., Pachpatte, D.B., New fractional inequalities via Hadamard fractional integral, Int. J. Funct. Anal., 5 (2013), 165-176.
  • Dahmani, Z., On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Ann. Funct. Anal., 1(1) (2010), 51–58. https://doi.org/10.15352/afa/1399900993
  • Ekinci, A., Özdemir, M.E., Some new integral inequalities via Riemann-Liouville integral operators, App. Comp. Math., 3(18) (2019), 288–295.
  • Ekinci, A., Özdemir, M.E., Set, E., New integral inequalities of Ostrowski type for quasi−convex functions with applications, Turkish Jour. Sci., 5(3) (2020), 290-304.
  • Hardy, G.H., Littewood, J.E., P`olya, G., Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, 1988, pp.30-32, 123, 146-150.
  • Jarad, F., Abdeljawad, T., Shah, K., On the weighted fractional operators of a function with respect to another function, Fractals, 28(08) (2020), 2040011. https://doi.org/10.1142/S0218348X20400113
  • Jarad, F., Ugurlu, U., Abdeljawad, T., Baleanu, D., On a new class of fractional operators, Adv. Differ. Equ, 2017(247) (2017), 1-16. https://doi.org/10.1186/s13662-017-1306-z
  • Katugampola, U.N., New approach to generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865. https://doi.org/10.3390/math10040573
  • Katugampola, U.N., A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1-15.
  • Khalil, R., Al Horani, M., Yousef, A., Sababheh, M., A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002
  • Khan, T.U., Khan, M.A., Generalized conformable fractional integral operators, J. Comput. Appl. Math., 346 (2019), 378–389. https://doi.org/10.1016/j.cam.2018.07.018
  • Kızıl, S¸., Ardı¸c, M.A., Inequalities for strongly convex functions via Atangana-Baleanu integral operators, Turkish Jour. Sci., 6(2) (2021), 96-109.
  • Liouville, J., Memoire sur quelques questions de geometrie et de mecanique, et sur un nouveau genre de calcul pour resoudre ces questions, Journal de l’ Ecole Polytechnique, 13 (1832), 1–69.
  • Miller, K.S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
  • Mitrinovic, D.S., Peˇcari´c, J.E., Fink, A.M., Classical and New Inequalities in Analysis, Kluwer Academic Publishers, London, 1993.
  • Mohammed, P.O., Abdeljawad, T., Alqudah, M.A., Jarad, F., New discrete inequalities of Hermite–Hadamard type for convex functions, Adv. Diff. Equat., 1 (2021), 1-10. https://doi.org/10.1186/s13662-021-03290-3
  • Mohammed, P.O., Abdeljawad, T., Kashuri, A., Fractional Hermite-Hadamard-Fejer inequalities for a convex function with respect to an increasing function involving a positive weighted symmetric function, Symmetry, 12(9) (2020), 1503. https://doi.org/10.3390/sym12091503
  • Mohammed, P.O., Abdeljawad, T., Zeng, S., Kashuri, A., Fractional Hermite-Hadamard integral inequalities for a new class of convex functions, Symmetry, 12(9) (2020), 1485. https://doi.org/10.3390/sym12091485
  • Özdemir, M.E., Dragomir, S.S., Yıldız, Ç., The Hadamard inequality for convex function via fractional integrals, Acta Math. Sci., 33B(5) (2013), 1293-1299. https://doi.org/10.1016/S0252-9602(13)60081-8
  • Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, 1999.
  • Sarıkaya, M.Z., Yaldız, H., On generalization integral inequalities for fractional inegrals, Nihonkai Math. J., 25 (2014), 93-104.
  • Set, E., Akdemir, A.O., Gözpınar, A., Jarad, F., Ostrowski type inequalities via new fractional conformable integrals, AIMS Mathematics, 4(6) (2019), 1684–1697. https://doi.org/10.3934/math.2019.6.1684
  • Set, E., Özdemir, M.E., Dragomir, S.S., On the Hermite–Hadamard inequality and other integral inequalities involving two functions, J. Ineq. Appl., 2010(148102) (2010), 1-9. https://doi.org/10.1155/2010/148102
  • Sousa, J.V.D.C., de Oliveira, E.C., The Minkowski’s inequality by means of a generalized fractional integral, AIMS Mathematics, 3(1) (2018), 131-142. https://doi.org/10.3934/Math.2018.1.131
  • Yavuz, M., Özdemir, N., Comparing the new fractional derivative operators involving exponential and Mittag-Leffler kernel, Discrete Contin. Dyn. Syst., Ser., 13(3) (2020), 995. https://doi.org/10.3934/dcdss.2020058
There are 32 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Çetin Yıldız 0000-0002-8302-343X

Mustafa Gürbüz 0000-0002-7092-4298

Publication Date September 30, 2022
Submission Date January 6, 2022
Acceptance Date April 21, 2022
Published in Issue Year 2022 Volume: 71 Issue: 3

Cite

APA Yıldız, Ç., & Gürbüz, M. (2022). The Minkowski type inequalities for weighted fractional operators. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(3), 884-897. https://doi.org/10.31801/cfsuasmas.1054069
AMA Yıldız Ç, Gürbüz M. The Minkowski type inequalities for weighted fractional operators. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. September 2022;71(3):884-897. doi:10.31801/cfsuasmas.1054069
Chicago Yıldız, Çetin, and Mustafa Gürbüz. “The Minkowski Type Inequalities for Weighted Fractional Operators”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71, no. 3 (September 2022): 884-97. https://doi.org/10.31801/cfsuasmas.1054069.
EndNote Yıldız Ç, Gürbüz M (September 1, 2022) The Minkowski type inequalities for weighted fractional operators. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71 3 884–897.
IEEE Ç. Yıldız and M. Gürbüz, “The Minkowski type inequalities for weighted fractional operators”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 71, no. 3, pp. 884–897, 2022, doi: 10.31801/cfsuasmas.1054069.
ISNAD Yıldız, Çetin - Gürbüz, Mustafa. “The Minkowski Type Inequalities for Weighted Fractional Operators”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71/3 (September 2022), 884-897. https://doi.org/10.31801/cfsuasmas.1054069.
JAMA Yıldız Ç, Gürbüz M. The Minkowski type inequalities for weighted fractional operators. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71:884–897.
MLA Yıldız, Çetin and Mustafa Gürbüz. “The Minkowski Type Inequalities for Weighted Fractional Operators”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 71, no. 3, 2022, pp. 884-97, doi:10.31801/cfsuasmas.1054069.
Vancouver Yıldız Ç, Gürbüz M. The Minkowski type inequalities for weighted fractional operators. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71(3):884-97.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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