Для цитирования:
Nápoles J. E., Guzmán P. M., Bayraktar B. New integral inequalities in the class of functions (h, m)-convex [Наполес Х. Э., Гузман П. М., Байрактар Б. Новые интегральные неравенства в классе (h, m)-выпуклых функций] // Известия Саратовского университета. Новая серия. Серия: Математика. Механика. Информатика. 2024. Т. 24, вып. 2. С. 173-183. DOI: 10.18500/1816-9791-2024-24-2-173-183, EDN: WYDLVW
New integral inequalities in the class of functions (h, m)-convex
[Новые интегральные неравенства в классе (h, m)-выпуклых функций]
В статье определены новые взвешенные интегральные операторы. Сформулирована лемма, в которой получено обобщенное тождество через эти интегральные операторы. С использованием данного тождества получены некоторые новые обобщенные неравенства типа Симпсона для $(h,m)$-выпуклых функций. Эти результаты получены на основе свойства выпуклости, классического неравенства Гельдера и его другой формы — неравенства степенного среднего. Общность результатов статьи заключается в двух основных моментах. Первый — используемый интегральный оператор, так как «вес» позволяет охватить многие известные интегральные операторы, в том числе классические Римана и Римана – Лиувилля. Второй момент — используемое понятие выпуклости, при адекватном выборе параметров оно содержит несколько уже известных понятий выпуклости. Это позволяет сделать заключение, что многие известные в литературе результаты являются частными случаями рассматриваемых в статье.
- Napoles J. E., Rabossi F., Samaniego A. D. Convex functions: Ariadne’s thread or Sharlotte’s spiderweb? Advanced Mathematical Models & Applications, 2020, vol. 5, iss. 2, pp. 176–191.
- Alomari M., Hussain S. Two inequalities of Simpson type for quasi-convex functions and applications. Applied Mathematics E-Notes, 2011, vol. 11, pp. 110–117.
- Set E., Ozdemir E., Sarıkaya M. Z. On new inequalities of Simpson’s type for quasi-convex functions with applications. Tamkang Journal of Mathematics, 2012, vol. 43, iss. 3, pp. 357–364. https://doi.org/10.5556/j.tkjm.43.2012.616
- Bayraktar B. Some integral inequalities for functions whose absolute values of the third derivative is concave and r-convex. Turkish Journal of Inequalities, 2020, vol. 4, iss. 2, pp. 59–78.
- Bayraktar B., Napoles J. E., Rabossi F. On generalizations of integral inequalities. Problemy Analiza – Issues of Analysis, 2022, vol. 11 (29), iss. 2, pp. 3–23. https://doi.org/10.15393/j3.art.2022.11190
- Dragomir S. S., Agarwal R. P., Cerone P. On Simpson’s inequality and applications. Journal of Inequalities and Applications, 2000, vol. 5, iss. 6, pp. 533–579. https://doi.org/10.1155/S102558340000031X
- Liu Z. An inequality of Simpson type. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 2005, vol. 461, iss. 2059, pp. 2155–2158. https://doi.org/10.1098/rspa.2005.1505
- Hussain S., Qaisar S. Generalizations of Simpson’s type inequalities through preinvexity and prequasiinvexity. Punjab University Journal of Mathematics, 2014, vol. 46, iss. 2, pp. 1–9.
- Park J. Hermite – Hadamard type and Simpson’s type inequalities for the decreasing (α, m)- geometrically convex functions. Applied Mathematical Sciences, 2014, vol. 61–64, pp. 3181–3195.
- Sarıkaya M. Z., Set E., Ozdemir M. E. On new inequalities of Simpson’s type for s-convex functions. Computers & Mathematics with Applications, 2010, vol. 60, iss. 8, pp. 2191–2199. https://doi.org/10.1016/j.camwa.2010.07.033
- Desalegn H., Mijena J. B., Nwaeze E. R., Abdi T. Simpson’s type inequalities for s-convex functions via a generalized proportional fractional integral. Foundations, 2022, vol. 2, pp. 607–616. https://doi.org/10.3390/foundations2030041
- Hua J., Xi B.-Y., Qi F. Some new inequalities of Simpson type for strongly s-convex functions. Afrika Matematika, 2015, vol. 26, pp. 741–752 http://dx.doi.org/10.1007/s13370-014-0242-2
- Kashuri A., Meftah B., Mohammed P. O. Some weighted Simpson type inequalities for differentiable s-convex functions and their applications. Journal of Fractional Calculus and Nonlinear Systems, 2021, vol. 1, iss. 1, pp. 75–94. http://dx.doi.org/10.48185/jfcns.v1i1.150
- Du T. S., Li Y. J., Yang Z. Q. A generalization of Simpson’s inequality via differentiable mapping using extended (s, m)-convex functions. Applied Mathematics and Computation, 2017, vol. 293, pp. 358–369. https://doi.org/10.1016/j.amc.2016.08.045
- Du T. S., Liao J. G., Li Y. J. Properties and integral inequalities of Hadamard – Simpson type for the generalized (s, m)-preinvex functions, Journal of Nonlinear Sciences and Applications, 2016, vol. 9, iss. 5, pp. 3112–3126. http://dx.doi.org/10.22436/jnsa.009.05.102
- Luo C., Du T. Generalized Simpson type inequalities involving Riemann – Liouville fractional integrals and their applications. Filomat, 2020, vol. 34, iss. 3, pp. 751–760. https://doi.org/10.2298/FIL2003751L
- Hsu K. C., Hwang S. R., Tseng K. L. Some extended Simpson type inequalities and applications. Bulletin of the Iranian Mathematical Society, 2017, vol. 43, iss. 2, pp. 409-425.
- Ujevic N. Double integral inequalities of Simpson type and applications. Journal of Applied Mathematics and Computing, 2004, vol. 14, pp. 213–223. https://doi.org/10.1007/BF02936109
- Bayraktar B., Napoles J. E. Hermite – Hadamard weighted integral inequalities for (h, m)-convex modified functions. Fractional Differential Calculus, 2022, vol. 12, iss. 2, pp. 235–248. https://doi.org/10.7153/fdc-2022-12-15
- Bayraktar B., Napoles J. E. New generalized integral inequalities via (h, m)-convex modified functions. Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 2022, vol. 60, pp. 3–15. https://doi.org/10.35634/2226-3594-2022-60-01
- Bayraktar B., Napoles J. E. Integral inequalities for mappings whose derivatives are (h, m, s)- convex modified of second type via Katugampola integrals. Annals of the University of Craiova, Mathematics and Computer Science Series, 2022, vol. 49, iss. 2, pp. 371–383. https://doi.org/10.52846/ami.v49i2.1596
- Rainville E. D. Special Functions. New York, Macmillan Co., 1960. 365 p.
- D´ıaz R., Pariguan E. On hypergeometric functions and Pochhammer k-symbol. Divulgaciones Matematicas, 2007, vol. 15, iss. 2, pp. 179–192. https://doi.org/10.48550/arXiv.math/0405596
- Mubeen S., Habibullah G. M. k-fractional integrals and application. International Journal of Contemporary Mathematical Sciences, 2012, vol. 7, iss. 2, pp. 89–94.
- Akkurt A. E., Yildirim M., Yildirim H. On some integral inequalities for (k, h)-Riemann – Liouville fractional integral. New Trends in Mathematical Sciences, 2016, vol. 4, iss. 1, pp. 138–146. http://dx.doi.org/10.20852/ntmsci.2016217824
- Jarad F., Abdeljawad T., Shah T. On the weighted fractional operators of a function with respect to another function. Fractals, 2020, vol. 28, iss. 8, art. 2040011. http://dx.doi.org/10.1142/S0218348X20400113
- Sarikaya M. Z., Ertugral F. On the generalized Hermite – Hadamard inequalities. Annals of the University of Craiova, Mathematics and Computer Science Series, 2020, vol. 47, iss. 1, pp. 193–213. https://doi.org/10.52846/ami.v47i1.1139
- Jarad F., Ugurlu U., Abdeljawad T., Baleanu D. On a new class of fractional operators. Advances in Difference Equations, 2017, vol. 2017, iss. 247, pp. 1–16. https://doi.org/10.1186/s13662-017-1306-z
- Khan T. U., Khan M. A. Generalized conformable fractional integral operators. Journal of Computational and Applied Mathematics 2019, vol. 346, pp. 378–389. http://dx.doi.org/10.1016/j.cam.2018.07.018
- Ozdemir M. E., Kavurmaci H., Yildiz C. Fractional integral inequalities via s-convex functions. Turkish Journal of Analysis and Number Theory, 2017, vol. 5, iss. 1, pp. 18–22. https://doi.org/10. 48550/arXiv.1201.4915
- Dragomir S. S., Agarwal R. P. Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula. Applied Mathematics Letters, 1998, vol. 11, iss. 5, pp. 91–95. https://doi.org/10.1016/S0893-9659(98)00086-X
- Kirmaci U. S., Bakula M. K., Ozdemir M. E., Pecaric J. Hadamard-type inequalities for s-convex functions. Applied Mathematics and Computation, 2007, vol. 193, iss. 1, pp. 26–35. https://doi.org/10.1016/j.amc.2007.03.030
- Pearce C. E. M., Pecaric J. Inequalities for differentiable mappings with application to special means and quadrature formulae. Applied Mathematics Letters, 2000, vol. 13, pp. 51–55. https://doi.org/10.1016/S0893-9659(99)00164-0
- Hudzik H., Maligranda L. Some remarks on s-convex functions. Aequationes Mathematicae, 1994, vol. 48, iss. 1, pp. 100–111. https://doi.org/10.1007/BF01837981
- Wu S., Iqbal S., Aamir M., Samraiz M., Younus A. On some Hermite – Hadamard inequalities involving k-fractional operators. Journal of Inequalities and Applications, 2021, vol. 2021, iss. 32. https://doi.org/10.1186/s13660-020-02527-1
- Aljaaidia T. A., Pachpatte D. New generalization of reverse Minkowski’s inequality for fractional integral. Advances in the Theory of Nonlinear Analysis and its Applications, 2021, vol. 5, iss. 1, pp. 72–81. https://doi.org/10.31197/atnaa.756605
- 400 просмотров