Новые обобщенные интегральные неравенства через $(h,m)$-выпуклые модифицированные функции
Ключевые слова:
неравенство Эрмита-Адамара, неравенство Гёльдера, степенное неравенство, взвешенные интегралы, $(m,h)$-выпуклые функции
Аннотация
В этой статье мы устанавливаем несколько неравенств для $(h,m)$-выпуклых отображений, связанных с взвешенными интегралами, которые использовались в предыдущих работах. На протяжении всей работы мы показываем, что наши результаты обобщают несколько известных из литературы интегральных неравенств.
Литература
1. Akdemir A.O., Butt S.I., Nadeem M., Ragusa M.A. New general variants of Chebyshev type inequalities via generalized fractional integral operators // Mathematics. 2021. Vol. 9. Issue 2. 122. https://doi.org/10.3390/math9020122
2. Akkurt A., Yildirim M.E., Yildirim H. On some integral inequalities for $(k,h)$-Riemann-Liouville fractional integral // New Trends in Mathematical Science. 2016. Vol. 4. Issue 2. P. 138-146. https://doi.org/10.20852/ntmsci.2016217824
3. Aljaaidi 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. No. 1. P. 72-81. https://doi.org/10.31197/atnaa.756605
4. Байрактаров Б.Р., Аттаев А.Х., Кудаев В.Ч. Некоторые обобщенные неравенства типа Адамара через дробные интегралы // Известия высших учебных заведений. Математика. 2021. № 2. С. 3-18. https://doi.org/10.26907/0021-3446-2021-2-3-18
5. Bayraktar B., Emin Özdemir M. Generalization of Hadamard-type trapezoid inequalities for fractional integral operators // Уфимский математический журнал. 2021. Т. 13. Вып. 1. С. 119-130. http://mi.mathnet.ru/ufa548
6. Bessenyei M., Páles Z. On generalized higher-order convexity and Hermite-Hadamard-type inequalities // Acta Scientiarum Mathematicarum (Szeged). 2004. Vol. 70. Nos. 1-2. P. 13-24.
7. Breckner W.W. Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Räumen // Publications de l'Institut Mathématique. Nouvelle série. 1978. Vol. 23 (37). Issue 43. P. 13-20 (in German). http://eudml.org/doc/257486
8. Bruckner A.M., Ostrow E. Some function classes related to the class of convex functions // Pacific Journal of Mathematics. 1962. Vol. 12. No. 4. P. 1203-1215. https://doi.org/10.2140/pjm.1962.12.1203
9. Butt S.I., Bayraktar B., Umar M. Several new integral inequalities via $k$-Riemann-Liouville fractional integrals operators // Проблемы анализа - Issues of Analysis. 2021. Т. 10 (28). Вып. 1. С. 3-22. https://doi.org/10.15393/j3.art.2021.8770
10. Díaz R., Pariguan E. On hypergeometric functions and Pochhammer $k$-symbol // Divulgaciones Matemáticas. 2007. Vol. 15. No. 2. P. 179-192.
11. 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. Issue 5. P. 91-95. https://doi.org/10.1016/S0893-9659(98)00086-X
12. Du T., Awan M.U., Kashuri A., Zhao S.S. Some $k$-fractional extensions of the trapezium inequalities through generalized relative semi-$(m,h)$-preinvexity // Applicable Analysis. 2021. Vol. 100. Issue 3. P. 642-662. https://doi.org/10.1080/00036811.2019.1616083
13. Farid G., Rehman A.U., Ain Q.U. $k$-fractional integral inequalities of Hadamard type for $(h-m)$-convex functions // Computational Methods for Differential Equations. 2020. Vol. 8. Issue 1. P. 119-140.
14. Farid G., Rehman A.U., Zahra M. On Hadamard-type inequalities for $k$-fractional integrals // Nonlinear Functional Analysis and Applications. 2016. Vol. 21. No. 3. P. 463-478. http://nfaa.kyungnam.ac.kr/journal-nfaa/index.php/NFAA/article/view/902/832
15. Hudzik H., Maligranda L. Some remarks on $s$-convex functions // Aequationes Mathematicae. 1994. Vol. 48. Issue 1. P. 100-111. https://doi.org/10.1007/BF01837981
16. İlhan E. Analysis of the spread of Hookworm infection with Caputo-Fabrizio fractional derivative // Turkish Journal of Science. 2022. Vol. 7. Issue 1. P. 43-52.
17. Jarad F., Abdeljawad T., Shah K. On the weighted fractional operators of a function with respect to another function // Fractals. 2020. Vol. 28. No. 8. 2040011. https://doi.org/10.1142/S0218348X20400113
18. Jarad F., Uğurlu E., Abdeljawad T., Baleanu D. On a new class of fractional operators // Advances in Difference Equations. 2017. Vol. 2017. Issue 1. Article 247. https://doi.org/10.1186/s13662-017-1306-z
19. Khan T.U., Khan M.A. Generalized conformable fractional operators // Journal of Computational and Applied Mathematics. 2019. Vol. 346. P. 378-389. https://doi.org/10.1016/j.cam.2018.07.018
20. Kırmacı U.S., Klaričić Bakula M., Özdemir M.E., Pečarić J. Hadamard-type inequalities for $s$-convex functions // Applied Mathematics and Computation. 2007. Vol. 193. Issue 1. P. 26-35. https://doi.org/10.1016/j.amc.2007.03.030
21. Kızıl Ş., Ardıç M.A. Inequalities for strongly convex functions via Atangana-Baleanu integral operators // Turkish Journal of Science. 2021. Vol. 6. Issue 2. P. 96-109.
22. Koca İ., Akçetin E., Yaprakdal P. Numerical approximation for the spread of SIQR model with Caputo fractional order derivative // Turkish Journal of Science. 2020. Vol. 5. Issue 2. P. 124-139.
23. Mihesan V.G. A generalization of convexity // Seminar on functional equations, approximation and convexity. Cluj-Napoca, Romania: 1993.
24. Mubeen S., Habibullah G.M. $k$-fractional integrals and application // International Journal of Contemporary Mathematical Sciences. 2012. Vol. 7. No. 2. P. 89-94.
25. Nápoles Valdés J.E., Bayraktar B., Butt S.I. New integral inequalities of Hermite-Hadamard type in a generalized context // Punjab University Journal Of Mathematics. 2021. Vol. 53. Issue 11. P. 765-777. https://doi.org/10.52280/Pujm.2021.531101
26. Nápoles Valdés J.E., Bayraktar B. On the generalized inequalities of the Hermite-Hadamard type // Filomat. 2021. Vol. 35. Issue 14. P. 4917-4924. https://doi.org/10.2298/FIL2114917N
27. Nápoles Valdés J.E., Rabossi F., Samaniego A.D. Convex functions: Ariadne's thread or Charlotte's Spiderweb? // Advanced Mathematical Models and Applications. 2020. Vol. 5. No. 2. P. 176-191.
28. Özdemir M.E., Akdemir A.O., Set E. On $(h-m)$-convexity and Hadamard-type inequalities // Transylvanian Journal Of Mathematics And Mechanics. 2016. Vol. 8. No. 1. P. 51-58.
29. Özdemir M.E., Dragomir S.S., Yildiz Ç. The Hadamard inequality for convex function via fractional integrals // Acta Mathematica Scientia. 2013. Vol. 33B. Issue 5. P. 1293-1299. https://doi.org/10.1016/S0252-9602(13)60081-8
30. Park J. Generalization of Ostrowski-type inequalities for differentiable real $(s,m)$-convex mappings // Far East Journal of Mathematical Sciences. 2011. Vol. 49. No. 2. P. 157-171.
31. Pearce C.E.M., Pečarić J. Inequalities for differentiable mappings with application to special means and quadrature formulæ // Applied Mathematics Letters. 2000. Vol. 13. Issue 2. P. 51-55. https://doi.org/10.1016/S0893-9659(99)00164-0
32. Rainville E.D. Special functions. New York: Macmillan, 1960.
33. Sahoo S.K., Ahmad H., Tariq M., Kodamasingh B., Aydi H., de la Sen M. Hermite-Hadamard type inequalities involving $k$-fractional operator for $(\bar{h},m)$-convex functions // Symmetry. 2021. Vol. 13. Issue 9. 1686. https://doi.org/10.3390/sym13091686
34. Sarıkaya M.Z., Ertuğral F. On the generalized Hermite-Hadamard inequalities // Annals of the University of Craiova. Mathematics and Computer Science series. 2020. Vol. 47. No. 1. P. 193-213.
35. Sarıkaya M.Z., Set E., Yaldız H., Başak N. Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities // Mathematical and Computer Modelling. 2013. Vol. 57. Issues 9-10. P. 2403-2407. https://doi.org/10.1016/j.mcm.2011.12.048
36. Toader G. Some generalizations of the convexity // Proceedings of the Colloquium on Approximation and Optimization. Cluj-Napoca: University Cluj-Napoca, 1985. P. 329-338.
37. 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. Article number 32. https://doi.org/10.1186/s13660-020-02527-1
38. Xi B.-Y., Gao D.-D., Qi F.Integral inequalities of Hermite-Hadamard type for $(\alpha,s)$-convex and $(\alpha,s,m)$-convex functions // Italian Journal Of Pure And Applied Mathematics. 2020. No. 44. P. 499-510.
39. Xi B.-Y., Qi F. Inequalities of Hermite-Hadamard type for extended $s$-convex functions and applications to means // Journal of Nonlinear and Convex Analysis. 2015. Vol. 16. No. 5. P. 873-890.
40. Yildiz Ç., Özdemir M.E., Önalan H.K. Fractional integral inequalities via $s$-convex functions // Turkish Journal of Analysis and Number Theory. 2017. Vol. 5. No. 1. P. 18-22.
2. Akkurt A., Yildirim M.E., Yildirim H. On some integral inequalities for $(k,h)$-Riemann-Liouville fractional integral // New Trends in Mathematical Science. 2016. Vol. 4. Issue 2. P. 138-146. https://doi.org/10.20852/ntmsci.2016217824
3. Aljaaidi 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. No. 1. P. 72-81. https://doi.org/10.31197/atnaa.756605
4. Байрактаров Б.Р., Аттаев А.Х., Кудаев В.Ч. Некоторые обобщенные неравенства типа Адамара через дробные интегралы // Известия высших учебных заведений. Математика. 2021. № 2. С. 3-18. https://doi.org/10.26907/0021-3446-2021-2-3-18
5. Bayraktar B., Emin Özdemir M. Generalization of Hadamard-type trapezoid inequalities for fractional integral operators // Уфимский математический журнал. 2021. Т. 13. Вып. 1. С. 119-130. http://mi.mathnet.ru/ufa548
6. Bessenyei M., Páles Z. On generalized higher-order convexity and Hermite-Hadamard-type inequalities // Acta Scientiarum Mathematicarum (Szeged). 2004. Vol. 70. Nos. 1-2. P. 13-24.
7. Breckner W.W. Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Räumen // Publications de l'Institut Mathématique. Nouvelle série. 1978. Vol. 23 (37). Issue 43. P. 13-20 (in German). http://eudml.org/doc/257486
8. Bruckner A.M., Ostrow E. Some function classes related to the class of convex functions // Pacific Journal of Mathematics. 1962. Vol. 12. No. 4. P. 1203-1215. https://doi.org/10.2140/pjm.1962.12.1203
9. Butt S.I., Bayraktar B., Umar M. Several new integral inequalities via $k$-Riemann-Liouville fractional integrals operators // Проблемы анализа - Issues of Analysis. 2021. Т. 10 (28). Вып. 1. С. 3-22. https://doi.org/10.15393/j3.art.2021.8770
10. Díaz R., Pariguan E. On hypergeometric functions and Pochhammer $k$-symbol // Divulgaciones Matemáticas. 2007. Vol. 15. No. 2. P. 179-192.
11. 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. Issue 5. P. 91-95. https://doi.org/10.1016/S0893-9659(98)00086-X
12. Du T., Awan M.U., Kashuri A., Zhao S.S. Some $k$-fractional extensions of the trapezium inequalities through generalized relative semi-$(m,h)$-preinvexity // Applicable Analysis. 2021. Vol. 100. Issue 3. P. 642-662. https://doi.org/10.1080/00036811.2019.1616083
13. Farid G., Rehman A.U., Ain Q.U. $k$-fractional integral inequalities of Hadamard type for $(h-m)$-convex functions // Computational Methods for Differential Equations. 2020. Vol. 8. Issue 1. P. 119-140.
14. Farid G., Rehman A.U., Zahra M. On Hadamard-type inequalities for $k$-fractional integrals // Nonlinear Functional Analysis and Applications. 2016. Vol. 21. No. 3. P. 463-478. http://nfaa.kyungnam.ac.kr/journal-nfaa/index.php/NFAA/article/view/902/832
15. Hudzik H., Maligranda L. Some remarks on $s$-convex functions // Aequationes Mathematicae. 1994. Vol. 48. Issue 1. P. 100-111. https://doi.org/10.1007/BF01837981
16. İlhan E. Analysis of the spread of Hookworm infection with Caputo-Fabrizio fractional derivative // Turkish Journal of Science. 2022. Vol. 7. Issue 1. P. 43-52.
17. Jarad F., Abdeljawad T., Shah K. On the weighted fractional operators of a function with respect to another function // Fractals. 2020. Vol. 28. No. 8. 2040011. https://doi.org/10.1142/S0218348X20400113
18. Jarad F., Uğurlu E., Abdeljawad T., Baleanu D. On a new class of fractional operators // Advances in Difference Equations. 2017. Vol. 2017. Issue 1. Article 247. https://doi.org/10.1186/s13662-017-1306-z
19. Khan T.U., Khan M.A. Generalized conformable fractional operators // Journal of Computational and Applied Mathematics. 2019. Vol. 346. P. 378-389. https://doi.org/10.1016/j.cam.2018.07.018
20. Kırmacı U.S., Klaričić Bakula M., Özdemir M.E., Pečarić J. Hadamard-type inequalities for $s$-convex functions // Applied Mathematics and Computation. 2007. Vol. 193. Issue 1. P. 26-35. https://doi.org/10.1016/j.amc.2007.03.030
21. Kızıl Ş., Ardıç M.A. Inequalities for strongly convex functions via Atangana-Baleanu integral operators // Turkish Journal of Science. 2021. Vol. 6. Issue 2. P. 96-109.
22. Koca İ., Akçetin E., Yaprakdal P. Numerical approximation for the spread of SIQR model with Caputo fractional order derivative // Turkish Journal of Science. 2020. Vol. 5. Issue 2. P. 124-139.
23. Mihesan V.G. A generalization of convexity // Seminar on functional equations, approximation and convexity. Cluj-Napoca, Romania: 1993.
24. Mubeen S., Habibullah G.M. $k$-fractional integrals and application // International Journal of Contemporary Mathematical Sciences. 2012. Vol. 7. No. 2. P. 89-94.
25. Nápoles Valdés J.E., Bayraktar B., Butt S.I. New integral inequalities of Hermite-Hadamard type in a generalized context // Punjab University Journal Of Mathematics. 2021. Vol. 53. Issue 11. P. 765-777. https://doi.org/10.52280/Pujm.2021.531101
26. Nápoles Valdés J.E., Bayraktar B. On the generalized inequalities of the Hermite-Hadamard type // Filomat. 2021. Vol. 35. Issue 14. P. 4917-4924. https://doi.org/10.2298/FIL2114917N
27. Nápoles Valdés J.E., Rabossi F., Samaniego A.D. Convex functions: Ariadne's thread or Charlotte's Spiderweb? // Advanced Mathematical Models and Applications. 2020. Vol. 5. No. 2. P. 176-191.
28. Özdemir M.E., Akdemir A.O., Set E. On $(h-m)$-convexity and Hadamard-type inequalities // Transylvanian Journal Of Mathematics And Mechanics. 2016. Vol. 8. No. 1. P. 51-58.
29. Özdemir M.E., Dragomir S.S., Yildiz Ç. The Hadamard inequality for convex function via fractional integrals // Acta Mathematica Scientia. 2013. Vol. 33B. Issue 5. P. 1293-1299. https://doi.org/10.1016/S0252-9602(13)60081-8
30. Park J. Generalization of Ostrowski-type inequalities for differentiable real $(s,m)$-convex mappings // Far East Journal of Mathematical Sciences. 2011. Vol. 49. No. 2. P. 157-171.
31. Pearce C.E.M., Pečarić J. Inequalities for differentiable mappings with application to special means and quadrature formulæ // Applied Mathematics Letters. 2000. Vol. 13. Issue 2. P. 51-55. https://doi.org/10.1016/S0893-9659(99)00164-0
32. Rainville E.D. Special functions. New York: Macmillan, 1960.
33. Sahoo S.K., Ahmad H., Tariq M., Kodamasingh B., Aydi H., de la Sen M. Hermite-Hadamard type inequalities involving $k$-fractional operator for $(\bar{h},m)$-convex functions // Symmetry. 2021. Vol. 13. Issue 9. 1686. https://doi.org/10.3390/sym13091686
34. Sarıkaya M.Z., Ertuğral F. On the generalized Hermite-Hadamard inequalities // Annals of the University of Craiova. Mathematics and Computer Science series. 2020. Vol. 47. No. 1. P. 193-213.
35. Sarıkaya M.Z., Set E., Yaldız H., Başak N. Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities // Mathematical and Computer Modelling. 2013. Vol. 57. Issues 9-10. P. 2403-2407. https://doi.org/10.1016/j.mcm.2011.12.048
36. Toader G. Some generalizations of the convexity // Proceedings of the Colloquium on Approximation and Optimization. Cluj-Napoca: University Cluj-Napoca, 1985. P. 329-338.
37. 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. Article number 32. https://doi.org/10.1186/s13660-020-02527-1
38. Xi B.-Y., Gao D.-D., Qi F.Integral inequalities of Hermite-Hadamard type for $(\alpha,s)$-convex and $(\alpha,s,m)$-convex functions // Italian Journal Of Pure And Applied Mathematics. 2020. No. 44. P. 499-510.
39. Xi B.-Y., Qi F. Inequalities of Hermite-Hadamard type for extended $s$-convex functions and applications to means // Journal of Nonlinear and Convex Analysis. 2015. Vol. 16. No. 5. P. 873-890.
40. Yildiz Ç., Özdemir M.E., Önalan H.K. Fractional integral inequalities via $s$-convex functions // Turkish Journal of Analysis and Number Theory. 2017. Vol. 5. No. 1. P. 18-22.
Поступила в редакцию
2022-04-06
Опубликована 2022-11-20
Опубликована 2022-11-20