On local fractional integral inequalities via generalized ( h ˜ 1, h ˜ 2)-preinvexity involving local fractional integral operators with Mittag-Leffler kernel

• Autorzy: Vivas-Cortez Miguel , Bibi Maria , Muddassar Muhammad , Al-Sa’di Sa’ud

Identyfikatory

DOI

10.1515/dema-2022-0216

• Języki publikacji: EN

Abstrakty

EN

Local fractional integral inequalities of Hermite-Hadamard type involving local fractional integral operators with Mittag-Leffler kernel have been previously studied for generalized convexities and preinvexities. In this article, we analyze Hermite-Hadamard-type local fractional integral inequalities via generalized (h˜1, h˜2)-preinvex function comprising local fractional integral operators and Mittag-Leffler kernel. In addition, two examples are discussed to ensure that the derived consequences are correct. As an application, we construct an inequality to establish central moments of a random variable.

Słowa kluczowe

EN

generalized (h˜1, h˜2)-preinvex functions; local fractional integrals; generalized Hermite-Hadamard inequality; fractal sets; Mittag-Leffler kernel

PL

całki; fraktale; nierówność Hermite-Hadamarda; funkcja Mittag-Lefflera

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2023
• Tom: Vol. 56, nr 1
• Strony: art. no. 20220216
• Opis fizyczny: Bibliogr. 44 poz.

Twórcy

autor

Vivas-Cortez Miguel

• Pontificia Universidad Católica del Ecuador, Facultad de Ciencias Naturales y Exactas, Escuela de Ciencias Físicas y Matemáticas, Av. 12 de Octubre 1076, Sede Quito 17-01-2184, Ecuador

autor

Bibi Maria

• Department of Basic Sciences, University of Engineering and Technology, Taxila, Pakistan

autor

Muddassar Muhammad

• Department of Basic Sciences, University of Engineering and Technology, Taxila, Pakistan

autor

Al-Sa’di Sa’ud

• Department of Mathematics, Faculty of Science, The Hashemite University, P.O Box 330127, Zarqa 13133, Jordan

Bibliografia

• [1] J. Hadamard, Étude sur les propriétés des fonctions entières et en particulier daune fonction considérée par Riemann, J. de mathématiques pures et appliquées. 9 (1893), 171–216.
• [2] J. M. Viloria and M. Vivas-Cortez, Hermite-Hadamard type inequalities for harmonically convex functions on n-coordinates, Appl. Math. Inf. Sci. Lett. 6 (2018), no. 2, 1–6, DOI: https://doi.org/10.18576/amisl/060201.
• [3] M. Vivas-Cortez, Relative strongly h-convex functions and integral inequalities, Appl. Math. Inf. Sci. Lett. 4 (2016), no. 2, 39–45, DOI: https://doi.org/10.18576/amisl/040201.
• [4] H. H. Chu, S. Rashid, Z. Hammouch, and Y. M. Chu, New fractional estimates for Hermite-Hadamard-Mercer’s type inequalities, Alexandr. Eng. J. 59 (2020), no. 5, 3079–3089, DOI: https://doi.org/10.1016/j.aej.2020.06.040.
• [5] S. I. Butt, A. Kashuri, M. Tariq, J. Nasir, A. Aslam, and W. Gao, Hermite-Hadamard-type inequalities via n-polynomial exponential-type convexity and their applications, Adv. Difference Equ. 1 (2020), 1–25, DOI: https://doi.org/10.1186/s13662-020-02967-5.
• [6] T. Du, M. U. Awan, A. Kashuri, and S. Zhao, Some k -fractional extensions of the trapezium inequalities through generalized relative semi-(m, h)-preinvexity, Appl. Anal. 100 (2021), no. 3, 642–662, DOI: https://doi.org/10.1080/00036811.2019.1616083.
• [7] Y. Q. Song, S. I. Butt, A. Kashuri, J. Nasir, and M. Nadeem, New fractional integral inequalities pertaining 2D-approximately coordinate (r1 ,h1) (r2 ,h2)-convex functions, Alexandr. Eng. J. 61 (2022), no. 1, 563–573, DOI: https://doi.org/10.1016/j.aej.2021.06.044.
• [8] V. Stojiljković, R. Ramaswamy, F. Alshammari, O. A. Ashour, M. L. H. Alghazwani, and S. Radenović, Hermite-Hadamard type inequalities involving (kp) fractional operator for various types of convex functions, Fractal Fractional 6 (2022), no. 7, 376, DOI: https://doi.org/10.3390/fractalfract6070376.
• [9] A. Ben-IsraeL and B. Mond, What is invexity? ANZIAM J. 28 (1986), no. 1, 1–9, DOI: https://doi.org/10.1017/S0334270000005142.
• [10] T. Weir and B. Mond, Pre-invex functions in multiple objective optimization, J. Math. Anal. Appl. 136 (1988), no. 1, 29–38, DOI: https://doi.org/10.1016/0022-247X(88)90113-8.
• [11] M. A. Noor, K. I. Noor, and S. Rashid, Some new classes of preinvex functions and inequalities, Mathematics 7 (2019), no. 1, 29, DOI: https://doi.org/10.3390/math7010029.
• [12] X. J. Yang, Advanced Local Fractional Calculus and its Applications, World Science Publisher, New York, 2012.
• [13] X. J. Yang, D. Baleanu, and H. M. Srivastava, Local Fractional Integral Transforms and their Applications, Academic Press, New York, 2015.
• [14] A. Atangana, Modelling the spread of COVID-19 with new fractal-fractional operators: Can the lockdown save mankind before vaccination? Chaos Solitons Fractals 136 (2020), 109860, DOI: https://doi.org/10.1016/j.chaos.2020.109860.
• [15] S. Kumar and A. Atangana, A numerical study of the nonlinear fractional mathematical model of tumor cells in presence of chemotherapeutic treatment, Int. J. Biomath. 13 (2020), no. 3, 2050021, DOI: https://doi.org/10.1142/S1793524520500217.
• [16] K. J. Wang, Variational principle and approximate solution for the generalized Burgers-Huxley equation with fractal derivative, Fractals 29 (2021), no. 2, 2150044–1246, DOI: https://doi.org/10.1142/S0218348X21500444.
• [17] X. J. Yang, F. Gao, and H. M. Srivastava, Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations, Comput. Math. Appl. 73 (2017), no. 2, 203–210, DOI: https://doi.org/10.1016/j.camwa.2016.11.012.
• [18] N. D. Phuong, L. V. C. Hoan, E. Karapinar, J. Singh, H. D. Binh, N. H. Can, Fractional order continuity of a time semi-linear fractional diffusion-wave system, Alexandr. Eng. J. 59 (2020), no. 6, 4959–4968, DOI: https://doi.org/10.1016/j.aej.2020.08.054.
• [19] N. H. Luc, L. N. Huynh, D. Baleanu, and N. H. Can, Identifying the space source term problem for a generalization of the fractional diffusion equation with hyper-Bessel operator, Adv. Difference Equ. (2020), Article ID 261, DOI: https://doi.org/10.1186/s13662-020-02712-y.LFII involving LFI operators with Mittag-Leffler kernel  13
• [20] H. T. Nguyen, N. A. Tuan, and C. Yang, Global well-posedness for fractional Sobolev-Galpern type equations, Discrete Contin. Dyn. Syst. 42 (2022), no. 6, 2637–2665, DOI: https://doi.org/10.3934/dcds.2021206.
• [21] A. T. Nguyen, N. H. Tuan, and C. Yang, On Cauchy problem for fractional parabolic-elliptic Keller-Segel model, Adv. Nonlinear Anal. 12 (2023), no. 1, 97–116, DOI: https://doi.org/10.1515/anona-2022-0256.
• [22] N. H. Tuan, V. V. Au, and A. T. Nguyen, Mild solutions to a time-fractional Cauchy problem with nonlocal nonlinearity in Besov spaces, Archiv der Mathematik 118 (2022), no. 3, 305–314, DOI: https://doi.org/10.1007/s00013-022-01702-8.
• [23] N. H. Tuan, M. Foondun, T. Ngoc Thach, and R. Wang, On backward problems for stochastic fractional reaction equations with standard and fractional Brownian motion, Bulletin des Sciences Mathématiques 179 (2022), Article no. 103158, DOI: https://doi.org/10.1016/j.bulsci.2022.103158.
• [24] T. Caraballo Garrido, N. H. Tuan, T. B. Ngoc, and Y. Zhou, Existence and regularity results for terminal value problem for nonlinear fractional wave equations, Nonlinearity 34 (2021), no. 3, 1448–1502, DOI: https://doi.org/10.1088/1361-6544/abc4d9.
• [25] A. Tuan Nguyen, T. Caraballo, and N. Tuan, On the initial value problem for a class of nonlinear biharmonic equation with time-fractional derivative, Proc. Roy. Soc. Edinburgh Sect. A Math. 152 (2022), no. 4, 989–1031, DOI: https://doi.org/doi:10.1017/prm.2021.44.
• [26] S. Al-Sa’di, M. Bibi, and M. Muddassar, Some Hermite-Hadamard’s type local fractional integral inequalities for generalized γ-preinvex function with applications, Math. Meth. Appl. Sci. 46 (2023), no. 2, 2941–2954, DOI: https://doi.org/10.1002/mma.8680.
• [27] M. Bibi and M. Muddassar, Integral inequalities for generalized approximately h-convex functions on fractal sets via generalized local fractional integrals, Innovative J. Math. (IJM) 1 (2022), no. 3, 1–12, DOI: https://doi.org/10.55059/ijm.2022.1.3/48.
• [28] M. Vivas-Cortez, J. Hernández, and N. Merentes, New Hermite-Hadamard and Jensen-type inequalities for h-convex functions on fractal sets, Revista Colombiana de Matemáticas 50 (2016), no. 2, 145–164, DOI: https://doi.org/10.15446/recolma.v50n2.62207.
• [29] O. Almutairi and A. Kilicman, Generalized Fejér-Hermite-Hadamard type via generalized h m−( )-convexity on fractal sets and applications, Chaos Solitons Fractals 147 (2021), 110938, DOI: https://doi.org/10.1016/j.chaos.2021.110938.
• [30] T. Du, H. Wang, M. A. Khan, and Y. Zhang, Certain integral inequalities considering generalized m-convexity on fractal sets and their applications, Fractals. 27 (2019), no. 07, 1950117, DOI: https://doi.org/10.1142/S0218348X19501172.
• [31] S. Iftikhar, S. Erden, P. Kumam, and M. U. Awan, Local fractional Newton’s inequalities involving generalized harmonic convex functions, Adv. Difference Equ. 2020 (2020), 1–14, DOI: https://doi.org/10.1186/s13662-020-02637-6.
• [32] M. Sarikaya and H. Budak, Generalized Ostrowski type inequalities for local fractional integrals, Proc. Amer. Math. Soc. 145 (2017), no. 4, 1527–1538, DOI: https://doi.org/10.1090/proc/13488.
• [33] H. Mo, X. Sui, and D. Yu, Generalized convex functions on fractal sets and two related inequalities, Abstract and Applied Analysis. 2014 (2014), Article ID 636751, DOI: https://doi.org/10.1155/2014/636751.
• [34] W. Sun, Hermite-Hadamard type local fractional integral inequalities for generalized s-preinvex functions and their generalization, Fractals 29 (2021), no. 04, 2150098, DOI: https://doi.org/10.1142/S0218348X21500985.
• [35] W. Sun, Generalized h-convexity on fractal sets and some generalized Hadamard-type inequalities, Fractals 28 (2020), no. 02, 2050021, DOI: https://doi.org/10.1142/S0218348X20500218.
• [36] W. Sun, Some Hermite-Hadamard type inequalities for generalized h-preinvex function via local fractional integrals and their applications, Adv. Difference Equ. 2020 (2020), no. 1, 1–14, DOI: https://doi.org/10.1186/s13662-020-02812-9.
• [37] W. Sun, Some local fractional integral inequalities for generalized preinvex functions and applications to numerical quadrature, Fractals 27 (2019), no. 05, 1950071, DOI: https://doi.org/10.1142/S0218348X19500713.
• [38] W. Sun, On generalization of some inequalities for generalized harmonically convex functions via local fractional integrals, Quaest. Math. 42 (2019), no. 9, 1159–1183, DOI: https://doi.org/10.2989/16073606.2018.1509242.
• [39] W. Sun and Q. Liu, Hadamard type local fractional integral inequalities for generalized harmonically convex functions and applications, Math. Meth. Appl. Sci. 43 (2020), no. 9, 5776–5787, DOI: https://doi.org/10.1002/mma.6319.
• [40] W. Sun, Local fractional Ostrowski-type inequalities involving generalized h-convex functions and some applications for generalized moments, Fractals 29 (2021), no. 01, 2150006, DOI: https://doi.org/10.1142/S0218348X21500067.
• [41] B. Ahmad, A. Alsaedi, M. Kirane, and B. T. Torebek, Hermite-Hadamard, Hermite-Hadamard-Fejér, Dragomir-Agarwal and Pachpatte type inequalities for convex functions via new fractional integrals, J. Comput. Appl. Math. 353 (2019), 120–129, DOI: https://doi.org/10.1016/j.cam.2018.12.030.
• [42] W. Sun, Some new inequalities for generalized h-convex functions involving local fractional integral operators with Mittag-Leffler kernel, Math. Meth. Appl. Sci. 44 (2021), no. 06, 4985–4998, DOI: https://doi.org/10.1002/mma.7081.
• [43] W. Sun, Hermite-Hadamard type local fractional integral inequalities with Mittag-Leffler kernel for generalized preinvex functions, Fractals 29 (2021), no. 08, 2150253, DOI: https://doi.org/10.1142/S0218348X21502534.
• [44] S. Rashid, M. A. Noor, K. I. Noor, and F. Safdar, Integral inequalities for generalized preinvex functions, Punjab Univ. J. Math. 51 (2019), no. 10, 77–91.