On some Simpson's and Newton's type of inequalities in multiplicative calculus with applications

Saowaluck Chasreechai; Muhammad Aamir Ali; Surapol Naowarat; Thanin Sitthiwirattham; Kamsing Nonlaopon; Saowaluck Chasreechai; Muhammad Aamir Ali; Surapol Naowarat; Thanin Sitthiwirattham; Kamsing Nonlaopon

doi:10.3934/math.2023193

AIMS Mathematics

2023, Volume 8, Issue 2: 3885-3896. doi: 10.3934/math.2023193

Previous Article Next Article

Research article

On some Simpson's and Newton's type of inequalities in multiplicative calculus with applications

1.
Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand
2.
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China
3.
Department of Mathematics, Faculty of Science and Technology, Suratthani Rajabhat University, Surat Thani 84100, Thailand
4.
Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok 10300, Thailand
5.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Received: 04 October 2022 Revised: 04 November 2022 Accepted: 09 November 2022 Published: 29 November 2022
MSC : 26D07, 26D10, 26D15

In this paper, we establish an integral equality involving a multiplicative differentiable function for the multiplicative integral. Then, we use the newly established equality to prove some new Simpson's and Newton's inequalities for multiplicative differentiable functions. Finally, we give some mathematical examples to show the validation of newly established inequalities.
- Simpson's inequalities,
- Newton's inequalities,
- convex functions,
- multiplicative calculus
Citation: Saowaluck Chasreechai, Muhammad Aamir Ali, Surapol Naowarat, Thanin Sitthiwirattham, Kamsing Nonlaopon. On some Simpson's and Newton's type of inequalities in multiplicative calculus with applications[J]. AIMS Mathematics, 2023, 8(2): 3885-3896. doi: 10.3934/math.2023193

Related Papers:

Abstract

In this paper, we establish an integral equality involving a multiplicative differentiable function for the multiplicative integral. Then, we use the newly established equality to prove some new Simpson's and Newton's inequalities for multiplicative differentiable functions. Finally, we give some mathematical examples to show the validation of newly established inequalities.

References

[1]	U. S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput., 147 (2004), 137–146. https://doi.org/10.1016/S0096-3003(02)00657-4 doi: 10.1016/S0096-3003(02)00657-4
[2]	S. S. Dragomir, R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11 (1998), 91–95. https://doi.org/10.1016/S0893-9659(98)00086-X doi: 10.1016/S0893-9659(98)00086-X
[3]	N. Alp, M. Z. Sarikaya, M. Kunt, İ. İşcan, $q$ -Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions, J. King Saud Univ. Sci., 30 (2018), 193–203. https://doi.org/10.1016/j.jksus.2016.09.007 doi: 10.1016/j.jksus.2016.09.007
[4]	S. Bermudo, P. Kórus, J. N. Valdés, On $q$ -Hermite-Hadamard inequalities for general convex functions, Acta Math. Hung., 162 (2020), 364–374. https://doi.org/10.1007/s10474-020-01025-6 doi: 10.1007/s10474-020-01025-6
[5]	S. Ali, S. Mubeen, R. S. Ali, G. Rahman, A. Morsy, K. S. Nisar, et al., Dynamical significance of generalized fractional integral inequalities via convexity, AIMS Math., 6 (2021), 9705–9730. https://doi.org/10.3934/math.2021565 doi: 10.3934/math.2021565
[6]	S. Saker, M. Kenawy, G. AlNemer, M. Zakarya, Some fractional dynamic inequalities of Hardy's type via conformable calculus, Mathematics, 8 (2020), 434. https://doi.org/10.3390/math8030434 doi: 10.3390/math8030434
[7]	M. A. Ali, M. Abbas, Z. Zhang, I. B. Sial, R. Arif, On integral inequalities for product and quotient of two multiplicatively convex functions, Asian Res. J. Math., 12 (2019), 1–11. https://doi.org/10.9734/arjom/2019/v12i330084 doi: 10.9734/arjom/2019/v12i330084
[8]	M. A. Ali, M. Abbas, A. A. Zafar, On some Hermite-Hadamard integral inequalities in multiplicative calculus, J. Inequal. Spec. Funct., 10 (2019), 111–122.
[9]	S. Özcan, Some integral inequalities of Hermite-Hadamard type for multiplicatively preinvex functions, AIMS Math., 5 (2020), 1505–1518. https://doi.org/10.3934/math.2020103 doi: 10.3934/math.2020103
[10]	S. Özcan, Hermite-Hadamard type ınequalities for multiplicatively $s$-convex functions, Cumhuriyet Sci. J., 41 (2020), 245–259. https://doi.org/10.17776/csj.663559 doi: 10.17776/csj.663559
[11]	S. Özcan, Some integral inequalities of Hermite-Hadamard type for multiplicatively $s$-preinvex functions, Int. J. Math. Model. Comput., 9 (2019), 253–266.
[12]	S. Özcan, Hermite-Hadamard type inequalities for multiplicatively $h$-preinvex functions, Turkish J. Math. Anal. Number Theory, 9 (2021), 65–70. https://doi.org/10.12691/tjant-9-3-5 doi: 10.12691/tjant-9-3-5
[13]	M. A. Ali, H. Budak, M. Z. Sarikaya, Z. Zhang, Ostrowski and Simpson type inequalities for multiplicative integrals, Proyecciones, 40 (2021), 743–763. https://doi.org/10.22199/issn.0717-6279-4136 doi: 10.22199/issn.0717-6279-4136
[14]	H. Budak, K. Özçelik, On Hermite-Hadamard type inequalities for multiplicative fractional integrals, Miskolc Math. Notes, 21 (2020), 91–99. https://doi.org/10.18514/MMN.2020.3129 doi: 10.18514/MMN.2020.3129
[15]	H. Fu, Y. Peng, T. Du, Some inequalities for multiplicative tempered fractional integrals involving the $\lambda $-incomplete gamma functions, AIMS Math., 6 (2021), 7456–7478. https://doi.org/10.3934/math.2021436 doi: 10.3934/math.2021436
[16]	M. A. Ali, Z. Zhang, H. Budak, M. Z. Sarikaya, On Hermite-Hadamard type inequalities for interval-valued multiplicative integrals, Commun. Fac. Sci. Univ., 69 (2020), 1428–1448. https://doi.org/10.31801/cfsuasmas.754842 doi: 10.31801/cfsuasmas.754842
[17]	F. Başar, Summability theory and its applications, 2Eds., CRC Press/Taylor and Francis Group, Boca, Raton, London, New York, 2022.
[18]	M. Mursaleen, F. Başar, Sequence spaces: Topics in modern summability theory, CRC Press/Taylor and Francis Group, Series: Mathematics and Its Applications, Boca, Raton, London, New York, 2020.
[19]	Z. Çakir, Spaces of continuous and bounded functions over the field of geometric complex numbers, J. Inequal. Appl., 2013 (2013), 363. https://doi.org/10.1186/1029-242X-2013-363 doi: 10.1186/1029-242X-2013-363
[20]	A. F. Çakmak, F. Başar, On line and double integrals in the non-Newtonian sense, AIP Conf. Proc., 1611 (2014), 415–423. https://doi.org/10.1063/1.4893869 doi: 10.1063/1.4893869
[21]	A. F. Çakmak, F. Başar, On the classical sequence spaces and non-newtonian calculus, J. Inequal. Appl., 2012.
[22]	A. F. Çakmak, F. Başar, Certain spaces of functions over the field of non-Newtonian complex numbers, Abstr. Appl. Anal., 2014 (2014). https://doi.org/10.1155/2014/236124 doi: 10.1155/2014/236124
[23]	A. F. Çakmak, F. Başar, Some sequence spaces and matrix transformations in multiplicative sense, TWMS J. Pure Appl. Math., 6 (2015), 27–37.
[24]	S. Tekin, F. Başar, Certain sequence spaces over the non-Newtonian complex field, Abstr. Appl. Anal., 2013 (2013). https://doi.org/10.1155/2013/739319 doi: 10.1155/2013/739319
[25]	C. Türkmen, F. Başar, Some basic results on the sets of sequences with geometric calculus, Commun. Fac. Fci. Univ. Ank. Series A, 61 (2012), 17–34. https://doi.org/10.1063/1.4747648 doi: 10.1063/1.4747648
[26]	C. Türkmen, F. Başar, Some basic results on the sets of sequences with geometric calculus, AIP Conf. Proc., 1470 (2012), 95–98. https://doi.org/10.1063/1.4747648 doi: 10.1063/1.4747648
[27]	A. Uzer, Multiplicative type Complex Calculus as an alternative to the classical calculus, Comput. Math. Appl., 60 (2010), 2725–2737. https://doi.org/10.1016/j.camwa.2010.08.089 doi: 10.1016/j.camwa.2010.08.089
[28]	A. Uzer, Exact solution of conducting half plane problems as a rapidly convergent series and an application of the multiplicative calculus, Turk. J. Electr. Eng. Co., 23 (2015), 1294–1311. https://doi.org/10.3906/elk-1306-163 doi: 10.3906/elk-1306-163
[29]	S. Rashid, R. Ashraf, E. Bonyah, Nonlinear dynamics of the media addiction model using the fractal-fractional derivative technique, Complexity, 2022 (2022). https://doi.org/10.1155/2022/2140649 doi: 10.1155/2022/2140649
[30]	S. Rashid, B. Kanwal, M. Attique, E. Bonyah, An efficient technique for time-fractional water dynamics arising in physical systems pertaining to generalized fractional derivative operators, Math. Probl. Eng., 2022 (2022). https://doi.org/10.1155/2022/7852507 doi: 10.1155/2022/7852507
[31]	S. Rashid, A. G. Ahmad, F. Jarad, A. Alsaadi, Nonlinear fractional differential equations and their existence via fixed point theory concerning to Hilfer generalized proportional fractional derivative, AIMS Math., 8 (2023), 382–403. https://doi.org/10.3934/math.2023018 doi: 10.3934/math.2023018
[32]	M. A. Qureshi, S. Rashid, F. Jarad, A computational study of a stochastic fractal-fractional hepatitis B virus infection incorporating delayed immune reactions via the exponential decay, Math. Biosci. Eng., 19 (2022), 12950–12980. https://doi.org/10.3934/mbe.2022605 doi: 10.3934/mbe.2022605
[33]	S. W. Yao, S. Rashid, E. E. Elattar, On fuzzy numerical model dealing with the control of glucose in insulin therapies for diabetes via nonsingular kernel in the fuzzy sense, AIMS Math., 7 (2022), 17913–17941. https://doi.org/10.3934/math.2022987 doi: 10.3934/math.2022987
[34]	A. E. Bashirov, E. M Kurpınar, A. Özyapıcı, Multiplicative calculus and its applications, J. Math. Anal. Appl., 337 (2008), 36–48. https://doi.org/10.1016/j.jmaa.2007.03.081 doi: 10.1016/j.jmaa.2007.03.081
[35]	C. Niculescu, L. E. Persson, Convex functions and their applications, New York: Springer, 2006.

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)