The generalization of Hermite-Hadamard type Inequality with exp-convexity involving non-singular fractional operator

Muhammad Imran Asjad; Waqas Ali Faridi; Mohammed M. Al-Shomrani; Abdullahi Yusuf; Muhammad Imran Asjad; Waqas Ali Faridi; Mohammed M. Al-Shomrani; Abdullahi Yusuf

doi:10.3934/math.2022392

AIMS Mathematics

2022, Volume 7, Issue 4: 7040-7055. doi: 10.3934/math.2022392

Previous Article Next Article

Research article Special Issues

The generalization of Hermite-Hadamard type Inequality with exp-convexity involving non-singular fractional operator

1.
Department of Mathematics, University of Management and Technology, Lahore, Pakistan
2.
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
3.
Department of Computer Engineering, Biruni University, Istanbul, Turkey
4.
Department of Mathematics, Federal University Dutse, Jigawa, Nigeria

Received: 07 October 2021 Revised: 12 January 2022 Accepted: 13 January 2022 Published: 07 February 2022
MSC : 26D10, 26D15, 26A51

The theory of convex function has a lot of applications in the field of applied mathematics and engineering. The Caputo-Fabrizio non-singular operator is the most significant operator of fractional theory which permits to generalize the classical theory of differentiation. This study consider the well known Hermite-Hadamard type and associated inequalities to generalize further. To full fill this mileage, we use the exponential convexity and fractional-order differential operator and also apply some existing inequalities like Holder, power mean, and Holder-Iscan type inequalities for further extension. The generalized exponential type fractional integral Hermite-Hadamard type inequalities establish involving the global integral. The applications of the developed results are displayed to verify the applicability. The establish results of this paper can be considered an extension and generalization of the existing results of convex function and inequality in literature and we hope that will be more helpful for the researcher in future work.
- Caputo-Fabrizio fractional integral,
- Hermite-Hadamard inequality,
- exp-convexity
Citation: Muhammad Imran Asjad, Waqas Ali Faridi, Mohammed M. Al-Shomrani, Abdullahi Yusuf. The generalization of Hermite-Hadamard type Inequality with exp-convexity involving non-singular fractional operator[J]. AIMS Mathematics, 2022, 7(4): 7040-7055. doi: 10.3934/math.2022392

Related Papers:

Abstract

The theory of convex function has a lot of applications in the field of applied mathematics and engineering. The Caputo-Fabrizio non-singular operator is the most significant operator of fractional theory which permits to generalize the classical theory of differentiation. This study consider the well known Hermite-Hadamard type and associated inequalities to generalize further. To full fill this mileage, we use the exponential convexity and fractional-order differential operator and also apply some existing inequalities like Holder, power mean, and Holder-Iscan type inequalities for further extension. The generalized exponential type fractional integral Hermite-Hadamard type inequalities establish involving the global integral. The applications of the developed results are displayed to verify the applicability. The establish results of this paper can be considered an extension and generalization of the existing results of convex function and inequality in literature and we hope that will be more helpful for the researcher in future work.

References

[1]	M. Caputo, Modeling social and economic cycles, Altern. Public Econ., 2014.
[2]	G. Iaffaldano, M. Caputo, S. Martino, Experimental and theoretical memory diffusion of water in the sand, Hydrol. Earth Syst. Sc., 10 (2006), 93–100. https://doi.org/10.5194/hess-10-93-2006 doi: 10.5194/hess-10-93-2006
[3]	R. L. Magin, Fractional calculus in bioengineering, Begell House Publishers, 2006.
[4]	F. Cesarone, M. Caputo, C. Cametti, Memory formalism in the passive diffusion across a biological membrane, J. Membr. Sci., 250 (2004), 79–84.
[5]	M. El-Shaed, Fractional calculus model of the semilunar heart valve vibrations, Proceedings of DETC'03 ASME 2003 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Chicago, Illinois, USA, 2003. https://doi.org/10.1115/DETC2003/VIB-48384
[6]	G. Jumarie, New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations, Math. Comput. Model., 44 (2006), 231–254. https://doi.org/10.1016/j.mcm.2005.10.003 doi: 10.1016/j.mcm.2005.10.003
[7]	Z. Temur, Kalanov vector calculus and Maxwells equations: Logic errors in mathematics and electrodynamics, Open J. Math. Sci., 4 (2020), 343–355.
[8]	E. K. Ghiasi, S. Noeiaghdam, Truncating the series expansion for unsteady velocity-dependent Eyring-Powell fluid, Eng. Appl. Sci. Lett., 3 (2020), 28–34. https://doi.org/10.30538/psrp-easl2020.0049 doi: 10.30538/psrp-easl2020.0049
[9]	S. M. Kang, G. Farid, W. Nazeer, B. Tariq, Hadamard and Fejé-Hadamard inequalities for extended generalized fractional integrals involving special functions, J. Inequal. Appl., 2018 (2018), 119. https://doi.org/10.1186/s13660-018-1701-3 doi: 10.1186/s13660-018-1701-3
[10]	Y. C. Kwun, G. Farid, S. Ullah, W. Nazeer, K. Mahreen, S. M. Kang, Inequalities for a unified integral operator and associated results in fractional calculus, IEEE Access, 7 (2019), 126283–126292. https://doi.org/10.1109/ACCESS.2019.2939166 doi: 10.1109/ACCESS.2019.2939166
[11]	Y. C. Kwun, M. S. Saleem, M. Ghafoor, W. Nazeer, S. M. Kang, Hermite-Hadamard-type inequalities for functions whose derivatives are convex via fractional integrals, J. Inequal. Appl., 2019 (2019), 44. https://doi.org/10.1186/s13660-019-1993-y doi: 10.1186/s13660-019-1993-y
[12]	N. Saba, A. Boussayoud, Complete homogeneous symmetric functions of Gauss Fibonacci polynomials and bivariate Pell polynomials, Open J. Math. Sci., 4 (2020), 179–185. https://doi.org/10.30538/oms2020.0108 doi: 10.30538/oms2020.0108
[13]	K. M. Lélén, T. Alowou-Egnim, G. N'gniamessan, T. Kokou, Second mixed problem for an Euler-Poisson- Darboux equation with Dirac potential, Open J. Math. Sci., 4 (2020), 174–178. https://doi.org/10.30538/oms2020.0107 doi: 10.30538/oms2020.0107
[14]	M. G. Sobamowo, Transient free convection heat and mass transfer of Casson nanofluid over a vertical porous plate subjected to magnetic field and thermal radiation, Eng. Appl. Sci. Lett., 3 (2020), 35–54. https://doi.org/10.30538/psrp-easl2020.0050 doi: 10.30538/psrp-easl2020.0050
[15]	R. Hassan, M. El-Agamy, M. S. A. Latif, H. Nour, On Backlund transformation of Riccati equation method and its application to nonlinear partial differential equations and differential-difference equations, Open J. Math. Sci., 4 (2020), 56–62. https://doi.org/10.30538/oms2020.0094 doi: 10.30538/oms2020.0094
[16]	S. A. Salawu, M. G. Sobamowo, O. M. Sadiq, Dynamic analysis of non-homogenous varying thickness rectangular plates resting on Pasternak and Winkler foundations, Eng. Appl. Sci. Lett., 3 (2020), 1–20. https://doi.org/10.30538/psrp-easl2020.0031 doi: 10.30538/psrp-easl2020.0031
[17]	H. Zhang, J. S. Cheng, H. M. Zhang, W. W. Zhang, J. D. Cao, Quasi-uniform synchronization of Caputo type fractional neural networks with leakage and discrete delays, Chaos Solitons Fractals, 152 (2021), 111432. https://doi.org/10.1016/j.chaos.2021.111432 doi: 10.1016/j.chaos.2021.111432
[18]	C. Wang, H. Zhang, H. M. Zhang, W. W. Zhang, Globally projective synchronization for Caputo fractional quaternion-valued neural networks with discrete and distributed delays, AIMS Math., 6 (2020), 14000–14012. http://dx.doi.org/10.3934/math.2021809 doi: 10.3934/math.2021809
[19]	H. Zhang, R. Y. Ye, J. D. Cao, A. Alsaedi, Delay-independent stability of Riemann-Liouville fractional neutral-type delayed neural networks, Neural Process. Lett., 47 (2018), 427–442. https://doi.org/10.1007/s11063-017-9658-7 doi: 10.1007/s11063-017-9658-7
[20]	W. Sun, Some local fractional integral inequalities for generalized preinvex functions and applications to numerical quadrature, Fractals, 27 (2019), 1950071. https://doi.org/10.1142/S0218348X19500713 doi: 10.1142/S0218348X19500713
[21]	W. Sun, On generalization of some inequalities for generalized harmonically convex functions via local fractional integrals, Quaest. Math., 42 (2019), 1159–1183. https://doi.org/10.2989/16073606.2018.1509242 doi: 10.2989/16073606.2018.1509242
[22]	W. Sun, Q. Liu, Hadamard type local fractional integral inequalities for generalized harmonically convex functions and applications, Math. Methods Appl. Sci., 43 (2020), 5776–5787. https://doi.org/10.1002/mma.6319 doi: 10.1002/mma.6319
[23]	D. Barrera, A. Guessab, M. J. Ibáñez, O. Nouisser, Increasing the approximation order of spline quasi-interpolants, J. Comput. Appl. Math., 252 (2013), 27–39. https://doi.org/10.1016/j.cam.2013.01.015 doi: 10.1016/j.cam.2013.01.015
[24]	A. Guessab, Approximations of differentiable convex functions on arbitrary convex polytopes, J. Comput. Appl. Math., 240 (2014), 326–338. https://doi.org/10.1016/j.amc.2014.04.075 doi: 10.1016/j.amc.2014.04.075
[25]	A. Guessab, Generalized barycentric coordinates and Jensen type inequalities on convex polytopes, J. Nonlinear Convex Anal. Yokohama, 17 (2016), 1–20.
[26]	A. Guessab, Generalized barycentric coordinates and approximations of convex functions on arbitrary convex polytopes, Comput. Math. Appl., 66 (2013), 1120–1136. https://doi.org/10.1016/j.camwa.2013.07.014 doi: 10.1016/j.camwa.2013.07.014
[27]	A. Guessab, G. Schmeisser, Two Krovkin type theorems in multivariate approximation, Banach J. Math. Anal., 2 (2008), 121–128. https://doi.org/10.15352/bjma/1240336298 doi: 10.15352/bjma/1240336298
[28]	R. Kumar, V. Gupta, I. A. Abbas, Plane deformation due to thermal source in fractional order thermoelastic media, J. Comput. Theor. Nanos., 10 (2013), 2520–2525. https://doi.org/10.1166/jctn.2013.3241 doi: 10.1166/jctn.2013.3241
[29]	A. Hobiny, I. A. Abbas, Fractional order thermoelastic wave assessment in a two-dimension medium with voids, Geomech. Eng., 21 (2020), 85–93. https://doi.org/10.12989/gae.2020.21.1.085 doi: 10.12989/gae.2020.21.1.085
[30]	S. Horrigue, I. A. Abbas, Fractional-order thermoelastic wave assessment in a two-dimensional fiber-reinforced anisotropic material, Mathematics, 8 (2020), 1609. https://doi.org/10.3390/math8091609 doi: 10.3390/math8091609
[31]	A. Hobiny, I. A. Abbas, Analytical solutions of fractional bio-heat model in a spherical tissue, Mech. Based Des. Struc. Mach., 49 (2021), 430–439. https://doi.org/10.1080/15397734.2019.1702055 doi: 10.1080/15397734.2019.1702055
[32]	T. Saeed, I. A. Abbas, The effect of fractional time derivative on two-dimension porous materials due to pulse heat flux, Mathematics, 9 (2021), 207. https://doi.org/10.3390/math9030207 doi: 10.3390/math9030207
[33]	G. Farid, W. Nazeer, M. S. Saleem, S. Mehmood, S. Kang, Bounds of Riemann-Liouville fractional integrals in general form via convex functions and their applications, Mathematics, 6 (2018), 248. https://doi.org/10.3390/math6110248 doi: 10.3390/math6110248
[34]	Y. C. Kwun, G. Farid, W. Nazeer, S. Ullah, S. M. Kang, Generalized Riemann-liouville $k$-fractional integrals associated with ostrowski type inequalities and error bounds of hadamard inequalities, IEEE Access, 6 (2018), 64946–64953. https://doi.org/10.1109/ACCESS.2018.2878266 doi: 10.1109/ACCESS.2018.2878266
[35]	X. Yang, G. Farid, W. Nazeer, M. Yussouf, Y. M. Chu, C. Dong, Fractional generalized Hadamard and FejÃ©r-Hadamard inequalities for $m$-convex functions, AIMS Math., 5 (2020), 6325–6340. https://doi.org/10.3934/math.2020407 doi: 10.3934/math.2020407
[36]	X. Wang, M. S. Saleem, K. N. Aslam, X. Wu, T. Zhou, On Caputo-Fabrizio fractional integral inequalities of Hermite-Hadamard type for modifiedh-convex functions, J. Math., 2020 (2020), 8829140. https://doi.org/10.1155/2020/8829140 doi: 10.1155/2020/8829140
[37]	T. Abdeljawad, M. A. Ali, P. O. Mohammed, A. Kashuri, On inequalities of Hermite-Hadamard-Mercer type involving Riemann-Liouville fractional integrals, AIMS Math., 6 (2021), 712–725. https://doi.org/10.3934/math.2021043 doi: 10.3934/math.2021043
[38]	T. Abdeljawad, Fractional operators with exponential kernelsand a Lyapunov type inequality, Adv. Differ. Equ., 2017 (2017), 313. https://doi.org/10.1186/s13662-017-1285-0 doi: 10.1186/s13662-017-1285-0
[39]	M. Kadakal, İ. İşcan, Exponential type convexity and some related inequalities, J. Inequal. Appl., 2020 (2020), 82. https://doi.org/10.1186/s13660-020-02349-1 doi: 10.1186/s13660-020-02349-1
[40]	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
[41]	M. Gürbüz, A. O. Akdemir, S. Rashid, E. Set, Hermite-Hadamard inequality for fractional integrals of Caputo-Fabrizio type and related inequalities, J. Inequal. Appl., 2020 (2020), 172. https://doi.org/10.1186/s13660-020-02438-1 doi: 10.1186/s13660-020-02438-1
[42]	İ. İşcan, New refinements for integral and sum forms of Holder inequality, J. Inequal. Appl., 2019 (2019), 304. https://doi.org/10.1186/s13660-019-2258-5 doi: 10.1186/s13660-019-2258-5
[43]	M. Kadakal, İ. İşcan, H. Kadakal, K. Bekar, On improvements of some integral inequalities, Honam Math. Soc., 43 (2021), 441–452. https://doi.org/10.5831/HMJ.2021.43.3.441 doi: 10.5831/HMJ.2021.43.3.441

Reader Comments

Your name:*

Email:*
© 2022 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)