Ostrowski type inequalities in the sense of generalized $\mathcal{K}$-fractional integral operator for exponentially convex functions

Saima Rashid; Muhammad Aslam Noor; Khalida Inayat Noor; Yu-Ming Chu; Saima Rashid; Muhammad Aslam Noor; Khalida Inayat Noor; Yu-Ming Chu

doi:10.3934/math.2020171

AIMS Mathematics

2020, Volume 5, Issue 3: 2629-2645. doi: 10.3934/math.2020171

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Research article

Ostrowski type inequalities in the sense of generalized $\mathcal{K}$-fractional integral operator for exponentially convex functions

1.
Department of Mathematics, Government College (GC) University, Faisalabad 38000, Pakistan
2.
Department of Mathematics, COMSATS University Islamabad, Islamabad 45550, Pakistan
3.
Department of Mathematics, Huzhou University, Huzhou 313000, P. R. China
4.
School of Mathematics and Statistics, Changsha University of Science & Technology, Changsha 413000, P. R. China

Received: 01 January 2020 Accepted: 04 March 2020 Published: 13 March 2020
MSC : 26D07, 26D15, 26D20

The investigation of the proposed methods is effective and convenient for solving the integrodifferential and difference equations. In this note, we introduce the generalized $\mathcal{K}$-fractional integral in terms of a new parameter $\mathcal{K}>0$ for exponentially convex functions. This paper offers some novel inequalities of Ostrowski-type using the generalized $\mathcal{K}$-fractional integral. In the application viewpoint, we proved several corollaries that investigate for proving Hermite-Hadamard inequalities for generalized $\mathcal{K}$-fractional integral operator. Some numerical examples are offered to explain the obtained results. Moreover, some applications of proposed results are presented to the demonstration of the efficiency of the proposed technique. The numerical results show that our approach is superior to some related methodologies.
- integral inequality,
- exponentially convex functions,
- generalized $\mathcal{K}$-fractional integral operator,
- Hermite-Hadamard inequality
Citation: Saima Rashid, Muhammad Aslam Noor, Khalida Inayat Noor, Yu-Ming Chu. Ostrowski type inequalities in the sense of generalized $\mathcal{K}$-fractional integral operator for exponentially convex functions[J]. AIMS Mathematics, 2020, 5(3): 2629-2645. doi: 10.3934/math.2020171

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Abstract

The investigation of the proposed methods is effective and convenient for solving the integrodifferential and difference equations. In this note, we introduce the generalized $\mathcal{K}$-fractional integral in terms of a new parameter $\mathcal{K}>0$ for exponentially convex functions. This paper offers some novel inequalities of Ostrowski-type using the generalized $\mathcal{K}$-fractional integral. In the application viewpoint, we proved several corollaries that investigate for proving Hermite-Hadamard inequalities for generalized $\mathcal{K}$-fractional integral operator. Some numerical examples are offered to explain the obtained results. Moreover, some applications of proposed results are presented to the demonstration of the efficiency of the proposed technique. The numerical results show that our approach is superior to some related methodologies.

References

[1]	I. Abbas Baloch, Y. M. Chu, Petrović-type inequalities for harmonic h-convex functions, J. Funct. Space., 2020 (2020), 1-7.
[2]	M. Adil Khan, M. Hanif, Z. A. Khan, et al. Association of Jensen's inequality for s-convex function with Csiszár divergence, J. Inequal. Appl., 2019 (2019), 1-14. doi: 10.1186/s13660-019-1955-4
[3]	M. Adil Khan, A. Iqbal, M. Suleman, et al. Hermite-Hadamard type inequalities for fractional integrals via Green's function, J. Inequal. Appl., 2018 (2018), 1-15. doi: 10.1186/s13660-017-1594-6
[4]	M. Adil Khan, Y. Khurshid, T. S. Du, et al. Generalization of Hermite-Hadamard type inequalities via conformable fractional integrals, J. Funct. Space., 2018 (2018), 1-12.
[5]	M. Adil Khan, N. Mohammad, E. R. Nwaeze, et al. Quantum Hermite-Hadamard inequality by means of a Green function, Adv. Differ. Equ., 2020 (2020), 1-20. doi: 10.1186/s13662-019-2438-0
[6]	M. Adil Khan, S. H. Wu, H. Ullah, et al. Discrete majorization type inequalities for convex functions on rectangles, J. Inequal. Appl., 2019 (2019), 1-18. doi: 10.1186/s13660-019-1955-4
[7]	M. Adil Khan, S. Zaheer Ullah, Y. M. Chu, The concept of coordinate strongly convex functions and related inequalities, RACSAM, 113 (2019), 2235-2251. doi: 10.1007/s13398-018-0615-8
[8]	Y. Adjabi, F. Jarad, D. Baleanu, et al. On Cauchy problems with Caputo Hadamard fractional derivatives, J. Comput. Anal. Appl., 21 (2016), 661-681.
[9]	G. Alirezaei, R. Mathar, On exponentially concave functions and their impact in information theory, J. Inf. Theory Appl., 9 (2018), 265-274.
[10]	M. Andrić, A. Barbir, S. Iqbal, et al. An Opial-type integral inequality and exponentially convex functions, Fract. Differ. Calc., 5 (2015), 25-42.
[11]	M. Avriel, r-convex functions, Math. Program., 2 (1972), 309-323. doi: 10.1007/BF01584551
[12]	M. U. Awan, M. A. Noor, K. I. Noor, Hermite-Hadamard inequalities for exponentially convex functions, Appl. Math. Inf. Sci., 12 (2018), 405-409. doi: 10.18576/amis/120215
[13]	D. Baleanu, K. Diethelm, E. Scalas, et al. Fractional Calculus, World Scientific Publishing, Hackensack, 2012.
[14]	S. N. Bernstein, Sur les fonctions absolument monotones, Acta Math., 52 (1929), 1-66. doi: 10.1007/BF02592679
[15]	P. S. Bullen, D. S. Mitrinović, P. M. Vasić, Means and Their Inequalities, D. Reidel Publishing Co., Dordrecht, 1988.
[16]	Y. M. Chu, M. Adil Khan, T. Ali, et al. Inequalities for α-fractional differentiable functions, J. Inequal. Appl., 2017 (2017), 1-12. doi: 10.1186/s13660-016-1272-0
[17]	Y. M. Chu, M. K. Wang, S. L. Qiu, Optimal combinations bounds of root-square and arithmetic means for Toader mean, Proc. Indian Acad. Sci. Math. Sci., 122 (2012), 41-51. doi: 10.1007/s12044-012-0062-y
[18]	R. Díaz, E. Pariguan, On hypergeometric functions and Pochhammer k-symbol, Divulg. Mat., 15 (2007), 179-192.
[19]	S. S. Dragomir, Operator Inequalities of Ostrowski and Trapezoidal Type, Springer, New York, 2012.
[20]	S. S. Dragomir, I. Gomm, Some Hermite-Hadamard type inequalities for functions whose exponentials are convex, Stud. Univ. Babeş-Bolyai Math., 60 (2015), 527-534.
[21]	S. S. Dragomir, T. M. Rassias, Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publishers, Dordrecht, 2002.
[22]	X. H. He, W. M. Qian, H. Z. Xu, et al. Sharp power mean bounds for two Sándor-Yang means, RACSAM, 113 (2019), 2627-2638. doi: 10.1007/s13398-019-00643-2
[23]	T. R. Huang, B. W. Han, X. Y. Ma, et al. Optimal bounds for the generalized Euler-Mascheroni constant, J. Inequal. Appl., 2018 (2018), 1-9. doi: 10.1186/s13660-017-1594-6
[24]	C. X. Huang, L. Z. Liu, Sharp function inequalities and boundness for Toeplitz type operator related to general fractional singular integral operator, Publ. Inst. Math., 92 (2012), 165-176. doi: 10.2298/PIM1206165H
[25]	T. R. Huang, S. Y. Tan, X. Y. Ma, et al. Monotonicity properties and bounds for the complete p-elliptic integrals, J. Inequal. Appl., 2018 (2018), 1-11. doi: 10.1186/s13660-017-1594-6
[26]	C. X. Huang, H. Zhang, L. H. Huang, Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term, Commun. Pure Appl. Anal., 18 (2019), 3337-3349. doi: 10.3934/cpaa.2019150
[27]	A. Iqbal, M. Adil Khan, S. Ullah, et al. Some new Hermite-Hadamard-type inequalities associated with conformable fractional integrals and their applications, J. Funct. Space., 2020 (2020), 1-18.
[28]	J. Jakšetić, J. Pečarić, Exponential convexity method, J. Convex Anal., 20 (2013), 181-197.
[29]	F. Jarad, E. Uǧurlu, T. Abdeljawad, et al. On a new class of fractional operators, Adv. Differ. Equ., 2017 (2017), 1-16. doi: 10.1186/s13662-016-1057-2
[30]	F. Jarad, T. Abdeljawad, D. Baleanu, On the generalized fractional derivatives and their Caputo modification, J. Nonlinear Sci. Appl., 10 (2017), 2607-2619. doi: 10.22436/jnsa.010.05.27
[31]	Y. J. Jiang, X. J. Xu, A monotone finite volume method for time fractional Fokker-Planck equations, Sci. China Math., 62 (2019), 783-794. doi: 10.1007/s11425-017-9179-x
[32]	E. Kacar, Z. Kacar, H. Yildirim, Integral inequalities for Riemann-Liouville fractional integrals of a function with respect to another function, Iran. J. Math. Sci. Inform., 13 (2018), 1-13. doi: 10.22457/jmi.v13a1
[33]	U. N. Katugampola, New fractional integral unifying six existing fractional integrals, arXiv:1612.08596.
[34]	R. Khalil, M. Al Horani, A. Yousef, et al. A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70. doi: 10.1016/j.cam.2014.01.002
[35]	T. U. Khan, M. Adil Khan, Generalized conformable fractional operators, J. Comput. Appl. Math., 346 (2019), 378-389. doi: 10.1016/j.cam.2018.07.018
[36]	S. Khan, M. Adil Khan, Y. M. Chu, Converses of the Jensen inequality derived from the Green functions with applications in information theory, Math. Method. Appl. Sci., 30 (2020), 2577-2587.
[37]	Y. Khurshid, M. Adil Khan, Y. M. Chu, et al. Hermite-Hadamard-Fejér inequalities for conformable fractional integrals via preinvex functions, J. Funct. Space., 2019 (2019), 1-9.
[38]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006.
[39]	Y. C. Kwun, G. Farid, W. Nazeer, et al. Generalized Riemann-Liouville k-fractional integrals associated with Ostrowski type inequalities and error bounds of Hadamard inequalities, IEEE Access, 6 (2018), 64946-64953. doi: 10.1109/ACCESS.2018.2878266
[40]	M. A. Latif, S. Rashid, S. S. Dragomir, et al. Hermite-Hadamard type inequalities for co-ordinated convex and qausi-convex functions and their applications, J. Inequal. Appl., 2019 (2019), 1-33. doi: 10.1186/s13660-019-1955-4
[41]	F. W. Liu, L. B. Feng, V. Anh, et al. Unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time-space fractional Bloch-Torrey equations on irregular convex domains, Comput. Math. Appl., 78 (2019), 1637-1650. doi: 10.1016/j.camwa.2019.01.007
[42]	F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010.
[43]	S. Mubeen, G. M. Habibullah, k-fractional integrals and application, Int. J. Contemp. Math. Sci., 7 (2012), 89-94.
[44]	K. S. Nisar, G. Rahman, J. Choi, et al. Certain Gronwall type inequalities associated with Riemann-Liouville k- and Hadamard k-fractional derivatives and their applications, East Asian Math. J., 34 (2018), 249-263.
[45]	D. S. Oliveira, E. Capelas de Oliveira, Hilfer-Katugampola fractional derivative, arXiv:1705.07733.
[46]	A. Ostrowski, Über die Absolutabweichung einer differentiierbaren Funktion von ihrem Integralmittelwert, Comment. Math. Helv., 10 (1937), 226-227. doi: 10.1007/BF01214290
[47]	S. Pal, T. K. L. Wong, Exponentially concave functions and a new information geometry, Ann. Probab., 46 (2018), 1070-1113. doi: 10.1214/17-AOP1201
[48]	W. M. Qian, Z. Y. He, Y. M. Chu, Approximation for the complete elliptic integral of the first kind, RACSAM, 114 (2020), 1-12. doi: 10.1007/s13398-019-00732-2
[49]	W. M. Qian, Z. Y. He, H. W. Zhang, et al. Sharp bounds for Neuman means in terms of two-parameter contraharmonic and arithmetic mean, J. Inequal. Appl., 2019 (2019), 1-13. doi: 10.1186/s13660-019-1955-4
[50]	W. M. Qian, Y. Y. Yang, H. W. Zhang, et al. Optimal two-parameter geometric and arithmetic mean bounds for the Sándor-Yang mean, J. Inequal. Appl., 2019 (2019), 1-12. doi: 10.1186/s13660-019-1955-4
[51]	W. M. Qian, W. Zhang, Y. M. Chu, Bounding the convex combination of arithmetic and integral means in terms of one-parameter harmonic and geometric means, Miskolc Math. Notes, 20 (2019), 1157-1166.
[52]	S. Rashid, F. Jarad, H. Kalsoom, et al. On Pólya-Szegö and Čebyšev type inequalities via generalized k-fractional integral, Adv. Differ. Equ., 2020 (2020).
[53]	S. Rashid, F. Jarad, M. A. Noor, et al. Inequalities by means of generalized proportional fractional integral operators with respect to another function, Mathematics, 7 (2019), 1-18.
[54]	S. Rashid, F. Safdar, A. O. Akdemir, et al. Some new fractional integral inequalities for exponentially m-convex functions via extended generalized Mittag-Leffler function, J. Inequal. Appl., 2019 (2019), 1-17. doi: 10.1186/s13660-019-1955-4
[55]	E. Set, New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals, Comput. Math. Appl., 63 (2012), 1147-1154. doi: 10.1016/j.camwa.2011.12.023
[56]	J. F. Wang, X. Y. Chen, L. H. Huang, The number and stability of limit cycles for planar piecewise linear systems of node-saddle type, J. Math. Anal. Appl., 469 (2019), 405-427. doi: 10.1016/j.jmaa.2018.09.024
[57]	M. K. Wang, H. H. Chu, Y. M. Chu, Precise bounds for the weighted Hölder mean of the complete p-elliptic integrals, J. Math. Anal. Appl., 480 (2019), 123388.
[58]	M. K. Wang, Y. M. Chu, W. Zhang, Monotonicity and inequalities involving zero-balanced hypergeometric function, Math. Inequal. Appl., 22 (2019), 601-617.
[59]	J. F. Wang, C. X. Huang, L. H. Huang, Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type, Nonlinear Anal. Hybrid Syst., 33 (2019), 162-178. doi: 10.1016/j.nahs.2019.03.004
[60]	M. K. Wang, M. Y. Hong, Y. F. Xu, et al. Inequalities for generalized trigonometric and hyperbolic functions with one parameter, J. Math. Inequal., 14 (2020), 1-21.
[61]	B. Wang, C. L. Luo, S. H. Li, et al. Sharp one-parameter geometric and quadratic means bounds for the Sándor-Yang means, RACSAM, 114 (2020), 7.
[62]	M. K. Wang, W. Zhang, Y. M. Chu, Monotonicity, convexity and inequalities involving the generalized elliptic integrals, Acta Math. Sci., 39B (2019), 1440-1450.
[63]	J. Wu, Y. C. Liu, Uniqueness results and convergence of successive approximations for fractional differential equations, Hacet. J. Math. Stat., 42 (2013), 149-158.
[64]	S. H. Wu, Y. M. Chu, Schur m-power convexity of generalized geometric Bonferroni mean involving three parameters, J. Inequal. Appl., 2019 (2019), 1-11. doi: 10.1186/s13660-019-1955-4
[65]	Z. H. Yang, W. M. Qian, W. Zhang, et al. Notes on the complete elliptic integral of the first kind, Math. Inequal. Appl., 23 (2020), 77-93.
[66]	S. Zaheer Ullah, M. Adil Khan, et al. Majorization theorems for strongly convex functions, J. Inequal. Appl., 2019 (2019), 1-13. doi: 10.1186/s13660-019-1955-4
[67]	S. Zaheer Ullah, M. Adil Khan, Y. M. Chu, A note on generalized convex functions, J. Inequal. Appl., 2019 (2019), 1-10. doi: 10.1186/s13660-019-1955-4
[68]	T. H. Zhao, Y. M. Chu, H. Wang, Logarithmically complete monotonicity properties relating to the gamma function, Abstr. Appl. Anal., 2011 (2011), 1-13.
[69]	T. H. Zhao, L. Shi, Y. M. Chu, Convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means, RACSAM, 114 (2020), 1-14. doi: 10.1007/s13398-019-00732-2
[70]	S. H. Zhou, Y. J. Jiang, Finite volume methods for N-dimensional time fractional Fokker-Planck equations, Malays. Math. Sci. Soc., 42 (2019), 3167-3186. doi: 10.1007/s40840-018-0652-7

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