Different type parameterized inequalities for preinvex functions with respect to another function via generalized fractional integral operators and their applications

A. Kashuri; M. Z. Sarikaya

doi:10.37863/umzh.v73i9.805

A. Kashuri Univ. Ismail Qemali, Vlora, Albania
M. Z. Sarikaya Düzce Univ., Turkey

DOI: https://doi.org/10.37863/umzh.v73i9.805

Keywords: Trapezium inequality, Simpson inequality, preinvexity, general fractional integrals

Abstract

UDC 517.5
The authors have proved an identity with two parameters for differentiable function with respect to another function via generalized integral operator. By applying the established identity, the generalized trapezium, midpoint and Simpson type integral inequalities have been discovered. It is pointed out that the results of this research provide integral inequalities for almost all fractional integrals discovered in recent past decades. Various special cases have been identified. Some applications of presented results to special means and new error estimates for the trapezium and midpoint quadrature formula have been analyzed. The ideas and techniques of this paper may stimulate further research in the field of integral inequalities.

References

S. M. Aslani, M. R. Delavar, S. M. Vaezpour, Inequalities of Fej´er type related to generalized convex functions with applications, Int. J. Anal. and Appl., 16, № 1, 38 – 49 (2018).

F. X. Chen, S. H. Wu, Several complementary inequalities to inequalities of Hermite – Hadamard type for $s$-convex functions, J. Nonlinear Sci. and Appl., 9, № 2, 705 – 716 (2016), https://doi.org/10.22436/jnsa.009.02.32 DOI: https://doi.org/10.22436/jnsa.009.02.32

Y. M. Chu, M. A. Khan, T. U. Khan, T. Ali, Generalizations of Hermite – Hadamard type inequalities for MT -convex functions, J. Nonlinear Sci. and Appl., 9, № 5, 4305 – 4316 (2016), https://doi.org/10.22436/jnsa.009.06.72 DOI: https://doi.org/10.22436/jnsa.009.06.72

M. R. Delavar, S. S. Dragomir, On $η$ -convexity, Math. Inequal. and Appl., 20, 203 – 216 (2017), https://doi.org/10.7153/mia-20-14 DOI: https://doi.org/10.7153/mia-20-14

M. R. Delavar, M. De La Sen, Some generalizations of Hermite – Hadamard type inequalities, vol. 5, SpringerPlus (2016). DOI: https://doi.org/10.1186/s40064-016-3301-3

S. S. Dragomir, R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and trapezoidal formula, Appl. Math. Lett., 11, № 5, 91 – 95 (1998), https://doi.org/10.1016/S0893-9659(98)00086-X DOI: https://doi.org/10.1016/S0893-9659(98)00086-X

G. Farid, Existence of an unified integral operator and its consequences in fractional calculus, Sci. Bull. Politech. Univ. Buchar. Ser. A (to appear).

G. Farid, A. U. Rehman, Generalizations of some integral inequalities for fractional integrals, Ann. Math. Sil., 31, 14 (2017), https://doi.org/10.1515/amsil-2017-0010 DOI: https://doi.org/10.1515/amsil-2017-0010

J. Hristov, Response functions in linear viscoelastic constitutive equations and related fractional operators, Math. Model. Nat. Phenom., 14, № 3, 1 – 34 (2019), https://doi.org/10.1051/mmnp/2018067 DOI: https://doi.org/10.1051/mmnp/2018067

M. Jleli, B. Samet, On Hermite – Hadamard type inequalities via fractional integral of a function with respect to another function, J. Nonlinear Sci. and Appl., 9, 1252 – 1260 (2016), https://doi.org/10.1007/s40065-016-0159-8 DOI: https://doi.org/10.22436/jnsa.009.03.50

A. Kashuri, R. Liko, Some new Hermite – Hadamard type inequalities and their applications, Stud. Sci. Math. Hung., 56, № 1, 103 – 142 (2019), https://doi.org/10.1556/012.2019.56.1.1418 DOI: https://doi.org/10.1556/012.2019.56.1.1418

U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. and Comput., 218, №3, 860 – 865 (2011), https://doi.org/10.1016/j.amc.2011.03.062 DOI: https://doi.org/10.1016/j.amc.2011.03.062

M. A. Khan, Y. M. Chu, A. Kashuri, R. Liko, Hermite – Hadamard type fractional integral inequalities for

$MT_{(r;g,m,φ )}$ -preinvex functions, J. Comput. Anal. and Appl., 26, № 8, 1487 – 1503 (2019).

M. A. Khan, Y. M. Chu, A. Kashuri, R. Liko, G. Ali, Conformable fractional integrals versions of Hermite – Hadamard inequalities and their generalizations, J. Funct. Spaces, Article ID 6928130 (2018), 9 p., https://doi.org/10.1155/2018/6928130 DOI: https://doi.org/10.1155/2018/6928130

A. A. Kilbas, O. I. Marichev, S. G. Samko, Fractional integrals and derivatives. Theory and applications, Gordon and Breach, Switzerland (1993).

A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Sci. B.V., Amsterdam (2006).

R. Khalil, M. A. Horani, A. Yousef, M. Sababheh, A new definition of fractional derivatives, J. Comput. and Appl. Math., 264, 65 – 70 (2014), https://doi.org/10.1016/j.cam.2014.01.002 DOI: https://doi.org/10.1016/j.cam.2014.01.002

B. Ahmad, A. Alsaedi, M. Kirane, B. T. Torebek, Hermite – Hadamard, Hermite – Hadamard – Fej´er, Dragomir – Agarwal and Pachpatti type inequalities for convex functions via fractional integrals, J. Comput. and Appl. Math., 353, 120 – 129 (2019), https://doi.org/10.1016/j.cam.2018.12.030 DOI: https://doi.org/10.1016/j.cam.2018.12.030

W. J. Liu, Some Simpson type inequalities for $h$-convex and $(alpha,m)$-convex functions, J. Comput. Anal. and Appl., 16, № 5, 1005 – 1012 (2014).

W. Liu, W. Wen, J. Park, Hermite – Hadamard type inequalities for MT -convex functions via classical integrals and fractional integrals, J. Nonlinear Sci. and Appl., 9, 766 – 777 (2016), https://doi.org/10.22436/jnsa.009.03.05 DOI: https://doi.org/10.22436/jnsa.009.03.05

C. Luo, T. S. Du, M. A. Khan, A. Kashuri, Y. Shen, Some $k$-fractional integrals inequalities through generalized $lambda phi m$-MT -preinvexity, J. Comput. Anal. and Appl., 27, № 4, 690 – 705 (2019).

M. V. Mihai, Some Hermite – Hadamard type inequalities via Riemann – Liouville fractional calculus, Tamkang J. Math., 44, № 4, 411 – 416 (2013), https://doi.org/10.5556/j.tkjm.44.2013.1218 DOI: https://doi.org/10.5556/j.tkjm.44.2013.1218

S. Mubeen, G. M. Habibullah, $k$-Fractional integrals and applications, Int. J. Contemp. Math. Sci., 7, 89 – 94 (2012).

O. Omotoyinbo, A. Mogbodemu, Some new Hermite – Hadamard integral inequalities for convex functions, Int. J. Sci. Innovation Tech., 1, № 1, 1 – 12 (2014).

M. E. Özdemir, S. S. Dragomir, C. Yildiz, The Hadamard’s inequality for convex function via fractional integrals, Acta Math. Sci., 33, № 5, 153 – 164 (2013), https://doi.org/10.1016/S0252-9602(13)60081-8 DOI: https://doi.org/10.1016/S0252-9602(13)60081-8

F. Qi, B. Y. Xi, Some integral inequalities of Simpson type for GA $varepsilon$ -convex functions, Georgian Math. J., 20, № 5, 775 – 788 (2013), https://doi.org/10.1515/gmj-2013-0043 DOI: https://doi.org/10.1515/gmj-2013-0043

M. Z. Sarikaya, F. Ertu˘gral, On the generalized Hermite – Hadamard inequalities, https://www.researchgate.net/publication/321760443.

M. Z. Sarikaya, H. Yildirim, On generalization of the Riesz potential, Indian J. Math. and Math. Sci., 3, № 2, 231 – 235 (2007).

E. Set, M. A. Noor, M. U. Awan, A. G¨ozpinar, Generalized Hermite – Hadamard type inequalities involving fractional integral operators, J. Inequal. and Appl., 169, 1 – 10 (2017), https://doi.org/10.1186/s13660-017-1444-6 DOI: https://doi.org/10.1186/s13660-017-1444-6

H. Wang, T. S. Du, Y. Zhang, $k$-Fractional integral trapezium-like inequalities through $(h,m)$-convex and $(alpha ,m)$-convex mappings, J. Inequal. and Appl., 2017, Article 311 (2017), 20 p., https://doi.org/10.1186/s13660-017-1586-6 DOI: https://doi.org/10.1186/s13660-017-1586-6

R. Y. Xi, F. Qi, Some integral inequalities of Hermite – Hadamard type for convex functions with applications to means, J. Funct. Spaces and Appl., 2012, Article ID 980438 (2012), 14 p., https://doi.org/10.1155/2012/980438 DOI: https://doi.org/10.1155/2012/980438

X. M. Zhang,Y. M. Chu, X. H. Zhang, The Hermite – Hadamard type inequality of GA-convex functions and its applications, J. Inequal. and Appl., 2010, Article ID 507560 (2010), 11 p., https://doi.org/10.1155/2010/507560 DOI: https://doi.org/10.1155/2010/507560

Y. Zhang, T. S. Du, H. Wang, Y. J. Shen, A. Kashuri, Extensions of different type parameterized inequalities for generalized $(m, h)$-preinvex mappings via $k$-fractional integrals, J. Inequal. and Appl., 2018, № 1 (2018), 30 p., https://doi.org/10.1186/s13660-018-1639-5 DOI: https://doi.org/10.1186/s13660-018-1639-5