Some new inequalities for generalized convex functions pertaining generalized fractional integral operators and their applications

Kashuri, A.; Ali, M.A.; Abbas, M.; Toseef, M.

doi:10.2478/jamsi-2021-0003

Abstract

In this paper, authors establish a new identity for a differentiable function using generic integral operators. By applying it, some new integral inequalities of trapezium, Ostrowski and Simpson type are obtained. Moreover, several special cases have been studied in detail. Finally, many useful applications have been found.

References

R.P. Agarwal, M.J. Luo and R.K. Raina, On Ostrowski type inequalities. Fasc. Math. 204 (2016), 5–27.
Search in Google Scholar Back to article
M. Ahmadmir and R. Ullah, Some inequalities of Ostrowski and Grüss type for triple integrals on time scales. Tamkang J. Math. 42(4) (2011), 415–426.
Search in Google Scholar Back to article
M. Alomari, M. Darus, S.S. Dragomir and P. Cerone, Ostrowski type inequalities for functions whose derivatives are s–convex in the second sense. Appl. Math. Lett. 23 (2010), 1071–1076.
Search in Google Scholar Back to article
S.M. Aslani, M.R. Delavar and S.M. Vaezpour, Inequalities of Fejér type related to generalized convex functions with applications. Int. J. Anal. Appl. 16(1) (2018), 38–49.
Search in Google Scholar Back to article
F.X. Chen and S.H. Wu, Several complementary inequalities to inequalities of Hermite–Hadamard type for s–convex functions. J. Nonlinear Sci. Appl. 9(2) (2016), 705–716.
Search in Google Scholar Back to article
M.R. Delavar and M. De La Sen, Some generalizations of Hermite–Hadamard type inequalities. SpringerPlus 5(1661) (2016).
Search in Google Scholar Back to article
S.S. Dragomir, On the Ostrowski’s integral inequality for mappings with bounded variation and applications. Math. Ineq. & Appl. 1(2) (1998).
Search in Google Scholar Back to article
S.S. Dragomir, The Ostrowski integral inequality for Lipschitzian mappings and applications. Comput. Math. Appl. 38 (1999), 33–37.
Search in Google Scholar Back to article
S.S. Dragomir, Ostrowski-type inequalities for Lebesgue integral: A survey of recent results. Aust. J. Math. Anal. Appl. 14(1) (2017), 1–287.
Search in Google Scholar Back to article
S.S. Dragomir, Further Ostrowski and trapezoid type inequalities for the generalized Riemann–Liouville fractional integrals of functions with bounded variation. RGMIA Res. Rep. Coll. 20 (2017), Art. 84.
Search in Google Scholar Back to article
S.S. Dragomir, Ostrowski and trapezoid type inequalities for the generalized k-g–fractional integrals of functions with bounded variation. RGMIA Res. Rep. Coll. 20 (2017), Art. 111.
Search in Google Scholar Back to article
S.S. Dragomir and R.P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and trapezoidal formula. Appl. Math. Lett. 11(5) (1998), 91–95.
Search in Google Scholar Back to article
S.S. Dragomir and S. Wang, An inequality of Ostrowski–Grüss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules. Comput. Math. Appl. 13(11) (1997), 15–20.
Search in Google Scholar Back to article
S.S. Dragomir and S. Wang, A new inequality of Ostrowski’s type in L₁–norm and applications to some special means and to some numerical quadrature rules. Tamkang J. Math. 28 (1997), 239–244.
Search in Google Scholar Back to article
T.S. 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. (2019).
Search in Google Scholar Back to article
G. Farid, Existence of an integral operator and its consequences in fractional calculus. Submitted.
Search in Google Scholar Back to article
G. Farid, Some new Ostrowski type inequalities via fractional integrals. Int. J. Anal. Appl. 14(1) (2017), 64–68.
Search in Google Scholar Back to article
J. Hristov, Response functions in linear viscoelastic constitutive equations and related fractional operators. Math. Model. Nat. Phenom. 14(3) (2019), 1–34.
Search in Google Scholar Back to article
M. Jleli and B. Samet, On Hermite–Hadamard type inequalities via fractional integral of a function with respect to another function. J. Nonlinear Sci. Appl. 9 (2016), 1252–1260.
Search in Google Scholar Back to article
A. Kashuri and R. Liko, Some new Hermite–Hadamard type inequalities and their applications. Stud. Sci. Math. Hung. 56(1) (2019), 103–142.
Search in Google Scholar Back to article
U.N. Katugampola, New approach to a generalized fractional integral. Appl. Math. Comput. 218(3) (2011), 860–865.
Search in Google Scholar Back to article
M.A. Khan, Y.M. Chu, A. Kashuri, R. Liko and G. Ali, Conformable fractional integrals versions of Hermite–Hadamard inequalities and their generalizations. J. Funct. Spaces (2018), Article ID 6928130, pp. 9.
Search in Google Scholar Back to article
A.A. Kilbas, O.I. Marichev and S.G. Samko, Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Switzerland, (1993).
Search in Google Scholar Back to article
A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science B.V., Amsterdam, (2006).
Search in Google Scholar Back to article
R. Khalil, M.A. Horani, A. Yousef and M. Sababheh, A new de finition of fractional derivatives. J. Comput. Appli. Math. 264 (2014), 65–70.
Search in Google Scholar Back to article
M. Kirane and B.T. Torebek, Hermite–Hadamard, Hermite–Hadamard–Fejér, Dragomir–Agarwal and Pachpatti type inequalities for convex functions via fractional integrals. arXive:1701.00092.
Search in Google Scholar Back to article
W.J. Liu, Some Simpson type inequalities for h–convex and (a,m)–convex functions. J. Comput. Anal. Appl. 16(5) (2014), 1005–1012.
Search in Google Scholar Back to article
W., Liu, W. Wen and J. Park, Hermite–Hadamard type inequalities for MT –convex functions via classical integrals and fractional integrals. J. Nonlinear Sci. Appl. 9 (2016), 766–777.
Search in Google Scholar Back to article
W. Liu, W. Wen and J. Park, Ostrowski type fractional integral inequalities for MT –convex functions. Miskolc Math. Notes 16(1) (2015), 249–256.
Search in Google Scholar Back to article
Z. Liu, Some Ostrowski–Grüss type inequalities and applications. Comput. Math. Appl. 53 (2007), 73–79.
Search in Google Scholar Back to article
Z. Liu, Some companions of an Ostrowski type inequality and applications. J. Inequal. in Pure and Appl. Math. 10(2) (2009), Art. 52, pp. 12.
Search in Google Scholar Back to article
M. Matloka, Ostrowski type inequalities for functions whose derivatives are h–convex via fractional integrals. Journal of Scienti fic Research and Reports 3(12) (2014), 1633–1641.
Search in Google Scholar Back to article
M.V. Mihai, Some Hermite–Hadamard type inequalities via Riemann–Liouville fractional calculus. Tamkang J. Math. 44(4) (2013), 411–416.
Search in Google Scholar Back to article
S. Mubeen and G.M. Habibullah, k–Fractional integrals and applications. Int. J. Contemp. Math. Sci. 7 (2012), 89–94.
Search in Google Scholar Back to article
M.A. Noor, Some new classes of nonconvex functions. Nonlinear Funct. Anal. Appl. 11 (2006), 165–171.
Search in Google Scholar Back to article
M.A. Noor, Some properties of nonconvex functions. Nonlinear Funct. Anal. Appl. 23(3) (2018), 575–583.
Search in Google Scholar Back to article
O. Omotoyinbo and A. Mogbodemu, Some new Hermite–Hadamard integral inequalities for convex functions. Int. J. Sci. Innovation Tech. 1(1) (2014), 1–12.
Search in Google Scholar Back to article
M.E.Özdemir, S.S. Dragomir and C. Yildiz, The Hadamard’s inequality for convex function via fractional integrals. Acta Math. Sci., Ser. B, Engl. Ed. 33(5) (2013), 153–164.
Search in Google Scholar Back to article
M.E.Özdemir, H. Kavurmac and E. Set, Ostrowski’s type inequalities for (a,m)–convex functions. Kyungpook Math. J. 50 (2010), 371–378.
Search in Google Scholar Back to article
B.G. Pachpatte, On an inequality of Ostrowski type in three independent variables. J. Math. Anal. Appl. 249 (2000), 583–591.
Search in Google Scholar Back to article
B.G. Pachpatte, On a new Ostrowski type inequality in two independent variables. Tamkang J. Math. 32(1) (2001), 45–49.
Search in Google Scholar Back to article
F. Qi and B.Y. Xi, Some integral inequalities of Simpson type for GA–e–convex functions. Georgian Math. J. 20(5) (2013), 775–788.
Search in Google Scholar Back to article
A. Ra fiq, N. A. Mir and F. Ahmad, WeightedČebyšev–Ostrowski type inequalities. Applied Math. Mechanics (English Ed.) 28(7) (2007), 901–906.
Search in Google Scholar Back to article
M.Z. Sarikaya, On the Ostrowski type integral inequality. Acta Math. Univ. Comenianae 79(1) (2010), 129–134.
Search in Google Scholar Back to article
M.Z. Sarikaya and F. Ertuğral, On the generalized Hermite–Hadamard inequalities. https://www.researchgate.net/publication/321760443.
Search in Google Scholar Back to article
M.Z. Sarikaya and H. Yildirim, On generalization of the Riesz potential. Indian Jour. of Math. and Mathematical Sci. 3(2), (2007), 231–235.
Search in Google Scholar Back to article
M.A. Noor, M.U. Awan and A. Gözpinar, Generalized Hermite–Hadamard type inequalities involving fractional integral operators. J. Inequal. Appl. 169 (2017), 1–10.
Search in Google Scholar Back to article
R.K. Raina, On generalized Wright’s hypergeometric functions and fractional calculus operators. EAMJ, East Asian Math. J. 21(2) (2005), 191–203.
Search in Google Scholar Back to article
M. Tomar, P. Agarwal, M. Jleli and B. Samet, Certain Ostrowski type inequalities for generalized s–convex functions. J. Nonlinear Sci. Appl. 10 (2017), 5947–5957.
Search in Google Scholar Back to article
M. Tunç, Ostrowski type inequalities for functions whose derivatives are MT –convex. J. Comput. Anal. Appl. 17(4) (2014), 691–696.
Search in Google Scholar Back to article
N. Ujević, Sharp inequalities of Simpson type and Ostrowski type. Comput. Math. Appl. 48 (2004), 145–151.
Search in Google Scholar Back to article
H. Wang, T.S. Du and Y. Zhang, k–fractional integral trapezium-like inequalities through (h,m)–convex and (a,m)–convex mappings. J. Inequal. Appl. 2017(311) (2017), pp. 20.
Search in Google Scholar Back to article
B.Y. Xi and F. Qi, Some integral inequalities of Hermite–Hadamard type for convex functions with applications to means. J. Funct. Spaces Appl. 2012 (2012), Article ID 980438, pp. 14.
Search in Google Scholar Back to article
Ç. Yildiz, M.E.Özdemir and M.Z. Sarikaya, New generalizations of Ostrowski-like type inequalities for fractional integrals. Kyungpook Math. J. 56 (2016), 161–172.
Search in Google Scholar Back to article
X.M. Zhang, Y.M. Chu and X.H. Zhang, The Hermite–Hadamard type inequality of GA–convex functions and its applications. J. Inequal. Appl. (2010), Article ID 507560, pp. 11.
Search in Google Scholar Back to article
Y. Zhang, T.S. Du, H. Wang, Y.J. Shen and A. Kashuri, Extensions of different type parameterized inequalities for generalized (m,h)–preinvex mappings via k–fractional integrals. J. Inequal. Appl. 2018 (49) (2018), pp. 30.
Search in Google Scholar Back to article
L. Zhongxue, On sharp inequalities of Simpson type and Ostrowski type in two independent variables. Comput. Math. Appl. 56 (2008), 2043–2047.
Search in Google Scholar Back to article

Some new inequalities for generalized convex functions pertaining generalized fractional integral operators and their applications

Abstract

Paradigm

My account