Tbilisi Mathematical Journal

Some new inequalities involving the Katugampola fractional integrals for strongly $\eta$-convex functions

Seth Kermausuor and Eze R. Nwaeze

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We introduced several new integral inequalities of the Hermite-Hadamard type for strongly $\eta$-convex functions via the Katugampola fractional integrals. Some results in the literature are particular cases of our results.

Note

The authors are very grateful to the anonymous referee for his/her valuable comments and suggestions.

Article information

Source
Tbilisi Math. J., Volume 12, Issue 1 (2019), 117-130.

Dates
Received: 28 November 2018
Accepted: 20 January 2019
First available in Project Euclid: 26 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.tbilisi/1553565631

Digital Object Identifier
doi:10.32513/tbilisi/1553565631

Mathematical Reviews number (MathSciNet)
MR3954224

Subjects
Primary: 26D15: Inequalities for sums, series and integrals
Secondary: 26A33: Fractional derivatives and integrals 26A51: Convexity, generalizations 26D10: Inequalities involving derivatives and differential and integral operators

Keywords
Hermite-Hadamard type inequality strongly $\eta$-convex functions Hölder's inequality Katugampola fractional integrals Riemann-Liouville fractional integrals Hadamard fractional integrals

Citation

Kermausuor, Seth; Nwaeze, Eze R. Some new inequalities involving the Katugampola fractional integrals for strongly $\eta$-convex functions. Tbilisi Math. J. 12 (2019), no. 1, 117--130. doi:10.32513/tbilisi/1553565631. https://projecteuclid.org/euclid.tbilisi/1553565631


Export citation

References

  • M. Alomari, M. Darus and S. S. Dragomir, New inequalities of Hermite–Hadamard's type for functions whose second derivatives absolute values are quasiconvex, Tamkang. J. Math. 41 (2010), 353–359.
  • M. U. Awan, M. A. Noor, K. I. Noor and F. Safdar, On strongly generalized convex functions, Filomat 31(18)(2017), 5783–5790.
  • H. Chen and U. N. Katugampola, Hermite–Hadamard and Hermite–Hadamard–Fejér type inequalities for generalized fractional integrals, J. Math. Anal. Appl. 446(2) (2017), 1274–1291.
  • L. Chun and F. Qi, Integral inequalities of Hermite–Hadamard type for functions whose third derivatives are convex, J. Inequal. Appl. 2013, 2013:451
  • L. Chun and F. Qi, Integral inequalities of Hermite–Hadamard type for functions whose 3rd derivatives are $s$-convex, Appl. Math. 3 (2012), 1680–1685.
  • S. S. Dragomir, Two mappings in connection to Hadamard's inequalities, J. Math. Anal. Appl. 167 (1992), 49–56.
  • S. S. Dragomir and R. P. Agarwal, Two inequalities for differentiable mappings and their applications to special means for real numbers and to trapezoidal formula, Appl. Math. Lett. 11(5)(1998), 91–95.
  • G. Farid, A. U. Rehman and M. Zahra, On Hadamard-type inequalities for $k$-fractional integrals, Konulrap J. Math. 4(2)(2016), 79–86.
  • M. E. Gordji, M. R. Delavar and S. S. Dragomir, Some inequalities related to $\eta$-convex functions, RGMIA 18 (2015), Art. 8.
  • M. E. Gordji, M. R. Delavar and M. De La Sen, On $\varphi$-convex functions, J. Math. Inequal. 10(1)(2016), 173–183.
  • M. E. Gordji, S. S. Dragomir and M. R. Delavar, An inequality related to $\eta$-convex functions (II), Int. J. Nonlinear Anal. Appl. 6(2)(2015), 27–33.
  • J. Hadamard, Etude sur les properties des fonctions entries et an particular d'une fonction considree par, Riemann, J. Math. Pures. et Appl. 58 (1893), 171–215.
  • U.N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput. 218(3)(2011), 860–865.
  • U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 6(4)(2014), 1–15.
  • M. A. Khan, Y. Khurshid and T. Ali, Hermite–Hadamard inequality for fractional integrals via $\eta$-convex functions, Acta Math. Univ. Comen. LXXXVI (1)(2017), 153–164.
  • E. R. Nwaeze, Inequalities of the Hermite–Hadamard type for quasi-convex functions via the $(k,s)$-Riemann–Liouville fractional integrals, Fractional Differ. Calc. 8(2)(2018), 327–336.
  • E. R. Nwaeze, S. Kermausuor and A. M. Tameru, Some new $k$-Riemann–Liouville Fractional integral inequalities associated with the strongly $\eta$-quasiconvex functions with modulus $\mu\geq0$, J. Inequal. Appl. 2018:139 (2018).
  • E. R. Nwaeze and D. F. M. Torres, Novel results on the Hermite–Hadamard kind inequality for $\eta$-convex functions by means of the $(k, r)$-fractional integral operators, Advances in Mathematical Inequalities and Applications (AMIA); Trends in Mathematics; Dragomir, S.S., Agarwal, P., Jleli, M., Samet, B., Eds.; Birkhäuser: Singapore, 2018, 311–321.
  • I. Podlubny, Fractional differential equations: Mathematics in Science and Engineering, Academic Press, San Diego, CA. 1999.
  • A. P. Prudnikov, Y. A. Brychkov and O. I. Marichev, Integral and series. In: Elementary Functions, vol. 1. Nauka, Moscow (1981).
  • S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Amsterdam, 1993.
  • M. Z. Sarikaya, E. Set, H. Yaldiz and N. Basak, Hermite–Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model 57(9–10) (2013), 2403–2407.