Research Article
BibTex RIS Cite

Fractional Hermite-Hadamard Type Inequalities for Functions Whose Derivatives are s-Preinvex

Year 2019, , 128 - 138, 15.10.2019
https://doi.org/10.36753/mathenot.618335

Abstract

In this paper, we establish a new fractional integral identity, and then we
derive some new fractional Hermite-Hadamard type inequalities for functions
whose derivatives are s-preinvex.

References

  • \bibitem{1} W. W. Breckner, Stetigkeitsaussagen f\"{u}r eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen R\"{a}umen. (German) \emph{Publ. Inst. Math}. (Beograd) (N.S.) 23(37) (1978), 13--20.
  • \bibitem{2} S. S. Dragomir, J. E. Pe\v{c}ari\'{c}, and L. E. Persson, Some inequalities of Hadamard type. \emph{Soochow J. Math}. 21 (1995), no. 3, 335--341.
  • \bibitem{3} S. S. Dragomir, inequalities of Hermite-Hadamard type for $h$% -convex functions on linear spaces. \emph{Proyecciones} 34 (2015), no. 4, 323--341.
  • \bibitem{4} M. A. Hanson, On su ciency of the Kuhn-Tucker conditions. \emph{J. Math. Anal. Appl}. 80 (1981), no. 2, 545-550.
  • \bibitem{5} A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
  • \bibitem{6} U. S. Kirmaci and M. E. \"{O}zdemir, Some inequalities for mappings whose derivatives are bounded and applications to special means of real numbers. \emph{Appl. Math. Lett}. 17 (2004), no. 6, 641--645.
  • \bibitem{7} J.-Y. Li, On Hadamard-type inequalities for $s$-preinvex functions. \emph{Journal of Chongqing Normal University (Natural Science)} 27(2010), no. 4, p. 003.
  • \bibitem{8} B. Meftah and K. Boukerrioua and T. Chiheb, Hadamard type inequalities for $(s,r)$-preinvex functions in the first sense. \emph{Electronic Journal of Mathematical Analysis and Applications} Vol. 5 (2017) no. 2, pp. 170-190.
  • \bibitem{9} B. Meftah and K. Boukerrioua and T. Chiheb , On some new Hadamard type inequalities for $(s,r)$-preinvex functions in the second sense. \emph{Konuralp Journal of Mathematics}. 5 (2017), no. 1, 24-42.
  • \bibitem{10} B. Meftah and M. Merad, Hermite-Hadamard type inequalities for functions whose $n^{th}$ order of derivatives are $s$-convex in the second sense. \emph{Revista De Matem\'{a}ticas De la Universidad del Atl\'{a}ntico P\'{a}% ginas}. 4 (2017), no. 2, 87--99.
  • \bibitem{11} D. S. Mitrinovi\'{c}, J. E. Pe\v{c}ari\'{c} and A. M. Fink, Classical and new inequalities in analysis. Mathematics and its Applications (East European Series), 61. Kluwer Academic Publishers Group, Dordrecht, 1993.
  • \bibitem{12} M. A. Noor, Variational-like inequalities. \emph{Optimization} 30 (1994), no. 4, 323-330.
  • \bibitem{13} M. A. Noor, Invex equilibrium problems. \emph{J. Math. Anal. Appl}. 302 (2005), no. 2, 463-475.
  • \bibitem{14} M. A. Noor, K. I. Noor, M. U. Awan and S. Khan, Hermite-Hadamard inequalities for $s$-Godunova-Levin preinvex functions, \emph{J. Adv. Math. Stud.}, 7(2014), no. 2, 12-19.
  • \bibitem{15} M. A. Noor, K. I. Noor, M. U. Awan and J. Li, On Hermite-Hadamard inequalities for $h$-preinvex functions, \emph{Filomat}, 28 (2014), no. 7, 1463-1474.
  • \bibitem{16} M. A. Noor, Hermite-Hadamard integral inequalities for $\log $% -preinvex functions. \emph{J. Math. Anal. Approx. Theory} 2 (2007), no. 2, 126--131.?
  • \bibitem{17} M. A. Noor, Hadamard integral inequalities for product of two preinvex functions. \emph{Nonlinear Anal. Forum} 14 (2009), 167--173.?
  • \bibitem{18} M. A. Noor, On Hadamard integral inequalities involving two $\log $-preinvex functions. \emph{JIPAM. J. Inequal. Pure Appl. Math}. 8 (2007), no. 3, Article 75, 6 pp.?
  • \bibitem{19} M. Noor, K. Noor, S. Rashid, Some New Classes of Preinvex Functions and Inequalities. \emph{Mathematics}, 7 (2019) (1), 29.
  • \bibitem{20} J. Park, Hermite-Hadamard-like type inequalities for $s$-convex function and $s$-Godunova-Levin functions of two kinds, \emph{Int. Math. Forum}, 9 (2015), 3431-3447.
  • \bibitem{21} J. E. Pe\v{c}ari\'{c}, F. Proschan and Y. L. Tong, Convex functions, partial orderings, and statistical applications. Mathematics in Science and Engineering, 187. Academic Press, Inc., Boston, MA, 1992.
  • \bibitem{22} R. Pini, Invexity and generalized convexity. \emph{Optimization} 22 (1991), no. 4, 513-525.
  • \bibitem{23} M. Z. Sarikaya, E. Set, H. Yaldiz and N. Ba\c{s}ak, Hermite--Hadamard's inequalities for fractional integrals and related fractional inequalities. \emph{Mathematical and Computer Modelling}. 57 (2013), no. 9, 2403-2407.
  • \bibitem{24} E. Set, M. Z. Sarikaya and F. Karako\c{c}, Hermite-Hadamard type inequalities for $h$-convex functions via fractional integrals. \emph{Konuralp J. Math}. 4 (2016), no. 1, 254--260.
  • \bibitem{25} E. Set, M. E. \"{O}zdemir, M. Z. Sarikaya and F. Karako\c{c}, Hermite-Hadamard type inequalities for $(\alpha ,m)$-convex functions via fractional integrals, \emph{Moroccan J. Pure and Appl. Anal}. 3 (2017), no. 1, 15-21.
  • \bibitem{26} T. Weir and B. Mond, Pre-invex functions in multiple objective optimization. \emph{J. Math. Anal. Appl}. 136 (1988), no. 1, 29--38.
  • \bibitem{27} C. Zhu, M. Fe\v{c}kan and J. Wang, Fractional integral inequalities for differentiable convex mappings and applications to special means and a midpoint formula. \emph{J. Appl. Math. Stat. Inf}. 8 (2012), no. 2, 21-28.
Year 2019, , 128 - 138, 15.10.2019
https://doi.org/10.36753/mathenot.618335

Abstract

References

  • \bibitem{1} W. W. Breckner, Stetigkeitsaussagen f\"{u}r eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen R\"{a}umen. (German) \emph{Publ. Inst. Math}. (Beograd) (N.S.) 23(37) (1978), 13--20.
  • \bibitem{2} S. S. Dragomir, J. E. Pe\v{c}ari\'{c}, and L. E. Persson, Some inequalities of Hadamard type. \emph{Soochow J. Math}. 21 (1995), no. 3, 335--341.
  • \bibitem{3} S. S. Dragomir, inequalities of Hermite-Hadamard type for $h$% -convex functions on linear spaces. \emph{Proyecciones} 34 (2015), no. 4, 323--341.
  • \bibitem{4} M. A. Hanson, On su ciency of the Kuhn-Tucker conditions. \emph{J. Math. Anal. Appl}. 80 (1981), no. 2, 545-550.
  • \bibitem{5} A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
  • \bibitem{6} U. S. Kirmaci and M. E. \"{O}zdemir, Some inequalities for mappings whose derivatives are bounded and applications to special means of real numbers. \emph{Appl. Math. Lett}. 17 (2004), no. 6, 641--645.
  • \bibitem{7} J.-Y. Li, On Hadamard-type inequalities for $s$-preinvex functions. \emph{Journal of Chongqing Normal University (Natural Science)} 27(2010), no. 4, p. 003.
  • \bibitem{8} B. Meftah and K. Boukerrioua and T. Chiheb, Hadamard type inequalities for $(s,r)$-preinvex functions in the first sense. \emph{Electronic Journal of Mathematical Analysis and Applications} Vol. 5 (2017) no. 2, pp. 170-190.
  • \bibitem{9} B. Meftah and K. Boukerrioua and T. Chiheb , On some new Hadamard type inequalities for $(s,r)$-preinvex functions in the second sense. \emph{Konuralp Journal of Mathematics}. 5 (2017), no. 1, 24-42.
  • \bibitem{10} B. Meftah and M. Merad, Hermite-Hadamard type inequalities for functions whose $n^{th}$ order of derivatives are $s$-convex in the second sense. \emph{Revista De Matem\'{a}ticas De la Universidad del Atl\'{a}ntico P\'{a}% ginas}. 4 (2017), no. 2, 87--99.
  • \bibitem{11} D. S. Mitrinovi\'{c}, J. E. Pe\v{c}ari\'{c} and A. M. Fink, Classical and new inequalities in analysis. Mathematics and its Applications (East European Series), 61. Kluwer Academic Publishers Group, Dordrecht, 1993.
  • \bibitem{12} M. A. Noor, Variational-like inequalities. \emph{Optimization} 30 (1994), no. 4, 323-330.
  • \bibitem{13} M. A. Noor, Invex equilibrium problems. \emph{J. Math. Anal. Appl}. 302 (2005), no. 2, 463-475.
  • \bibitem{14} M. A. Noor, K. I. Noor, M. U. Awan and S. Khan, Hermite-Hadamard inequalities for $s$-Godunova-Levin preinvex functions, \emph{J. Adv. Math. Stud.}, 7(2014), no. 2, 12-19.
  • \bibitem{15} M. A. Noor, K. I. Noor, M. U. Awan and J. Li, On Hermite-Hadamard inequalities for $h$-preinvex functions, \emph{Filomat}, 28 (2014), no. 7, 1463-1474.
  • \bibitem{16} M. A. Noor, Hermite-Hadamard integral inequalities for $\log $% -preinvex functions. \emph{J. Math. Anal. Approx. Theory} 2 (2007), no. 2, 126--131.?
  • \bibitem{17} M. A. Noor, Hadamard integral inequalities for product of two preinvex functions. \emph{Nonlinear Anal. Forum} 14 (2009), 167--173.?
  • \bibitem{18} M. A. Noor, On Hadamard integral inequalities involving two $\log $-preinvex functions. \emph{JIPAM. J. Inequal. Pure Appl. Math}. 8 (2007), no. 3, Article 75, 6 pp.?
  • \bibitem{19} M. Noor, K. Noor, S. Rashid, Some New Classes of Preinvex Functions and Inequalities. \emph{Mathematics}, 7 (2019) (1), 29.
  • \bibitem{20} J. Park, Hermite-Hadamard-like type inequalities for $s$-convex function and $s$-Godunova-Levin functions of two kinds, \emph{Int. Math. Forum}, 9 (2015), 3431-3447.
  • \bibitem{21} J. E. Pe\v{c}ari\'{c}, F. Proschan and Y. L. Tong, Convex functions, partial orderings, and statistical applications. Mathematics in Science and Engineering, 187. Academic Press, Inc., Boston, MA, 1992.
  • \bibitem{22} R. Pini, Invexity and generalized convexity. \emph{Optimization} 22 (1991), no. 4, 513-525.
  • \bibitem{23} M. Z. Sarikaya, E. Set, H. Yaldiz and N. Ba\c{s}ak, Hermite--Hadamard's inequalities for fractional integrals and related fractional inequalities. \emph{Mathematical and Computer Modelling}. 57 (2013), no. 9, 2403-2407.
  • \bibitem{24} E. Set, M. Z. Sarikaya and F. Karako\c{c}, Hermite-Hadamard type inequalities for $h$-convex functions via fractional integrals. \emph{Konuralp J. Math}. 4 (2016), no. 1, 254--260.
  • \bibitem{25} E. Set, M. E. \"{O}zdemir, M. Z. Sarikaya and F. Karako\c{c}, Hermite-Hadamard type inequalities for $(\alpha ,m)$-convex functions via fractional integrals, \emph{Moroccan J. Pure and Appl. Anal}. 3 (2017), no. 1, 15-21.
  • \bibitem{26} T. Weir and B. Mond, Pre-invex functions in multiple objective optimization. \emph{J. Math. Anal. Appl}. 136 (1988), no. 1, 29--38.
  • \bibitem{27} C. Zhu, M. Fe\v{c}kan and J. Wang, Fractional integral inequalities for differentiable convex mappings and applications to special means and a midpoint formula. \emph{J. Appl. Math. Stat. Inf}. 8 (2012), no. 2, 21-28.
There are 27 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Badreddine Meftah

Abdourazek Souahi This is me

Publication Date October 15, 2019
Submission Date January 16, 2019
Acceptance Date October 3, 2019
Published in Issue Year 2019

Cite

APA Meftah, B., & Souahi, A. (2019). Fractional Hermite-Hadamard Type Inequalities for Functions Whose Derivatives are s-Preinvex. Mathematical Sciences and Applications E-Notes, 7(2), 128-138. https://doi.org/10.36753/mathenot.618335
AMA Meftah B, Souahi A. Fractional Hermite-Hadamard Type Inequalities for Functions Whose Derivatives are s-Preinvex. Math. Sci. Appl. E-Notes. October 2019;7(2):128-138. doi:10.36753/mathenot.618335
Chicago Meftah, Badreddine, and Abdourazek Souahi. “Fractional Hermite-Hadamard Type Inequalities for Functions Whose Derivatives Are S-Preinvex”. Mathematical Sciences and Applications E-Notes 7, no. 2 (October 2019): 128-38. https://doi.org/10.36753/mathenot.618335.
EndNote Meftah B, Souahi A (October 1, 2019) Fractional Hermite-Hadamard Type Inequalities for Functions Whose Derivatives are s-Preinvex. Mathematical Sciences and Applications E-Notes 7 2 128–138.
IEEE B. Meftah and A. Souahi, “Fractional Hermite-Hadamard Type Inequalities for Functions Whose Derivatives are s-Preinvex”, Math. Sci. Appl. E-Notes, vol. 7, no. 2, pp. 128–138, 2019, doi: 10.36753/mathenot.618335.
ISNAD Meftah, Badreddine - Souahi, Abdourazek. “Fractional Hermite-Hadamard Type Inequalities for Functions Whose Derivatives Are S-Preinvex”. Mathematical Sciences and Applications E-Notes 7/2 (October 2019), 128-138. https://doi.org/10.36753/mathenot.618335.
JAMA Meftah B, Souahi A. Fractional Hermite-Hadamard Type Inequalities for Functions Whose Derivatives are s-Preinvex. Math. Sci. Appl. E-Notes. 2019;7:128–138.
MLA Meftah, Badreddine and Abdourazek Souahi. “Fractional Hermite-Hadamard Type Inequalities for Functions Whose Derivatives Are S-Preinvex”. Mathematical Sciences and Applications E-Notes, vol. 7, no. 2, 2019, pp. 128-3, doi:10.36753/mathenot.618335.
Vancouver Meftah B, Souahi A. Fractional Hermite-Hadamard Type Inequalities for Functions Whose Derivatives are s-Preinvex. Math. Sci. Appl. E-Notes. 2019;7(2):128-3.

20477

The published articles in MSAEN are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.