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

In this paper, we establish a new fractional integral identity, and then we

Keywords:

integral inequality, s-preinvex function Hölder inequality, power mean inequality,

PDF

___

\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.