Hermite-Hadamard type inequalities for F-convex functions involving generalized fractional integrals

Huseyin Budak, Muhammad Aamir Ali, Artion Kashuri

Abstract


In this paper, we firstly summarize some properties of the family F and F-convex functions which are defined by B. Samet. Utilizing generalized fractional integrals new Hermite-Hadamard type inequalities for F-convex functions have been provided. Some results given earlier works are also as special cases of our results.


Keywords


Hermite-Hadamard inequality; F-convex; general fractional integral.

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References


bibitem{Ali2} M.A. Ali, H. Budak, M. Abbas, M.Z. Sarikaya, and A. Kashuri,

textit{Hermite-Hadamard type inequalities for $h$-convex functions via

generalized fractional integrals}, Submitted, (2019).

bibitem{budak} H. Budak, M.Z. Sarikaya, and M.K. Yildiz, textit{%

Hermite--Hadamard type inequalities for }$mathit{F}$textit{-convex

function involving fractional integrals} Filomat 32(16) (2018), 5509--5518.

bibitem{budak2} H. Budak and M.Z. Sarikaya, textit{On Ostrowski type

inequalities for }$mathit{F}$textit{-convex function}, AIP Conference

Proceedings, 1833, 020057 (2017), doi: 10.1063/1.4981705.

bibitem{budak3} H. Budak, T. Tunc{c}, and M.Z. Sarikaya, textit{On

Hermite-Hadamard type inequalities for }$mathit{F}$textit{-convex functions%

}, Miskolc Math. Notes, 20(1) (2019), 169--191.

bibitem{definetti} B. Defnetti, textit{Sulla strati cazioni convesse},

Ann. Math. Pura. Appl. 30 (1949), 173--183.

bibitem{dragomir} S.S. Dragomir and C.E.M. Pearce, textit{Selected topics

on Hermite--Hadamard inequalities and applications}, RGMIA Monographs,

Victoria University, (2000).

bibitem{dragomir2} S.S. Dragomir and R.P. Agarwal, textit{Two inequalities

for differentiable mappings and applications to special means of real

numbers and to trapezoidal formula}, Appl. Math. Lett. 11(5) (1998), 91--95.

bibitem{Gorenflo} R. Gorenflo and F. Mainardi, textit{Fractional calculus:

integral and differential equations of fractional order}, Springer Verlag,

Wien (1997), 223--276.

bibitem{hudzik} H. Hudzik and L. Maligranda, textit{Some remarks on }$s$%

textit{-convex functions}, Aequationes Math. 48 (1994), 100--111.

bibitem{hyers} D.H. Hyers and S.M. Ulam, textit{Approximately convex

functions}, Proc. Amer. Math. Soc. 3 (1952), 821--828.

bibitem{kilbas} A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, textit{%

Theory and Applications of Fractional Differential Equations}, North-Holland

Mathematics Studies, 204 (2006).

bibitem{kirmaci} U.S. Kirmaci, textit{Inequalities for differentiable

mappings and applications to special means of real numbers and to midpoint

formula}, Appl. Math. Comput. 147 (2004), 91--95.

bibitem{man} O.L. Mangasarian, textit{Pseudo-convex functions}, SIAM

Journal on Control. 3 (1965), 281--290.

bibitem{Miller} S. Miller and B. Ross, textit{An introduction to the

Fractional Calculus and Fractional Differential Equations}, John Wiley &

Sons, USA, (1993), pp. 2.

bibitem{mohammed} P.O. Mohammed and M.Z. Sarikaya,textit{Hermite--Hadamard

type inequalities for} $F$textit{-convex function involving fractional

integrals}, J. Inequal. Appl., 2018(359) (2018).

bibitem{pecaric} J.E. Pev{c}ari'{c}, F. Proschan, and Y.L. Tong, textit{%

Convex functions, partial orderings and statistical applications}, Academic

Press, Boston, (1992).

bibitem{pearce} C.E.M. Pearce and J. Pecaric, textit{Inequalities for

differentiable mappings with application to special means and quadrature

formula}, Appl. Math. Lett. 13 (2000), 51--55.

bibitem{Podlubni} I. Podlubni, textit{Fractional Differential Equations},

Academic Press, San Diego, (1999).

bibitem{polyak} B. T. Polyak, textit{Existence theorems and convergence of

minimizing sequences in extremum problems with restrictions}, Soviet Math.

Dokl. 7 (1966), 72--75.

bibitem{samet} B. Samet, textit{On an implicit convexity concept and some

integral inequalities}, J. Inequal. Appl., 2016(308) (2016).

bibitem{sarikaya} M.Z. Sarikaya and F. Ertuu{g}ral, textit{On the

generalized Hermite-Hadamard inequalities}, Annals of the University of

Craiova--Mathematics and Computer Science Series, In Press.

bibitem{sarikaya2} M.Z. Sarikaya, A. Saglam, and H. Yildirim, textit{New

inequalities of Hermite-Hadamard type for functions whose second derivatives

absolute values are convex and quasi-convex}, Int. J. Open Probl. Comput.

Sci. Math., IJOPCM, 5(3) (2012), 1--14.

bibitem{sarikaya1} M.Z. Sarikaya and N. Aktan, textit{On the

generalization some integral inequalities and their applications,} Math.

Comput. Modelling, 54(9-10) (2011), 2175--2182.

bibitem{sarikaya3} M.Z. Sarikaya, A. Saglam, and H. Yildirim, textit{On

some Hadamard-type inequalities for }$h-$textit{convex functions}, J. Math.

Inequal., 2(3) (2008), 335--341.

bibitem{sarikaya4} M.Z. Sarikaya, E. Set, H. Yaldiz, and N. Basak, textit{%

Hermite--Hadamard's inequalities for fractional integrals and related

fractional inequalities}, Math. Comput. Modelling, 57 (2013), 2403--2407.

bibitem{sarikaya5} M.Z. Sarikaya, T. Tunc{c}, and H. Budak, textit{%

Simpson's type inequality for} $mathit{F}$textit{-convex function}. Facta

Univ., Ser. Math. Inf., 32(5) (2018), 747--753.

bibitem{set} E. Set and I. Mumcu, textit{Hermite--Hadamard type

inequalities for} $F$textit{-convex functions via Katukampola fractional

integral}, Submitted, (2018).

bibitem{varo} S. Varosanec, textit{On} $h$textit{-convexity}, J. Math.

Anal. Appl. 326(1) (2007), 303--311.




DOI: http://dx.doi.org/10.24193/subbmath.2022.1.11

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