HERMITE--HADAMARD TYPE INEQUALITIES VIA DIFFERENTIABLE hφ−h_{\varphi }-hφ− PREINVEX FUNCTIONS FOR FRACTIONAL INTEGRALS

Abdullah Akkurt; Hüseyin Yıldırım

Research Article

Year 2021, Volume: 4 Issue: 1, 87 - 101, 31.01.2021

Abdullah Akkurt , Hüseyin Yıldırım

Abstract

References

Barani, A.; Ghazanfari, A.G.; Dragomir, S.S.: Hermite-Hadamardinequality for functions whose derivatives absolute values are preinvex. J.Inequal. Appl. 2012, 247 (2012)
Dragomir, S.S.; Pearce, C.E.M.: Selected topics onHermite--Hadamard inequalities and applications. Victoria University (2000)
Dragomir, S.S.; Pecaric, J.; Persson, L.E.: Some inequalities ofHadamard type. Soochow J. Math. 21, 335--341, (1995)
\.{I}\c{s}can, \.{I}.: Hermite--Hadamard's inequalities forpreinvex function via fractional integrals and related functionalinequalities, Am. J. Math. Anal. 1 (3) (2013) 33--38.
Latif, M.A.: Some inequalities for differentiable prequasiinvexfunctions with applications. Konuralp J. Math. 1(2), 17--29 (2013)
Latif, M.A.; Dragomir, S.S.: Some Hermite-Hadamard typeinequalities for functions whose partial derivatives in absloute value arepreinvex on the co-oordinates. Facta Universitatis (NIS) Ser. Math. Inform.28(3), 257--270, (2013)
Latif, M.A.; Dragomir, S. S.; Momoniat, E.: Some weightedintegral inequalities for differentiable preinvex and prequasiinvexfunctions. RGMIA (2014)
Noor, M.A.: On Hadamard integral inequalities involving twolog-preinvex functions. J. Inequal. Pure Appl. Math. 8, 1--6 (2007)
Noor, M.A.: On Hermite-Hadamard integral inequalities forproduct of two nonconvex functions. J. Adv. Math. Studies 2(1), 53--62 (2009)
Noor, M.A.: Some new classes of nonconvex functions. Nonl.Funct. Anal. Appl. 11(1), 165--171 (2006)
Noor, M.A.; Awan, M.U.; Noor, K.I.: On some inequalities forrelative semi-convex functions. J. Inequal. Appl., 2013, 332 (2013)
Noor, M.A.; Noor, K.I.: Generalized preinvex functions andtheir properties. Journal of Appl. Math. Stochastic Anal., 2006(12736),1--13, doi:10.1155/JAMSA/2006/12736
Noor, M.A.; Noor, K.I.; Ashraf, M.A.; Awan, M.U.; Bashir, B.:Hermite--Hadamard inequalities for $h_{\varphi }-$convex functions. Nonl.Anal. Formum. 18, 65--76 (2013)
Noor, M.A.; Noor, K.I.; Awan, M.U.: Hermite--Hadamardinequalities for relative semi-convex functions and applications. Filomat,28(2), 221--230, (2014)
Noor, M.A.; Noor, K.I.; Awan, M.U.: Generalized convexity andintegral inequalities. Appl. Math. Inf. Sci. 9(1), 233--243, (2015)
Noor, M.A.; Noor, K.I.; Awan, M.U.; Khan, S.: FractionalHermite-Hadamard inequalities for some new classes of Godunova--Levinfunctions. Appl. Math. Inf. Sci. 8(6), 2865--2872, (2014)
Noor, M.A.; Noor, K.I.; Awan, M.U.; Li, J.: OnHermite--Hadamard type Inequalities for h-preinvex functions, Filomat 28:7(2014), 1463--1474, DOI 10.2298/FIL1407463N
Noor, M.A.; Noor, K.I.; Al-Said, E.: Iterative methods forsolving nonconvex equilibrium variational inequalities. Appl. Math. Inf.Sci. 6(1), 65--69, (2012).
Noor, M.A.; Postolache, M.; Noor, K.I.; Awan, M.U.:Geometrically nonconvex functions and integral inequalities. Appl. Math.Inf. Sci. 9, (2015).
Noor, M.A.; Noor, K.I.; Awan, M.A.; Khan, S.: Hermite--Hadamardtype inequalities for differantiable $h_{\varphi }-$preinvex functions.Arab. J. Math.4:63-76 (2015).
Sarikaya, M.Z.; Set, E.: \"{O}zdemir M.E.: On some newinequalities of Hadamard type involving h-convex functions. Acta Math. Univ.Comenianae. 2, 265--272 (2010).
Sarikaya, M.Z.; Alp, N.; Bozkurt, H.: On Hermite-Hadamard TypeIntegral Inequalities for preinvex and log-preinvex functions, ContemporaryAnalysis and Applied Mathematics, Vol.1, No.2, 237-252, 2013.
Varosanec, S.: On h-convexity. J. Math. Anal. Appl. 326,303--311 (2007)
Weir, T.; Mond, B.: Preinvex functions in multiple objectiveoptimization. J. Math. Anal. Appl. 136, 29--38 (1998)
Samko, S.G.; Kilbas, A.A.; Marichev, O.I.: Fractional Integralsand Derivatives, Theory and Applications, Gordon and Breach, Yverdon,Switzerland, 1993

HERMITE--HADAMARD TYPE INEQUALITIES VIA DIFFERENTIABLE hφ−h_{\varphi }-hφ− PREINVEX FUNCTIONS FOR FRACTIONAL INTEGRALS

Year 2021, Volume: 4 Issue: 1, 87 - 101, 31.01.2021

Abdullah Akkurt , Hüseyin Yıldırım

Abstract

In this paper, we consider a new class of convex functions which is called h_varphi-preinvex functions. We prove several Hermite--Hadamard type

inequalities for differentiable h_varphi-preinvex functions via Fractional Integrals. Some special cases are also discussed. Our results

extend and improve the corresponding ones in the literature.

Keywords

integral inequalities , fractional integrals , Hermite-Hadamard Inequality

References

Barani, A.; Ghazanfari, A.G.; Dragomir, S.S.: Hermite-Hadamardinequality for functions whose derivatives absolute values are preinvex. J.Inequal. Appl. 2012, 247 (2012)
Dragomir, S.S.; Pearce, C.E.M.: Selected topics onHermite--Hadamard inequalities and applications. Victoria University (2000)
Dragomir, S.S.; Pecaric, J.; Persson, L.E.: Some inequalities ofHadamard type. Soochow J. Math. 21, 335--341, (1995)
\.{I}\c{s}can, \.{I}.: Hermite--Hadamard's inequalities forpreinvex function via fractional integrals and related functionalinequalities, Am. J. Math. Anal. 1 (3) (2013) 33--38.
Latif, M.A.: Some inequalities for differentiable prequasiinvexfunctions with applications. Konuralp J. Math. 1(2), 17--29 (2013)
Latif, M.A.; Dragomir, S.S.: Some Hermite-Hadamard typeinequalities for functions whose partial derivatives in absloute value arepreinvex on the co-oordinates. Facta Universitatis (NIS) Ser. Math. Inform.28(3), 257--270, (2013)
Latif, M.A.; Dragomir, S. S.; Momoniat, E.: Some weightedintegral inequalities for differentiable preinvex and prequasiinvexfunctions. RGMIA (2014)
Noor, M.A.: On Hadamard integral inequalities involving twolog-preinvex functions. J. Inequal. Pure Appl. Math. 8, 1--6 (2007)
Noor, M.A.: On Hermite-Hadamard integral inequalities forproduct of two nonconvex functions. J. Adv. Math. Studies 2(1), 53--62 (2009)
Noor, M.A.: Some new classes of nonconvex functions. Nonl.Funct. Anal. Appl. 11(1), 165--171 (2006)
Noor, M.A.; Awan, M.U.; Noor, K.I.: On some inequalities forrelative semi-convex functions. J. Inequal. Appl., 2013, 332 (2013)
Noor, M.A.; Noor, K.I.: Generalized preinvex functions andtheir properties. Journal of Appl. Math. Stochastic Anal., 2006(12736),1--13, doi:10.1155/JAMSA/2006/12736
Noor, M.A.; Noor, K.I.; Ashraf, M.A.; Awan, M.U.; Bashir, B.:Hermite--Hadamard inequalities for $h_{\varphi }-$convex functions. Nonl.Anal. Formum. 18, 65--76 (2013)
Noor, M.A.; Noor, K.I.; Awan, M.U.: Hermite--Hadamardinequalities for relative semi-convex functions and applications. Filomat,28(2), 221--230, (2014)
Noor, M.A.; Noor, K.I.; Awan, M.U.: Generalized convexity andintegral inequalities. Appl. Math. Inf. Sci. 9(1), 233--243, (2015)
Noor, M.A.; Noor, K.I.; Awan, M.U.; Khan, S.: FractionalHermite-Hadamard inequalities for some new classes of Godunova--Levinfunctions. Appl. Math. Inf. Sci. 8(6), 2865--2872, (2014)
Noor, M.A.; Noor, K.I.; Awan, M.U.; Li, J.: OnHermite--Hadamard type Inequalities for h-preinvex functions, Filomat 28:7(2014), 1463--1474, DOI 10.2298/FIL1407463N
Noor, M.A.; Noor, K.I.; Al-Said, E.: Iterative methods forsolving nonconvex equilibrium variational inequalities. Appl. Math. Inf.Sci. 6(1), 65--69, (2012).
Noor, M.A.; Postolache, M.; Noor, K.I.; Awan, M.U.:Geometrically nonconvex functions and integral inequalities. Appl. Math.Inf. Sci. 9, (2015).
Noor, M.A.; Noor, K.I.; Awan, M.A.; Khan, S.: Hermite--Hadamardtype inequalities for differantiable $h_{\varphi }-$preinvex functions.Arab. J. Math.4:63-76 (2015).
Sarikaya, M.Z.; Set, E.: \"{O}zdemir M.E.: On some newinequalities of Hadamard type involving h-convex functions. Acta Math. Univ.Comenianae. 2, 265--272 (2010).
Sarikaya, M.Z.; Alp, N.; Bozkurt, H.: On Hermite-Hadamard TypeIntegral Inequalities for preinvex and log-preinvex functions, ContemporaryAnalysis and Applied Mathematics, Vol.1, No.2, 237-252, 2013.
Varosanec, S.: On h-convexity. J. Math. Anal. Appl. 326,303--311 (2007)
Weir, T.; Mond, B.: Preinvex functions in multiple objectiveoptimization. J. Math. Anal. Appl. 136, 29--38 (1998)
Samko, S.G.; Kilbas, A.A.; Marichev, O.I.: Fractional Integralsand Derivatives, Theory and Applications, Gordon and Breach, Yverdon,Switzerland, 1993

There are 25 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Research Article
Authors	Abdullah Akkurt 0000-0001-5644-1276 Hüseyin Yıldırım
Submission Date	September 6, 2019
Acceptance Date	February 27, 2020
Publication Date	January 31, 2021
Published in Issue	Year 2021 Volume: 4 Issue: 1

Cite

APA	Akkurt, A., & Yıldırım, H. (2021). HERMITE--HADAMARD TYPE INEQUALITIES VIA DIFFERENTIABLE hφ−h_{\varphi }-hφ− PREINVEX FUNCTIONS FOR FRACTIONAL INTEGRALS. Journal of Universal Mathematics, 4(1), 87-101.
AMA	Akkurt A, Yıldırım H. HERMITE--HADAMARD TYPE INEQUALITIES VIA DIFFERENTIABLE hφ−h_{\varphi }-hφ− PREINVEX FUNCTIONS FOR FRACTIONAL INTEGRALS. JUM. January 2021;4(1):87-101.
Chicago	Akkurt, Abdullah, and Hüseyin Yıldırım. “HERMITE--HADAMARD TYPE INEQUALITIES VIA DIFFERENTIABLE Hφ−h_{\varphi }-Hφ− PREINVEX FUNCTIONS FOR FRACTIONAL INTEGRALS”. Journal of Universal Mathematics 4, no. 1 (January 2021): 87-101.
EndNote	Akkurt A, Yıldırım H (January 1, 2021) HERMITE--HADAMARD TYPE INEQUALITIES VIA DIFFERENTIABLE hφ−h_{\varphi }-hφ− PREINVEX FUNCTIONS FOR FRACTIONAL INTEGRALS. Journal of Universal Mathematics 4 1 87–101.
IEEE	A. Akkurt and H. Yıldırım, “HERMITE--HADAMARD TYPE INEQUALITIES VIA DIFFERENTIABLE hφ−h_{\varphi }-hφ− PREINVEX FUNCTIONS FOR FRACTIONAL INTEGRALS”, JUM, vol. 4, no. 1, pp. 87–101, 2021.
ISNAD	Akkurt, Abdullah - Yıldırım, Hüseyin. “HERMITE--HADAMARD TYPE INEQUALITIES VIA DIFFERENTIABLE Hφ−h_{\varphi }-Hφ− PREINVEX FUNCTIONS FOR FRACTIONAL INTEGRALS”. Journal of Universal Mathematics 4/1 (January2021), 87-101.
JAMA	Akkurt A, Yıldırım H. HERMITE--HADAMARD TYPE INEQUALITIES VIA DIFFERENTIABLE hφ−h_{\varphi }-hφ− PREINVEX FUNCTIONS FOR FRACTIONAL INTEGRALS. JUM. 2021;4:87–101.
MLA	Akkurt, Abdullah and Hüseyin Yıldırım. “HERMITE--HADAMARD TYPE INEQUALITIES VIA DIFFERENTIABLE Hφ−h_{\varphi }-Hφ− PREINVEX FUNCTIONS FOR FRACTIONAL INTEGRALS”. Journal of Universal Mathematics, vol. 4, no. 1, 2021, pp. 87-101.
Vancouver	Akkurt A, Yıldırım H. HERMITE--HADAMARD TYPE INEQUALITIES VIA DIFFERENTIABLE hφ−h_{\varphi }-hφ− PREINVEX FUNCTIONS FOR FRACTIONAL INTEGRALS. JUM. 2021;4(1):87-101.

Article Files

Full Text

33564
https://creativecommons.org/licenses/by-nc-nd/4.0/