Yousaf Khurshid, Mohammad Adil Khan

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Many recent results have been. This inequality has many applications in the area of pure and applied mathematics. In this paper, our main aim is to give results for conformable integral version of Hermite-Hadamard inequality for η-convex functions. First, we prove an identity associated with the Hermite-Hadamard inequality for conformable integrals using η-convex functions. By using this identity and η-convexity of function and some well-known inequalities, we obtain several results for the inequality.


η-convex functions, Hermite-Hadamard inequality, Conformable derivative, Conformable integrals

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{sc T. Abdeljawad}: textit{On conformable fractional calculus}. J. Comput. Appl. Math. textbf{279}(2015), 57--66.


{sc T. Abdeljawad, R. P. Agarwal, J. Alzabut, F. Jarad {rm and} A. "{O}zbekler}: textit{Lyapunov-type inequalties for mixed non-linear forced differential

equations within conformalbe derivatives}. J. Inequal. Appl. textbf{2018}(2018), 1--17.


{sc T. Abdeljawad, J. Alzabut {rm and} F. Jarad}: textit{A generalized Lyapunov-type inequality in the frame of conformable derivatives}. Adv. Difference Equ.

textbf{2017}(2017), 1--10.


{sc D. R. Anderson}: textit{Taylor's formula and integral inequalities for conformable fractional derivatives, Contributions

in Mathematics and Engineering}. Springer, Cham, (2016).


{sc A. O. Akdemir, A. Ekinci {rm and} E. Set}: textit{Conformable fractional integrals and related new integral inequalities}. J. Nonlinear convex Anal.

textbf{18}(4) (2017), 661--674.


{sc Y. M. Chu, M. A. khan, T. Ali {rm and} S. S. Dragomir}: textit{Inequalities for $alpha$-fractional differentiable functions}. J. Inequal. Appl.

textbf{2017}(2017), 1--12.


{sc W. S. Chung}: textit{ Fractional Newton mechanics with conformable fractional derivative}. J. Comput. Appl. Math.

textbf{290}(2015), 150--158.


{sc S. S. Dragomir}: textit{Two mappings in connection to Hadamard's inequalities}. J. Math. Anal. Appl. textbf{167}(1) (1992), 49--56.


{sc S. S. Dragomir {rm 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. textbf{11}(5) (1998), 91--95.


{sc S. S. Dragomir {rm and} S. Fitzpatrick}: textit{The Hadamard inequalities for s-convex functions in the second sense}. Demonstratio Math.

textbf{32}(4) (1999), 687--696.


{sc S. S. Dragomir {rm and} A. McAndrew}: textit{Refinements of the Hermite-Hadamard inequality for convex functions}. JIPAM. J. Inequal. Pure Appl. Math.

textbf{6}(5) (2005), 1--6.


{sc L. Fej'{e}r}: textit{ Uberdie Fourierreihen II}. Math. Natur. Anz Ungar. Akad. Wiss., 24(1906), 369--390.


{sc M. E. Gordji, S. S. Dragomir {rm and} M. R. Delavar}: textit{ An inequality related to $eta$-convex

functions (II)}. Int. J. Nonlinear Anal. Appl. textbf{6}(2), (2016) 26--32.


{sc M. E. Gordji, M. R. Delavar {rm and} M. D. L. Sen}: textit{ On $varphi$-convex functions}. J. Math. Inequal. textbf{10}(1) 2016, 173--183.


{sc J. Hadamard}:textit{ '{E}tude sur les propri'{e}t'{e}s des fonctions enti`{e}res et en particulier `{d}une fonction consid'{e}r'{e}e par Riemann}. J. Math. Pures Appl. textbf{58}(1893), 171--215.


{sc M. S. Hashemi}: textit{Invariant subspaces admitted by fractional differential equations with conformable derivatives}.

Chaos Solitons Fractals textbf{107}(2018), 161--169.


{sc .{I}. .{I}c{s}can}: textit{Hermite-Hadamard's inequalities for Preinvex function

via fractional integrals and related fractional inequalities}. Amer. J. Math. Anal. textbf{1}(3) (2013), 33--38.


{sc A. Iqbal, M. A. khan, M. Suleman {rm and} Y. M. Chu}: textit{The right Riemann-Liouville fractional Hermite-Hadamard type inequalities derived from Green's function}. AIP Advances textbf{10}(2020) 1--9.


{sc A. Iqbal, M. A. khan, S. Ullah {rm and} Y. M. Chu}: textit{Some new Hermite-Hadamard type inequalities associated with conformable fractional integrals and their applications}. J. Funct. Spaces textbf{2020}(2020), 1--18.


{sc R. Khalil, M. Al Horani, A. Yousef {rm and} M. Sababheh}: textit{A new

definition of fractional derivative}. J. Comput. Appl. Math. textbf{264} (2014), 65--70.


{sc M. A. khan, S. Begum, Y. Khurshid {rm and} Y. M. Chu}: textit{Ostrowski type inequalities involving conformable fractional integrals}. J. Inequal. Appl.

textbf{2018}(2018), 1--14.


{sc M. A. khan, Y. M. Chu, A. Kashuri, R. Liko {rm and} G. Ali}:textit{ Conformable fractional integrals version of Hermite-Hadamard inequalities and their generalizations}.J. Funct. Spaces textbf{2018}(2018), 1--9.


{sc M. A. khan, Y. M. Chu, T. U. Khan {rm and} J. Khan}: textit{Some new inequalities of Hermite-Hadamard type for s-convex functions with applications}. Open Math. textbf{15}(2017), 1414--1430.

bibitem{[AYT]} {sc M. A. khan, Y. Khurshid {rm and} T. Ali}: textit{Hermite-Hadamard inequality for

fractional integrals Via $eta$-convex functions}. Acta Math. Univ. Comenian. textbf{86}(1) (2017), 153--164.

bibitem{AYTN}{sc M. A. khan, Y. Khurshid, T. Ali {rm and} N. Rehman}: textit{Inequalities for three times differentiable functions}. Punjab Univ. J. Math.

textbf{48}(2) (2016), 35--48.


{sc M. A. khan, T. Ali {rm and} T. U. Khan}: textit{Hermite-Hadamard Type Inequalities with Applications}. Fasciculi Mathematici textbf{59}(2017), 57-74.


{sc S. Khan, M. A. khan {rm and} Y. M. Chu}: textit{New converses of Jensen inequality via Green functions with

applications}. RACSAM textbf{114}(3) (2020) 1--14.


{sc M. A. khan, N. Mohammad, E. R. Nwaeze {rm and} Y. M. Chu}: textit{Quantum Hermite-Hadamard inequality by means of a green function}. Adv. Difference Equ.

textbf{2020}(2020), 1--20.


{sc M. A. khan, S. Khan {rm and} Y. M. Chu}: textit{A new bound for the Jensen gap with applications in information theory}. IEEE Access textbf{20}(2020), 98001--98008.


{sc M. A. khan, J. Pev{c}ari'{c} {rm and} Y. M. Chu}: textit{Refinements of Jensen's and McShane's inequalities with applications}. AIMS Mathematics, 5(5) (2020),



{sc S. Khan, M. A. khan, S. I. Butt {rm and} Y. M. Chu}:textit{ A new bound for the Jensen gap pertaining twice differentiable functions with applications}. Adv. Difference Equ. textbf{2020}(2020), 1--11.

bibitem {usaf1} {sc Y. Khurshid, M. A. khan, Y. M. Chu {rm and} Z. A. Khan}: textit{ Hermite-Hadamard-Fej'{e}r inequalities

for conformable fractional integrals via preinvex functions}. J. Funct. Spaces textbf{2019} (2019), 1--9.

bibitem {usaf2}{sc Y. Khurshid, M. A. khan {rm and} Y. M. Chu}: textit{Conformable integral inequalities of the Hermite-Hadamard type in terms of

$GG$- and $GA$-convexities}. J. Funct. Spaces textbf{2019} (2019), 1--8.

bibitem {usaf3} {sc Y. Khurshid, M. A. khan {rm and} Y. M. Chu}: textit{ Ostrowski type inequalities involving conformable

integrals via preinvex functions}. AIP Advances textbf{10}(055204) (2020), 1--9.


{sc Y. Khurshid, M. A. khan {rm and} Y. M. Chu}: textit{ Conformable fractional integral inequalities for GG- and

GA-convex function}. AIMS Mathematics textbf{5}(5) (2020), 5012-5030.


{sc Y. Khurshid, M. A. khan {rm and} Y. M. Chu}: textit{ Conformable integral version of Hermite-Hadamard-Fej´er inequalities via

$eta$-convex functions}. AIMS Mathematics textbf{5}(5) (2020), 5106-5120.


{sc Y. Khurshid {rm and} M. A. khan}: textit{ Hermite-Hadamard's inequalities for

$eta$-convex functions via conformable fractional integrals and related inequalities}. Acta Math. Univ. Comenianae textbf{2} (2021), 157-169.


{sc Y. Khurshid {rm and} M. A. khan}: textit{ Hermite-Hadamard type inequalities for conformable integrals

via preinvex functions}. Appl. Math. E-Notes textbf{21} (2021), 437-450.


{sc U. S. Kirmaci}: textit{Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula}. Appl. Math. Comput.

textbf{147}(2004), 137-146.


{sc M. A. Refai {rm and} T. Abdeljawad}: textit{Fundamental results of conformable Sturm-Liouville engenvalue problems}.

Complexity textbf{2017}(2017), 1--7.


{sc M. Z. Sarikaya, H. Yaldiz {rm and} H. Budak}: textit{On weighted Iyengar-type inequalities for conformable fractional integrals}. Math. Sci.

textbf{11}(4) (2017), 327--331.


{sc M. Z. Sarikaya, A. Akkurt, H. Budak, M. E. Yildirim {rm and} H. Yildirim}: textit{ Hermite-Hadamards inequalities for conformable fractional

integrals}. An International Journal of Optimization and Control: Theories & Applications textbf{9}(1) (2019), 49--59.


{sc E. Set, .{I}. Mumcu {rm and} M. E. "{O}zdemir}: textit{On the mor general Hermite-Hadamard type inequalities for convex functions via

conformable fractional integrals}. Topol. Algebra Appl. textbf{5}(1) (2017), 67--73.


{sc Y. Q. Song, M. A. khan, S. Z. Ullah {rm and} Y. M. Chu}: textit{Integral inequalities involving strongly convex functions}. J. Funct. Spaces

textbf{2018}(2018), 1--8.


{sc Z. H. Yang {rm and} Y. M. Chu}:textit{ A monotonicity properties involving the generalized elliptic integral of the first kind}. Math.

Inequal. Appl. textbf{20}(3) (2017), 729--735.


{sc Z. H. Yang, W. M. Qian, Y. M. Chu {rm and} W. Zhang}: textit{On rational bounds for the gamma function}. J. Inequal. Appl.

textbf{2017}(2017), 1--17.


{sc Z. H. Yang, W. M. Qian, Y. M. Chu {rm and} W. Zhang}: textit{On approximating the arithmetic-geometric mean and complete elliptic integral of the

first kind}. J. Math. Anal. Appl. textbf{462}(2) (2018), 1714--1726.


{sc Z. H. Yang, W. M. Qian, Y. M. Chu {rm and} W. Zhang}:textit{ On approximating the error function}. Math. Inequal. Appl.

textbf{21}(2) (2018), 469--479.


{sc Z. H. Yang, W. Zhang {rm and} Y. M. Chu}: textit{Sharp Gautschi inequality for parameter $0

textbf{20}(4) (2017), 1107--1120.


{sc X. M. Zhang, Y. M. Chu {rm and} X. H. Zhang}: textit{The Hermite-Hadamard type inequality of GA-convex functions and its applications}.

J. Inequal. Appl. textbf{2010} (2010), 1--11.

DOI: https://doi.org/10.22190/FUMI220914005K


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