Main Article Content
Abstract
In this paper, the authors introduce two new classes of generalized convex
functions of two independent variables, and establish a new integral
identity, from which they derive some new fractional Ostrowski's integral
inequalities for functions whose second derivatives are in these new classes
of functions.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
References
- W. Alshanti, and A. Qayyum, A Note On New Ostrowski Type
- Inequalities Using A Generalized Kernel, Bulletin of Mathematical Analysis
- and Applications. 9 (2017), no.1, 1-18.
- N. S. Barnett and S. S. Dragomir, An Ostrowski type inequality
- for double integrals and applications for cubature formulae. Soochow J.
- Math. 27 (2001), no. 1, 1--10.
- G. Farid, Some new Ostrowski type inequalities via fractional
- integrals. Int. J. Anal. App., 14 (2017), no. 1, 64-68.
- 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.
- M. A. Latif, S. Hussain and S. S. Dragomir, Refinements of
- Hermite-Hadamard-type inequalities for co-ordinated quasi-convex functions.
- International Journal of Mathematical Archive. 3 (2012), no 1, 161-171.
- M. A. Latif and S. Hussain, New inequalities of Ostrowski type
- for co-ordineted convex functions via fractional integrals. J. Fract. Calc.
- Appl. 2 (2012), no. 9, 1-15.
- M. Matloka, On some Hadamard-type inequalities for $(h_{1},h_{2})$%
- -preinvex functions on the co-ordinates. J. Inequal. Appl. 2013, 2013:227,
- pp.
- B. Meftah, Some New Ostrwoski's Inequalities for Functions Whose
- nth Derivatives are $r$-Convex. International Journal of Analysis, 2016, 7
- pages.
- B. Meftah, Ostrowski inequalities for functions whose first
- derivatives are logarithmically preinvex. Chin. J. Math. (N.Y.) 2016, Art.
- ID 5292603, 10 pp.
- B. Meftah, New Ostrowski's inequalities. Rev. Colombiana Mat. 51
- (2017), no. 1, 57--69.
- B. Meftah, Ostrowski inequality for functions whose first
- derivatives are $s$-preinvex in the second sense. Khayyam J. Math. 3 (2017),
- no. 1, 61--80.
- B. Meftah, Fractional Ostrowski type inequalities for functions
- whose first derivatives are $s$-preinvex in the second sense. International
- Journal of Analysis and Applications 15 (2017), no. 2, 146--154.
- bibitem{} B. Meftah, Some new Ostrowski's inequalities for n-times
- differentiable mappings which are quasi-convex. Facta Univ. Ser. Math.
- Inform. 32 (2017), no. 3, 319--327.
- B. Meftah, Fractional Ostrowski type inequalities for functions
- whose first derivatives are $varphi $-preinvex. J. Adv. Math. Stud. 10
- (2017), no. 3, 335-347.
- B. Meftah , Some new Ostrowski inequalities for functions whose $%
- n^{th}$ derivatives are logarithmically convex. Ann. Math. Sil. De Gruyter
- Open.
- A. Ostrowski, Alexander, "{U}ber die Absolutabweichung einer
- differentiierbaren Funktion von ihrem Integralmittelwert. (German) Comment.
- Math. Helv. 10 (1937), no. 1, 226--227.
- M. E. "{O}zdemir, H. Kavurmaci and E. Set, Ostrowski's type
- inequalities for $(alpha ,m)$-convex functions. Kyungpook Math. J. 50
- (2010), no. 3, 371--378.
- B. G. Pachpatte, A new Ostrowski type inequality for double
- integrals. Soochow J. Math. 32 (2006), no. 2, 317--322.
- M. Z. Sarikaya, On the Ostrowski type integral inequality. Acta
- Math. Univ. Comenian. (N.S.) 79 (2010), no. 1, 129--134.
- M. Z. Sarikaya and H. Ogunmez, On the weighted Ostrowski-type
- integral inequality for double integrals. Arab. J. Sci. Eng. 36 (2011), no.
- , 1153--1160.
- M. Z. Sari kaya, On the Hermite-Hadamard-type inequalities for
- co-ordinated convex function via fractional integrals. Integral Transforms
- Spec. Funct. 25 (2014), no. 2, 134--147.
- E. Set, New inequalities of Ostrowski type for mappings whose
- derivatives are $s$-convex in the second sense via fractional integrals.
- Comput. Math. Appl. 63 (2012), no. 7, 1147--1154.
- B. -Y. Xi, J. Sun, and S. -P. Bai, On some Hermite-Hadamard-type
- integral inequalities for co-ordinated $left( alpha ,QCright) $- and $%
- left( alpha ,CJright) $-convex functions. Tbilisi Math. J. 8 (2015), no.
- , 75--86.
- Q. Xue, J. Zhu and W. Liu, A new generalization of Ostrowski-type
- inequality involving functions of two independent variables. Comput. Math.
- Appl. 60 (2010), no. 8, 2219--2224.
References
W. Alshanti, and A. Qayyum, A Note On New Ostrowski Type
Inequalities Using A Generalized Kernel, Bulletin of Mathematical Analysis
and Applications. 9 (2017), no.1, 1-18.
N. S. Barnett and S. S. Dragomir, An Ostrowski type inequality
for double integrals and applications for cubature formulae. Soochow J.
Math. 27 (2001), no. 1, 1--10.
G. Farid, Some new Ostrowski type inequalities via fractional
integrals. Int. J. Anal. App., 14 (2017), no. 1, 64-68.
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.
M. A. Latif, S. Hussain and S. S. Dragomir, Refinements of
Hermite-Hadamard-type inequalities for co-ordinated quasi-convex functions.
International Journal of Mathematical Archive. 3 (2012), no 1, 161-171.
M. A. Latif and S. Hussain, New inequalities of Ostrowski type
for co-ordineted convex functions via fractional integrals. J. Fract. Calc.
Appl. 2 (2012), no. 9, 1-15.
M. Matloka, On some Hadamard-type inequalities for $(h_{1},h_{2})$%
-preinvex functions on the co-ordinates. J. Inequal. Appl. 2013, 2013:227,
pp.
B. Meftah, Some New Ostrwoski's Inequalities for Functions Whose
nth Derivatives are $r$-Convex. International Journal of Analysis, 2016, 7
pages.
B. Meftah, Ostrowski inequalities for functions whose first
derivatives are logarithmically preinvex. Chin. J. Math. (N.Y.) 2016, Art.
ID 5292603, 10 pp.
B. Meftah, New Ostrowski's inequalities. Rev. Colombiana Mat. 51
(2017), no. 1, 57--69.
B. Meftah, Ostrowski inequality for functions whose first
derivatives are $s$-preinvex in the second sense. Khayyam J. Math. 3 (2017),
no. 1, 61--80.
B. Meftah, Fractional Ostrowski type inequalities for functions
whose first derivatives are $s$-preinvex in the second sense. International
Journal of Analysis and Applications 15 (2017), no. 2, 146--154.
bibitem{} B. Meftah, Some new Ostrowski's inequalities for n-times
differentiable mappings which are quasi-convex. Facta Univ. Ser. Math.
Inform. 32 (2017), no. 3, 319--327.
B. Meftah, Fractional Ostrowski type inequalities for functions
whose first derivatives are $varphi $-preinvex. J. Adv. Math. Stud. 10
(2017), no. 3, 335-347.
B. Meftah , Some new Ostrowski inequalities for functions whose $%
n^{th}$ derivatives are logarithmically convex. Ann. Math. Sil. De Gruyter
Open.
A. Ostrowski, Alexander, "{U}ber die Absolutabweichung einer
differentiierbaren Funktion von ihrem Integralmittelwert. (German) Comment.
Math. Helv. 10 (1937), no. 1, 226--227.
M. E. "{O}zdemir, H. Kavurmaci and E. Set, Ostrowski's type
inequalities for $(alpha ,m)$-convex functions. Kyungpook Math. J. 50
(2010), no. 3, 371--378.
B. G. Pachpatte, A new Ostrowski type inequality for double
integrals. Soochow J. Math. 32 (2006), no. 2, 317--322.
M. Z. Sarikaya, On the Ostrowski type integral inequality. Acta
Math. Univ. Comenian. (N.S.) 79 (2010), no. 1, 129--134.
M. Z. Sarikaya and H. Ogunmez, On the weighted Ostrowski-type
integral inequality for double integrals. Arab. J. Sci. Eng. 36 (2011), no.
, 1153--1160.
M. Z. Sari kaya, On the Hermite-Hadamard-type inequalities for
co-ordinated convex function via fractional integrals. Integral Transforms
Spec. Funct. 25 (2014), no. 2, 134--147.
E. Set, New inequalities of Ostrowski type for mappings whose
derivatives are $s$-convex in the second sense via fractional integrals.
Comput. Math. Appl. 63 (2012), no. 7, 1147--1154.
B. -Y. Xi, J. Sun, and S. -P. Bai, On some Hermite-Hadamard-type
integral inequalities for co-ordinated $left( alpha ,QCright) $- and $%
left( alpha ,CJright) $-convex functions. Tbilisi Math. J. 8 (2015), no.
, 75--86.
Q. Xue, J. Zhu and W. Liu, A new generalization of Ostrowski-type
inequality involving functions of two independent variables. Comput. Math.
Appl. 60 (2010), no. 8, 2219--2224.