Generalized Ostrowski type inequalities for local fractional integrals

Sarikaya, Mehmet; Budak, Hüseyin

doi:10.1090/proc/13488

Generalized Ostrowski type inequalities for local fractional integrals
HTML articles powered by AMS MathViewer

by Mehmet Zeki Sarikaya and Hüseyin Budak

Proc. Amer. Math. Soc. 145 (2017), 1527-1538

DOI: https://doi.org/10.1090/proc/13488

Published electronically: December 27, 2016

PDF | Request permission

Abstract:

First, we establish the generalized Ostrowski inequality for local fractional integrals on fractal sets $R^{\alpha }$ $\left ( 0<\alpha \leq 1\right )$ of real line numbers. Secondly, we obtain some new inequalities using the generalized convex function on fractal sets $R^{\alpha }$.

References

M. Alomari, M. Darus, S. S. Dragomir, and P. Cerone, Ostrowski type inequalities for functions whose derivatives are $s$-convex in the second sense, Appl. Math. Lett. 23 (2010), no. 9, 1071–1076. MR 2659140, DOI 10.1016/j.aml.2010.04.038
Mohammad W. Alomari, M. Emin Özdemir, and Havva Kavurmac, On companion of Ostrowski inequality for mappings whose first derivatives absolute value are convex with applications, Miskolc Math. Notes 13 (2012), no. 2, 233–248. MR 3002626, DOI 10.18514/mmn.2012.480
M. W. Alomari and M. Darus, Some Ostrowski’s type inequalities for convex functions with application, RGMIA Res. Rep. Coll. 13(1) 2010, Art. 3.
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. MR 1821346
Pietro Cerone and Sever S. Dragomir, Ostrowski type inequalities for functions whose derivatives satisfy certain convexity assumptions, Demonstratio Math. 37 (2004), no. 2, 299–308. MR 2057852, DOI 10.1515/dema-2004-0208
S. S. Dragomir, Ostrowski type inequalities for functions whose derivatives are h-convex in absolute value, RGMIA Research Report Collection, 16(2013), Article 71, 15 pp.
Sever S. Dragomir, Ostrowski type inequalities for functions whose derivatives are $h$-convex in absolute value, Tbilisi Math. J. 7 (2014), no. 1, 1–17. MR 3313040, DOI 10.2478/tmj-2014-0001
S. S. Dragomir and R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett. 11 (1998), no. 5, 91–95. MR 1638774, DOI 10.1016/S0893-9659(98)00086-X
U. S. Kirmaci and M. E. Özdemir, On some inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput. 153 (2004), no. 2, 361–368. MR 2064663, DOI 10.1016/S0096-3003(03)00637-4
Uǧur S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput. 147 (2004), no. 1, 137–146. MR 2007685, DOI 10.1016/S0096-3003(02)00657-4
Huixia Mo, Xin Sui, and Dongyan Yu, Generalized convex functions on fractal sets and two related inequalities, Abstr. Appl. Anal. , posted on (2014), Art. ID 636751, 7. MR 3226218, DOI 10.1155/2014/636751
Alexander Ostrowski, Über die Absolutabweichung einer differentiierbaren Funktion von ihrem Integralmittelwert, Comment. Math. Helv. 10 (1937), no. 1, 226–227 (German). MR 1509574, DOI 10.1007/BF01214290
M. Emin Özdemir, Havva Kavurmacı, and Merve Avcı, Ostrowski type inequalities for convex functions, Tamkang J. Math. 45 (2014), no. 4, 335–340. MR 3302307, DOI 10.5556/j.tkjm.45.2014.1143
Josip E. Pečarić, Frank Proschan, and Y. L. Tong, Convex functions, partial orderings, and statistical applications, Mathematics in Science and Engineering, vol. 187, Academic Press, Inc., Boston, MA, 1992. MR 1162312
M. Z. Sarikaya, On the Ostrowski type integral inequality, Acta Math. Univ. Comenian. (N.S.) 79 (2010), no. 1, 129–134. MR 2684215
Mehmet Zeki Sarikaya, On the Ostrowski type integral inequality for double integrals, Demonstratio Math. 45 (2012), no. 3, 533–540. MR 2987183
Mehmet Zeki Sarikaya and Hasan Ogunmez, On the weighted Ostrowski-type integral inequality for double integrals, Arab. J. Sci. Eng. 36 (2011), no. 6, 1153–1160 (English, with English and Arabic summaries). MR 2845540, DOI 10.1007/s13369-011-0102-4
Mehmet Zeki Sarıkaya, Erhan. Set, M. Emin Ozdemir, and Sever S. Dragomir, New some Hadamard’s type inequalities for co-ordinated convex functions, Tamsui Oxf. J. Inf. Math. Sci. 28 (2012), no. 2, 137–152. MR 3087182
M. Z. Sarikaya and H. Yaldiz, On the Hadamard’s type inequalities for $L$-Lipschitzian mapping, Konuralp Journal of Mathematics, Volume 1, No. 2, pp. 33-40 (2013)
X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, 2012.
Yong-Ju Yang, Dumitru Baleanu, and Xiao-Jun Yang, Analysis of fractal wave equations by local fractional Fourier series method, Adv. Math. Phys. , posted on (2013), Art. ID 632309, 6. MR 3071601, DOI 10.1155/2013/632309