Fractional Integral Inequalities for Differentiable Convex Mappings and Applications to Special Means and a Midpoint Formula

Zhu, Chun; Fečkan, Michal; Wang, Jinrong

doi:10.2478/v10294-012-0011-5

Abstract

In this paper, Riemann-Liouville type fractional integral identity and inequality for differentiable convex mappings are studied. Some applications to special means of real numbers are given. Finally, error estimates for a midpoint formula are also obtained.

References

[1] D. Baleanu, J. A. T. Machado, A.C.-J. Luo, Fractional dynamics and control, Springer, 2012.10.1007/978-1-4614-0457-6
Search in Google Scholar
[2] K. Diethelm, The analysis of fractional difierential equations, Lecture Notes in Mathematics, 2010.10.1007/978-3-642-14574-2
Search in Google Scholar
[3] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional difierential equations, Elsevier Science B.V., 2006.
Search in Google Scholar
[4] V. Lakshmikantham, S. Leela, J. Vasundhara Devi, Theory of fractional dynamic systems, Cambridge Scientific Publishers, 2009.
Search in Google Scholar
[5] K. S. Miller, B. Ross, An introduction to the fractional calculus and difierential equations, John Wiley, 1993.
Search in Google Scholar
[6] M. W. Michalski, Derivatives of noninteger order and their applications, Dissertationes Mathematicae, CCCXXVIII, Inst. Math., Polish Acad. Sci., 1993.
Search in Google Scholar
[7] I. Podlubny, Fractional difierential equations, Academic Press, 1999.
Search in Google Scholar
[8] V. E. Tarasov, Fractional dynamics: Application of fractional calculus to dynamics of particles, fields and media, Springer, HEP, 2011.10.1007/978-3-642-14003-7_11
Search in Google Scholar
[9] M. Z. Sarikaya, E. Set, H. Yaldiz, N. Bafisak, Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., (2012), doi:10.1016/j.mcm.2011.12.048.10.1016/j.mcm.2011.12.048
Search in Google Scholar
[10] S. Abramovich, J. Barific, Josip Peficarific, Fejer and Hermite-Hadamard type inequalities for superquadratic functions, J. Math. Anal. Appl., 344(2008), 1048-1056.10.1016/j.jmaa.2008.03.051
Search in Google Scholar
[11] J. Cal, J. Carcamob, L. Escauriaza, A general multidimensional Hermite-Hadamard type inequality, J. Math. Anal. Appl., 356(2009), 659-663.10.1016/j.jmaa.2009.03.044
Search in Google Scholar
[12] M. Avci, H. Kavurmaci, M. E. Ödemir, New inequalities of HermiteCHadamard type via s-convex functions in the second sense with applications, Appl. Math. Comput., 217(2011), 5171-5176.
Search in Google Scholar
[13] M. E. Ödemir, M. Avci, E. Set, On some inequalities of Hermite-Hadamard type via m- convexity, Appl. Math. Lett., 23(2010), 1065-1070.10.1016/j.aml.2010.04.037
Search in Google Scholar
[14] M. E. Ödemir, M. Avci, H. Kavurmaci, Hermite-Hadamard-type inequalities via (fi;m)- convexity, Comput. Math. Appl., 61(2011), 2614-2620.10.1016/j.camwa.2011.02.053
Search in Google Scholar
[15] S. S. Dragomir, Hermite-Hadamard's type inequalities for operator convex functions, Appl. Math. Comput., 218(2011), 766-772.
Search in Google Scholar
[16] S. S. Dragomir, Hermite-Hadamard's type inequalities for convex functions of selfadjoint operators in Hilbert spaces, Linear Algebra Appl., 436(2012), 1503-1515.10.1016/j.laa.2011.08.050
Search in Google Scholar
[17] M. Z. Sarikaya, N. Aktan, On the generalization of some integral inequalities and their applications, Math. Comput. Model., 54(2011), 2175-218210.1016/j.mcm.2011.05.026
Search in Google Scholar
[18] Z. Xiao, Z. Zhang, Y. Wu, On weighted Hermite-Hadamard inequalities, Appl. Math. Comput., 218(2011), 1147-1152.
Search in Google Scholar
[19] B. Xi, R. Bai, F. Qi, Hermite-Hadamard type inequalities for the m-and (fi;m)- geometrically convex functions, Aequat. Math., (2012) doi:10.1007/s00010-011-0114-x.10.1007/s00010-011-0114-x
Search in Google Scholar
[20] M. Bessenyei, The Hermite-Hadamard inequality in Beckenbach's setting, J. Math. Anal. Appl., 364(2010), 366-383.10.1016/j.jmaa.2009.11.015
Search in Google Scholar
[21] K. Tseng, S. Hwang, K. Hsu, Hadamard-type and Bullen-type inequalities for Lipschitzian functions and their applications, Comput. Math. Appl., (2012), doi:10.1016/j.camwa.2011.12.076.10.1016/j.camwa.2011.12.076
Search in Google Scholar
[22] C. P. Niculescu, The Hermite-Hadamard inequality for log-convex functions, Nonlinear Anal.:TMA, 75(2012), 662-669.10.1016/j.na.2011.08.066
Search in Google Scholar
[23] U. S. Kirmaci, Inequalities for difierentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput., 147(2004), 137-146.
Search in Google Scholar
[24] U. S. Kirmaci, M. E. Özdemir, On some inequalities for difierentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput., 153(2004), 361-368.
Search in Google Scholar
[25] C. E. M. Pearce, J. E. Peficarific, Inequalities for difierentiable mappings with application to special means and quadrature formula, Appl. Math. Lett., 13(2000), 51-55.10.1016/S0893-9659(99)00164-0
Search in Google Scholar

Fractional Integral Inequalities for Differentiable Convex Mappings and Applications to Special Means and a Midpoint Formula

Abstract

Paradigm

My account