On some inequalities for generalized s-convex functions and applications on fractal sets
- Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia.
- Department of Mathematics, Putra University of Malaysia (UPM), Serdang, Malaysia.
The authors present some new inequalities of generalized Hermite-Hadamard’s type for the class of functions whose second
local fractional derivatives of order \(\alpha\) in absolute value at certain powers are generalized s-convex functions in the second sense.
Moreover, some applications are given.
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Adem Kilicman, Wedad Saleh, On some inequalities for generalized s-convex functions and applications on fractal sets, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 2, 583--594
Kilicman Adem, Saleh Wedad, On some inequalities for generalized s-convex functions and applications on fractal sets. J. Nonlinear Sci. Appl. (2017); 10(2):583--594
Kilicman, Adem, Saleh, Wedad. "On some inequalities for generalized s-convex functions and applications on fractal sets." Journal of Nonlinear Sciences and Applications, 10, no. 2 (2017): 583--594
- s-convex functions
- fractal space
- local fractional derivative.
A. Atangana, S. B. Belhaouari , Solving partial differential equation with space- and time-fractional derivatives via homotopy decomposition method, Math. Probl. Eng., 2013 (2013 ), 9 pages.
A. Atangana, E. F. Doungmo-Goufo, Solution of diffusion equation with local derivative with new parameter, Therm. Sci., 19 (2015), 231–238.
D. Baleanu, H. M. Srivastava, X.-J. Yang, Local fractional variational iteration algorithms for the parabolic Fokker-Planck equation defined on Cantor sets, Prog. Fract. Differ. Appl., 1 (2015), 1–11.
S. S. Dragomir, On the Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwanese J. Math., 5 (2001), 775–788.
S. S. Dragomir, S. Fitzpatrick, The Hadamard inequalities for s-convex functions in the second sense, Demonstratio Math., 32 (1999), 687–696.
M. Grinblatt, J. T. Linnainmaa, Jensen’s inequality, parameter uncertainty, and multi-period investment, Rev. Asset Pric. Stud., 1 (2011), 1–34.
L. Hörmander, Notions of convexity, Progress in Mathematics, Birkhäuser Boston, Inc., Boston, MA (1994)
J. Hua, B.-Y. Xi, F. Qi, Inequalities of Hermite-Hadamard type involving an s-convex function with applications, Appl. Math. Comput., 246 (2014), 752–760.
A. Kılıçman, W. Saleh, Notions of generalized s-convex functions on fractal sets, J. Inequal. Appl., 2015 (2015 ), 16 pages.
A. Kılıçman, W. Saleh, Some generalized Hermite-Hadamard type integral inequalities for generalized s-convex functions on fractal sets, Adv. Difference Equ., 2015 (2015 ), 15 pages.
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (2006)
H.-X. Mo, X. Sui, Generalized s-convex functions on fractal sets, Abstr. Appl. Anal., 2014 (2014 ), 8 pages .
H.-X. Mo, X. Sui, Hermite-Hadamard type inequalities for generalized s-convex functions on real linear fractal set \(R^\alpha(0 < 1)\), ArXiv, 2015 (2015 ), 10 pages.
H.-X. Mo, X. Sui, D.-Y. Yu, Generalized convex functions on fractal sets and two related inequalities, Abstr. Appl. Anal., 2014 (2014 ), 7 pages.
M. E. Özdemir, Ç . Yıldız, A. O. Akdemir, E. Set, On some inequalities for s-convex functions and applications, J. Inequal. Appl., 2013 (2013 ), 11 pages.
C. E. M. Pearce, J. Pečarić, Inequalities for differentiable mappings with application to special means and quadrature formulæ, Appl. Math. Lett., 13 (2000), 51–55.
J. J. Ruel, M. P. Ayres, Jensen’s inequality predicts effects of environmental variation, Trends Ecol. Evol., 14 (1999), 361–366.
M. Z. Sarikaya, T. Tunc, H. Budak, On generalized some integral inequalities for local fractional integrals, Appl. Math. Comput., 276 (2016), 316–323.
X.-J. Yang, Local fractional integral transforms, Prog. Nonlinear Sci., 4 (2011), 1–225.
X.-J. Yang, Advanced local fractional calculus and its applications, World Science Publ., New York (2012)
X.-J. Yang, D. Baleanu, H. M. Srivastava, Local fractional similarity solution for the diffusion equation defined on Cantor sets, Appl. Math. Lett., 47 (2015), 54–60.
A.-M. Yang, X.-J. Yang, Z.-B. Li, Local fractional series expansion method for solving wave and diffusion equations on Cantor sets, Abstr. Appl. Anal., 2013 (2013 ), 5 pages.