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Convex Functions on Discrete Time Domains

Published online by Cambridge University Press:  20 November 2018

Ferhan M. Atıcı
Affiliation:
Department of Mathematics, Western Kentucky University, Bowling Green, Kentucky 42101-3576, USA e-mail: [email protected]
Hatice Yaldız
Affiliation:
Department of Mathematics, Düzce University, Düzce, Turkey e-mail: [email protected]
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Abstract

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In this paper, we introduce the definition of a convex real valued function $f$ defined on the set of integers, $\mathbb{Z}$ . We prove that $f$ is convex on $\mathbb{Z}$ if and only if ${{\Delta }^{2}}f\,\ge \,0$ on $\mathbb{Z}$ . As a first application of this new concept, we state and prove discrete Hermite–Hadamard inequality using the basics of discrete calculus (i.e., the calculus on $\mathbb{Z}$ ). Second, we state and prove the discrete fractional Hermite–Hadamard inequality using the basics of discrete fractional calculus. We close the paper by defining the convexity of a real valued function on any time scale.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Abdeljawad, T., Baleanu, D., Jarad, F., and Agarwal, R., Fractional sums and differences with binomial coefficients. Discrete Dyn. Nat. Soc. 2013, 104173.Google Scholar
[2] Agarwal, R. P., Difference equations and inequalities. Theory, methods and applications.Second éd., Monographs and Textbooks in Pure and Applied Mathematics, 228, Marcel Dekker, Inc., New York, 2000.Google Scholar
[3] Atici, F. M. and Eloe, P. W., Discrete fractional calculus with the nabla operator. Electron.J. Qual. Theory Differ. Equ. Spec. Ed 1(2009), no.3,1-12.Google Scholar
[4] Bohner, M. and Peterson, A., Dynamic equations on time scales.An introduction with applications. Birkhâuser, Boston, 2001. http://dx.doi.org/10.1007/978-1-4612-0201-1 Google Scholar
[5] Caputo, M. C., Time scales: from nabla calculus to delta calculus and vice versa via duality. Int. J. Difference Equ. 5(2010), no. 1, 2540.Google Scholar
[6] Eloe, P. W., Sheng, Q., and Henderson, J., Notes on crossed symmetry solutions of the two-point boundary value problems on time scales. J. Difference Equ. Appl. 9(2003), no. 1, 2948. http://dx.doi.org/10.1080/10236100309487533 Google Scholar
[7] Miller, K. S. and Ross, B., Fractional difference calculus.In: Univalent functions, fractional calculus, and their applications (Kôriyama, 1988), Ellis Horwood Ser. Math. Appl. Horwood, Chichester, 1989, pp. 139152.Google Scholar
[8] Dragomir, S. S. and Pearce, C. E. M., Selected topics on Hermite-Hadamard inequalities and applications. RGMIA Monographs, Victoria University, 2000. http://rgmia.org/monographs.php Google Scholar
[9] Kelley, W. and Peterson, A. C., Difference equations: an introduction with applications. Academic Press, Harcourt/Academic Press, San Diego, 2001.Google Scholar
[10] Sarikaya, M. Z., Set, E., Yaldiz, H., and Basak, N., Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities. Math.Comput.Modelling 57(2013), 24032407. http://dx.doi.org/10.1016/j.mcm.2O11.12.048 Google Scholar
[11] Sarikaya, M. Z. and Yaldiz, H., On weighted montogomery identities for Riemann-Liouville fractional integrals. Konuralp J. Math. 1(2013), 4853.Google Scholar
[12] Sarikaya, M. Z. and Yaldiz, H., On generalization integral inequalities for fractional integrals. Nihonkai Math. J. 25(2014), 93104.Google Scholar