Fractional Hermite-Hadamard-type inequalities for interval-valued functions

Budak, Hüseyin; Tunç, Tuba; Sarikaya, Mehmet

doi:10.1090/proc/14741

Fractional Hermite-Hadamard-type inequalities for interval-valued functions
HTML articles powered by AMS MathViewer

by Hüseyin Budak, Tuba Tunç and Mehmet Zeki Sarikaya

Proc. Amer. Math. Soc. 148 (2020), 705-718

DOI: https://doi.org/10.1090/proc/14741

Published electronically: August 7, 2019

PDF | Request permission

Abstract:

In this paper, we define interval-valued right-sided Riemann- Liouville fractional integrals. Later, we handle Hermite-Hadamard inequality and Hermite-Hadamard-type inequalities via interval-valued Riemann-Liouville fractional integrals.

References

Jean-Pierre Aubin and Arrigo Cellina, Differential inclusions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 264, Springer-Verlag, Berlin, 1984. Set-valued maps and viability theory. MR 755330, DOI 10.1007/978-3-642-69512-4
Alfonso G. Azpeitia, Convex functions and the Hadamard inequality, Rev. Colombiana Mat. 28 (1994), no. 1, 7–12. MR 1304041
Wolfgang W. Breckner, Continuity of generalized convex and generalized concave set-valued functions, Rev. Anal. Numér. Théor. Approx. 22 (1993), no. 1, 39–51. MR 1621546
Y. Chalco-Cano, A. Flores-Franulič, and H. Román-Flores, Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative, Comput. Appl. Math. 31 (2012), no. 3, 457–472. MR 3009184
Y. Chalco-Cano, W. A. Lodwick, and W. Condori-Equice, Ostrowski type inequalities and applications in numerical integration for interval-valued functions, Soft Comput. 19 (2015), no.11, 3293–3300. DOI:10.1007/s00500-014-1483-6.
Feixiang Chen, A note on Hermite-Hadamard inequalities for products of convex functions via Riemann-Liouville fractional integrals, Ital. J. Pure Appl. Math. 33 (2014), 299–306. MR 3310752
Feixiang Chen, A note on Hermite-Hadamard inequalities for products of convex functions, J. Appl. Math. , posted on (2013), Art. ID 935020, 5. MR 3133978, DOI 10.1155/2013/935020
Feixiang Chen and Shanhe Wu, Several complementary inequalities to inequalities of Hermite-Hadamard type for $s$-convex functions, J. Nonlinear Sci. Appl. 9 (2016), no. 2, 705–716. MR 3416283, DOI 10.22436/jnsa.009.02.32
T. M. Costa, Jensen’s inequality type integral for fuzzy-interval-valued functions, Fuzzy Sets and Systems 327 (2017), 31–47. MR 3704668, DOI 10.1016/j.fss.2017.02.001
T. M. Costa and H. Román-Flores, Some integral inequalities for fuzzy-interval-valued functions, Inform. Sci. 420 (2017), 110–125. MR 3694425, DOI 10.1016/j.ins.2017.08.055
Alexander Dinghas, Zum Minkowskischen Integralbegriff abgeschlossener Mengen, Math. Z. 66 (1956), 173–188 (German). MR 92170, DOI 10.1007/BF01186606
S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, (2000).
S. S. Dragomir, Inequalities of Hermite-Hadamard type for $h$-convex functions on linear spaces, Proyecciones 34 (2015), no. 4, 323–341. MR 3437222, DOI 10.4067/S0716-09172015000400002
Sever Silvestru Dragomir, Two mappings in connection to Hadamard’s inequalities, J. Math. Anal. Appl. 167 (1992), no. 1, 49–56. MR 1165255, DOI 10.1016/0022-247X(92)90233-4
S. S. Dragomir, J. Pečarić, and L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math. 21 (1995), no. 3, 335–341. MR 1348130
H. Román-Flores, Y. Chalco-Cano, and W. A. Lodwick, Some integral inequalities for interval-valued functions, Comput. Appl. Math. 37 (2018), no. 2, 1306–1318. MR 3804129, DOI 10.1007/s40314-016-0396-7
H. Roman-Flores, Y. Chalco-Cano, and G. N. Silva, A note on Gronwall type inequality for interval-valued functions, IFSA World Congress and NAFIPS Annual Meeting IEEE, 35 (2013), 1455–1458, DOI:10.1109/IFSA-NAFIPS.2013.6608616.
A. Flores-Franulic, Y. Chalco-Cano, and H. Roman-Flores, An Ostrowski type inequality for interval-valued functions, IFSA World Congress and NAFIPS Annual Meeting IEEE, 35 (2013), 1459–1462, DOI: 10.1109/IFSA-NAFIPS.2013.6608617.
R. Gorenflo and F. Mainardi, Fractional calculus: integral and differential equations of fractional order, Fractals and fractional calculus in continuum mechanics (Udine, 1996) CISM Courses and Lect., vol. 378, Springer, Vienna, 1997, pp. 223–276. MR 1611585
Luc Jaulin, Michel Kieffer, Olivier Didrit, and Éric Walter, Applied interval analysis, Springer-Verlag London, Ltd., London, 2001. With examples in parameter and state estimation, robust control and robotics; With 1 CD-ROM (UNIX, Sun Solaris). MR 1989308, DOI 10.1007/978-1-4471-0249-6
Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. MR 2218073
U. S. Kirmaci, M. Klaričić Bakula, M. E. Özdemir, and J. Pečarić, Hadamard-type inequalities for $s$-convex functions, Appl. Math. Comput. 193 (2007), no. 1, 26–35. MR 2385762, DOI 10.1016/j.amc.2007.03.030
Vasile Lupulescu, Fractional calculus for interval-valued functions, Fuzzy Sets and Systems 265 (2015), 63–85. MR 3310327, DOI 10.1016/j.fss.2014.04.005
Svetoslav Markov, On the algebraic properties of convex bodies and some applications, J. Convex Anal. 7 (2000), no. 1, 129–166. MR 1773180
S. Markov, Calculus for interval functions of a real variable, Computing 22 (1979), no. 4, 325–337 (English, with German summary). MR 620060, DOI 10.1007/BF02265313
J.P. Merlet, Interval analysis for certified numerical solution of problems in robotics, International Journal of Applied Mathematics and Computer Science 19 (2009), no.3, 399-412, DOI: 10.2478/v10006-009-0033-3.
Kenneth S. Miller and Bertram Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1993. MR 1219954
Flavia-Corina Mitroi, Kazimierz Nikodem, and Szymon Wąsowicz, Hermite-Hadamard inequalities for convex set-valued functions, Demonstratio Math. 46 (2013), no. 4, 655–662. MR 3136183
Ramon E. Moore, Interval analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. MR 0231516
Ramon E. Moore, R. Baker Kearfott, and Michael J. Cloud, Introduction to interval analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009. MR 2482682, DOI 10.1137/1.9780898717716
Kazimierz Nikodem, On midpoint convex set-valued functions, Aequationes Math. 33 (1987), no. 1, 46–56. MR 901799, DOI 10.1007/BF01836150
K. Nikodem, J. L. Snchez, and L. Snchez, Jensen and Hermite-Hadamard inequalities for strongly convex set-valued maps, Mathematica Aeterna 4 (2014), no. 8, 979-987.
Y. Chalco-Cano, M. A. Rojas-Medar, and R. Osuna-Gómez, $s$-convex fuzzy processes, Comput. Math. Appl. 47 (2004), no. 8-9, 1411–1418. MR 2070994, DOI 10.1016/S0898-1221(04)90133-2
B. G. Pachpatte, On some inequalities for convex functions, RGMIA Res. Rep. Coll. 6 (2003).
Choonkil Park, Set-valued additive $\rho$-functional inequalities, J. Fixed Point Theory Appl. 20 (2018), no. 2, Paper No. 70, 12. MR 3787891, DOI 10.1007/s11784-018-0547-0
Zlatko Pavić, Improvements of the Hermite-Hadamard inequality, J. Inequal. Appl. , posted on (2015), 2015:222, 11. MR 3368212, DOI 10.1186/s13660-015-0742-0
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
B. Piatek, On the Riemann integral of set-valued functions, Zeszyty Naukowe. Matematyka Stosowana/Politechnika Ślaska, (2012).
Bożena Piątek, On the Sincov functional equation, Demonstratio Math. 38 (2005), no. 4, 875–881. MR 2199140, DOI 10.1515/dema-2005-0411
Igor Podlubny, Fractional differential equations, Mathematics in Science and Engineering, vol. 198, Academic Press, Inc., San Diego, CA, 1999. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. MR 1658022
Elżbieta Sadowska, Hadamard inequality and a refinement of Jensen inequality for set-valued functions, Results Math. 32 (1997), no. 3-4, 332–337. MR 1487604, DOI 10.1007/BF03322144
M. Z. Sarikaya, E. Set, H. Yaldiz, and N. Basak, Hermite -Hadamard’s inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling 57 (2013), 2403–2407, DOI:10.1016/j.mcm.2011.12.048.
J. M. Snyder, Interval analysis for computer graphics, SIGGRAPH Comput.Graph, 26 (1992), 121-130, http://dx.doi.org/10.1145/142920.134024.
Kuei-Lin Tseng and Shiow-Ru Hwang, New Hermite-Hadamard-type inequalities and their applications, Filomat 30 (2016), no. 14, 3667–3680. MR 3593739, DOI 10.2298/FIL1614667T
Gou-Sheng Yang and Kuei-Lin Tseng, On certain integral inequalities related to Hermite-Hadamard inequalities, J. Math. Anal. Appl. 239 (1999), no. 1, 180–187. MR 1719056, DOI 10.1006/jmaa.1999.6506
Gou-Sheng Yang and Min-Chung Hong, A note on Hadamard’s inequality, Tamkang J. Math. 28 (1997), no. 1, 33–37. MR 1457248, DOI 10.5556/j.tkjm.28.1997.4331
Dafang Zhao, Tianqing An, Guoju Ye, and Wei Liu, New Jensen and Hermite-Hadamard type inequalities for $h$-convex interval-valued functions, J. Inequal. Appl. , posted on (2018), Paper No. 302, 14. MR 3874054, DOI 10.1186/s13660-018-1896-3
D. Zhao, G. Ye, W. Liu, D.F.M. Tores, Some inequalities for interval-valued functions on time scales, Soft Computing (2018), 1-11, http://dx.doi.org/10.1007/s00500-018-3538-6.