# Article

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Keywords:
co-ordinate; quasi-convex; Wright-quasi-convex; Jensen-quasi-convex
Summary:
A function $f\colon I\rightarrow \mathbb {R}$, where $I\subseteq \mathbb {R}$ is an interval, is said to be a convex function on $I$ if $$f( tx+( 1-t) y) \leq tf( x) +(1-t) f( y)$$ holds for all $x,y\in I$ and $t\in [ 0,1]$. There are several papers in the literature which discuss properties of convexity and contain integral inequalities. Furthermore, new classes of convex functions have been introduced in order to generalize the results and to obtain new estimations. \endgraf We define some new classes of convex functions that we name quasi-convex, Jensen-convex, Wright-convex, Jensen-quasi-convex and Wright-quasi-convex functions on the co-ordinates. We also prove some inequalities of Hadamard-type as Dragomir's results in Theorem 5, but now for Jensen-quasi-convex and Wright-quasi-convex functions. Finally, we give some inclusions which clarify the relationship between these new classes of functions.
References:
[1] Alomari, M., Darus, M., Dragomir, S. S.: New inequalities of Hermite-Hadamard type for functions whose second derivatives absolute values are quasi-convex. Tamkang J. Math. 41 353-359 (2010). MR 2789971 | Zbl 1214.26003
[2] Alomari, M. W., Darus, M., Kirmaci, U. S.: Refinements of Hadamard-type inequalities for quasi-convex functions with applications to trapezoidal formula and to special means. Comput. Math. Appl. 59 225-232 (2010). DOI 10.1016/j.camwa.2009.08.002 | MR 2575509 | Zbl 1189.26037
[3] Alomari, M., Darus, M.: On some inequalities Simpson-type via quasi-convex functions with applications. Transylv. J. Math. Mech. 2 (2010), 15-24. MR 2817188
[4] Alomari, M., Darus, M.: Hadamard-type inequalities for $s$-convex functions. Int. Math. Forum 3 (2008), 1965-1975. MR 2470655 | Zbl 1163.26325
[5] Alomari, M., Darus, M.: The Hadamard's inequality for $s$-convex function of $2$-variables on the co-ordinates. Int. J. Math. Anal., Ruse 2 (2008), 629-638. MR 2482668 | Zbl 1178.26017
[6] Alomari, M., Darus, M.: Coordinated $s$-convex function in the first sense with some Hadamard-type inequalities. Int. J. Contemp. Math. Sci. 3 (2008), 1557-1567. MR 2514034 | Zbl 1178.26015
[7] Dragomir, S. S.: 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. MR 1870047 | Zbl 1002.26017
[8] Dragomir, S. S., Pearce, C. E. M.: Quasi-convex functions and Hadamard's inequality. Bull. Aust. Math. Soc. 57 (1998), 377-385. DOI 10.1017/S0004972700031786 | MR 1623227 | Zbl 0908.26015
[9] Hwang, D. Y., Tseng, K. L., Yang, G. S.: Some Hadamard's inequalities for co-ordinated convex functions in a rectangle from the plane. Taiwanese J. Math. 11 (2007), 63-73. MR 2304005 | Zbl 1132.26360
[10] Ion, D. A.: Some estimates on the Hermite-Hadamard inequality through quasi-convex functions. An. Univ. Craiova, Ser. Mat. Inf. 34 (2007), 83-88. MR 2517875 | Zbl 1174.26321
[11] Bakula, M. Klaričić, Pečarić, J.: On the Jensen's inequality for convex functions on the coordinates in a rectangle from the plane. Taiwanese J. Math. 10 (2006), 1271-1292. MR 2253378
[12] Latif, M. A., Alomari, M.: On Hadamard-type inequalities for $h$-convex functions on the co-ordinates. Int. J. Math. Anal., Ruse 3 (2009), 1645-1656. MR 2657722
[13] Latif, M. A., Alomari, M.: Hadamard-type inequalities for product two convex functions on the co-ordinates. Int. Math. Forum 4 (2009), 2327-2338. MR 2579666 | Zbl 1197.26029
[14] Özdemir, M. E., Kavurmacı, H., Akdemir, A. O., Avcı, M.: Inequalities for convex and $s$-convex functions on $\Delta =[a,b]\times[c,d]$. J. Inequal. Appl. 2012:20 (2012), 19 pp doi:10.1186/1029-242X-2012-20. DOI 10.1186/1029-242X-2012-20
[15] Özdemir, M. E., Latif, M. A., Akdemir, A. O.: On some Hadamard-type inequalities for product of two $s$-convex functions on the co-ordinates. J. Inequal. Appl. 2012:21 (2012), 13 pp doi:10.1186/1029-242X-2012-21. DOI 10.1186/1029-242X-2012-21 | MR 2892628
[16] Özdemir, M. E., Set, E., kaya, M. Z. Sarı: Some new Hadamard type inequalities for co-ordinated $m$-convex and $(\alpha ,m)$-convex functions. Hacet. J. Math. Stat. 40 219-229 (2011).
[17] Pečarić, J. E., Proschan, F., Tong, Y. L.: Convex Functions, Partial Orderings and Statistical Applications. Academic Press, Boston (1992). MR 1162312 | Zbl 0749.26004
[18] Tseng, K.-L., Yang, G.-S., Dragomir, S. S.: On quasi convex functions and Hadamard's inequality. Demonstr. Math. 41 323-336 (2008). MR 2419910 | Zbl 1151.26333
[19] Wright, E. M.: An inequality for convex functions. Amer. Math. Monthly 61 (1954), 620-622. DOI 10.2307/2307675 | MR 0064828 | Zbl 0057.04801

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