A study on the continuity of some classes of $ E $-$\mathbb {Q} $-convex functions

Document Type : Research Paper


1 Department of Mathematics, Payame Noor University (PNU), P. O. Box 19395-4697, Tehran, Iran.

2 Department of Mathematics‎, ‎Faculty of Science, Shahid Rajaee Teacher Training University, P.O‎. ‎Box 16785-136‎, ‎Tehran‎, ‎Iran.



As a generalization of convexity,  $ E $-convexity has been defined and studied in many publications. In this study, we recall the class of $ E $-$\mathbb {Q} $-convex sets, $ E $-$ \mathbb {Q}  $-convex and $ E $-additive functions and proved some properties of $ E $-$ \mathbb {Q}  $-convex functions.  Also, we develop the classical theorems of Jensen and Bernstein-Doetsch on $ E $-$ \mathbb {Q}  $-convex functions when vector spaces are over the rational numbers $ \mathbb {Q} $.


[1] M. Aghajani and K. Nourouzi, The continuity of $\mathbb {Q_{+}} $- homogeneous superadditive correspondences, J. 
     Nonlinear Convex Anal., 16 (2015), 1899-1904.
[2] X. Chen, Some properties of semi-$E$-convex functions, J. Math. Anal. Appl., 275(1) (2002), 251-262.
[3] A. Hussain and A. Iqbal, Quasi strongly -convex functions with applications, Non-linear Funct. Anal. Appl., 26 (2021), 
[4] A. Iqbal, and I. Ahmad, Strong geodesic convex functions of order m, Numer. Funct. Anal. Optim., 40 (2019), 
[5] M.E. Kuczma, On discontinuous additive function, Fund. Math., 66 (1970), 383-392.
[6] S.N. Majeed and  M.I. Abd Al-Majeed, On convex functions, E-convex functions and their generalizations: applications 
     to non-linear optimization problems, Int. J. Pure Appl. Math., 116 (2017), 655-673.
[7] S.N. Majeed, On strongly E-convex sets and strongly E-convex cone sets, J. AL-Qadisiyah Comput. Sci. Math., 11 (2019),
[8] G.G. Magaril-Ilyaev and V.M. Tikhomirov, Convex Analysis: Theory and Applications, AMS, Providence, R.I., Transl. of 
     Math. Monographs, 2003.
[9] M.R. Mehdi, On convex functions, J. London Math. Soc., 39 (1964), 321-326.
[10] F. Mirzapour, A. Mirzapour and M. Meghdadi, Generalization of some important theorems to $E$-midconvex 
       functions, Appl. Math. Lett., 24(8) (2011), 1384-1388.
[11] P. Najmadi and M. Aghajani, Some families of sublinear correspondences, J. Appl. Anal., 25(1) (2019), 91-95.
[12] W. Saleh, Hermite–Hadamard type inequality for (E, F)-convex functions and geodesic (E, F)-convex functions, Rairo-
       Oper. Res., 56 (2022), 4181-4189.
[13] M. Soleimani-damaneh, E-convexity and its generalizations, Int. J. Comput. Math., 88 (2011), 3335-3349.
[14] Y.R. Syau and E.S. Lee, Some properties of $E$-convex functions, Appl. Math. Lett., 18 (2005), 1074-1080.
[15] X.M. Yang, $E$-convex sets, $E$-convex functions and $E$-convex programming, J. Optim. Theory Appl., 109 (2001), 
[16] E.A. Youness, $E$-convex functions and $E$-convex programming, J. Optim. Theory Appl., 102(2) (1999), 439-450.
[17] E.A. Youness, Optimality criteria in E-convex programming, Chaos Solitons Fractals, 12 (2001), 1737-1745.
[18] B.C. Joshi and Pankaj, Mathematical programs involving E-convex functions, Scientific Bulletin. upb. ro., 83(2) (2011),