中文摘要：

广义E-凸函数在优化理论中有着重要的作用.通过对比凸函数的相关性质,得到了广义E-凸函数与凸函数的关系,对文献[5]中引理的证明举出了反例,并对E-凸函数与E-单调的等价性给出了重新证明,进一步论证了在有限维空间中,f在特定的开集E（M）上任意一点都有上界;最后,针对广义E-凸函数已有的性质和结论进行了相应的推广.

英文摘要：

The generalized E-convex function plays an important role in the optimization theory. The relationship between generalized E-convex function and convex function is obtained by comparing the correlation properties of convex functions. Then a counterexample is given for the proof of the lemma in the literature [ 5 ], and the equivalence between the E-convex function and the E-monotone is proved newly. It is further proved that f has an upper bound on a specific open set E（M） in a finite dimensional space. Finally, the properties and conclusions of the generalized E-convex function are generalized.