论文标题
在大约凸和仿射功能上
On approximately convex and affine functions
论文作者
论文摘要
在实际开放间隔$ i $上定义的真正有价值的函数$ f $称为$φ$ -CONVEX，如果对于所有$ x，y \ in I $，$ t \ in [0,1] $ IT满足$$ f（tx+（1-t）y）\ leq tf（x）+（1-t）f（y）+tφ\ big（（（1-t）| x-y | \ big）+（1-t）φ\ big（t | x-y | \ big），$φ：$φ：$φ：\ mathbb {r} _+r} _+\ to mathb to \ ismath \ = $ a__如果$ f $和$ -f $同时是$φ$ -CONVEX，则$ f $被称为$φ$ - 随身函数。在本文的主要结果中，我们描述了这两个类别的结构和包容性能。我们表征了这两类函数，并研究了它们与近似单调和近似Hölder函数的关系。我们还引入了一个享受所谓的$γ$属性的错误函数的子类，我们表明，对于$φ$ -CONVEX函数最佳的错误函数必须属于此子类。也研究了此错误函数子类的属性。然后，我们为下$φ$ -Convex信封提供两个公式。此外，还添加了三明治型定理。
A real valued function $f$ defined on a real open interval $I$ is called $Φ$-convex if, for all $x,y\in I$, $t\in[0,1]$ it satisfies $$ f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+tΦ\big((1-t)|x-y|\big)+(1-t)Φ\big(t|x-y|\big), $$ where $Φ:\mathbb{R}_+\to\mathbb{R}_+$ is a nonnegative error function. If $f$ and $-f$ are simultaneously $Φ$-convex, then $f$ is said to be a $Φ$-affine function. In the main results of the paper, we describe the structural and inclusion properties of these two classes. We characterize these two classes of functions and investigate their relationship with approximately monotone and approximately-Hölder functions. We also introduce a subclass of error functions which enjoy the so-called $Γ$ property and we show that the error function which is the most optimal for a $Φ$-convex function has to belong to this subclass. The properties of this subclass of error function are investigated as well. Then we offer two formulas for the lower $Φ$-convex envelop. Besides, a sandwich type theorem is also added.
