论文标题
大约单调和大约Hölder函数的表征
Characterization of approximately monotone and approximately Hölder functions
论文作者
论文摘要
在实际开放间隔$ i $上定义的真实有价值的功能$ f $，称为$φ$ - 单子酮，如果对于所有$ x，y \ in i $ in i $ with $ x \ leq y $ it满足$$ f（x）\ leq f（y）+φ（y-x），$$其中$φ：[0，\ ell（i）[\，\ to \ to \ mathbb {r} _+$是给定的非负错误函数，其中$ \ ell（i）$表示间隔$ i $ i $的长度。如果$ f $和$ -f $同时是$φ$ -Sonotone，则$ f $被称为$φ$-Hölder函数。在本文的主要结果中，使用上下插值的概念，我们为两类功能建立了一个表征。这允许一个人构建$φ$ -Sonotone和$φ$-Hölder函数，这些功能可以称为这些类别的构建块。在第二部分中，我们从$φ$ - 单调性和$φ$-Hölder属性中推断出Ostrowski-和hermite- HADAMARD-TYPE不平等现象，然后我们验证了这些含义的清晰度。我们还在反向方向上建立了含义。
A real valued function $f$ defined on a real open interval $I$ is called $Φ$-monotone if, for all $x,y\in I$ with $x\leq y$ it satisfies $$ f(x)\leq f(y)+Φ(y-x), $$ where $Φ:[0,\ell(I)[\,\to\mathbb{R}_+$ is a given nonnegative error function, where $\ell(I)$ denotes the length of the interval $I$. If $f$ and $-f$ are simultaneously $Φ$-monotone, then $f$ is said to be a $Φ$-Hölder function. In the main results of the paper, using the notions of upper and lower interpolations, we establish a characterization for both classes of functions. This allows one to construct $Φ$-monotone and $Φ$-Hölder functions from elementary ones, which could be termed the building blocks for those classes. In the second part, we deduce Ostrowski- and Hermite--Hadamard-type inequalities from the $Φ$-monotonicity and $Φ$-Hölder properties, and then we verify the sharpness of these implications. We also establish implications in the reversed direction.
