Characterization of approximately monotone and approximately Hölder functions

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.

No reviews yet.

No comments yet.