Characterization of approximately monotone and approximately Hölder functions

Abstract

A real valued function $f$ defined on a real open interval $I$ is called $\Phi$-monotone if, for all $x,y\in I$ with $x\le y$ it satisfies $$f(x)\le f(y)+\Phi (y-x),$$ where $\Phi :[0,\ell (I)[\to {\mathbb{R}}_{+}$ is a given nonnegative error function, where $\ell (I)$ denotes the length of the interval $I$. If $f$ and $-f$ are simultaneously $\Phi$-monotone, then $f$ is said to be a $\Phi$-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 $\Phi$-monotone and $\Phi$-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 $\Phi$-monotonicity and $\Phi$-Hölder properties, and then we verify the sharpness of these implications. We also establish implications in the reversed direction.