Characterization of approximately monotone and approximately H\"older functions
A real valued function $f$ defined on a real open interval $I$ is called $\Phi$-monotone if, for all $x,y\in I$ with $x\leq y$ it satisfies $$ f(x)\leq f(y)+\Phi(y-x), $$ where $\Phi:[0,\ell(I)[\,\to\mathbb{R}_+$ is a given nonnegative error function, where $\ell(I)$ denotes the length of the interv...
Saved in:
Main Authors | , |
---|---|
Format | Journal Article |
Language | English |
Published |
14.07.2020
|
Subjects | |
Online Access | Get full text |
Cover
Loading…
Summary: | A real valued function $f$ defined on a real open interval $I$ is called
$\Phi$-monotone if, for all $x,y\in I$ with $x\leq y$ it satisfies $$
f(x)\leq f(y)+\Phi(y-x), $$ 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 $\Phi$-H\"older 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
$\Phi$-H\"older 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
$\Phi$-H\"older properties, and then we verify the sharpness of these
implications. We also establish implications in the reversed direction. |
---|---|
DOI: | 10.48550/arxiv.2007.07114 |