Abstract
A real valued function f defined on a real open interval I is called Phi -monotone if, for all x,yin I with xle y it satisfies f(x)≤f(y)+Φ(y-x),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} f(x)\\le f(y)+\\Phi (y-x), \\end{aligned}$$\\end{document}where Phi :[0,ell (I) [ rightarrow 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, we describe structural properties of these function classes, determine the error function which is the most optimal one. We show that optimal error functions for Phi -monotonicity and Phi -Hölder property must be subadditive and absolutely subadditive, respectively. Then we offer a precise formula for the lower and upper Phi -monotone and Phi -Hölder envelopes. We also introduce a generalization of the classical notion of total variation and we prove an extension of the Jordan Decomposition Theorem known for functions of bounded total variations.
Highlights
The main concepts and results of this paper are distillated from the following elementary observations
F is nondecreasing with an error term described in terms of the pth power function
For p > 1 a function f : I → R satisfies (1.1) for some nonnegative ε if and only if f is nondecreasing. Another motivation for our paper comes from the theory of approximate convexity which has a rich literature, see for instance [1–34,36–39]
Summary
The main concepts and results of this paper are distillated from the following elementary observations. For p > 1 a function f : I → R satisfies (1.1) for some nonnegative ε if and only if f is nondecreasing Another motivation for our paper comes from the theory of approximate convexity which has a rich literature, see for instance [1–34,36–39]. In the paper [35], the particular case p = 0 of inequality (1.1) was considered and the following result was proved: A function f : I → R satisfies (1.1) for some ε ≥ 0 with p = 0 if and only if there exists a nondecreasing function g : I → R such that | f − g| ≤ ε/2 holds on I. The above described observations and results motivate the investigation of classes of functions that obey a more general approximate monotonicity and the related Hölder property. We introduce a generalization of the classical notion of total variation and we prove a generalization of the Jordan Decomposition Theorem known for functions of bounded variations
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