# A class of nowhere differentiable functions satisfying some concavity type estimate

## TL;DR

This paper introduces a class of continuous, nowhere differentiable functions characterized by a concavity-type estimate on second-order differences, exploring their geometric, analytic, and functional series properties.

## Contribution

It defines and characterizes the class P of functions via geometric inequalities and difference estimates, linking it to nowhere differentiability and functional series.

## Key findings

- Functions in P are nowhere differentiable.
- A geometric inequality characterizes functions in P.
- Explicit examples of functions in P are constructed from series.

## Abstract

In this paper, we introduce and investigate a class P of continuous and periodic functions on R. The class P is defined so that second-order central differences of a function satisfy some concavity-type estimate. Although this definition seems to be independent of nowhere differentiable character, it turns out that each function in P is nowhere differentiable. The class P naturally appear from both a geometrical viewpoint and an analytic viewpoint. In fact, we prove that a function belongs to P if and only if some geometrical inequality holds for a family of parabolas with vertexes on this function. As its application, we study the behavior of the Hamilton Jacobi flow starting from a function in P. A connection between P and some functional series is also investigated. In terms of second-order central differences, we give a necessary and sufficient condition so that a function given by the series belongs to P. This enables us to construct a large number of examples of functions in P through an explicit formula.