Product rules for the Fractional variation of H\"olderian functions

TL;DR

This paper develops product rules for fractional variation of H"olderian functions, revealing deviations from classical Leibniz rules and introducing fractional co-variation to describe these differences.

Contribution

It derives the product rules for fractional variation and introduces fractional co-variation, expanding understanding of non-differentiable functions.

Findings

Product rules for fractional variation are established.

Deviations from Leibniz rule are characterized by fractional co-variation.

Basic algebraic properties of fractional co-variation are demonstrated.

Abstract

Fractional variation is defined as the limit of the difference quotient of the increments of a function and its argument raised to a fractional power. Fractional velocity can be suitable for characterizing singular behavior of derivatives of H\"olderian functions and non differentiable functions. The manuscript derives the product rules for fractional variation. Correspondence with integer-order derivatives is discussed. It is demonstrated that for H\"older functions under certain conditions the product rules deviates from the Leibniz rule. This deviation is expressed by another quantity, fractional co-variation. Basic algebraic properties of the fractional co-variation are demonstrated.