Fractional order Taylor's series and the neo-classical inequality

TL;DR

This paper proves the neo-classical inequality with the best possible constant by introducing fractional order Taylor's series, providing a new analytical approach that confirms a longstanding conjecture.

Contribution

It introduces fractional order Taylor's series with residuals and applies them to establish the optimal neo-classical inequality, resolving a conjecture by T. J. Lyons.

Findings

Proved the neo-classical inequality with optimal constant

Developed fractional order Taylor's series with residuals

Provided a new identity for the inequality

Abstract

We prove the neo-classical inequality with the optimal constant, which was conjectured by T. J. Lyons [Rev. Mat. Iberoamericana 14 (1998) 215-310]. For the proof, we introduce the fractional order Taylor's series with residual terms. Their application to a particular function provides an identity that deduces the optimal neo-classical inequality.