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We prove the neo‐classical inequality with the optimal constant, which was conjectured by T. J. Lyons (‘Differential equations driven by rough signals’,  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.

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