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Fractional order Taylor’s series and the neo-classical inequality. (English) Zbl 1194.26027
T. J. Lyons [Rev. Mat. Iberoam. 14, No. 2, 215–310 (1998; Zbl 0923.34056)] introduced the following neo-classical inequality:
Let \(\alpha\in (0,1]\), \(n\in {\mathbb N}\), \(x\geq0\), and \(y\geq 0\). Then we have:
\[ \alpha^2\sum_{j=0}^{n}\binom {\alpha n}{\alpha j} x^{\alpha j}y^{\alpha(n-j)}\leq (x+y)^{\alpha n}. \]
Lyons conjectured that the coefficient \(\alpha^2\) in the left-hand side could be replaced by \(\alpha\). The authors provide an affirmative answer to Lyons conjecture, proving the neo-classical inequality with the optimal constant, by using fractional order Taylor’s series with residual terms.

MSC:
26D15 Inequalities for sums, series and integrals
30B10 Power series (including lacunary series) in one complex variable
26A33 Fractional derivatives and integrals
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