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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.

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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