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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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