Best Khintchine Type Inequalities for Sums of Independent, Rotationally Invariant Random Vectors

By H. König, S. Kwapień. 2001

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The Bohr radius of the $n$-dimensional polydisk is equivalent to $\sqrt{\frac{\log n}{n}} (2013, background): “y p ∈ [1,2], there exists a constant AC,p such that, for any n ≥ 1 and any a1,...,an ∈ C, Xn i=1 |ai|2 !1/2 ≤ AC,p Z Tn Xn i=1 aizi p dz !1/p . The best constant AR,p and AC,p are known (see [14] and [17]). Indeed, • AR,p = √1 2 Γ(1+p 2 ) √ π −1/p if p > p0 ≈ 1.847; • AC,p = Γ p+2 2 −1/p if p ∈ ”
Optimal Hardy-Littlewood type inequalities for polynomials and multilinear operators (2013, background): “The best constants AR,p and AC,p are known (see [15] and [18]): • AR,p = ”