×

The best constant in the Khintchine inequality for complex Steinhaus variables, the case \(p=1\). (English) Zbl 0612.60017

It is shown that \[ (\frac{1}{2\pi})^ n\int^{2\pi}_{0}...\int^{2\pi}_{0}| \sum^{n}_{i=1}a_ ie^{it_ i}| dt_ 1...dt_ n\geq \frac{\sqrt{\pi}}{2}(\sum^{n}_{i=1}| a_ i|^ 2)^{1/2} \] for arbitrary complex numbers \(a_ 1,a_ 2,...,a_ n\) and for \(n=1,2,...\). The constant \(\sqrt{\pi}/2\) is the largest possible.

MSC:

60E15 Inequalities; stochastic orderings
46E99 Linear function spaces and their duals
60F05 Central limit and other weak theorems
PDFBibTeX XMLCite
Full Text: DOI EuDML