The problem is raised of finding the best possible constant in the maximal Khintchine inequality for Rademacher sequence ε=(εk)k[ges ]1:

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being valid for all a1, …, an∈R with n[ges ]1, where 0<p<∞ is given and fixed. We conjecture that the best possible constant is

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where B=(Bt[ges ]0) is standard Brownian motion. For simplicity, we consider only the case p=1 and prove that this conjecture is as close to the truth as desired in the following asymptotic sense:

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being valid for all [mid ]a1[mid ][les ]1, …, [mid ]an[mid ][les ]1 and all n[ges ]1, where Sk =[sum ]ki=1aiεi and ∥a→n∥2 =([sum ]nk=1 [mid ]ak[mid ]2) ½[ges ]2. It should be noted here that

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The method of proof relies upon Skorohod's imbedding. Motivated by consequences of this result we deduce in a purely computational way that:

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whenever a1=1, a2=λ, a3=λ2, …, an=λn−1 and λ belongs to ]0, 1/2] with n[ges ]1. The constant 2/√3 is shown to be the best possible in this inequality.