Best Constants in Kahane-Khintchine Inequalities in Orlicz Spaces

Several inequalities of Kahane-Khintchine′s type in certain Orlicz spaces are proved. For this, the classical symmetrization technique is used and four basically different methods have been presented. The first two are based on the well-known estimates for subnormal random variables, the third one is a consequence of a certain Gaussian-Jensen′s majorization technique, and the fourth one is obtained by Haagerup-Young-Stechkin′s best possible constants in the classical Khintchine inequalities. Moreover, by using the central limit theorem it is shown that this fourth approach gives the best possible numerical constant in the inequality under consideration: If {ϵi | i ≥ 1} is a Bernoulli sequence, and || · ||ψ denotes the Orlicz norm induced by the function ψ(x) = ex2 − 1 for x ∈ R, then the best possible numerical constant C satisfying the inequality [formula] for all a1, ..., an ∈ R and all n ≥ 1, is equal to [formula]. Similarly, the best possible estimates of that type are also deduced for some other inequalities in Orlicz spaces, discovered in this paper.