An Extension of the Kahane-khinchine Inequality
نویسنده
چکیده
(2) r n N I I P Ï 1/p r n N \\} E j ; ^ J >cp ExyTM>*J fr )l lli==1 II ; l l l i = 1 II ) Recalling that in general {E| f ^ } 1 ^ decreases to exp E log \f\ as p decreases to zero, one sees that (1) is a strictly stronger statement than (2); in fact (1) says simply that cp may be taken bounded away from zero in (2). Note that the inequality obtained from (1) by replacing ej with the jth Rademacher function ry is false, even in the case B = C : If sn = n 1 / 2 ( r i H \-rn) then exp E log | s n | = 0 for even values of n, although sn is asymptotically normal. In other words: Suppose that X is a random variable; suppose even \X\ < 1 a.s. Then to say exp E log \X\ > c implies that the set where X is small must be small, while to say {E|X|}/ > c does not even preclude the possibility that X vanish on a set of positive measure! In the case B = C inequality (1) is proved in [UK], and various applications are given. In particular one may use (1) to show that the zero set of a Bloch function may be strictly larger than is possible for a function in the "little-oh" Bloch space, answering a question of Ahern and Rudin [AR]; this fact then gives a result analogous to Theorem 6.1 of [AR], with VMOA and H°° replaced by BMOA and VMOA, respectively. Inequality (1) also allows one to construct new and improved Ryll-Wojtaszczyk polynomials [RW]: There exists a sequence Pi, P2,..., of polynomials in C n such that Pj is homogeneous of degree j and satisfies |P>(^)| < 1 (z G C n , \z\ < 1) while
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