We define the k:th moment of a Banach space valued random variable as the expectation of its k:th tensor power; thus the moment (if it exists) is an element of a tensor power of the original Banach space.We study both the projective and injective tensor products, and their relation. Moreover, in order to be general and flexible, we study three different types of expectations: Bochner integrals, Pettis integrals and Dunford integrals.One of the problems studied is whether two random variables with the same injective moments (of a given order) necessarily have the same projective moments; this is of interest in applications. We show that this holds if the Banach space has the approximation property, but not in general.Several sections are devoted to results in special Banach spaces, including Hilbert spaces, CpKq and Dr0, 1s. The latter space is nonseparable, which complicates the arguments, and we prove various preliminary results on e.g. measurability in Dr0, 1s that we need.One of the main motivations of this paper is the application to Zolotarev metrics and their use in the contraction method. This is sketched in an appendix. 1 NotationsWe will use the following standard notations, usually without comment. LpB 1 , . . . , B k ; B 1 q is the space of bounded k-linear maps B 1ˆ¨¨¨ˆBk Ñ B 1 . In particular, with B 1 " R, we have the space of bounded k-linear forms. When B 1 "¨¨¨" B k " B we also write LpB k ; B 1 q.B˚denotes the dual space of the Banach space B, i.e., the space LpB; Rq of bounded linear functionals B Ñ R. If x P B and x˚P B˚, we use the

A sharpened version of Carleman's inequality is proved. This result unifies and generalizes some recent results of this type. Also the ''ordinary'' sum that serves as the upper bound is replaced by the corresponding Cesaro sum. Moreover, a Carleman-type inequality with a more general measure is proved and this result may also be seen as a generalization of a continuous variant of Carleman's inequality, which is usually referred to as Knopp's inequality. A new elementary proof of (Carleman-)Knopp's inequality and a new inequality of Hardy-Knopp type is pointed out. # 2002 Elsevier Science (USA)

Abstract.A recently discovered Hardy-Pólya type inequality described by a convex function is considered and further developed both in weighted and unweighted cases. Also some corresponding multidimensional and reversed inequalities are pointed out. In particular, some new multidimensional Hardy and Pólya-Knopp type inequalities and some new integral inequalities with general integral operators (without additional restrictions on the kernel) are derived.Mathematics subject classification (2000): 26D10, 26D15.

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In this paper we shall study Hardy spaces of analytic functions in a strip S. Our main result is on one hand an intrinsic characterization of the spaces and on the second that polynomials are dense. We also present an orthogonal (in H 2 (S)) basis of polynomials.