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We unify Littlewood's classical 4/3-inequality (a forerunner of Grothendieck's inequality) together with its m-linear extension due to Bohnenblust and Hille (which originally settled Bohr's absolute convergence problem for Dirichlet series) with a scale of inequalties of Bennett and Carl in p -spaces (which are of fundamental importance in the theory of eigenvalue distribution of power compact operators). As an application we give estimates for the monomial coefficients of homogeneous p -valued polynomials on c 0 .

ABSTRACT. We investigate the summability of the coefficients of m-homogeneous polynomials and m-linear mappings defined on ℓ p -spaces. In our research we obtain results on the summability of the coefficients of m-linear mappings defined on ℓ p 1 × · · · × ℓ p m . The first results in this respect go back to Littlewood and Bohnenblust and Hille (for bilinear and mlinear forms on c 0 ) and Hardy and Littlewood and Praciano-Pereira (for bilinear and m-linear forms on arbitrary ℓ p -spaces). Our results recover and in some case complete these old results through a general approach on vector valued m-linear mappings.

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