Let, ( ) be the best constant that fulfills the following inequality: for every -homogeneous polynomial ( ) = ∑ | |= in complex variables,For every degree , and a wide range of values of , ∈ [1, ∞] (including any in the case ∈ [1, 2], and any and for the 2-homogeneous case), we give the correct asymptotic behavior of these constants as (the number of variables) tends to infinity.Remarkably, in many cases, extremal polynomials for these inequalities are not (as traditionally expected) found using classical random unimodular polynomials, and special combinatorial configurations of monomials are needed. Namely, we show that Steiner polynomials (i.e., -homogeneous polynomials such that the multi-indices corresponding to the nonzero coefficients form partial Steiner systems), do the work for certain range of values of , . As a byproduct, we present some applications of these estimates to the interpolation of tensor products of Banach spaces, to the study of (mixed) unconditionality in spaces of polynomials and to the multivariable von Neumann's inequality. K E Y W O R D SHardy-Littlewood inequalities, multivariable von Neumann's inequality, unconditionality in spaces of polynomials, unimodular polynomials M S C ( 2 0 1 0 ) 11C08, 15A60, 15A69, 46G25, 47A30, 47H60 Problem 1.3. Let , ( ) and , ( ) be the smallest constants that fulfill the following inequalities: for every -homogeneous polynomial in complex variables,

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Let K (B ℓ n p , B ℓ n q ) be the n-dimensional (p, q)-Bohr radius for holomorphic functions on C n . That is, K (B ℓ n p , B ℓ n q ) denotes the greatest constant r ≥ 0 such that for every entire function f (z) = α c α z α in n-complex variables, we have the following (mixed) Bohr-type inequalitywhere B ℓ n r denotes the closed unit ball of the n-dimensional sequence space ℓ n r . For every 1 ≤ p, q ≤ ∞, we exhibit the exact asymptotic growth of the (p, q)-Bohr radius as n (the number of variables) goes to infinity.where a n ∈ C and s = σ + i t is a complex variable. The regions of convergence, absolute convergence and uniform convergence of these series define half-planes of the form [Re(s) > σ 0 ] in the complex field. Bohr was mainly interested in controlling the region of convergence of a series. To achieve this, he related different types of convergence and focused on finding the width of the greatest strip for which a Dirichlet series can converge uniformly but not absolutely. This question is popular and known nowadays as the Bohr's absolute convergence problem.Although the solution of this problem problem appeared two decades after it was proposed (given by Bohnenblust and Hille [BH31] who showed that the maximum width of this strip is 1 2 ), Bohr made major contributions in the area (arguably, even more important than the solution of the problem 2010 Mathematics Subject Classification. 32A05,32A22 (primary),46B15,46B20,46G25,46E50 (secondary).