Carlson type inequalities for finite sum and integrals on bounded intervals

We investigate Carlson type inequalities for finite sums, that is, inequalities of the form (m)Sigma(k=1) ak<C((m)Sigma(k=1) k(alpha1) alpha(k)(r+1))(mu) ((m)Sigma (k=1) k (alpha2) alpha (r+1) (k)) (lambda) to hold for some constant C independent of the finite, non-zero set a(1),...,a(m) of non-negative numbers. We find constants C which are strictly smaller than the sharp constants in the corresponding infinite series case. Moreover, corresponding results for integrals over bounded intervals are given and a case with any finite number of factors on the right-hand side is proved.