We investigate Carlson type inequalities for finite sums, that is, inequalities of the form ∑mk=1ak < C (∑mk=1ka1akr+1) μ(∑mk=1ka2akr+1)λ, to hold for some constant C independent of the finite, non-zero set a1,⋯,am 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