Inequalities for Minkowski sums of convex bodies play a central role in convex geometry and additive combinatorics. A particular  example is the Plünnecke–Ruzsa-type inequality, which also has a geometric analogue for convex bodies. In this talk, we begin by introducing such geometric inequalities and highlighting some of the open problems related to them. We then present functional analogues of Plünnecke–Ruzsa-type inequalities using the alpha-sum, a generalized sup-convolution. Our focus is on integral inequalities for alpha-concave functions, a class that extends log-concave functions. We conclude by discussing sharp constants in the case of decreasing log-concave functions supported on the positive orthant.