The paper offers an overview of Aumann integral inequalities for interval-valued functions, such as the interval Minkowski inequality, or interval Beckenbach inequality. Similarly, some Sugeno integral-based inequalities for non-negative real functions well known from the literature are added. The main results of the paper deal with fuzzy-interval-valued functions and are based on Aumann level integral, denoted by the authors as FA-integral (Fuzzy Aumann integral). They have proved related versions of Minkowski's and Beckenbach's FA-integral inequalities for fuzzy-interval-valued functions and exemplified these inequalities by some examples. Note also that their introduction of p-integrability for fuzzy-interval-valued functions considered in presented inequalities allows to introduce a new type of \(L_p\)-type fuzzy-intervals spaces.