Abstract: Using a new simpler analytic method, we prove that the K_n-functioon class （n=1, 2, …） and also p-function class defined on an additive semigroup T are respectively closed under pointwise multiplication. At the same time we obtain the following inequalities: if p is a K_n-function defined on T, then \begingatheredF_k\left(t_1, \cdots, t_k ; \boldsymbolp^r\right) \geqslant\leftF_k\left(t_1, \cdots, t_k ; \boldsymbolp\right)\right^r,(k=1,2, \cdots, n) \\ \sum_k=1^n F_k\left(t_1, \cdots, t_k ; \boldsymbolp^r\right) \leqslant\left\sum_k=1^n F_k\left(t_2, \cdots, t_k ; \boldsymbolp\right)\right^r,\endgathered, where r is a positive integer and t₁, …, t_n∈T. Moreover, for the k_n-function p defined ou T, these inequalities are also true for every real number r≥1.