The strict upper bound of ranks of commutator subgroups of finite $p$-groups

All groups in the abstract are finite.We define rank d(G) of a p-group G as the minimal number of generators of G. Let p be any prime number, k1, . . ., kn -positive integers, n ≥ 2. By D(k1, . . ., kn) we denote the number of sequences i1, . . ., i k in which k ≥ 2, i1, . . ., i k are positive integers from [1, n], i1 > i2, i2 ≤ • • • ≤ i k and for any j ∈ [1, n] number j may not occur in such sequences more than (p k j -1) times.We prove that for any p-group G generated by elements a1, . . ., an of orders p k 1 1 , . . ., p kn n (n ≥ 2) the inequality d(G ′ ) ≤ D(k1, . . ., kn, p) is true and the equality in this inequality is attainable.Also, we prove that for any p-group G generated by elements a1, . . ., an of orders p k 1 1 , . . ., p kn n (n ≥ 2), with elementary abelian commutator subgroup G ′ the class of nilpotency of G ′ does not exceed p k 1 1 +• • •+p kn n -n and this upper bound is also attainable.