Bounding the maximal size of independent generating sets of finite groups

Denote by $m(G)$ the largest size of a minimal generating set of a finite group $G$. We estimate $m(G)$ in terms of $sum_{pin pi(G)}d_p(G),$ where we are denoting by $d_p(G)$ the minimal number of generators of a Sylow $p$-subgroup of $G$ and by $pi(G)$ the set of prime numbers dividing the order of $G$.