On the number of characters in blocks of quasi-simple groups

We prove, for primes $p\ge 5$, two inequalities between the fundamental invariants of Brauer $p$-blocks of finite quasi-simple groups: the number of characters in the block, the number of modular characters, the number of height zero characters, and the number of conjugacy classes of a defect group and of its derived subgroup. For this, we determine these invariants explicitly, or at least give bounds for them for several classes of classical groups.