An inequality of block invariants (Q1364315)

Let \(B\) be a block of a finite group \(G\) with normal abelian defect group \(D\). The author proves that \(k(B)+|D|\leq(e_B+1)m_B\) where \(k(B)\) is the number of irreducible ordinary characters in \(B\), \(e_B\) is the inertial index of \(B\) and \(m_B\) is the number of (major) \(B\)-subsections. Moreover, if the inertial factor group of \(B\) is cyclic then \(2m_B\sqrt{e_B}\leq k(B)+|D|\).