Eigenvalue inequalities for positive block matrices with the inradius of the numerical range

We prove the operator norm inequality, for a positive matrix partitioned into four blocks in [Formula: see text], text] where is diameter of largest possible disc numerical range text]. This shows that inradius satisfies Several eigenvalue inequalities are derived. In particular, if normal whose spectrum lies radius third full bounded by second sum diagonal block, think optimal and we propose conjecture related to inequality Hayashi.