List of top Mathematics Questions on Absolute maxima and Absolute minima asked in GATE MA

Let \( {H} = \{ z \in {C} : \operatorname{Im}(z)>0 \ \) and \( {D} = \{ z \in {C} : |z|<1 \} \). Then \[ \sup \{ |f'(0)| : f { is an analytic function from } {D} { to } {H} { and } f(0) = \frac{i}{2} \} \] is equal to:}

Let \( A \in M_n({C}) \) be a normal matrix. Consider the following statements:
1. If all the eigenvalues of \( A \) are real, then \( A \) is Hermitian.
2. If all the eigenvalues of \( A \) have absolute value 1, then \( A \) is unitary.
Which one of the following is correct?