This paper discusses the relationships among the \(H^ \infty\) norm \(\| f\|_ \infty\), BMOA norm \(\| f\|_ *\), Bloch norm \(\| f\|_ B\) and \(H^ 2\) norm \(\| f\|_ 2\) of a function \(f\) analytic in the unit disk \(D\). Two classes of BMOA functions \(f\) in the form of power series were found to have the properties \(\| f\|_ *= \| f\|_ 2\) and \(\| f\|_ *= \| f\|_ \infty\) respectively. A sufficient and necessary condition for a function \(f\) analytic in \(D\) and continuous on \(\overline D\) to have property \(\| f\|_ *= \| f\|_ B\) is given, and a Blaschke product which has the same property is constructed.