Let $f$ be a non-constant entire function with $f ( 0 ) = 0$. Let $u$ and $v$ be the real and imaginary parts of $f$ respectively. Let $R > 0$ and
$$B = \sup \{ u ( z ) : | z | = R \}$$
(A) (2 marks) Show that $B > 0$.\\
(B) (2 marks) Consider the function
$$F ( z ) : = \frac { f ( z ) } { z ( 2 B - f ( z ) ) }$$
Show that $F$ is analytic on the open ball with radius $R$ and continuous on the boundary $\{ z : | z | = R \}$.\\
(C) (3 marks) Show that $\sup \{ | F ( z ) | : | z | = R \} \leq \frac { 1 } { R }$.\\
(D) (3 marks) Show that
$$\sup \left\{ | f ( z ) | : | z | = \frac { R } { 2 } \right\} \leq 2 B$$