the question is from Conway's Functions of One Complex Variable, volume I,second edition, chapter VI section 1, exercise 7.
Let $f$ be analytic in the disk $B(0,R)$ and for $0 \leq r \leq R$ define $A(r)=\max\{\operatorname{Re} f(z) : |z|=r\}.$
Show that unless $f$ is constant, $A(r)$ is strictly increasing function of $r$.
Now obviously from the maximum modulus we must have for any $r_1< r_2$ and $|z|=r_1$,$|\zeta|=r_2$, $|f(z)|\geq |f(\zeta)|\geq \operatorname{Re} f(\zeta)$, but don't see how use for the real parts here.
Only hints if you can.
Thanks.