which one can be taken as the upper bound for $|f'(0)|$?

Question

If $f(z)$ is analytic on $D$ where $D=\{(x,y):|x|\leq a, |y|\leq b, a \ge b\}$. If $f(z)$satisffies the inequality $|f(z)|\leq M$ on the boundary of $D$ then which one can be taken as the upper bound for $|f'(0)|$?

(A)$\frac{2M(a+b)}{\pi a b}$

(B)$\frac{2\pi a b}{M^2}$

(C)$\frac{2M^2}{ 2\pi a b}$

(D)$\frac{2M(a+b)}{ \pi b^2}$

My ideas on this problem.

$|f'(0)|=|\frac{1}{2\pi i}\int_{\gamma}\frac{f(z)}{z^2}dz|\leq \frac{1}{2\pi }\int_{\gamma}|\frac{f(z)}{z^2}||dz|\leq \frac{M}{2\pi }\frac{\int_{\gamma}|dz|}{|ab|}$

($\because |z|\ge |b|$ and $|z|\ge |a| \implies |z|^2\ge |ab|$)

Answer 1

This is not correct for all $z$ on the boundary of $D$:

$\because |z|\ge |b|$ and $|z|\ge |a| \implies |z|^2\ge |ab|$

We can say that $|z|\ge |b|$ or $|z|\ge |a|$. Since $a \ge b$ it follows that $|z^2| \ge b^2$ in any case, and this leads to estimate (D).