a) Is true the following statement. Let $h$ be analytic in the unit disk such that $$|h(z)|\le \frac{|z|^2}{1-|z|^2},$$ then $$|h'(z)|\le \frac{2}{(1-|z|^2)^2}.$$ 
a') Is true the following statement. Let $h$ be analytic in the unit disk such that $$|h(z)|\le \frac{|z|^2}{1-|z|^2},$$ then the inequality $$|h'(z)|\le \frac{8}{\pi(1-|z|^2)^2}$$ is sharp. The inequality can be proved by using Schur test, and Riesz-Thorin convexity type theorem (Dunford & Schwartz 1958, §VI.10.11).


b) If $$|h(z)|\le \frac{|z|^2}{|1-z^2|}$$ then we have better conclusion $$|h'|\le \frac{2|z|}{(1-|z|^2)|1-z^2|}$$ and this follows by using Schwarz lemma. Namely in this case $$|H(z)|=|(1-z^2) h(z)/z^2|\le 1.$$ Then $$|H'(z)|\le \frac{1-|H(z)|^2}{1-|z|^2}.$$


As $$H'(z)=(1-z^2) h'(z)/z^2-2/z^3 h(z),$$ it follows that $$|(1-z^2) h'(z)/z^2|\le \frac{2(1-|z|^2)/|z|^3 h(z)+1-|H(z)|^2}{1-|z|^2}$$ $$\le \frac{2|H(z)|/|z| +1-|H(z)|^2}{1-|z|^2}\le \frac{2|z|^{-1}}{1-|z|^2}.$$


The question a) is related to precise estimation of norm of a Bergman projection into Bloch space and is far for being a homework.



Schwarz type inequality
Let $|z|=r$, apply the Cauchy estimate to the disc $|\zeta-z|<(1-r)/2$.
We obtain
$$|f'(z)|\leq \frac{2}{1-r}\frac{(1+r)^2}{(1-r)(3+r)}.$$
Maximizing the factor $(1+r)^2/(3+r)$ by Calculus, we obtain that is it at most $1$.


This gives 
$$|f'(z)|\leq\frac{2}{(1-|z|)^2}$$
which is worse than conjectured only by a factor of $(1+|z|)^2$, which is at most $4$.
Perhaps one can improve the constant by applying Cauchy to a disc of radius $t\in(0,1-r)$,
and then optimizing in $t$, which leads to solving a cubic equation.


It is not likely that a simple extremal function exists, and probably for each $z$
there will be a different extremal function.


