A problem on Schwarz Lemma

Question

I am currently solving problems on Schwarz's Lemma from Conway's book and stumbled in one problem (prob-3,Page-133).

Let $D=\{z\in \mathbb{C} \mid |z|<1\}$ be the open unit disk and $f:D\to \mathbb{C}$ be an analytic function such that $Ref(z)\geq 0$ for all $z\in D$.Then show that

1. Re$f(z)>0$ for all $z\in D$.

2. If $f(0)=1$,use appropriate Möbius Transformation and Schwarz's lemma to show that $|f(z)|\leq \frac{1+|z|}{1-|z|}$ for all $z\in D$.

3. Also show that $\big|f(z)\big|\geq \frac{1-|z|}{1+|z|}$ for all $z\in D$.

I could solve the first two problems but can't solve the third.

The first one I solved by Using Open Mapping theorem and the second one by using the Möbius Transform $T$ which maps the imaginary axis to $\partial D$ and right half plane to $D$ given by $T(z)=\frac{z-1}{z+1}$ and applying Schwarz's lemma to $T\circ f$. But can't think of the third one. Any suggestion will be appreciated.

Thanks.