Let $f$ be a holomorphic function on $|z|<1$ with $|f(z)|<1$ for all $|z|<1$.
(1) Find necessary and sufficient conditions for equality of $$\frac{|f'(z)|}{1-|f(z)|^2} \leq \frac{1}{1-|z|^2}$$ for all $|z| < 1.$
(2) If $f(\frac{1}{2}) = \frac{1}{3}$, find a sharp upper bound for $|f'(\frac{1}{2})|.$
I know that the inequality is Schwarz-Pick Lemma, and I think that it has something to do with Mobius Transformation (Linear fractional transformation) which I supposed that it will be introduced later in my textbook (Complex Analysis in Spirit of Lipman Bers, second edition). This problem is in chapter 6, and the Mobius should be introduced in chapter 8. So, actually, I do not know much about Mobius Transformation. So I do not have any clear idea for the condition concerning equality of the inequality above. Also, I do not know what is a sharp upper bound. I found the definition on http://en.wikipedia.org/wiki/Upper_and_lower_bounds ,but I do not fully understand what it means, and, according to the question, what I have to do with $|f'(\frac{1}{2})|$