I am trying to prove that for schlicht functions, that is, univalent functions $f$ from the unit disc to $\mathbb{C}$ of the form $$f(z) = z + a_2 z^2 + a_3 z^3 + \cdots,$$ we always have $|a_3| \leq 3$. This is a well-known result. The Bieberbach conjecture states that a much more general result is true: that $|a_n|\leq n$ for all $n\geq 2$. However, I'm only interested in the third coefficient, and I have read enough to know that the proof of the whole Bieberbach conjecture, originally by de Branges and simplified by others, is not the easiest way to bound $a_3$.
The standard proof of this particular bound was given by Loewner in 1923 and I'm familiar with it thanks to Duren's 1983 book Univalent Functions, where Loewner's proof is discussed in section 3.3.
However, Duren leaves out many details and I would like to find another source. It doesn't have to be more detailed, exactly, but it would be very useful to see another author's take on the proof. I'm looking into Pommerenke's Univalent Functions from 1975 but it will take a little while for me to obtain the book.
Loewner's original article is in German and besides I'm sure Duren's reproduction of it is already more detailed. I have found a paper by Richard Pell advertising itself as a simple proof of this third coefficient bound, but it's only a condensed version of Duren's presentation of Loewner's proof and it's therefore not at all useful.
No book in complex analysis that I've found (and I've checked many of the standard ones) that is not specifically about univalent functions or conformal maps has any proper focus on the Bieberbach conjecture, and even Goluzin's Geometric Theory of Functions of a Complex Variable only refers to the conjecture in a supplement, where there is a proof of the bound on $a_4$ but not $a_3$.
Other sources would be appreciated.