In his pioneering paper *An extension of Schwarz’s lemma*, Ahlfors proves the lower bound on the Bloch constant $B geq frac{sqrt{3}}{4}$. The proof of this lower bound proceeds as follows:

Let $W$ be a Riemann surface. For an arbitrary point $mathfrak{m}$ of $W$, let $rho(mathfrak{m})$ denote the radius of the largest simple circle of centre $mathfrak{m}$ contained in $W$. It is clear that $rho(mathfrak{m})$ is continuous, and is zero only at the branch points. We introduce the metric $ds = lambda | dw |$, with $$lambda = frac{A}{2 sqrt{rho} (A^2-rho)},$$ where $rho = rho(mathfrak{m})$ and $w$ denotes the variable on the function plane (not the uniformizing variable), and $A^2>B(f)$ (the Bloch constant of $f$).

If $mathfrak{a}$ is a branch point, then $rho = | w – mathfrak{a} |$. Let $n$ be the multiplicity of $mathfrak{a}$. Then $w_1 = (w-mathfrak{a})^{frac{1}{n}}$ is a uniformizing variable the corresponding $lambda_1$ is determined from $lambda_1 | dw_1 | = lambda | dw |$. Explicitly, $$lambda_1 = n rho^{frac{1}{2} – frac{1}{n}} / 2(A^2 – rho).$$ For $n=2$ the metric is regular, and for $n>2$, the metric is zero.

Ahlfors gives no explanation for why the $lambda$ above is chosen. Is it simply constructed so that, with respect to the uniformizing variable, we obtain a continuous pseudo metric? Can one choose another metric to start with?