This is a vague question, and I know nothing about this area.
We fix some $c\in\mathbb C$ and iterate the map $z\mapsto z^2+c$. This gives some filled Julia set, i.e. the set of points $z\in\mathbb C$ so that the orbit of $z$ is bounded. For example, for $c=0$ this is the closed unit disk. I'm wondering if anyone knows about any relationship between the area of the Julia set and the position of $c$ in the Mandelbrot set.
Problem A-1 in Milnor's book Dynamics in one complex variable contains a formula for the area expressed as a series based on Gronwall's area theorem: $$ \pi (1 - |a_2|^2 - 3|a_4|^2 - 5|a_6|^2 - \cdots) $$ The series is said to converge slowly. The coefficients of the series can be easily computed recursively though by solving $$ \psi(w^2) = \psi(w)^2+c $$ for $$ \def\F#1{\frac{a_{#1}}{w^{#1}}} \psi(w) = w(1 + \F2 + \F4 + \F6 + \cdots) $$
Here is graph of the area for $c \in [-2,0.25]$ obtained by truncating the series at several points (the better estimates are the inner ones):
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The topmost graph represents the estimate $A \le \pi (1 - |a_2|^2) = \pi (1 - |c|^2/4)$.
