The recent question [Area of the boundary of the Mandelbrot set ?](https://mathoverflow.net/questions/37229/area-of-the-boundary-of-the-mandelbrot-set) prompted me to ask this question.


There has been some work on estimates for the area of the Mandelbrot set, e.g., a [paper](https://doi.org/10.1007/BF01385497) by John H. Ewing and Glenn Schober in *Numerische Mathematik*. Is there similar work for the estimating the area of quadratic filled Julia sets as a function of the parameter $c$? Perhaps the material in the book [Computability of Julia Sets](https://link.springer.com/book/10.1007/978-3-540-68547-0) implies some estimates but I don't have the book handy and from what I recall of skimming it, there was none.



Area of filled Julia sets
This paper contains some information about the area of filled Julia sets, though not a formula:



> 
> Yang, Guoxiao, [Some geometric properties of Julia sets and filled-in Julia sets of polynomials](https://doi.org/10.1080/02781070290013811). Complex Var. Theory Appl. 47 (2002), no. 5, 383–391. [MR1906990 (2003c:37067)](https://mathscinet.ams.org/mathscinet-getitem?mr=1906990), [Zbl 1028.30019](https://zbmath.org/?q=an%3A1028.30019)
> 
> 
> 


There is also this more promising paper by the same author, but it is in Chinese and I can't get a copy anyway:



> 
> Yang, Guo Xiao, The area and diameter of filled-in Julia sets and Mandelbrot sets.
> Acta Math. Sinica 38 (1995), no. 5, 607–613. [MR1372560 (96m:30040)](https://mathscinet.ams.org/mathscinet-getitem?mr=1372560), [Zbl 0895.30018](https://zbmath.org/?q=an%3A0895.30018)
> 
> 
> 


If someone knows these papers, I'd be grateful for any insights.


Problem A-1 in Milnor's *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
$$
\def\F#1{\frac{a\_{#1}}{w^{#1}}}
 \psi(w) = w(1 + \F2 + \F4 + \F6 + \cdots)
$$


