The recent question 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 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 implies some estimates but I don't have the book handy and from what I recall of skimming it, there was none.