From: elkies@ramanujan.math.harvard.edu (Noam Elkies)
Newsgroups: sci.math.research
Subject: Re: area of image of z f(z) for analytic f
Date: 30 Apr 1995 17:52:19 GMT

In article <3nudod$a7l@nnrp.ucs.ubc.ca>
pruss@unixg.ubc.ca (Alexander Pruss) writes:
>Let f be an analytic function on the unit disc.  Let g(z)=zf(z), also on
>the unit disc.  Can the area of the image of f ever exceed that of the image 
>of g?  (We are not counting multiplicities, just the planar area of the 
>image domain.)

Yes.

If  f = sum(a_n * z^n, n=0..inf)  then (as you surely know) the area
of the image of f, with multiplicities, is pi times the sum of n*|a_n|^2.

Now let f(z) = z + c*z^3, with c nonzero but small enough that f is 1:1.
Then f has image of area (1+3|c|^2)*pi.  As to g(z) = z^2 + c*z^4, the
area of its image is the same as that of z + c*z^2, which is at most
(1+2|c|^2)*pi.

More generally f can be any 1:1 odd function other than a multiple of z,
using much the same proof.

--Noam D. Elkies (elkies@math.harvard.edu)
  Dept. of Mathematics, Harvard University