## Re: Problem 101, amended

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• Suppose f is analytic and bounded in the open unit disk D, admits a continuous extension to the closed disk minus {1}, and has norm <= 1 on the boundary
Message 1 of 4232 , Nov 1, 1999
Suppose f is analytic and bounded in the open
unit disk D, admits a continuous extension to the
closed disk minus {1}, and has norm <= 1 on the
boundary (minus {1}). Show |f| <= 1 in D.
<br><br>First, it suffices to show |f(0)| <= 1 since every
point of D can be carried to 0 by a disk automorphism,
and this normalization does not change the other
hypotheses (but makes the Cauchy integral formula much
easier to use :).<br>Next, consider a contour \gamma of
the following form: \gamma traces out most' of the
unit circle, but makes a small detour near 1 (so the
enclosed region is a notched' unit disk). Standard
estimates show that the contribution from the detour can be
made arbitrarily small (by taking the detour to be
short, since f is bounded), while the contribution from
the portion of \gamma on dD is no larger than 1 in
absolute value. Formally'<br>|f(0)| = 1/(2\pi) |int(0,
2\pi, f(e^{i\theta}), d\theta)| (Cauchy integral
formula)<br><= 1/(2\pi) int(0, 2\pi, |f(e^{i\theta})|, d\theta)
(triangle inequality)<br><= 1 (|f| bounded above by 1 on
dD);<br>the gymnastics are required to control the integral
near 1.<br><br>Don't remember the agreed-upon notation
for integrals is in this club, but have used a
modification of what I recall to be the notation for sums. As
an earlier post), and I forgot to answer. <br>TeX
(pron: tech, with a soft k) is a typesetting language
for mathematics, developed by Donald Knuth and now
(in various incarnations) widely used in mathematics
and physics publishing. A document is prepared with a
text editor, then compiled' with TeX into a printable
binary file (with the extension dvi, for DeVice
Independent). Mathematical symbols (sums, integrals, Greek
letters, exotic binary operators, etc) are produced by
short descriptive commands beginning with a backslash
(examples: \sum, \int, \alpha, \beta,..., \times, \otimes,
\oplus, \leq, \geq...dozens of them). Subscripts are
produced with an underscore, superscripts with a caret. On
sums and integrals, sub- and superscripts become lower
and upper limits. Thus<br>\sum_{i=1}^n (x_i)^2 = 1 is
the equation of the unit sphere in R^n. In actual
use, mathematical symbols are delimited (so that TeX
knows the scope of the mathematics in the input file),
usually with dollar signs (e.g. $R^n$).
• can t sleep (yawn), i spent around 3 hr.s on your your original post here. it seems this implicit function solution set is the semi-stable modular eliptical
Message 4232 of 4232 , Apr 25, 2011
can't sleep (yawn), i spent around 3 hr.s on your your original post here. it seems this implicit function solution set is the semi-stable modular eliptical curves A. Wiles gave in his proof of fermat's last theorem. Your function is way simpler. have to get some sleep before i got to go to work, but the proof of this is around 4 to 5 pages as you said. i'll have to look at it more this evening. also, what's the scoop of you being the source for the plot of tron legacy? we should all get royalties! heheh.
--- In mathforfun@yahoogroups.com, jeshields_98 wrote:
>
> I hit the wrong button...<br><br>It should be = c^n<br>not + c^n<br><br>very VEEERRRRRYYYYY important<br><br>sorry
>
I
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