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Re: Problem 101, amended

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  • adh_math
    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 aside, Jason asked about TeX (notation I used in
      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. $R^n$).
    • sum_what_ever
      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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