Thursday, May 29, 2014

Complex Analysis: #30 Jensen`s Formula

  • Complex Analysis: #30 Jensen`s Formula

But before doing that, we look at the easier Jensen’s inequality.


Theorem 51 (Jensen’s Inequality)
Let R > 0 be given and let the (non-constant) analytic function f be defined in a region containing the closed disc DR = {z ∈ ℂ : |z| ≤ R}. Assume f(0) ≠ 0 and also f(z) ≠ 0 for all z with |z| = R. Let the zeros of f in DR be z1, . . ., zn . (Here a zero is listed m times if it is a zero of order m.) We assume the zeros are ordered according to their increasing absolute value. Let ||f||R = max{|f(z)| : |z| = R}. Then we have

Complex Analysis: #30 Jensen`s Formula equation pic 1

More generally, thinking about various values of R, let vf(R) = n be the number of zeros of f in DR, where R, thus n, is allowed to vary. Then we have Jensen’ inequality:

Complex Analysis: #30 Jensen`s Formula equation pic 2

It is obviously analytic in DR. Furthermore, we have |g(z)| = |f(z)| when |z| = R. This implies that |g(w)| ≤ ||f||R for all w ∈ DR. [If we had a point w in the interior of DR with |g(w)| > ||f||R, then we can assume that it is maximal with respect to this property. However, that would contradict theorem 19.] Therefore

Complex Analysis: #30 Jensen`s Formula equation pic 3


Theorem 52 (Jensen’s Formula)
The same assumptions as in the previous theorem. Then we have

Complex Analysis: #30 Jensen`s Formula equation pic 4

Proof
In this proof, we will first look at two very simple cases:
1. We first prove Jensen’s formula in the simple case that there are no zeros of f in DR. Then again, as in exercise 12.1, we have an analytic function g, defined in a neighborhood of DR, with f = eg. Or put another way, g = log f. (To be definite, we could specify that log f(0) should be in the principle branch of the logarithm.) Then Cauchy’s formula is simply

Complex Analysis: #30 Jensen`s Formula equation pic 5

which establishes the theorem in the first case.

2. The second case is even simpler. Namely, let ζ be a complex number with |ζ| < R. That is, ζ is some point in the interior of DR. This second case is that the function f is simply f(z) = z − ζ. We then define a new function, namely

Complex Analysis: #30 Jensen`s Formula equation pic 6

2 comments:

  1. Please show the related Mittag-Leffler approach.

    ReplyDelete
    Replies
    1. It's been some time since I went through this. So, I don't think I'll be updating these sections.

      Sorry. Try to find it on the web.

      Delete

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