• 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 D_R = {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 D_R be z₁, . . ., z_n . (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

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

It is obviously analytic in D_R. Furthermore, we have |g(z)| = |f(z)| when |z| = R. This implies that |g(w)| ≤ ||f||_R for all w ∈ D_R. [If we had a point w_∗ in the interior of D_R 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

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

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 D_R. Then again, as in exercise 12.1, we have an analytic function g, defined in a neighborhood of D_R, with f = e^g. 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

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 D_R. This second case is that the function f is simply f(z) = z − ζ. We then define a new function, namely