ON JENSEN’S FORMULA

Daniel Girela

Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain e-mail: girela@uma.es

Abstract. Jensen’s formula establishes an intimate connection between the values of an analytic function on a circle and the distribution of its zeros inside the circle. The standard proof of this formula involves showing that a certain integral is zero by means of the residue theorem. We present a proof of this fact which avoids the use of complex integration.

We also show that an important result concerning the boundary behaviour of Blaschke products can be obtained as a direct consequence of Jensen’s formula.

Jensen’s formula is a very important theorem in Complex Analysis. It establishes an intimate connection between the values of an analytic function on a circle and the distribution of its zeros inside the circle.

Jensen’s formula. Let f be an analytic function in the disc {z ∈ ℂ : |z| < R} with f(0) ≠ 0. If 0 < r < R and z_{1}, z_{2}, …, z_{n} are the zeros of f in {|z| ≤ r}, repeated according to multiplicity, then

\frac{1}{2π} ∫_{−π}^{π} log |f(re^{iθ})| dθ = log |f(0)| + ∑_{k=1}^{n} log \frac{r}{|z_{k}|}.

The standard proof of Jensen’s involves showing that

(1)  ∫_{−π}^{π} log |1 − e^{iθ}| dθ = 0,

which is proved by means of the residue theorem (see e.g., [1], [2], [4]). An elementary proof of (1) which avoids the use of the residue theorem is presented in [5].

A. Our first objective in this note is presenting another proof of (1) independent of the residue theorem.

1991 Mathematics Subject Classification. 30D35.

Let Δ denote the unit disc, Δ = {z ∈ ℂ : |z| < 1}. The function

u(z) = log |1 − z|, z ∈ Δ,

is harmonic in Δ and u(0) = 0. Hence,

(2)  ∫_{−π}^{π} log |1 − re^{iθ}| dθ = 0, 0 < r < 1.

Now, since sin x ≥ \frac{2}{π} x if 0 ≤ x ≤ \frac{π}{2} , we have

(3) 4 ≥ |1−re^{iθ}|^{2} = (1−r)^{2} +4r sin^{2} \frac{θ}{2} ≥ \frac{2|θ|^{2}}{π^{2}} ≥ \frac{|θ|^{2}}{π^{2}} , \frac{1}{2} < r < 1, −π ≤ θ ≤ π.

Then, it follows that

(4)  |log |1 − re^{iθ}|| ≤ log 2 + log \frac{π}{|θ|}, \frac{1}{2} < r < 1, −π ≤ θ ≤ π.

Now, it is clear that the function log 2 + log \frac{π}{|θ|} belongs to L^{1} ([−π, π]). Then, using (4), the Lebesgue’s Dominated Convergence Theorem [4] and (2), we obtain

∫_{−π}^{π} log |1 − e^{iθ}| dθ = lim_{r→1} ∫_{−π}^{π} log |1 − re^{iθ}| dθ = 0.

B. One of the applications of Jensen’s formula is showing that if f is a bounded analytic function in Δ, f ≢ 0, and {a_{n}} is the sequence of zeros of f then {a_{n}} satisfies the Blaschke condition

(5)  ∑(1 − |a_{n}|) < ∞.

Nothing more can be said about the sequence of zeros of a bounded analytic function in Δ. Indeed, if {a_{n}} is a sequence of complex numbers with 0 < |a_{n}| < 1 for all n which satisfies (5) then the Blaschke product

(6)  B(z) = ∏_{n=1}^{∞} \frac{|a_{n}|}{a_{n}} \frac{a_{n} − z}{1 − \overline{a}_{n}z}

defines an analytic function in Δ with |B(z)| ≤ 1, for all z, whose zeros are precisely the a_{n}’s.

Since B is bounded and analytic in Δ, it has a non-tangential limit B(e^{iθ}) a.e. An important property is that

(7)  |B(e^{iθ})| = 1, a.e.

This is proved as a consequence of either

(8)  \frac{1}{2π} ∫_{−π}^{π} log |B(e^{iθ})| dθ = 0,

or

(9)  \frac{1}{2π} ∫_{−π}^{π} |B(e^{iθ})| dθ = 1.

The standard proofs or (8) and (9) make use of the sequence of products B_{N}, (N ≥ 1), where, B_{N}(z) = ∏_{n=N}^{∞} \frac{|a_{n}|}{a_{n}} \frac{a_{n}−z}{1−\overline{a}_{n}z} (see [3] and [4]). Next we shall prove that (8) can be obtained as a direct consequence of Jensen’s formula. Indeed, by Jensen’s formula, we have

\frac{1}{2π} ∫_{−π}^{π} log |B(re^{iθ})| dθ = log |B(0)| + ∑_{|a_{j}|≤r} log \frac{r}{|a_{j}|}.

Notice that the right hand side of this equality is clearly an increasing function of r. This and the Lebesgue’s Monotone Convergence Theorem (see [4]) implies that

the limit lim_{r→1} \frac{1}{2π} ∫_{−π}^{π} log |B(re^{iθ})| dθ exists and that

lim_{r→1} \frac{1}{2π} ∫_{−π}^{π} log |B(re^{iθ})| dθ = log |B(0)| + ∑_{j=1}^{∞} log \frac{1}{|a_{j}|}.

But, clearly, log |B(0)| = ∑_{j=1}^{∞} log |a_{j}|. Hence,

(10)  lim_{r→1} \frac{1}{2π} ∫_{−π}^{π} log |B(re^{iθ})| dθ = 0.

Now, using (10), Fatou’s Lemma [4], and the fact that |B| ≤ 1, we obtain

0 = lim_{r→1} \frac{1}{2π} ∫_{−π}^{π} log |B(re^{iθ})| dθ ≤ \frac{1}{2π} ∫_{−π}^{π} log |B(e^{iθ})| ≤ 0.

This gives (8).

Remark. The fact that \frac{1}{2π} ∫_{−π}^{π} log |B(re^{iθ})| dθ is an increasing function of r is usually deduced from the subharmonicity of log |B|. However, we have not used this fact at all.

References

[1] L. V. Ahlfors, Complex Analysis, Third edition, McGraw-Hill, 1979.

[2] J. B. Conway, Functions of One Complex Variable, Second edition, Springer-Verlag, 1978.

[3] P. L. Duren, Theory of H^{p} Spaces, Academic Press, 1970.

[4] W. Rudin, Real and Complex Analysis, Second edition, McGraw-Hill, 1973.

[5] R. M. Young, On Jensen’s formula and ∫_{0}^{2π} log |1 − e^{iθ}| dθ., Amer. Math. Month. 93 (1986), no. 1, 44-45.