1804 CHAPTER 57. INFINITE PRODUCTS

57.5 Blaschke ProductsThe Blaschke3 product is a way to produce a function which is bounded and analytic onB(0,1) which also has given zeros in B(0,1) . The interesting thing here is that there maybe infinitely many of these zeros. Thus, unlike the above case of Jensen’s inequality, thefunction is not analytic on B(0,1). Recall for purposes of comparison, Liouville’s theoremwhich says bounded entire functions are constant. The Blaschke product gives examples ofbounded functions on B(0,1) which are definitely not constant.

Theorem 57.5.1 Let {αn} be a sequence of nonzero points in B(0,1) with the propertythat

∞

∑n=1

(1−|αn|)< ∞.

Then for k ≥ 0, an integer

B(z)≡ zk∞

∏k=1

αn− z1−αnz

|αn|αn

is a bounded function which is analytic on B(0,1) which has zeros only at 0 if k > 0 and atthe αn.

Proof: From Theorem 57.0.2 the above product will converge uniformly on B(0,r) forr < 1 to an analytic function if

∞

∑k=1

∣∣∣∣ αn− z1−αnz

|αn|αn−1∣∣∣∣

converges uniformly on B(0,r) . But for |z|< r,∣∣∣∣ αn− z1−αnz

|αn|αn−1∣∣∣∣

=

∣∣∣∣ αn− z1−αnz

|αn|αn− αn (1−αnz)

αn (1−αnz)

∣∣∣∣=

∣∣∣∣∣ |αn|αn−|αn|z−αn + |αn|2 z(1−αnz)αn

∣∣∣∣∣=

∣∣∣∣∣ |αn|αn−αn−|αn|z+ |αn|2 z(1−αnz)αn

∣∣∣∣∣= ||αn|−1|

∣∣∣∣ αn + z |αn|(1−αnz)αn

∣∣∣∣= ||αn|−1|

∣∣∣∣1+ z(|αn|/αn)

(1−αnz)

∣∣∣∣≤ ||αn|−1|

∣∣∣∣1+ |z|1−|z|

∣∣∣∣≤ ||αn|−1|∣∣∣∣1+ r1− r

∣∣∣∣3Wilhelm Blaschke, 1915