Di erentiating Blaschke productsivrii/LAtalk.pdfDi erentiating Blaschke Products Consider the following curious di erentiation procedure: to a Blaschke product F(z) = ei Yd i=1 z a

Differentiating Blaschke products

Oleg Ivrii

April 14, 2017

Differentiating Blaschke Products

Consider the following curious differentiation procedure: to a

Blaschke product

F (z) = e iψd∏

i=1

z − ai1− aiz

, ai ∈ D,

of degree d ≥ 1,

one can associate a Blaschke product B of degree

d − 1 whose zeros are located at the critical points of F .

Theorem. (M. Heins) The above map is a bijection:

Bd/Aut(D)→ Bd−1/S1.



Blaschke product

F (z) = e iψd∏

i=1

z − ai1− aiz

, ai ∈ D,

of degree d ≥ 1, one can associate a Blaschke product B of degree






Blaschke product

F (z) = e iψd∏

i=1

z − ai1− aiz

, ai ∈ D,

of degree d ≥ 1, one can associate a Blaschke product B of degree




Inner functions

An inner function is a holomorphic self-map of D such that for

almost every θ ∈ [0, 2π), the radial limit

limr→1

F (re iθ)

exists and is unimodular (has absolute value 1).

We will denote the space of all inner function by Inn.

BS decomposition

An inner function can be represented as a (possibly infinite)

Blaschke product × singular inner function:

B = e iψ∏i

− ai|ai |· z − ai

1− aiz, ai ∈ D,

∑(1− |ai |) <∞.

S = exp

(−∫S1

ζ + z

ζ − zdσζ

), σ ⊥ m, σ ≥ 0.

Here, the Blaschke factor takes out the zeros, while the singular

inner factor is zero-free.

Nevanlinna class and B(S/S1)O decomposition

The Nevanlinna class N consists of all holomorphic functions f on

D which satisfy

sup0<r<1

1

2π

∫|z|=r

log+ |f (z)|dθ <∞.

B = e iψ∏i

− ai|ai |· z − ai

1− aiz,

S/S1 = exp

(−∫S1

ζ + z

ζ − zdσζ

), σ ⊥ m,

O = exp

(∫S1

ζ + z

ζ − zlog |f (ζ)|dmζ

).

Inner functions of finite entropy

We will also be concerned with the subclass J of inner functions

whose derivative lies in Nevanlinna class:

limr→1

1

2π

∫ 2π

0log+ |F ′(re iθ)|dθ <∞.

Jensen’s formula: the set of critical points ci of F satisfies the

Blaschke condition∑

(1− |ci |) <∞,

and is therefore the zero set of some Blaschke product.




limr→1

1

2π

∫ 2π




(1− |ci |) <∞,





limr→1

1

2π

∫ 2π




(1− |ci |) <∞,


Dyakonov’s conjecture

Conjecture. (K. Dyakonov) Is the following map a bijection?

J /Aut(D)→ Inn /S1, F → InnF ′.

Remark. This mapping generalizes the construction outlined for

finite Blaschke products,

however, in addition to recording the critical set of F , InnF ′ may

also contain a non-trivial singular factor.















Some progress



Theorem. (K. Dyakonov, 2013) z ←→ 1.

Theorem. (D. Kraus, 2007) MBP ∩J ←→ BP.

Theorem. (I, 2017) The map is injective. The image is closed

under multiplication and taking divisors.

Some progress







Some progress







Background on conformal metrics

The curvature of a conformal metric λ(z)|dz | is given by

kλ = −∆ log λ

λ2.

Examples. The hyperbolic metric

λD =|dz |

1− |z |2

has curvature ≡ −4,

while the Euclidean metric |dz | has curvature ≡ 0.


Theorem. (Gauss) The curvature is a holomorphic invariant, that is

kλ(f (z)) = kf ∗λ(z).

Theorem. (Ahlfors) The hyperbolic metric on D is (pointwise) the

largest metric which has curvature ≤ −4.

Theorem. (Liouville) Any metric of curvature ≡ −4 (whose zeros

have integer multiplicity) on a simply-connected domain Ω is of

the form f ∗λD, for some holomorphic function f : Ω→ D.



kλ(f (z)) = kf ∗λ(z).








kλ(f (z)) = kf ∗λ(z).






Heins theory of conformal metrics

An SK-metric is a conformal metric with continuous density and

curvature ≤ −4.

A Perron family Φ of SK-metrics on a domain Ω:

I (Taking maxima) λ, µ ∈ Φ =⇒ max(λ, µ) ∈ Φ.

I (Disk modification) λ ∈ Φ =⇒ MDλ ∈ Φ.

Here, D ⊂ Ω is a round disk and MDλ = λ on Ω \ D and has

curvature ≡ −4 in D.

Theorem. (Heins, 1962) Provided Φ 6= ∅, the metric

λΦ := supλ∈Φ λ has curvature ≡ −4.



curvature ≤ −4.










curvature ≤ −4.








Maximal Blaschke products

Given a countable set C in D, counted with multiplicity, construct

a Blaschke product with critical set C , if one exists.

Procedure:

I Let ΦC be the collection of all SK-metrics that vanish on C .

I Set λC := supλ∈ΦCλ.

I Liouville’s theorem gives a holomorphic map FC : D→ D.

Theorem. (D. Kraus, O. Roth, 2007) If ΦC 6= ∅, then FC is an

indestructible Blaschke product.




Procedure:









Procedure:







Theorem. (M. Heins, 1962) Finite Blaschke products are maximal.

Theorem. (D. Kraus, 2007) If C satisfies the Blaschke condition,

then |BC |λD is an SK-metric, where BC is the Blaschke product

with zero set C .

Theorem. (D. Kraus, O. Roth, 2007) The collection of all “critical

sets of Blaschke products” coincides with “zero sets of functions in

the weighted Bergman space A21.”

In particular, if C1 and C2 are critical sets, their union may not be

a critical set.





with zero set C .





a critical set.





with zero set C .





a critical set.

The fundamental lemma

(1) If FC ∈ MBP ∩J , then λFC≥ |BC |λD by maximality.

(2) In fact, for any F ∈J , λF = F ∗λD ≥ | InnF ′|λD.

I proved (2) by considering Nevanlinna-stable approximations

Fn → F by finite Blaschke products, using (1) for the Fn, and

taking the limit as n→∞.

K. Dyakonov proved this in 1992 using Julia’s lemma. A little

later, H. Hedenmalm came up with a different proof using yet

another approach.









another approach.









another approach.









another approach.


Lemma. (I, 2017) λF is the minimal metric of curvature ≡ −4

satisfying

λF ≥ | InnF ′|λD.

For the proof, suppose λ is the minimal metric with the above

property. Consider the function

u(z) = log(λF/λ), z ∈ D.

It is non-negative and subharmonic (∆u ≥ 0), yet

limr→1

1

2π

∫|z|=r

u(re iθ)dθ = 0.


Lemma. (I, 2017) λF is the minimal metric of curvature ≡ −4

satisfying

λF ≥ | InnF ′|λD.

For the proof, suppose λ is the minimal metric with the above

property. Consider the function

u(z) = log(λF/λ), z ∈ D.

It is non-negative and subharmonic (∆u ≥ 0), yet

limr→1

1

2π

∫|z|=r

u(re iθ)dθ = 0.

Ahern-Clark theory

Lemma. If f ∈ N is a Nevanlinna class function, then

gap(f ) :=1

2π

∫|z|=1

log |f (z)|dθ −

limr→1

1

2π

∫|z|=r

log |f (z)|dθ

= σ(S1).

Corollary. If F ∈J then

gap(F ′) = limr→1

1

2π

∫|z|=r

logλDλF

dθ.

Ahern-Clark theory

Lemma. If f ∈ N is a Nevanlinna class function, then

gap(f ) :=1

2π

∫|z|=1

log |f (z)|dθ −

limr→1

1

2π

∫|z|=r

log |f (z)|dθ

= σ(S1).

Corollary. If F ∈J then

gap(F ′) = limr→1

1

2π

∫|z|=r

logλDλF

dθ.

Computing the entropy of inner functions

Lemma. Suppose F ∈J is a maximal Blaschke product with

F (0) = 0 and F ′(0) 6= 0. We have

1

2π

∫|z|=1

log |F ′(z)|dθ =∑cp

log1

|ci |−∑zeros

log1

|zi |,

where we omit the trivial zero at the origin.

Remark. This property characterizes maximal Blaschke products.

Computing the entropy of inner functions

Lemma (Ch. Pommerenke, 1976). Suppose UP ∈J is the

universal covering map D \ p1, p2, p3, . . . , pk with the

normalization UP(0) = 0. Then,

1

2π

∫|z|=1

log |U ′P(z)|dθ =k∑

i=1

log1

|pi |−∑zeros

log1

|zi |.

where we omit the trivial zero at the origin.

See his fantastic paper On Green’s functions of Fuchsian groups.

Nevanlinna stability

Suppose F ∈J . A sequence Fk → F is Nevanlinna stable if

InnF ′k → InnF ′ and OutF ′k → OutF ′.

Not every sequence is Nevanlinna stable,

but given any F ∈J , a stable approximation exists.

(M. Craizer, 1991)

Theorem. (I, 2017) Suppose FCk→ F is a stable sequence and

C1,k ⊂ Ck such that FC1,k→ G . Then, FC1,k

→ G is stable.






(M. Craizer, 1991)



→ G is stable.






(M. Craizer, 1991)



→ G is stable.






(M. Craizer, 1991)



→ G is stable.

Divisor law



→ G is stable.

Call I ∈ Inn constructible if its in the image of the map


Corollary. If I is constructible and J|I , then J is constructible.

Corollary. If S ′µ ∈ N then Sµ is constructible (by the work of

M. Cullen from 1971, this holds if suppµ is a Carleson set).

Divisor law



→ G is stable.

Call I ∈ Inn constructible if its in the image of the map


Corollary. If I is constructible and J|I , then J is constructible.

Corollary. If S ′µ ∈ N then Sµ is constructible (by the work of

M. Cullen from 1971, this holds if suppµ is a Carleson set).

Product and decomposition laws

Lemma. If I and J are constructible, then so is I · J.

The proof uses the divisor law and the Solynin-type estimate

λFC1λFC2

≥ λFC1∪C2λFC1∩C2

.

Corollary. The inner function BCSµ is constructible if and only if

BC and Sµ are.

Thank you for your attention!

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Di erentiating Blaschke productsivrii/LAtalk.pdfDi erentiating Blaschke Products Consider the following curious di erentiation procedure: to a Blaschke product F(z) = ei Yd i=1 z a