# A note on connectedness of Blaschke products

Yue Xin  School of Mathematics, Jilin University, 130012, Changchun, P. R.
China<EMAIL_ADDRESS>Bingzhe Hou  School of Mathematics, Jilin University,
130012, Changchun, P. R. China<EMAIL_ADDRESS>
###### Abstract.

Consider the space $\mathcal{F}$ of all inner functions on the unit open disk
under the uniform topology, which is a metric topology induced by the
$H^{\infty}$-norm. In the present paper, a class of Blaschke products, denoted
by $\mathcal{H}_{SC}$, is introduced. We prove that for each
$B\in\mathcal{H}_{SC}$, $B$ and $zB$ belong to the same path-connected
component of $\mathcal{F}$. It plays an important role of a method to select a
fine subsequence of zeros. As a byproduct, we obtain that each Blaschke
product in $\mathcal{H}_{SC}$ has an interpolating and one-component factor.

###### Key words and phrases:

Blaschke products; path-connectedness; pseudo-hyperbolic distance;
interpolating; one-component.

###### 1991 Mathematics Subject Classification:

Primary 30J05, 30J10; Secondary 54C35.

##  1\. Introduction

Let $\mathbb{D}$ be the unit open disk in the complex plane $\mathbb{C}$ and
let $\partial\mathbb{D}$ be the boundary of $\mathbb{D}$, i.e., the unit
circle. The pseudo-hyperbolic distance on the unit open disk $\mathbb{D}$,
denoted by $\rho$, is given by

$\rho(z,w)={\big{|}}\frac{z-w}{1-\overline{z}w}{\big{|}},\ \ \ \text{for any}\
z,w\in\mathbb{D}.$

Let $H^{\infty}$ be the Banach algebra of bounded analytic functions on
$\mathbb{D}$ equipped with the norm
${\|}f{\|}_{\infty}=\sup_{z\in\mathbb{D}}{|}f(z){|}$. A bounded analytic
function $f$ on $\mathbb{D}$ is called an inner function if it has unimodular
radial limits almost everywhere on the boundary $\partial\mathbb{D}$ of
$\mathbb{D}$. Furthermore, denote by $\mathcal{F}$ the set of all inner
functions. We are interested in the space $\mathcal{F}$ under uniform
topology, which is a metric topology induced by the $H^{\infty}$-norm. Notice
that the uniform topology on $\mathcal{F}$ is very complicated and
interesting, see [6, 9, 10] for instance.

A Blaschke product is an inner function of the form

$B(z)=\lambda
z^{m}\prod_{n}\frac{{|}z_{n}{|}}{z_{n}}\frac{z_{n}-z}{1-\overline{z_{n}}z},$

where $m$ is a nonnegative integer, $\lambda$ is a complex number with
${|}\lambda{|}=1$, and $\\{z_{n}\\}$ is a sequence of points in
$\mathbb{D}\setminus\\{0\\}$ satisfying the Blaschke condition
$\sum_{n}(1-{|}z_{n}{|})<\infty$. Moreover, if $\lambda=1$, we say that $B$ is
normalized.

If for every bounded sequence of complex numbers $\\{w_{n}\\}_{n=1}^{\infty}$,
there exists $f$ in $H^{\infty}$ satisfying $f(z_{n})=w_{n}$ for every
$n\in\mathbb{N}$, then both the sequence $\\{z_{n}\\}_{n=1}^{\infty}$ and the
Blaschke product $B(z)$ are called interpolating. Following from a celebrated
result of Carleson [3], one can see that $B(z)$ is an interpolating Blaschke
product if and only if $\\{z_{n}\\}$ is a uniformly separated sequence, i.e.,

$\inf_{n\in\mathbb{N}}\prod_{k\neq
n}{\big{|}}\frac{z_{k}-z_{n}}{1-\overline{z_{k}}z_{n}}{\big{|}}>0.$

Moreover, if

$\lim_{n\rightarrow\infty}\prod_{k\neq
n}{\big{|}}\frac{z_{k}-z_{n}}{1-\overline{z_{k}}z_{n}}{\big{|}}=1,$

both the sequence $\\{z_{n}\\}_{n=1}^{\infty}$ and the Blaschke product $B(z)$
are called thin. In addition, A Blaschke product is called Carleson-Newman if
it is a product of finitely many interpolating Blaschke products.
Interpolating Blaschke products and Carleson-Newman Blaschke products play an
important role in the study of $H^{\infty}$. As well-known, inner functions
can be approximated uniformly by Blaschke products, and Carleson-Newman
Blaschke products can be approximated uniformly by interpolating Blaschke
products [8]. However, there is still an open problem whether the set of all
interpolation Blaschke products is dense in the inner function space
$\mathcal{F}$. There has been obtained some related results. For instance,
Marshall ([7]) proved that finite linear combinations of Blaschke products are
dense in $H^{\infty}$; Nicolau and Suárez [11] characterize the connected
components of the subset $CN^{*}$ of $H^{\infty}$ formed by the products $bh$,
where $b$ is a Carleson-Newman Blaschke product and $h\in H^{\infty}$ is an
invertible function.

In particular, a result of K. Tse [13] tells us that a sequence
$\\{z_{n}\\}_{n=1}^{\infty}$ of points contained in a Stolz domain

$\\{z\in\mathbb{D}:{|}1-\overline{\xi}z{|}\leq C(1-{|}z{|})\\},$

where $\xi$ is a constant with ${|}\xi{|}=1$, is interpolating if and only if
it is separated, i.e.,

$\inf_{m\neq n}\rho(z_{m},z_{n})>0.$

Then, if the zero points lie in a Stolz domain, we may say something more
about the Blaschke products. For example, A. Reijonen gave a sufficient
condition for a Blaschke product with zeros in a Stolz domain to be a one-
component inner function in [12]. An inner function $u$ in $H^{\infty}$ is
said to be one-component if there is $\eta\in(0,1)$ such that the level set
$\Omega_{u}(\eta):=\\{z\in\mathbb{D}:{|}u(z){|}<\eta\\}$ is connected. More
details of one-component inner functions, we refer to [1, 4, 12].

In this paper, we focus on the path-connected components of the space
$\mathcal{F}$ under the uniform topology. Recall that the topology of the
uniform convergence on the set $\mathcal{F}$ is induced by the following
metric

$d(f,g)={\|}f-g{\|}_{\infty}=\sup_{z\in\mathbb{D}}{|}f(z)-g(z){|}=\sup_{\theta\in\mathbb{R}}{\rm
ess}{|}f(e^{{\bf i}\theta})-g(e^{{\bf i}\theta}){|}.$

For any two inner functions $f$ and $g$, if they belong to the same path-
connected component in the space $\mathcal{F}$, we denote $f\sim g$.

The path of inner functions has always been of great interest and is related
to many important issues. D. Herrero [6] considered the path-connected
components of the space $\mathcal{F}$. He showed that a component of
$\mathcal{F}$ can contain nothing but Blaschke products with infinitely many
zeroes, exactly one (up to a constant factor) singular inner function or
infinitely many pairwise coprime singular inner functions, which answered a
problem of Douglas. Furthermore, V. Nestoridis studied the invariant and
noninvariant connected components of the inner functions space $\mathcal{F}$.
He proved that the inner functions $d(z)={\rm exp}\\{(z+1)/(z-1)\\}$ and $zd$
belong to the same connected component [9], and gave a family of inner
functions, denoted by $H$, such that for every $B\in H$, $B$ and $zB$ don’t
belong to the same component [10]. In particular, the family $H$ contains only
Blaschke products, and contains all of thin Blaschke products. In addition,
several authors have studied the connected component of inner functions in the
context of model spaces and operator theory (see [2, 5] for instance).

In the present paper, we aim to give a class of Blaschke products, denoted by
$\mathcal{H}_{SC}$, such that each $B\in\mathcal{H}_{SC}$, $B$ and $zB$ belong
to the same connected component. Firstly, let us define a class of subsets of
the unit open disk, named strip cones and denoted by
$SC(\xi,\theta,T_{1},T_{2})$. In this paper, we study the Blaschke products
with zeros lying in a strip cone.

######  Definition 1.1.

Let $\theta\in(0,\pi)$, $\xi\in\partial\mathbb{D}$, $T_{1}$ and $T_{2}$ be two
nonzero real numbers. Denote by $J_{i}$ the arc on the circle

${|}z-(1-T_{i}e^{{\bf i}\theta})\xi{|}={|}T_{i}{|}$

in the unit open disk, for $i=1,2$. Write $\xi_{i}$ as the intersection point
of $J_{i}$ and $\partial\mathbb{D}$ other than $\xi$, $i=1,2$. We define
$SC(\xi,\theta,T_{1},T_{2})$ is the region bounded by $J_{1}$, $J_{2}$ and
$\wideparen{\xi_{1}\xi_{2}}$ which is the arc on the unit circle
$\partial\mathbb{D}$ from $\xi_{1}$ to $\xi_{2}$ without $\xi$, and call it a
strip cone. If $T_{1}=T_{2}\in\mathbb{R}\setminus\\{0\\}$, then
$SC(\xi,\theta,T_{1},T_{2})$ is just the arc $J_{1}=J_{2}$. In particular, if
$T_{1}$ and $T_{2}$ are infinity, then $SC(\xi,\theta,T_{1},T_{2})$ is just
the segment $J_{1}=J_{2}=(-1,1)$.

One can see some examples of strip cones in Figure 1. Next, we explain why we
name it strip cone.

Figure 1. Examples of strip cones

######  Definition 1.2.

Let $\theta\in(-\frac{\pi}{2},\frac{\pi}{2})$, and $L_{1}$ and $L_{2}$ be two
parallel straight lines with an angle of $\theta$ to the real axis. Denote by
$SL(\theta,L_{1},L_{2})$ the strip region between $L_{1}$ and $L_{2}$ in the
right half plane.

Given any strip cone $SC(\xi,\theta,T_{1},T_{2})$. Consider the fractional
linear transformation $\varphi_{\xi}(z)=(\xi+z)/(\xi-z)$, where ${|}\xi{|}=1$.
It is easy to see that $\varphi_{\xi}$ maps the unit open disk onto the right
half plane, and map the arcs $J_{1}$ and $J_{2}$ in Definition 1.1 to some
parallel straight lines $L_{1}$ and $L_{2}$ with the angle of $\theta$ to the
imaginary axis in the right half plane. Then, $\varphi_{\xi}$ is an analytic
bijection from $SC(\xi,\theta,T_{1},T_{2})$ to
$SL(\frac{\pi}{2}-\theta,L_{1},L_{2})$ (see Figure 2 for instance). This is
the reason we call the subset $SC(\xi,\theta,T_{1},T_{2})$ strip cone. Without
loss of generality, we may assume that $\xi=1$, because it is no difference to
deal with $\xi=1$ and general $\xi$ with ${|}\xi{|}=1$. Moreover, we always
denote $\varphi(z)=(1+z)/(1-z)$ through this paper.

Figure 2. $\varphi(z)$ maps $SC(1,\theta,T_{1},T_{2})$ onto
$SL(\frac{\pi}{2}-\theta,L_{1},L_{2})$

######  Definition 1.3.

Denote by $\mathcal{H}_{SC}$ the family of all Blaschke products $B$
satisfying the following conditions,

  1. (i)

the zeros $\\{z_{n}\\}_{n=1}^{\infty}$ of $B$ lie in some certain strip cone
$SC(\xi,\theta,T_{1},T_{2})$;

  2. (ii)

${|}\xi-z_{n}{|}$ non-increasingly tends to $0$;

  3. (iii)

there exists a positive number $\delta<1$, such that
$\rho(z_{n},z_{n+1})\leq\delta$ for any $n\in\mathbb{N}$.

Now we show our main result as the following theorem.

${\bf Main\ \ Theorem}\ \ $ For any $B\in\mathcal{H}_{SC}$, $B$ and $zB$
belong to the same path-connected component of the inner functions space
$\mathcal{F}$ under the uniform topology.

In the next section, we will introduce a method to select a "fine" factor of a
Blaschke product in $\mathcal{H}_{SC}$, which plays an important role to prove
the main theorem. As a byproduct, we obtain that each Blaschke product in
$\mathcal{H}_{SC}$ has an interpolating and one-component factor. Then, we
will complete the proof of the main theorem in the last section.

##  2\. Preliminaries

First of all, let us start from the following simple result.

######  Lemma 2.1.

Let $f=\varphi_{1}\cdot\varphi_{2}$ and $g=\psi_{1}\cdot\psi_{2}$, where
$f,g,\varphi_{1},\varphi_{2},\psi_{1},\psi_{2}\in\mathcal{F}$. If
$\varphi_{1}\sim\psi_{1}$ and $\varphi_{2}\sim\psi_{2}$, then $f\sim g$. In
particular, if $\varphi_{1}\sim z\varphi_{1}$, then $f\sim zf$.

To prove ${B}\sim z{B}$, it suffices to prove $\widetilde{B}\sim
z\widetilde{B}$, if $\widetilde{B}$ is factor of $B$. In this section, it will
be shown that we can select a factor $\widetilde{B}$ of $B$ such that
$\widetilde{B}$ satisfies more conditions than $B$. The major is how to choose
a subsequence of the zeros sequence of $B$. Recall that for a Blaschke product
$B\in\mathcal{H}_{SC}$ with zeros $\\{z_{n}\\}_{n=1}^{\infty}$, there exists a
positive number $\delta<1$, such that $\rho(z_{n},z_{n+1})\leq\delta$ for any
$n\in\mathbb{N}$.

######  Lemma 2.2.

Let $a$, $a{{}^{\prime}}$, $b$, $b{{}^{\prime}}$ be real numbers satisfying

$0\leq a\leq a{{}^{\prime}}<1\ \ \text{and}\ \ 0\leq b\leq b{{}^{\prime}}<1.$

Then,

$\frac{a+b}{1+{a}b}\leq\frac{a{{}^{\prime}}+b{{}^{\prime}}}{1+{a{{}^{\prime}}}b{{}^{\prime}}}$

###### Proof.

$\displaystyle\frac{a{{}^{\prime}}+b{{}^{\prime}}}{1+{a{{}^{\prime}}}b{{}^{\prime}}}-\frac{a+b}{1+{a}b}$
$\displaystyle=\frac{(a{{}^{\prime}}+b{{}^{\prime}}+a{{}^{\prime}}ab+abb{{}^{\prime}})-(a+b+aa{{}^{\prime}}b{{}^{\prime}}+a{{}^{\prime}}bb{{}^{\prime}})}{(1+ab)(1+a{{}^{\prime}}b{{}^{\prime}})}$
$\displaystyle=\frac{(a{{}^{\prime}}-a)(1-bb{{}^{\prime}})+(b{{}^{\prime}}-b)(1-a{{}^{\prime}}a)}{(1+ab)(1+a{{}^{\prime}}b{{}^{\prime}})}$
$\displaystyle\geq 0.$

∎

######  Lemma 2.3.

Let $\\{z_{n}\\}_{n=1}^{\infty}$ be a sequence of complex numbers in the unit
open disk, satisfying that

  1. (1)

for any $m\in\mathbb{N}$, $\rho(z_{m},z_{n})\rightarrow 1$ as
$n\rightarrow\infty$;

  2. (2)

there exists a positive number $0<\delta<1$, such that
$\rho(z_{n},z_{n+1})\leq\delta$ for any $n\in\mathbb{N}$.

Then, for any $0<\varepsilon<1$, we can choose a subsequence
$\\{z_{n_{k}}\\}_{n_{k}=1}^{\infty}$ of $\\{z_{n}\\}_{n=1}^{\infty}$ such that

$0<\varepsilon\leq\rho(z_{n_{k}},z_{n_{k+1}})\leq\frac{\varepsilon+\delta}{1+\varepsilon\delta}<1.$

###### Proof.

Given any $0<\varepsilon<1$. Put $z_{n_{1}}=z_{1}$. Since
$\rho(z_{n_{1}},z_{i})\rightarrow 1$ as $i\rightarrow\infty$, we can choose

$n_{2}=\min\\{i;\ \rho(z_{n_{1}},z_{i})\geq\varepsilon\\}.$

It is obvious that $\rho(z_{n_{1}},z_{n_{2}})\geq\varepsilon$.

With the same method, we choose ${n_{k+1}}$ by

${n_{k+1}}=\min\\{{i};\ i>n_{k},\ \rho(z_{n_{k}},z_{i})\geq\varepsilon\\}.$

Then, $\rho(z_{n_{k}},z_{n_{k+1}})\geq\varepsilon$. Moreover, for each
$k=1,2,\ldots$,

$0\leq\rho(z_{n_{k}},z_{n_{k+1}-1})<\varepsilon<1\ \ \text{and}\ \
0\leq\rho(z_{n_{k+1}-1},z_{n_{k+1}})<\delta<1.$

By Lemma 2.2, we have

$\frac{\rho(z_{n_{k}},z_{n_{k+1}-1})+\rho(z_{n_{k+1}-1},z_{n_{k+1}})}{1+\rho(z_{n_{k}},z_{n_{k+1}-1})\rho(z_{n_{k+1}-1},z_{n_{k+1}})}\leq\frac{\varepsilon+\delta}{1+\varepsilon\delta}.$

Therefore,

$0<\varepsilon\leq\rho(z_{n_{k}},z_{n_{k+1}})\leq\frac{\rho(z_{n_{k}},z_{n_{k+1}-1})+\rho(z_{n_{k+1}-1},z_{n_{k+1}})}{1+\rho(z_{n_{k}},z_{n_{k+1}-1})\rho(z_{n_{k+1}-1},z_{n_{k+1}})}\leq\frac{\varepsilon+\delta}{1+\varepsilon\delta}<1.$

∎

######  Lemma 2.4.

Let $\alpha$ and $\beta$ be two complex numbers in the unit open disk with
${|}1-\alpha{|}<{|}1-\beta{|}$. Denote $\alpha=s_{1}+{\bf
i}(1-s_{1})\cot\theta_{1}$ and $\beta=s_{2}+{\bf i}(1-s_{2})\cot\theta_{2}$,
where $\theta_{1},\theta_{2}\in(0,\pi)$. For the pair of $\alpha$ and $\beta$,
let the positive numbers $\varepsilon$, $\delta$, $\theta_{0}$, $C$, $\tau$
and $\eta$ satisfy the following conditions,

  1. (1)

$0<\varepsilon\leq\rho(\alpha,\beta)\leq\delta<1$;

  2. (2)

$\theta_{0}\in(0,\pi)$ and $C={|}1-{\bf i}\cot\theta_{0}{|}$;

  3. (3)

${|}\cot\theta_{i}-\cot\theta_{0}{|}<\tau<\sqrt{\frac{3C^{2}\varepsilon^{2}}{16C^{2}(1-\varepsilon^{2})+3\varepsilon^{2}}}$,
for $i=1,2$;

  4. (4)

$3\leq
4-2\eta\leq(1+s_{1}-(1-s_{1})\cot^{2}\theta_{1})(1+s_{2}-(1-s_{2})\cot^{2}\theta_{2})\leq
4$.

Then

$0<C_{1}\leq{\big{|}}\frac{1-\alpha}{1-\beta}{\big{|}}\leq C_{2}<1.$

where

$\displaystyle
C_{1}=\frac{C-\tau}{C+\tau}\left(\frac{C-\tau}{C+\tau}+\frac{2\delta^{2}-2\sqrt{\delta^{4}+\delta^{2}(C^{2}-\tau^{2})(1-\delta^{2})}}{(C+\tau)^{2}(1-\delta^{2})}\right),$
$\displaystyle
C_{2}=\frac{C+\tau}{C-\tau}\left(\frac{C+\tau}{C-\tau}+\frac{(2-\eta)\varepsilon^{2}-\sqrt{(2-\eta)^{2}\varepsilon^{4}+2\varepsilon^{2}(2-\eta)(C^{2}-\tau^{2})(1-\varepsilon^{2})}}{(C-\tau)^{2}(1-\varepsilon^{2})}\right).$

###### Proof.

For $i=1,2$, it follows from condition $(3)$ that

${|}(1-{\bf i}\cot\theta_{i})-(1-{\bf i}\cot\theta_{0}){|}<\tau,$

and consequently,

$0<C-\tau<{|}1-{\bf i}\cot\theta_{i}{|}<C+\tau.$

Denote

$K=\frac{1-s_{1}}{1-s_{2}}.$

Notice that

$\displaystyle 0<(1-s_{1})(C-\tau)\leq{|}1-\alpha{|}=(1-s_{1}){|}1-{\bf
i}\cot\theta_{1}{|}\leq(1-s_{1})(C+\tau),$ $\displaystyle
0<(1-s_{2})(C-\tau)\leq{|}1-\beta{|}=(1-s_{2}){|}1-{\bf
i}\cot\theta_{2}{|}\leq(1-s_{2})(C+\tau).$

Then

$\frac{K(C-\tau)}{C+\tau}\leq{\big{|}}\frac{1-\alpha}{1-\beta}{\big{|}}\leq\frac{K(C+\tau)}{C-\tau}.$
(2.1)

Since

$\rho(\alpha,\beta)^{2}=\frac{{|}\alpha-\beta{|}^{2}}{(1-\overline{\alpha}\beta)(1-\overline{\beta}\alpha)}=\frac{1}{\frac{(1-{|}\alpha{|}^{2})(1-{|}\beta{|}^{2})}{{|}\alpha-\beta{|}^{2}}+1},$

it follows from condition $(1)$ that

$\displaystyle
0<\frac{1}{\delta^{2}}-1\leq\frac{(1-{|}\alpha{|}^{2})(1-{|}\beta{|}^{2})}{{|}\alpha-\beta{|}^{2}}\leq\frac{1}{\varepsilon^{2}}-1.$
(2.2)

Consider
$\frac{(1-{|}\alpha{|}^{2})(1-{|}\beta{|}^{2})}{{|}\alpha-\beta{|}^{2}}$. We
have

$\displaystyle\frac{(1-{|}\alpha{|}^{2})(1-{|}\beta{|}^{2})}{{|}\alpha-\beta{|}^{2}}$
$\displaystyle=$
$\displaystyle\frac{(1-(s_{1}^{2}+(1-s_{1})^{2}\cot^{2}\theta_{1}))\cdot(1-(s_{2}^{2}+(1-s_{2})^{2}\cot^{2}\theta_{2}))}{{|}(1-\beta)-(1-\alpha){|}^{2}}$
$\displaystyle=$
$\displaystyle\frac{(1-s_{1})(1-s_{2})(1+s_{1}-(1-s_{1})\cot^{2}\theta_{1})(1+s_{2}-(1-s_{2})\cot^{2}\theta_{2}))}{{|}(1-s_{1})(1-{\bf
i}\cot\theta_{1})-(1-s_{2})(1-{\bf i}\cot\theta_{2}){|}^{2}}.$

Denote

${W}_{1}=1+s_{1}-(1-s_{1})\cot^{2}\theta_{1}\ \ \text{and}\ \
{W}_{2}=1+s_{2}-(1-s_{2})\cot^{2}\theta_{2}.$

Then,

$3\leq 4-2\eta\leq{W}_{1}\cdot{W}_{2}\leq 4.$

Now we give the lower bound and upper bound of $K$, respectively. Then, by
inequality (2.1), we can complete the proof.

Lower bound of $K$. If $K\leq\frac{C-\tau}{C+\tau}$, we have

$\displaystyle\frac{(1-{|}\alpha{|}^{2})(1-{|}\beta{|}^{2})}{{|}\alpha-\beta{|}^{2}}=$
$\displaystyle\frac{(1-s_{1})(1-s_{2}){W}_{1}{W}_{2}}{{|}(1-s_{1})(1-{\bf
i}\cot\theta_{1})-(1-s_{2})(1-{\bf i}\cot\theta_{2}){|}^{2}}$ $\displaystyle=$
$\displaystyle\frac{K{W}_{1}{W}_{2}}{{|}K(1-{\bf i}\cot\theta_{1})-(1-{\bf
i}\cot\theta_{2}){|}^{2}}$ $\displaystyle\leq$
$\displaystyle\frac{4K}{({|}(1-{\bf i}\cot\theta_{2}){|}-{|}K(1-{\bf
i}\cot\theta_{1}){|})^{2}}$ $\displaystyle\leq$
$\displaystyle\frac{4K}{((C-\tau)-K(C+\tau))^{2}}.$

It follows from inequality (2.2) that

$\frac{4K}{((C-\tau)-K(C+\tau))^{2}}\geq\frac{(1-{|}\alpha{|}^{2})(1-{|}\beta{|}^{2})}{{|}\alpha-\beta{|}^{2}}\geq\frac{1}{\delta^{2}}-1.$

Then,

$K^{2}-2\left(\frac{C-\tau}{C+\tau}+\frac{2\delta^{2}}{(C+\tau)^{2}(1-\delta^{2})}\right)K+\frac{(C-\tau)^{2}}{(C+\tau)^{2}}\leq
0,$

Let $a_{1}$ and $a_{2}$ be the two roots of the above quadratic polynomial of
$K$,

$a_{1}=\frac{C-\tau}{C+\tau}+\frac{2\delta^{2}-2\sqrt{\delta^{4}+\delta^{2}(C^{2}-\tau^{2})(1-\delta^{2})}}{(C+\tau)^{2}(1-\delta^{2})},$
$a_{2}=\frac{C-\tau}{C+\tau}+\frac{2\delta^{2}+2\sqrt{\delta^{4}+\delta^{2}(C^{2}-\tau^{2})(1-\delta^{2})}}{(C+\tau)^{2}(1-\delta^{2})}.$

One can see that

$0<a_{1}<\frac{C-\tau}{C+\tau}<a_{2}.$

Then, we always have

$K\geq a_{1}>0.$

Moreover, put $C_{1}=\frac{C-\tau}{C+\tau}\cdot a_{1}$,

${\big{|}}\frac{1-\alpha}{1-\beta}{\big{|}}\geq\frac{K(C-\tau)}{C+\tau}\geq
C_{1}>0.$

Upper bound of $K$. Notice that

$\displaystyle\frac{(1-{|}\alpha{|}^{2})(1-{|}\beta{|}^{2})}{{|}\alpha-\beta{|}^{2}}$
$\displaystyle=$ $\displaystyle\frac{K{W}_{1}{W}_{2}}{{|}K(1-{\bf
i}\cot\theta_{1})-(1-{\bf i}\cot\theta_{2}){|}^{2}}$ $\displaystyle=$
$\displaystyle\frac{K{W}_{1}{W}_{2}}{{\big{|}}iK(\cot\theta_{0}-\cot\theta_{1})-i(\cot\theta_{0}-\cot\theta_{2})+(K-1)(1-{\bf
i}\cot\theta_{0}){\big{|}}^{2}}$ $\displaystyle\geq$
$\displaystyle\frac{(4-2\eta)K}{\left({\big{|}}K(\cot\theta_{0}-\cot\theta_{1}){\big{|}}+{\big{|}}\cot\theta_{0}-\cot\theta_{2}{\big{|}}+{\big{|}}(K-1)C{\big{|}}\right)^{2}}$
$\displaystyle\geq$
$\displaystyle\frac{(4-2\eta)K}{({|}(K-1){|}C+(K+1)\tau)^{2}}.$

Firstly, let us prove that $K$ must be less than $1$. If $K\geq 1$, we have

$\frac{(1-{|}\alpha{|}^{2})(1-{|}\beta{|}^{2})}{{|}\alpha-\beta{|}^{2}}\geq\frac{(4-2\eta)K}{((1-K)C+(K+1)\tau)^{2}}.$

It follows from inequality (2.2) that

$\frac{(4-2\eta)K}{((K-1)C+(K+1)\tau)^{2}}\leq\frac{(1-{|}\alpha{|}^{2})(1-{|}\beta{|}^{2})}{{|}\alpha-\beta{|}^{2}}\leq\frac{1}{\varepsilon^{2}}-1.$

Then,

$K^{2}-2\left(\frac{C-\tau}{C+\tau}+\frac{(2-\eta)\varepsilon^{2}}{(1-\varepsilon^{2})(C+\tau)^{2}}\right)K+\frac{(C-\tau)^{2}}{(C+\tau)^{2}}\geq
0,$

Let $\lambda_{1}$ and $\lambda_{2}$ be the two roots of the above quadratic
polynomial of $K$,

$\lambda_{1}=\frac{C-\tau}{C+\tau}+\frac{(2-\eta)\varepsilon^{2}-\sqrt{(2-\eta)^{2}\varepsilon^{4}+2\varepsilon^{2}(2-\eta)(C^{2}-\tau^{2})(1-\varepsilon^{2})}}{(C+\tau)^{2}(1-\varepsilon^{2})},$
$\lambda_{2}=\frac{C-\tau}{C+\tau}+\frac{(2-\eta)\varepsilon^{2}+\sqrt{(2-\eta)^{2}\varepsilon^{4}+2\varepsilon^{2}(2-\eta)(C^{2}-\tau^{2})(1-\varepsilon^{2})}}{(C+\tau)^{2}(1-\varepsilon^{2})}.$

Since

$\displaystyle\left(\frac{C+\tau}{C-\tau}\right)^{2}-2\left(\frac{C-\tau}{C+\tau}+\frac{(2-\eta)\varepsilon^{2}}{(1-\varepsilon^{2})(C+\tau)^{2}}\right)\left(\frac{C+\tau}{C-\tau}\right)+\frac{(C-\tau)^{2}}{(C+\tau)^{2}}$
$\displaystyle=\ $
$\displaystyle\left(\frac{C+\tau}{C-\tau}-\frac{C-\tau}{C+\tau}\right)^{2}-\frac{(4-2\eta)\varepsilon^{2}}{(1-\varepsilon^{2})(C+\tau)^{2}}\cdot\left(\frac{C+\tau}{C-\tau}\right)$
$\displaystyle=\ $
$\displaystyle\frac{16C^{2}\tau^{2}}{(C^{2}-\tau^{2})^{2}}-\frac{(4-2\eta)\varepsilon^{2}}{(1-\varepsilon^{2})(C^{2}-\tau^{2})}$
$\displaystyle=\ $
$\displaystyle\frac{16C^{2}}{(C^{2}-\tau^{2})^{2}}\left(\tau^{2}-\frac{3\varepsilon^{2}(C^{2}-\tau^{2})}{16C^{2}(1-\varepsilon^{2})}\right)$
$\displaystyle<\ $ $\displaystyle 0,$

one can see that

$0<\lambda_{1}<\frac{C-\tau}{C+\tau}<1<\frac{C+\tau}{C-\tau}<\lambda_{2}.$

Then, we have

$K\geq\lambda_{2}>\frac{C+\tau}{C-\tau}>1.$

However, by inequality (2.1),

${\big{|}}\frac{1-\alpha}{1-\beta}{\big{|}}\geq\frac{K(C-\tau)}{C+\tau}\geq\lambda_{2}\cdot\frac{C-\tau}{C+\tau}>1.$

It is a contradiction to ${|}1-\alpha{|}<{|}1-\beta{|}$.

Now we have known $K<1$. Then,

$\frac{(1-{|}\alpha{|}^{2})(1-{|}\beta{|}^{2})}{{|}\alpha-\beta{|}^{2}}\geq\frac{(4-2\eta)K}{((1-K)C+(K+1)\tau)^{2}}.$

It follows from inequality (2.2) that

$\frac{(4-2\eta)K}{((1-K)C+(K+1)\tau)^{2}}\leq\frac{(1-{|}\alpha{|}^{2})(1-{|}\beta{|}^{2})}{{|}\alpha-\beta{|}^{2}}\leq\frac{1}{\varepsilon^{2}}-1.$

Then,

$K^{2}-2\left(\frac{C+\tau}{C-\tau}+\frac{(2-\eta)\varepsilon^{2}}{(1-\varepsilon^{2})(C-\tau)^{2}}\right)K+\frac{(C+\tau)^{2}}{(C-\tau)^{2}}\geq
0.$

Let $A_{1}$ and $A_{2}$ be the two roots of the above quadratic polynomial of
$K$,

$A_{1}=\frac{C+\tau}{C-\tau}+\frac{(2-\eta)\varepsilon^{2}-\sqrt{(2-\eta)^{2}\varepsilon^{4}+2\varepsilon^{2}(2-\eta)(C^{2}-\tau^{2})(1-\varepsilon^{2})}}{(C-\tau)^{2}(1-\varepsilon^{2})},$
$A_{2}=\frac{C+\tau}{C-\tau}+\frac{(2-\eta)\varepsilon^{2}+\sqrt{(2-\eta)^{2}\varepsilon^{4}+2\varepsilon^{2}(2-\eta)(C^{2}-\tau^{2})(1-\varepsilon^{2})}}{(C-\tau)^{2}(1-\varepsilon^{2})}.$

Since

$\displaystyle\left(\frac{C-\tau}{C+\tau}\right)^{2}-2\left(\frac{C+\tau}{C-\tau}+\frac{(2-\eta)\varepsilon^{2}}{(1-\varepsilon^{2})(C-\tau)^{2}}\right)\left(\frac{C-\tau}{C+\tau}\right)+\frac{(C+\tau)^{2}}{(C-\tau)^{2}}$
$\displaystyle=\ $
$\displaystyle\left(\frac{C-\tau}{C+\tau}-\frac{C+\tau}{C-\tau}\right)^{2}-\frac{(4-2\eta)\varepsilon^{2}}{(1-\varepsilon^{2})(C-\tau)^{2}}\cdot\left(\frac{C-\tau}{C+\tau}\right)$
$\displaystyle=\ $
$\displaystyle\frac{16C^{2}\tau^{2}}{(C^{2}-\tau^{2})^{2}}-\frac{(4-2\eta)\varepsilon^{2}}{(1-\varepsilon^{2})(C^{2}-\tau^{2})}$
$\displaystyle=\ $
$\displaystyle\frac{16C^{2}}{(C^{2}-\tau^{2})^{2}}\left(\tau^{2}-\frac{3\varepsilon^{2}(C^{2}-\tau^{2})}{16C^{2}(1-\varepsilon^{2})}\right)$
$\displaystyle<\ $ $\displaystyle 0,$

one can see that

$0<A_{1}<\frac{C-\tau}{C+\tau}<1<A_{2}.$

Then, we always have

$K\leq A_{1}<\frac{C-\tau}{C+\tau}<1.$

Moreover, put $C_{2}=\frac{C+\tau}{C-\tau}\cdot A_{1}$,

${\big{|}}\frac{1-\alpha}{1-\beta}{\big{|}}\leq\frac{K(C+\tau)}{C-\tau}\leq
C_{2}<1.$

∎

######  Lemma 2.5.

Let $\\{z_{n}\\}_{n=1}^{\infty}$ be a sequence of complex numbers in a strip
cone $SC(\xi,\theta_{0},T_{1},T_{2})$, $\theta_{0}\in(0,\pi)$, satisfying that

  1. (1)

${|}1-z_{n}{|}$ non-increasingly tends to $1$, as $n\rightarrow\infty$;

  2. (2)

there exist two positive numbers $0<\varepsilon\leq\delta<1$, such that for
any $n\in\mathbb{N}$,

$0<\varepsilon\leq\rho(z_{n},z_{n+1})\leq\delta<1.$

Then there exists a positive integer $N$ and two positive constants $C_{1}$
and $C_{2}$, such that for any $n\geq N$,

$0<C_{1}\leq{\big{|}}\frac{1-z_{n+1}}{1-z_{n}}{\big{|}}\leq C_{2}<1.$

Furthermore, this implies $\sum_{n=1}^{\infty}{|}1-z_{n}{|}<\infty$.

###### Proof.

Without loss of generality, we may assume that $\\{z_{n}\\}_{n=1}^{\infty}$
lie in a strip cone $SC(1,\theta_{0},T_{1},T_{2})$ (see Figure 3).

Figure 3. Zeros $\\{z_{n}\\}_{n=1}^{\infty}$ of $B(z)$

Denote

$z_{n}=x_{n}+{\bf i}(1-x_{n})\cot\theta_{n},\ \ \ \ \text{for}\ n=1,2,\ldots,$

and

$C={|}1-{\bf i}\cot\theta_{0}{|}.$

Following from $z_{n}\in SC(1,\theta_{0},T_{1},T_{2})$, we could also denote

$z_{n}=(1-t_{n}{\rm e}^{{\bf i}\theta_{0}})+t_{n}{\rm e}^{{\bf i}\zeta_{n}},$

where $t_{n}\in[T_{1},T_{2}]$ if $T_{1}T_{2}>0$, and
$t_{n}\in(-\infty,T_{1}]\cup[T_{2},+\infty)$ if $T_{1}T_{2}<0$. Furthermore,

$\lim\limits_{n\rightarrow\infty}1-z_{n}=\lim\limits_{n\rightarrow\infty}t_{n}({\rm
e}^{{\bf i}\theta_{0}}-{\rm e}^{{\bf
i}\zeta_{n}})=0=\lim\limits_{n\rightarrow\infty}1-x_{n}.$

Notice that $t_{n}$ is uniformly far away from $0$ whenever $T_{1}T_{2}>0$ or
$T_{1}T_{2}<0$. Then, $\zeta_{n}\rightarrow\theta_{0}$, as
$n\rightarrow\infty$. Consequently,

$\lim\limits_{n\rightarrow\infty}\cot\theta_{n}=\lim\limits_{n\rightarrow\infty}\frac{{\rm
Imz_{n}}}{1-{\rm
Rez_{n}}}=\lim\limits_{n\rightarrow\infty}\frac{t_{n}(\sin\zeta_{n}-\sin\theta_{0})}{t_{n}(\cos\zeta_{n}-\cos\theta_{0})}=\cot\theta_{0}.$

Hence, for any two positive number

$0<\eta\leq\frac{1}{2}\ \ \ \text{and}\ \ \
0<\tau\leq\sqrt{\frac{3C^{2}\varepsilon^{2}}{16C^{2}(1-\varepsilon^{2})+3\varepsilon^{2}}},$

there exists $N\in\mathbb{N}$ such that, for all $n\geq N$,

$3\leq
4-2\eta\leq(1+x_{n}-(1-x_{n})\cot^{2}\theta_{n})(1+x_{n+1}-(1-x_{n+1})\cot^{2}\theta_{n+1})\leq
4$

and

${|}\cot\theta_{n}-\cot\theta_{0}{|}<\tau.$

Therefore, by Lemma 2.4, there exist two positive constants $C_{1}$ and
$C_{2}$, such that for any $n\geq N$,

$0<C_{1}\leq{\big{|}}\frac{1-z_{n+1}}{1-z_{n}}{\big{|}}\leq C_{2}<1.$

Furthermore, this implies

$\sum\limits_{n=1}^{\infty}{|}1-z_{n}{|}\leq\sum_{n=1}^{\infty}{|}1-z_{1}{|}C_{2}^{n-1}=\frac{{|}1-z_{1}{|}}{1-C_{2}}<\infty.$

∎

Now, we show that one can select a "fine" subsequence of zeros of each
Blaschke product $B\in\mathcal{H}_{SC}$.

######  Definition 2.6.

A sequence $\\{z_{n}\\}_{n=1}^{\infty}$ is said to be fine, if it satisfies
the following conditions.

  1. (1)

The sequence $\\{z_{n}\\}_{n=1}^{\infty}$ lies in some certain strip cone
$SC(\xi,\theta,T_{1},T_{2})$.

  2. (2)

${|}\xi-z_{n}{|}$ non-increasingly tends to $\xi$, as $n\rightarrow\infty$;

  3. (3)

There exist two positive numbers $0<\varepsilon\leq\delta<1$ such that for any
$n\in\mathbb{N}$,

$0<\varepsilon\leq\rho(z_{n},z_{n+1})\leq\delta.$

Moreover, there exists a positive integer $N$ and two positive constants
$C_{1}$ and $C_{2}$, such that for any $n\geq N$,

$0<C_{1}\leq{\big{|}}\frac{\xi-z_{n+1}}{\xi-z_{n}}{\big{|}}\leq C_{2}<1.$

In particular, $\sum\limits_{n=1}^{\infty}{|}\xi-z_{n}{|}<\infty$.

  4. (4)

${\rm Re}\varphi_{\xi}(z_{n})$ monotonically tends to $+\infty$, where
$\varphi_{\xi}(z)=(\xi+z)/(\xi-z)$. Moreover, there are two positive numbers
$\widetilde{C_{1}}$ and $\widetilde{C_{2}}$ such that

$0<\widetilde{C_{1}}\leq\frac{{\rm Re}\varphi_{\xi}(z_{n})}{{\rm
Re}\varphi_{\xi}(z_{n+1})}\leq\widetilde{C_{2}}<1.$

Furthermore, a Blaschke product is said to be fine if its zeros sequence
$\\{z_{n}\\}_{n=1}^{\infty}$ is fine. And denote by
$\mathcal{\widetilde{H}}_{SC}$ the family of all fine Blaschke products.

######  Lemma 2.7.

Let $B$ be a Blaschke product in $\mathcal{H}_{SC}$ with zeros
$\\{z_{n}\\}_{n=1}^{\infty}$. Then $B$ has a factor in
$\mathcal{\widetilde{H}}_{SC}$, i.e., the sequence
$\\{z_{n}\\}_{n=1}^{\infty}$ has a fine subsequence.

###### Proof.

Without loss of generality, assume $\\{z_{n}\\}_{n=1}^{\infty}$ lies in some
certain strip cone $SC(1,\theta_{0},T_{1},T_{2})$. Recall that
$\\{z_{n}\\}_{n=1}^{\infty}$ satisfies the following conditions.

  1. (1)

${|}1-z_{n}{|}$ non-increasingly tends to $1$, as $n\rightarrow\infty$.

  2. (2)

There exists a positive number $0<\delta<1$, such that
$\rho(z_{n},z_{n+1})\leq\delta$ for any $n\in\mathbb{N}$.

Denote by $SL(\frac{\pi}{2}-\theta_{0},L_{1},L_{2})$ the image of the strip
cone $SC(1,\theta_{0},T_{1},T_{2})$ under $\varphi$. We may write

$L_{1}:y-\tan(\frac{\pi}{2}-\theta_{0})\cdot x-c_{1}=0\ \ \ \text{and}\ \ \
L_{2}:y-\tan(\frac{\pi}{2}-\theta_{0})\cdot x-c_{2}=0.$

Since ${|}1-z_{n+1}{|}\leq{|}1-z_{n}{|}$ for each $n\in\mathbb{N}$, we have

$\displaystyle{|}\varphi(z_{n+1}){|}=$
$\displaystyle{\big{|}}\frac{1+z_{n+1}}{1-z_{n+1}}{\big{|}}$ $\displaystyle=$
$\displaystyle{\big{|}}1-\frac{2}{1-z_{n+1}}{\big{|}}$ $\displaystyle\geq$
$\displaystyle{\big{|}}\frac{2}{1-z_{n+1}}{\big{|}}-1$ $\displaystyle\geq$
$\displaystyle{\big{|}}\frac{2}{1-z_{n}}{\big{|}}-1$ $\displaystyle\geq$
$\displaystyle{\big{|}}\frac{2}{1-z_{n}}-1{\big{|}}-2$ $\displaystyle=$
$\displaystyle{|}\varphi(z_{n}){|}-2.$

By Lemma 2.3, there exists a subsequence $\\{z_{n_{k}}\\}_{k=1}^{\infty}$ of
$\\{z_{n}\\}_{n=1}^{\infty}$ and positive numbers $\varepsilon$ and $\delta$,
such that

$0<\varepsilon\leq\rho(z_{n_{k}},z_{n_{k+1}})\leq\delta<1.$

Consequently, by Lemma 2.5, there are positive numbers $C_{1}$ and $C_{2}$
such that

$0<C_{1}\leq{\big{|}}\frac{1-z_{n_{k+1}}}{1-z_{n_{k}}}{\big{|}}\leq C_{2}<1.$

Since $z_{n_{k}}\rightarrow 1$ as $k\rightarrow\infty$ and
$\varphi(z_{n_{k}})\in SL(\frac{\pi}{2}-\theta_{0},L_{1},L_{2})$, we have

$\lim\limits_{k\rightarrow\infty}{|}\varphi(z_{n_{k}}){|}=\lim\limits_{k\rightarrow\infty}{\big{|}}\frac{1+z_{n_{k}}}{1-z_{n_{k}}}{\big{|}}=+\infty,$
$\lim\limits_{k\rightarrow\infty}\frac{{\rm
Re}\varphi(z_{n_{k}})}{{|}\varphi(z_{n_{k}}){|}}=\cos(\frac{\pi}{2}-\theta_{0})>0$

and

$\lim\limits_{k\rightarrow\infty}{\big{|}}\frac{\varphi(z_{n_{k}})}{\varphi(z_{n_{k+1}})}{\big{|}}\bigg{/}{\big{|}}\frac{1-z_{n_{k+1}}}{1-z_{n_{k}}}{\big{|}}=\lim\limits_{k\rightarrow\infty}{\big{|}}\frac{1+z_{n_{k}}}{1+z_{n_{k+1}}}{\big{|}}=1.$

Furthermore,

$\lim\limits_{k\rightarrow\infty}\left(\frac{{\rm Re}\varphi(z_{n_{k}})}{{\rm
Re}\varphi(z_{n_{k+1}})}\right)\bigg{/}{\big{|}}\frac{1-z_{n_{k+1}}}{1-z_{n_{k}}}{\big{|}}=1.$

Then, there exists a positive integer $k_{0}$ such that for any $k\geq k_{0}$,

$0<\frac{C_{1}}{2}\leq\frac{{\rm Re}\varphi(z_{n_{k}})}{{\rm
Re}\varphi(z_{n_{k+1}})}\leq\frac{1+C_{2}}{2}<1.$

Therefore, the subsequence $\\{z_{n_{k}}\\}_{k=k_{0}}^{\infty}$ is as
required. In particular,

${\rm Re}\varphi(z_{n_{k+1}})>{\rm Re}\varphi(z_{n_{k}}).$

∎

Then, to prove our main theorem, it suffices to consider the Blaschke products
in $\widetilde{\mathcal{H}}_{SC}$.

By the way, we could also find that fine sequences implies some other
properties of Blaschke products. More precisely, we could obtain that each
$B\in\mathcal{H}_{SC}$ has a factor being an interpolating and one-component
Blaschke product.

######  Lemma 2.8 (Corollary 2.5 in [4]).

Let $B$ be a Blaschke product whose zeros ${z_{n}}$ are contained in a Stolz
domain and are separated. Suppose that $\rho(z_{n},z_{n+1})\leq\eta<1$. Then
$B$ is a one-component inner function.

######  Theorem 2.9.

Each $B\in\mathcal{H}_{SC}$ has an interpolating and one-component Blaschke
product factor.

###### Proof.

By Lemma 2.7, it suffices to prove that each
$B\in\mathcal{\widetilde{H}}_{SC}$ has an interpolating and one-component
Blaschke product factor. Obviously, any fine sequence has a tail contained in
some certain Stolz domain, and then a Blaschke product in
$\mathcal{\widetilde{H}}_{SC}$ is interpolating if and only if its zeros
sequence is separated, i.e.,

$\inf\limits_{m\neq n}\rho(z_{n},z_{m})>0.$

Furthermore, together with Lemma 2.8, we only need to prove any fine sequence
has a separated subsequence.

Without loss of generality, assume $\\{z_{n}\\}_{n=1}^{\infty}$ is a fine
sequence in a strip cone $SC(1,\theta,T_{1},T_{2})$. Since
$\varphi(z)=(1+z)/(1-z)$, we have $z=(\varphi(z)-1)/(\varphi(z)+1)$, and then

$\displaystyle{|}z_{m}-z_{n}{|}$
$\displaystyle={\big{|}}\frac{\varphi(z_{m})-1}{\varphi(z_{m})+1}-\frac{\varphi(z_{n})-1}{\varphi(z_{n})+1}{\big{|}}$
$\displaystyle={\big{|}}\frac{2(\varphi(z_{m})-\varphi(z_{n}))}{(\varphi(z_{m})+1)(\varphi(z_{n})+1)}{\big{|}}$

and

$\displaystyle{|}1-\overline{z_{m}}z_{n}{|}$
$\displaystyle={\big{|}}1-\frac{\overline{\varphi(z_{m})}-1}{\overline{\varphi(z_{m})}+1}\cdot\frac{\varphi(z_{n})-1}{\varphi(z_{n})+1}{\big{|}}$
$\displaystyle={\big{|}}\frac{2(\overline{\varphi(z_{m})}+\varphi(z_{n}))}{(\overline{\varphi(z_{m})}+1)(\varphi(z_{n})+1)}{\big{|}}$
$\displaystyle={\big{|}}\frac{2[\overline{\varphi(z_{m})}+\varphi(z_{n})]}{(\varphi(z_{m})+1)(\varphi(z_{n})+1)}{\big{|}}.$

Therefore,

$\displaystyle\rho(z_{m},z_{n})$
$\displaystyle={\big{|}}\frac{z_{m}-z_{n}}{1-\overline{z_{m}}z_{n}}{\big{|}}$
$\displaystyle={\big{|}}\frac{\varphi(z_{m})-\varphi(z_{n})}{\overline{\varphi(z_{m})}+\varphi(z_{n})}{\big{|}}$
$\displaystyle\geq{\big{|}}\frac{{|}\varphi(z_{m}){|}-{|}\varphi(z_{n}){|}}{{|}\varphi(z_{m}){|}+{|}\varphi(z_{n}){|}}{\big{|}}$
$\displaystyle={\big{|}}\frac{1-{|}\frac{\varphi(z_{n})}{\varphi(z_{m})}{|}}{1+{|}\frac{\varphi(z_{n})}{\varphi(z_{m})}{|}}{\big{|}}.$

In addition, there are positive numbers $C_{1}$ and $C_{2}$ such that

$0<C_{1}\leq{\big{|}}\frac{1-z_{n_{k+1}}}{1-z_{n_{k}}}{\big{|}}\leq C_{2}<1.$

Since ${|}1-z_{n}{|}$ non-increasingly tends to $1$, as $n\to\infty$, there
exists a positive integer $N\in\mathbb{N}$ such that for any $n\geq N$,

${\big{|}}\frac{1+z_{n}}{1+z_{n+1}}{\big{|}}\leq\frac{1+C_{2}}{2C_{2}}.$

Then, we have

$\displaystyle{\big{|}}\frac{\varphi(z_{n})}{\varphi(z_{n+1})}{\big{|}}$
$\displaystyle={\big{|}}\frac{1+z_{n}}{1-z_{n}}\cdot\frac{1-z_{n+1}}{1+z_{n+1}}{\big{|}}$
$\displaystyle={\big{|}}\frac{1+z_{n}}{1+z_{n+1}}{\big{|}}{\big{|}}\frac{1-z_{n+1}}{1-z_{n}}{\big{|}}$
$\displaystyle\leq\frac{C_{2}+1}{2}$ $\displaystyle<1,$

Consequently, for any $m>n$,

$\displaystyle{\big{|}}\frac{\varphi(z_{n})}{\varphi(z_{m})}{\big{|}}\leq$
$\displaystyle\left(\frac{C_{2}+1}{2}\right)^{m-n}$ $\displaystyle\leq$
$\displaystyle\frac{C_{2}+1}{2}$ $\displaystyle<$ $\displaystyle 1.$

Then,

$\displaystyle\rho(z_{m},z_{n})$
$\displaystyle\geq{\big{|}}\frac{1-{|}\frac{\varphi(z_{n})}{\varphi(z_{m})}{|}}{1+{|}\frac{\varphi(z_{n})}{\varphi(z_{m})}{|}}{\big{|}}$
$\displaystyle=1-\frac{2}{\frac{1}{{\big{|}}\frac{\varphi(z_{n})}{\varphi(z_{m})}{\big{|}}}+1}$
$\displaystyle\geq 1-\frac{2}{\frac{1}{\frac{C_{2}+1}{2}}+1}$
$\displaystyle=\frac{1-C_{2}}{3+C_{2}}.$

Therefore, we have

$\inf\limits_{m\neq n}\rho(z_{n},z_{m})\geq\frac{1-C_{2}}{3+C_{2}}>0.$

This finishes the proof. ∎

##  3\. Proof of the main theorem

In this section, we would prove our main theorem. Consider a Blaschke product

$B=\lambda z^{m}\prod^{\infty}_{n=1}\frac{\overline{z_{n}}}{\mid
z_{n}\mid}\cdot\frac{z_{n}-z}{1-\overline{z_{n}}z}.$

where ${|}\lambda{|}=1$. It is well known that the finite Blaschke products
with the same order are in the same component. Then, $B\sim zB$ if and only if
for some certain $N\in\mathbb{N}$,

$\prod^{\infty}_{n=N}\frac{\overline{z_{n}}}{\mid
z_{n}\mid}\cdot\frac{z_{n}-z}{1-\overline{z_{n}}z}\ \sim\
\prod^{\infty}_{n=N+1}\frac{\overline{z_{n}}}{\mid
z_{n}\mid}\cdot\frac{z_{n}-z}{1-\overline{z_{n}}z}.$ (3.1)

Recall that $\varphi(z)=(1+z)/(1-z)$ is the fractional linear transformation
from the unit open disk to the right half plane. Let $\alpha_{n}(t)$ be the
unique point in $\mathbb{D}$ such that

$\varphi(\alpha_{n}(t))=(1-t)\varphi(z_{n})+t\varphi(z_{n+1}),$

for $t\in[0,1]$ and $n=1,2,\cdots$. Furthermore, following from the idea of
Nestoridis [9], define the map $B_{t}$ from $[0,1]$ to $\mathcal{F}$ by

$t\longmapsto
B_{t}=\prod^{\infty}_{n=N}\frac{\overline{\alpha_{n}(t)}}{\mid\alpha_{n}(t)\mid}\cdot\frac{\alpha_{n}(t)-z}{1-\overline{\alpha_{n}(t)}z}.$
(3.2)

If the above map is continuous, then the relation (3.1) holds and consequently
$B\sim zB$. To prove the continuity of $B_{t}$, the following theorem plays an
important role.

######  Lemma 3.1 (Lemma $1$ in [9]).

Let

$K_{1}=\prod^{\infty}_{n=1}\frac{\overline{\alpha_{n}}}{{|}\alpha_{n}{|}}\frac{\alpha_{n}-z}{1-\overline{\alpha_{n}}z}\
\ \ \text{and}\ \ \
K_{2}=\prod^{\infty}_{n=1}\frac{\overline{\beta_{n}}}{{|}\beta_{n}{|}}\frac{\beta_{n}-z}{1-\overline{\beta_{n}}z}$

be two infinite Blaschke products such that $K_{1}(0)>0$ and $K_{2}(0)>0$,
then we have the following inequality,

$\displaystyle{\|}K_{1}-K_{2}{\|}_{\infty}\leq$
$\displaystyle\sum_{n}{\big{|}}\textrm{arg}\frac{\alpha_{n}}{\beta_{n}}{\big{|}}+2\sum_{n}{\big{|}}\textrm{arg}\frac{1-\alpha_{n}}{1-\beta_{n}}{\big{|}}$
(3.3) $\displaystyle+2\sup_{y\in\mathbb{R}}{\rm
ess}\sum_{n}{\big{|}}\textrm{arg}\frac{\varphi(\alpha_{n})-{\bf
i}y}{\varphi(\beta_{n})-{\bf i}y}{\big{|}}.$

Now, we would apply Lemma 3.1 to verify the continuity of the path $B_{t}$.
More precisely,

$\displaystyle{\|}B_{t}-B_{t+\Delta t}{\|}_{\infty}\leq$
$\displaystyle\sum_{n}{\big{|}}\textrm{arg}\frac{\alpha_{n}(t)}{\alpha_{n}(t+\Delta
t)}{\big{|}}+2\sum_{n}{\big{|}}\textrm{arg}\frac{1-\alpha_{n}(t)}{1-\alpha_{n}(t+\Delta
t)}{\big{|}}$ (3.4)
$\displaystyle+2\sup_{y\in\mathbb{R}}\textrm{ess}\sum_{n}{\big{|}}\textrm{arg}\frac{\varphi(\alpha_{n}(t))-{\bf
i}y}{\varphi(\alpha_{n}(t+\Delta t))-{\bf i}y}{\big{|}}.$

Then, to prove $\lim\limits_{\Delta t\to 0}{\|}B_{t}-B_{t+\Delta t}{\|}=0$, it
suffices to prove the three items in the right of the inequality (3.4) tend to
$0$ as $\Delta t\to 0$.

######  Proposition 3.2.

For any $B\in\mathcal{\widetilde{H}}_{SC}$, there exists a positive integer
$N_{1}\in\mathbb{N}$ and a positive number $K_{1}$ such that for any
$t\in[0,1]$ and small positive number $\Delta t$,

$\sum\limits_{n=N_{1}}^{\infty}{\big{|}}\textrm{arg}\frac{\alpha_{n}(t)}{\alpha_{n}(t+\Delta
t)}{\big{|}}\leq K_{1}\Delta t.$

###### Proof.

Without loss of generality, suppose that the zeros sequence of
$B\in\mathcal{\widetilde{H}}_{SC}$ is fine in a strip cone
$SC(1,\theta,T_{1},T_{2})$. By the definition of $\alpha_{n}(t)$, we have

$\displaystyle 1-\alpha_{n}(t)=$ $\displaystyle
1-\varphi^{-1}(\varphi(\alpha_{n}(t)))$ $\displaystyle=$ $\displaystyle
1-\frac{\varphi(\alpha_{n}(t))-1}{\varphi(\alpha_{n}(t))+1}$ $\displaystyle=$
$\displaystyle\frac{2}{\varphi(\alpha_{n}(t))+1}$

Furthermore,

$\displaystyle\alpha_{n}(t+\Delta t)-\alpha_{n}(t)$ $\displaystyle=$
$\displaystyle(1-\alpha_{n}(t))-(1-\alpha_{n}(t+\Delta t))$ $\displaystyle=$
$\displaystyle\frac{2\Delta
t(\varphi(z_{n+1})-\varphi(z_{n}))}{(\varphi(\alpha_{n}(t))+1)(\varphi(\alpha_{n}(t+\Delta
t))+1)}$ $\displaystyle=$ $\displaystyle\frac{4\Delta
t(z_{n+1}-z_{n})}{(1-z_{n})(1-z_{n+1})}\cdot\frac{1}{(\varphi(\alpha_{n}(t))+1)(\varphi(\alpha_{n}(t+\Delta
t))+1)}.$

Since ${\rm Re}\varphi(z_{n})$ increasingly tends to $+\infty$ and

$\lim\limits_{k\rightarrow\infty}\frac{{\rm
Re}\varphi(z_{n})}{{|}\varphi(z_{n}){|}}=\cos(\frac{\pi}{2}-\theta)>0,$

there is a positive number $R_{1}>0$ such that

${|}\varphi(\alpha_{n}(t))+1{|}\geq{|}{\rm
Re}(\varphi(\alpha_{n}(t))+1){|}\geq{|}{\rm
Re}(\varphi(z_{n})+1){|}\geq\frac{{|}\varphi(z_{n})+1{|}}{R_{1}}.$

In addition, the followings hold.

  1. (1)

The sequence ${|}1-z_{n}{|}$ non-increasingly tends to $1$, as
$n\rightarrow\infty$.

  2. (2)

by Lemma 2.5, there exists a positive constants $C_{1}$ such that

$0<C_{1}\leq{\big{|}}\frac{1-z_{n+1}}{1-z_{n}}{\big{|}}.$

Then, we have

$\displaystyle{|}\alpha_{n}(t+\Delta t)-\alpha_{n}(t){|}$ $\displaystyle=$
$\displaystyle{\big{|}}\frac{4\Delta
t(z_{n+1}-z_{n})}{(1-z_{n})(1-z_{n+1})}{\big{|}}\cdot{\big{|}}\frac{1}{(\varphi(\alpha_{n}(t))+1)(\varphi(\alpha_{n}(t+\Delta
t))+1)}{\big{|}}$ $\displaystyle\leq$ $\displaystyle\frac{4\Delta
t({|}1-z_{n+1}{|}+{|}1-z_{n}{|})}{{|}1-z_{n}{|}{|}1-z_{n+1}{|}}\cdot\frac{R_{1}^{2}}{{|}\varphi(z_{n})+1{|}^{2}}$
$\displaystyle\leq$ $\displaystyle\frac{8\Delta
t{|}1-z_{n}{|}}{{|}1-z_{n}{|}{|}1-z_{n+1}{|}}\cdot\frac{R_{1}^{2}}{{|}\frac{2}{1-z_{n}}{|}^{2}}$
$\displaystyle=$ $\displaystyle 2R_{1}^{2}\cdot\Delta
t\cdot\frac{{|}1-z_{n}{|}}{{|}1-z_{n+1}{|}}\cdot{|}{1-z_{n}}{|}$
$\displaystyle\leq$ $\displaystyle\frac{2R_{1}^{2}}{C_{1}}\cdot{|}\Delta
t{|}\cdot{|}{1-z_{n}}{|}.$

Given a positive number $0<r<1$, it follows from $z_{n}\rightarrow 1$ that
there exists a positive integer $N_{1}\in\mathbb{N}$ such that, for any $n\geq
N_{1}$ and any $t\in[0,1]$,

${|}\alpha_{n}(t){|}\geq r.$

Thus, considering the area of the triangle with vertices $0$,
$\alpha_{n}(t+\Delta t)$ and $\alpha_{n}(t)$, we have

$\displaystyle{\big{|}}\textrm{arg}\frac{\alpha_{n}(t)}{\alpha_{n}(t+\Delta
t)}{\big{|}}$ $\displaystyle\leq$
$\displaystyle\frac{\pi}{2}\cdot\sin{\big{|}}\textrm{arg}\frac{\alpha_{n}(t)}{\alpha_{n}(t+\Delta
t)}{\big{|}}$ $\displaystyle\leq$
$\displaystyle\frac{\pi}{2}\cdot\frac{{|}\alpha_{n}(t+\Delta
t)-\alpha_{n}(t){|}\cdot 1}{{|}\alpha_{n}(t+\Delta
t){|}\cdot{|}\alpha_{n}(t){|}}$ $\displaystyle\leq$
$\displaystyle\frac{R_{1}^{2}\pi}{r^{2}C_{1}}\cdot\Delta
t\cdot{|}{1-z_{n}}{|}.$

Consequently, by $\sum_{n}{|}{1-z_{n}}{|}<\infty$, for any $t\in[0,1]$,

$\sum\limits_{n=N_{1}}^{\infty}{\big{|}}\textrm{arg}\frac{\alpha_{n}(t)}{\alpha_{n}(t+\Delta
t)}{\big{|}}\leq\Delta
t\cdot\frac{R_{1}^{2}\pi}{r^{2}C_{1}}\sum\limits_{n=N_{1}}^{\infty}{|}{1-z_{n}}{|}.$

Moreover, write

$K_{1}=\frac{R_{1}^{2}\pi}{r^{2}C_{1}}\sum\limits_{n=1}^{\infty}{|}{1-z_{n}}{|}$

as required. ∎

######  Proposition 3.3.

For any $B\in\mathcal{\widetilde{H}}_{SC}$, there exists a positive integer
$N_{2}\in\mathbb{N}$ and a positive number $K_{2}$ such that for any
$t\in[0,1]$ and small positive number $\Delta t$,

$\sup\limits_{y\in\mathbb{R}}\textrm{ess}\sum\limits_{n=N_{2}}^{\infty}{\big{|}}\textrm{arg}\frac{\varphi(\alpha_{n}(t))-{\bf
i}y}{\varphi(\alpha_{n}(t+\Delta t))-{\bf i}y}{\big{|}}\leq K_{2}\Delta t.$

###### Proof.

Without loss of generality, suppose that the zeros sequence of
$B\in\mathcal{\widetilde{H}}_{SC}$ is fine in a strip cone
$SC(1,\theta,T_{1},T_{2})$. Let the strip
$SL(\frac{\pi}{2}-\theta,L_{1},L_{2})$ be the image of the strip cone
$SC(1,\theta,T_{1},T_{2})$ under the map $\varphi(z)$, where

$L_{1}:y-\tan(\frac{\pi}{2}-\theta)\cdot x-c_{1}=0\ \ \ \text{and}\ \ \
L_{2}:y-\tan(\frac{\pi}{2}-\theta)\cdot x-c_{2}=0.$

Denote by $L$ the straight line passing $\varphi(z_{n})$ and parallel $L_{1}$,
and denote by $\omega_{n}$ the angle between $L$ and the straight line passing
through $\varphi(z_{n})$ and $\varphi(z_{n+1})$. Then

$\sin\omega_{n}\leq\frac{{|}c_{1}-c_{2}{|}}{{|}\varphi(z_{n+1})-\varphi(z_{n}){|}}.$

Since there are positive numbers $C_{1}$ and $C_{2}$ such that

$0<C_{1}\leq{\big{|}}\frac{1-z_{n+1}}{1-z_{n}}{\big{|}}\leq C_{2}<1,$

and

$\lim\limits_{n\rightarrow\infty}{\big{|}}\frac{\varphi(z_{n})}{\varphi(z_{n+1})}{\big{|}}\bigg{/}{\big{|}}\frac{1-z_{n+1}}{1-z_{n}}{\big{|}}=\lim\limits_{k\rightarrow\infty}{\big{|}}\frac{1+z_{n}}{1+z_{n+1}}{\big{|}}=1,$

one can see that

$\displaystyle\lim\limits_{n\rightarrow\infty}{|}\varphi(z_{n+1})-\varphi(z_{n}){|}=$
$\displaystyle\lim\limits_{n\rightarrow\infty}{|}\varphi(z_{n}){|}\left(\frac{{|}\varphi(z_{n+1}){|}}{{|}\varphi(z_{n}){|}}-1\right)$
$\displaystyle\leq$
$\displaystyle\lim\limits_{n\rightarrow\infty}\left(\frac{1}{C_{2}}-1\right){|}\varphi(z_{n}){|}$
$\displaystyle=$ $\displaystyle\infty.$

Consequently,

$\lim\limits_{n\rightarrow\infty}\omega_{n}\leq\frac{\pi}{2}\lim\limits_{n\rightarrow\infty}\sin\omega_{n}\leq\frac{\pi}{2}\lim\limits_{n\rightarrow\infty}\frac{{|}c_{1}-c_{2}{|}}{{|}\varphi(z_{n+1})-\varphi(z_{n}){|}}=0,$

that is

$\lim\limits_{n\rightarrow\infty}\textrm{arg}(\varphi(z_{n+1})-\varphi(z_{n}))=\frac{\pi}{2}-\theta.$

Denote by $\vartheta_{n}$ the angle between the imaginary axis and the
straight line passing through $\varphi(z_{n})$ and $\varphi(z_{n+1})$. Then,
for any $\epsilon>0$, there exists a positive integer $N_{2}\in\mathbb{N}$
such that for every $n\geq N_{2}$

$(1-\epsilon)\sin\theta\leq\sin\vartheta_{n}\leq(1+\epsilon)\sin\theta\ \ \
\text{and}\ \ \ \textrm{Re}z_{n}>0.$

Denote by $h_{n}$ the distance from $\mathbf{i}y$ to the straight line passing
through $\varphi(\alpha_{n}(t))$ and $\varphi(\alpha_{n}(t+\Delta t))$.
Consider the area of the triangle with vertices $\mathbf{i}y$,
$\varphi(\alpha_{n}(t))$ and $\varphi(\alpha_{n}(t+\Delta t))$, one can see
that

$\displaystyle\sin\left(\textrm{arg}\frac{\varphi(\alpha_{n}(t+\Delta t))-{\bf
i}y}{\varphi(\alpha_{n}(t))-{\bf i}y}\right)$ $\displaystyle=$
$\displaystyle\frac{{|}\varphi(\alpha_{n}(t+\Delta
t))-\varphi(\alpha_{n}(t)){|}\cdot h_{n}}{{|}\varphi(\alpha_{n}(t+\Delta
t))-{\bf i}y{|}\cdot{|}\varphi(\alpha_{n}(t))-{\bf i}y{|}}$
$\displaystyle\leq$
$\displaystyle\frac{\frac{\textrm{Re}\varphi(\alpha_{n}(t+\Delta
t))-\textrm{Re}\varphi(\alpha_{n}(t))}{(1-\epsilon)\sin\theta}\cdot\max\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}(1+\epsilon)\sin\theta}{{|}\varphi(\alpha_{n}(t+\Delta
t))-{\bf i}y{|}\cdot{|}\varphi(\alpha_{n}(t))-{\bf i}y{|}}$ $\displaystyle=$
$\displaystyle\Delta
t\cdot\frac{1+\epsilon}{1-\epsilon}\cdot\frac{(\textrm{Re}\varphi(z_{n+1})-\textrm{Re}\varphi(z_{n}))\cdot\max\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}}{{|}\varphi(\alpha_{n}(t+\Delta
t))-{\bf i}y{|}\cdot{|}\varphi(\alpha_{n}(t))-{\bf i}y{|}}.$

Since $\\{\textrm{Re}\varphi(z_{n})\\}$ is an increasing sequence of positive
numbers tending to $+\infty$, we have

$\displaystyle{\big{|}}\textrm{arg}\frac{\varphi(\alpha_{n}(t))-{\bf
i}y}{\varphi(\alpha_{n}(t+\Delta t))-{\bf i}y}{\big{|}}$ $\displaystyle\leq$
$\displaystyle\frac{\pi}{2}\sin\left(\textrm{arg}\frac{\varphi(\alpha_{n}(t+\Delta
t))-{\bf i}y}{\varphi(\alpha_{n}(t))-{\bf i}y}\right)$ $\displaystyle\leq$
$\displaystyle\Delta
t\cdot\frac{\pi(1+\epsilon)}{2(1-\epsilon)}\cdot\frac{(\textrm{Re}\varphi(z_{n+1})-\textrm{Re}\varphi(z_{n}))\cdot\max\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}}{{|}\varphi(\alpha_{n}(t+\Delta
t))-{\bf i}y{|}\cdot{|}\varphi(\alpha_{n}(t))-{\bf i}y{|}}.$

Moreover,

$\displaystyle{\big{|}}\textrm{arg}\frac{\varphi(\alpha_{n}(t))-{\bf
i}y}{\varphi(\alpha_{n}(t+\Delta t))-{\bf i}y}{\big{|}}$ $\displaystyle\leq$
$\displaystyle\Delta
t\cdot\frac{\pi(1+\epsilon)}{2(1-\epsilon)}\cdot\frac{(\textrm{Re}\varphi(z_{n+1})-\textrm{Re}\varphi(z_{n}))\cdot\max\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}}{{|}\textrm{Re}\varphi(z_{n}){|}\cdot{|}\textrm{Re}\varphi(z_{n}){|}}$
$\displaystyle=$ $\displaystyle\Delta
t\cdot\frac{\pi(1+\epsilon)}{2(1-\epsilon)}\cdot\left(\frac{\textrm{Re}\varphi(z_{n+1})}{\textrm{Re}\varphi(z_{n})}-1\right)\cdot\frac{\max\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}}{\textrm{Re}\varphi(z_{n})}$
$\displaystyle\leq$ $\displaystyle\Delta
t\cdot\frac{\pi(1+\epsilon)(1-\widetilde{C_{1}})}{2\widetilde{C_{1}}(1-\epsilon)}\cdot\frac{\max\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}}{\textrm{Re}\varphi(z_{n})}.$

$(1)$ Suppose that $y$ satisfies
$\min\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}\leq{|}c_{1}-c_{2}{|}$.

In this case,

$\max\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}\leq\min\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}+{|}c_{1}-c_{2}{|}\leq
2{|}c_{1}-c_{2}{|}.$

Then,

$\displaystyle\sum\limits_{n=N_{2}}^{\infty}{\big{|}}\textrm{arg}\frac{\varphi(\alpha_{n}(t))-{\bf
i}y}{\varphi(\alpha_{n}(t+\Delta t))-{\bf i}y}{\big{|}}$ $\displaystyle\leq$
$\displaystyle\sum\limits_{n=N_{2}}^{\infty}\Delta
t\cdot\frac{\pi(1+\epsilon)(1-\widetilde{C_{1}})}{2\widetilde{C_{1}}(1-\epsilon)}\cdot\frac{2{|}c_{1}-c_{2}{|}}{\textrm{Re}\varphi(z_{n})}$
$\displaystyle\leq$ $\displaystyle\Delta
t\cdot\frac{\pi(1+\epsilon)(1-\widetilde{C_{1}}){|}c_{1}-c_{2}{|}}{\widetilde{C_{1}}(1-\epsilon)}\cdot\sum\limits_{n=N_{2}}^{\infty}\frac{1}{{|}\varphi(z_{n}){|}(1-\epsilon)\sin\theta}$
$\displaystyle=$ $\displaystyle\Delta
t\cdot\frac{\pi(1+\epsilon)(1-\widetilde{C_{1}}){|}c_{1}-c_{2}{|}}{\widetilde{C_{1}}(1-\epsilon)^{2}\sin\theta}\cdot\sum\limits_{n=N_{2}}^{\infty}\frac{{|}1-z_{n}{|}}{{|}1+z_{n}{|}}$
$\displaystyle\leq$ $\displaystyle\Delta
t\cdot\frac{\pi(1+\epsilon)(1-\widetilde{C_{1}}){|}c_{1}-c_{2}{|}}{\widetilde{C_{1}}(1-\epsilon)^{2}\sin\theta}\cdot\sum\limits_{n=N_{2}}^{\infty}{|}1-z_{n}{|}.$

$(2)$ Suppose that $y$ satisfies
$\min\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}\geq{|}c_{1}-c_{2}{|}$.

Let $N\geq N_{2}$ be the first positive such that

$\textrm{Re}\varphi(z_{N})\geq\max\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}.$

Then,

$\displaystyle\sum\limits_{n=N}^{\infty}{\big{|}}\textrm{arg}\frac{\varphi(\alpha_{n}(t))-{\bf
i}y}{\varphi(\alpha_{n}(t+\Delta t))-{\bf i}y}{\big{|}}$ $\displaystyle\leq$
$\displaystyle\sum\limits_{n=N}^{\infty}\Delta
t\cdot\frac{\pi(1+\epsilon)(1-\widetilde{C_{1}})}{\widetilde{C_{1}}(1-\epsilon)}\cdot\frac{\max\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}}{\textrm{Re}\varphi(z_{n})}$
$\displaystyle=$ $\displaystyle\Delta
t\cdot\frac{\pi(1+\epsilon)(1-\widetilde{C_{1}})}{\widetilde{C_{1}}(1-\epsilon)}\cdot\frac{\max\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}}{\textrm{Re}\varphi(z_{N})}\cdot\sum\limits_{n=N}^{\infty}\frac{\textrm{Re}\varphi(z_{n})}{\textrm{Re}\varphi(z_{N})}$
$\displaystyle\leq$ $\displaystyle\Delta
t\cdot\frac{\pi(1+\epsilon)(1-\widetilde{C_{1}})}{\widetilde{C_{1}}(1-\epsilon)}\cdot\sum\limits_{k=0}^{\infty}\widetilde{C_{2}}^{k}$
$\displaystyle=$ $\displaystyle\Delta
t\cdot\frac{\pi(1+\epsilon)(1-\widetilde{C_{1}})}{\widetilde{C_{1}}(1-\epsilon)(1-\widetilde{C_{2}})}.$

Notice that for $t\in[0,1]$

${|}\varphi(\alpha_{n}(t))-{\bf
i}y{|}\geq\min\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}\sin\theta\geq{|}c_{1}-c_{2}{|}\sin\theta.$

It easy to see that

$\displaystyle\frac{\max\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}}{{|}\varphi(\alpha_{n}(t))-{\bf
i}y{|}}\leq$
$\displaystyle\frac{\min\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}+{|}c_{1}-c_{2}{|}}{\min\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}\sin\theta}$
$\displaystyle=$
$\displaystyle\left(1+\frac{{|}c_{1}-c_{2}{|}}{\min\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}}\right)\cdot\frac{1}{\sin\theta}$
$\displaystyle\leq$ $\displaystyle\frac{2}{\sin\theta}.$

Then, for any $N_{2}\leq n\leq N-1$,

$\displaystyle\sum\limits_{n=N_{2}}^{N-1}{\big{|}}\textrm{arg}\frac{\varphi(\alpha_{n}(t))-{\bf
i}y}{\varphi(\alpha_{n}(t+\Delta t))-{\bf i}y}{\big{|}}$ $\displaystyle\leq$
$\displaystyle\sum\limits_{n=N_{2}}^{N-1}\Delta
t\cdot\frac{\pi(1+\epsilon)}{2(1-\epsilon)}\cdot\frac{(\textrm{Re}\varphi(z_{n+1})-\textrm{Re}\varphi(z_{n}))\cdot\max\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}}{{|}\varphi(\alpha_{n}(t+\Delta
t))-{\bf i}y{|}\cdot{|}\varphi(\alpha_{n}(t))-{\bf i}y{|}}$
$\displaystyle\leq$ $\displaystyle\Delta
t\cdot\frac{\pi(1+\epsilon)}{2(1-\epsilon)}\cdot\sum\limits_{n=N_{2}}^{N-1}\frac{(\textrm{Re}\varphi(z_{n+1})-\textrm{Re}\varphi(z_{n}))\cdot\max\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}}{\min\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}\sin\theta\cdot{|}\varphi(\alpha_{n}(t))-{\bf
i}y{|}}$ $\displaystyle\leq$ $\displaystyle\Delta
t\cdot\frac{\pi(1+\epsilon)}{(1-\epsilon)\min\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}\sin^{2}\theta}\cdot\sum\limits_{n=N_{2}}^{N-1}(\textrm{Re}\varphi(z_{n+1})-\textrm{Re}\varphi(z_{n}))$
$\displaystyle=$ $\displaystyle\Delta
t\cdot\frac{\pi(1+\epsilon)}{(1-\epsilon)\sin^{2}\theta}\cdot\frac{\textrm{Re}\varphi(z_{N})-\textrm{Re}\varphi(z_{N_{2}})}{\min\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}}$
$\displaystyle\leq$ $\displaystyle\Delta
t\cdot\frac{\pi(1+\epsilon)}{(1-\epsilon)\sin^{2}\theta}\cdot\frac{\frac{\textrm{Re}\varphi(z_{N})}{\textrm{Re}\varphi(z_{N-1})}\cdot\textrm{Re}\varphi(z_{N-1})}{\min\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}}$
$\displaystyle\leq$ $\displaystyle\Delta
t\cdot\frac{\pi(1+\epsilon)}{(1-\epsilon)\sin^{2}\theta}\cdot\frac{\frac{1}{\widetilde{C_{1}}}\max\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}}{\min\\{{|}y-c_{1}{|},{|}y-c_{2}{|}\\}}$
$\displaystyle=$ $\displaystyle\Delta
t\cdot\frac{2\pi(1+\epsilon)}{(1-\epsilon)\widetilde{C_{1}}\sin^{2}\theta}.$

Thus, in this case,

$\displaystyle\sum\limits_{n=N_{2}}^{\infty}{\big{|}}\textrm{arg}\frac{\varphi(\alpha_{n}(t))-{\bf
i}y}{\varphi(\alpha_{n}(t+\Delta t))-{\bf i}y}{\big{|}}$ $\displaystyle\leq$
$\displaystyle\Delta
t\cdot\left(\frac{2\pi(1+\epsilon)}{(1-\epsilon)\widetilde{C_{1}}\sin^{2}\theta}+\frac{\pi(1+\epsilon)(1-\widetilde{C_{1}})}{\widetilde{C_{1}}(1-\epsilon)(1-\widetilde{C_{2}})}\right).$

Therefore, we could write

$K_{2}=\left\\{\begin{matrix}\frac{\pi(1+\epsilon)(1-\widetilde{C_{1}}){|}c_{1}-c_{2}{|}}{\widetilde{C_{1}}(1-\epsilon)^{2}\sin\theta}\cdot\sum\limits_{n=N_{2}}^{\infty}{|}1-z_{n}{|},\\\
\frac{2\pi(1+\epsilon)}{(1-\epsilon)\widetilde{C_{1}}\sin^{2}\theta}+\frac{\pi(1+\epsilon)(1-\widetilde{C_{1}})}{\widetilde{C_{1}}(1-\epsilon)(1-\widetilde{C_{2}})}\end{matrix}\right\\}$

as required. ∎

######  Proposition 3.4.

For any $B\in\mathcal{\widetilde{H}}_{SC}$, there exists a positive integer
$N_{3}\in\mathbb{N}$ and a positive number $K_{3}$ such that for any
$t\in[0,1]$ and small positive number $\Delta t$,

$\sum_{n}{\big{|}}\textrm{arg}\frac{1-\alpha_{n}(t)}{1-\alpha_{n}(t+\Delta
t)}{\big{|}}\leq K_{3}\Delta t.$

###### Proof.

Without loss of generality, suppose that the zeros sequence of
$B\in\mathcal{\widetilde{H}}_{SC}$ is fine in a strip cone
$SC(1,\theta,T_{1},T_{2})$. Let the strip
$SL(\frac{\pi}{2}-\theta,L_{1},L_{2})$ be the image of
$SC(1,\theta,T_{1},T_{2})$ under the map $\varphi(z)$, where

$L_{1}:y-\tan(\frac{\pi}{2}-\theta)\cdot x-c_{1}=0\ \ \ \text{and}\ \ \
L_{2}:y-\tan(\frac{\pi}{2}-\theta)\cdot x-c_{2}=0.$

For convenience, assume the line $L_{2}$ is on the right of the line $L_{1}$.
By translating $L_{2}$ one unit to the right, we obtain a new line
$\widehat{L_{2}}$, more precisely,

$\widehat{L_{2}}:\ y-\tan(\frac{\pi}{2}-\theta)\cdot(x-1)-c_{2}=0.$

Let

$\varphi(\widehat{z_{n}})=\varphi(z_{n})+1\ \ \text{and}\ \
\varphi(\widehat{\alpha}_{n}(t))=\varphi(\alpha_{n}(t))+1.$

It is not difficult to see that
$\\{\varphi(\widehat{z_{n}})\\}_{n=1}^{\infty}$ also satisfies the following
conditions as well as a fine sequence.

  1. (1)

The sequence $\\{\varphi(\widehat{z_{n}})\\}_{n=1}^{\infty}$ lies in the strip
$SL(\frac{\pi}{2}-\theta,L_{1},\widehat{L_{2}})$.

  2. (2)

$\lim\limits_{n\rightarrow\infty}\textrm{arg}(\varphi(z_{n+1})-\varphi(z_{n}))=\frac{\pi}{2}-\theta$.

  3. (3)

${\rm Re}\varphi(\widehat{z_{n}})$ monotonically tends to $+\infty$, and there
are two positive numbers $\widetilde{D_{1}}$ and $\widetilde{D_{2}}$ such that

$0<\widetilde{D_{1}}\leq\frac{{\rm Re}\varphi(\widehat{z_{n}})}{{\rm
Re}\varphi(\widehat{z_{n+1}})}\leq\widetilde{D_{2}}<1.$

Then, the Proposition 3.3 also holds for the sequence
$\\{\varphi(\widehat{z_{n}})\\}_{n=1}^{\infty}$. That is, there exists a
positive integer $N_{3}\in\mathbb{N}$ and a positive number $K_{3}$ such that
for any $t\in[0,1]$ and small positive number $\Delta t$,

$\sup\limits_{y\in\mathbb{R}}\textrm{ess}\sum\limits_{n=N_{3}}^{\infty}{\big{|}}\textrm{arg}\frac{\varphi(\widehat{\alpha}_{n}(t))-{\bf
i}y}{\varphi(\widehat{\alpha}_{n}(t+\Delta t))-{\bf i}y}{\big{|}}\leq
K_{3}\Delta t.$

In particular, the above inequality holds for $y=0$, and hence

$\displaystyle K_{3}\Delta t\geq$
$\displaystyle\sum\limits_{n=N_{3}}^{\infty}{\big{|}}\textrm{arg}\frac{\varphi(\widehat{\alpha}_{n}(t))}{\varphi(\widehat{\alpha}_{n}(t+\Delta
t))}{\big{|}}$ $\displaystyle=$
$\displaystyle\sum\limits_{n=N_{3}}^{\infty}{\big{|}}\textrm{arg}\frac{\varphi(\alpha_{n}(t))+1}{\varphi(\alpha_{n}(t+\Delta
t))+1}{\big{|}}$ $\displaystyle=$
$\displaystyle\sum\limits_{n=N_{3}}^{\infty}{\big{|}}\textrm{arg}\frac{\frac{1+\alpha_{n}(t)}{1-\alpha_{n}(t)}+1}{\frac{1+\alpha_{n}(t+\Delta
t)}{1-\alpha_{n}(t+\Delta t)}+1}{\big{|}}$ $\displaystyle=$
$\displaystyle\sum\limits_{n=N_{3}}^{\infty}{\big{|}}\textrm{arg}\frac{1-\alpha_{n}(t+\Delta
t)}{1-\alpha_{n}(t)}{\big{|}}.$

This completes the proof. ∎

Proof of Main Theorem : Given any $B\in\mathcal{{H}}_{SC}$. Without loss of
generality, we may assume that its zeros lie in a strip cone
$SC(1,\theta,T_{1},T_{2})$, $\theta\in(0,\pi)$. By Lemma 2.7, it has a factor
$\widetilde{B}\in\mathcal{\widetilde{H}_{SC}}$, denoted by

$\widetilde{B}(z)=\prod\limits_{n=1}^{\infty}\frac{{|}z_{n}{|}}{z_{n}}\frac{z_{n}-z}{1-\overline{z_{n}}z}.$

Following from Proposition 3.2, Proposition 3.3, Proposition 3.4 and Lemma
3.1, we could choose $N\geq\max\\{N_{1},N_{2},N_{3}\\}$ and then there is a
continuous path from

$\prod\limits_{n=N}^{\infty}\frac{{|}z_{n}{|}}{z_{n}}\frac{z_{n}-z}{1-\overline{z_{n}}z}\
\ \ \text{to}\ \ \
\prod\limits_{n=N+1}^{\infty}\frac{{|}z_{n}{|}}{z_{n}}\frac{z_{n}-z}{1-\overline{z_{n}}z}.$

Therefore, by the path-connectedness of Möbius transformations and Lemma 2.1,
we have $B\sim zB$.

## Declarations

  * •

Ethics approval

Not applicable.

  * •

Competing interests

The author declares that there are no conflict of interest or competing
interests.

  * •

Authors’ contributions

All authors reviewed this paper.

  * •

Funding

There is no funding source for this manuscript.

  * •

Availability of data and materials

Data sharing is not applicable to this article as no datasets were generated
or analyzed during the current study.

## References

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# Resonance: Replacing Software Constants with
Context-Aware Models in Real-time Communication

Jayant Gupchup, Ashkan Aazami, Yaran Fan, Senja Filipi, Tom Finley, Scott
Inglis,
Marcus Asteborg, Luke Caroll, Rajan Chari, Markus Cozowicz, Vishak Gopal,
Vinod Prakash,
Sasikanth Bendapudi, Jack Gerrits, Eric Lau, Huazhou Liu , Marco Rossi, Dima
Slobodianyk,
Dmitri Birjukov, Matty Cooper, Nilesh Javar, Dmitriy Perednya,
Sriram Srinivasan, John Langford, Ross Cutler, Johannes Gehrke

Microsoft Presently affiliated with Outreach.io. Author performed work while
at Microsoft

###### Abstract

Large software systems tune hundreds of ‘constants’ to optimize their runtime
performance. These values are commonly derived through intuition, lab tests,
or A/B tests. A ‘one-size-fits-all’ approach is often sub-optimal as the best
value depends on runtime context. In this paper, we provide an experimental
approach to replace constants with learned contextual functions for Skype\- a
widely used real-time communication (RTC) application. We present Resonance, a
system based on contextual bandits (CB). We describe experiences from three
real-world experiments: applying it to the audio, video, and transport
components in Skype. We surface a unique and practical challenge of performing
machine learning (ML) inference in large software systems written using
encapsulation principles. Finally, we open-source _FeatureBroker_ , a library
to reduce the friction in adopting ML models in such development environments.

##  1 Introduction

Hyperparameter tuning of ML models using automation has a very rich literature
golovin2017google ; akiba2019optuna ; perrone2019learning ; elsken2019neural ;
jin2019auto ; paul2019fast . However, tuning of application constants such as
buffer sizes, thresholds, or timeout settings is commonly done manually.
Tuning these constants in live production systems requires overcoming many
practical hurdles. In this paper, we study these challenges for a widely used
RTC application.

Motivating Example: Consider a video conference call in Skype where one of the
participants is using an unstable network. In Skype, if the endpoint stops
receiving media packets, the application attempts to “reconnect” the call on
an alternate interface (e.g., WiFi interface stops working, but 4G network is
active). The reconnect threshold constant represents a careful trade-off. If
the switch is triggered too early, the user’s connection is unnecessarily
moved into the “call reconnecting” status and starts to renegotiate all call
parameters, leading to a false positive (a disruptive experience). If the
application waits too long, the user loses patience and hangs up due to
frustration. Furthermore, we found mobile users tend to hang up more quickly
than desktop users, indicating the contextual nature of this problem as shown
in Figure 2.

Literature: Recently, a few solutions and systems have emerged to address some
of these challenges. Carbine et al. present _SmartChoices_ , an API to replace
heuristics in traditional algorithms with ML models built using standard
reinforcement learning (RL) methods smartchoices . _SmartChoices_ demonstrated
the effectiveness on search, sort (pivot index), and cache algorithms.
mao2016resource  and krishnan2018learning  are examples of applying RL based
methods in query optimization and job scheduling systems respectively. To
replace runtime constants, we employ a simpler approach of CB as we found it
to scale effectively in our production scenarios compared to RL. This
observation has been echoed in bakshy2018ae ; eytantalk ; dulac2019challenges
. In eytantalk  Bakshy et al. describe _Ax_ , a CB-based optimization and
experimentation system for learning parameters at Facebook. In Resonance, we
address a novel and practical challenge of ML inference in real-world systems
when context information is trapped behind API surfaces.

Inference Challenges: Large production software systems comprise of components
interacting via strict API boundaries (or contracts). This encapsulation
creates isolation, making it difficult to surface context (or features) to be
consumed by other components for runtime inference (detailed example in
Section 2). A naive approach requires several layers of API changes and
alignment that result in the breaking of established contracts. We find that
these high development costs were a big driver for the lack of adoption (or
integration) of CB models relying on distributed context. It is noteworthy
that this challenge is not unique to CB models, but rather any model requiring
access to features distributed in siloed components. To scale adoption, we
created a library called _FeatureBroker_ that was able to address these
challenges and bring down integration costs from 2 months to 2 days.

Contributions: This paper makes the following contributions:

  * •

We outline _Resonance_ , an experimental system to replace constants with CB
models for RTC scenarios.

  * •

We detail _FeatureBroker_ , a library for managing the conversation between an
inference engine and distributed features produced by components. We make this
library open source for the benefit of the community featurebroker .

  * •

We present results from applying _FeatureBroker_ to optimize Skype’s audio,
video, and transport components in production (Section 3).

Figure 1: The left figure shows the _Network Reconnect_ feature in Skype. The
right figure shows the impact of trigger threshold on average call duration
(ACD). 222Figure drawn for illustration by modeling data gathered from real
telemetry.

##  2 Resonance

System Overview: Resonance comprises of the following components: _Skype App_
, _A/B test system_ , _Client Inference (CI)_ and _Learning Service (LS)_ as
described in agarwal2016multiworld . The benefit of replacing an _App_
constant is evaluated as an A/B test; the control variant represents the fixed
constant, whereas the treatment variant is represented by a CB model built
using _VowpalWabbit (VW)_ langford2007vowpal  – a well known open source
library for CB models. The _CI_ component performs runtime inference based on
the _App_ context as specified by the model. The _CI_ component captures
training logs comprising of {context value, recommended action, action
probability, reward metric} and transmits them to the _LS_. The _LS_ uses
these training logs to produce a _VW_ models (or policies) that get delivered
to the client endpoints periodically (e.g., every 4 hours) completing the
inference-log-learn loop.

In this paper, we focus our attention on the novel _FeatureBroker_ client
library. In our explanations, we define context as a collection of feature
values (e.g., network type = WiFi, platform = mobile).

⬇

// Inference without FeatureBroker

jbHysteresis = model.predict([

FeatureStore.GetValue("nwType"),

FeatureStore.GetValue("platform"),

FeatureStore.GetValue("callType")]);

Figure 2: Left figure shows the organization of components within Skype’s RTC
stack; illustrating the distributed nature of features in a well-encapsulated
system. The right figure shows the inference performed in the absence of a
library like _FeatureBroker_.

Feature Broker: We motivate the need for such a library by way of an example.
Consider we want to replace a constant _jbHysteris_ in the audio component
with the output of a CB model. A simple API to do that is shown in Figure 2
(right side).

While this API looks simple, practical problems emerge when performing
inference using this approach. In this scenario, the context depends on the
global state of the system, such as the platform (e.g., desktop/wired), the
network type (e.g., wired/WiFi), and call modality (e.g., audio/video). Figure
2 (left side) shows the logical organization of components within our RTC
application. While call modality information is available within the audio
component, platform and network type information were produced in different
components. Moreover, the audio and platform/transport components were
separated by many other components and API surfaces. This situation where the
features of interest are encapsulated in well-separated components was found
to be a common pattern. Component owners want well-defined, stable public API
surfaces for maintenance reasons. In addition, expanding them on a feature-by-
feature basis is onerous. Furthermore, the components producing the features
may not be running in the same thread as the component performing inference,
and lastly, the changes in feature values at runtime (e.g., a mid-call change
in network type) would require updating the prediction gracefully.

The idea of a simple shared thread-safe key-value feature store proved
insufficient for two primary reasons:

  1. 1.

_Coherence_ : Inference libraries need feature values to be stable and
consistent. However, as seen above, feature values can update asynchronously
(e.g., network type can change mid-call).

  2. 2.

_Hierarchy_ : Features may be consumed by multiple models and inference
components. For example, three models (_JBHysteresis_ , _ScreenshareEncoding_
and _NetworkReconnect_) may consume platform and network type features while
having their own local features (e.g., call modality). The most graceful way
to handle this was to make the structure hierarchical and allow it to be
“forked” for sub-components.

The shared structure responsible for handling these problems is termed
_FeatureBroker_. It manages the conversation between the client code that
provides features (inputs) and consumes inference results (outputs) from a CB
model. In this paper, it would be impossible to describe the library in
detail. Instead, we highlight the main elements of the solution. Interested
readers can see a detailed real-world example in the open-source version of
the library _FeatureBroker_ featurebroker . The key operations of the
_FeatureBroker_ are:

  1. 1.

_Binding Inputs_ : Given a name and type returns an “input pipe” into which a
feature providing component can feed values.

  2. 2.

_Associating Models_ : Register and describe a scheme for transforming inputs
to outputs.

  3. 3.

_Binding Outputs_ : Returns an “output pipe” in which inference consuming
models can query for updated values.

The notion of binding draws parallels to binding values for form elements
obtained from a database lerman2010programming . Finally, it is noteworthy
that _FeatureBroker_ can be used with any inference library (e.g., ONNX
onnxruntime ) and the concept even generalizes beyond CB models - the problem
is most amplified in CB inference due to the distributed nature of context
generation. As stated earlier, detailed examples are presented in
featurebroker .

##  3 Experimental Results

Experiment Name |  | _Audio Jitter Buffer_
---
| _Screen Share Encoding_
---
| _Network Reconnect_
---
Platforms | Desktop |  | {Desktop, Mobile}
---
| {Desktop, Mobile}
---
| Impressions (Millions)
---
4M | 7.6M | 19M
| RTC Component
---
Audio | Video | Transport
Number of Actions | 10 | 10 | 5
Unique Contexts | 14 | 440 | 390
Metric |  | Poor-Audio-rating
---
| Poor-Video-rating
---
| Call Duration (CD)
---
Model Size | 4 KB | 22 KB | 10 KB
| Metric Improvement (%)
---
1.1% | 9.9% | 4.2%
Table 1: The table captures the overall Summary results from experiments
conducted on the representative scenarios.

We present results from three scenarios in Skype. The scenario related to
learning the reconnect threshold was described in Section 1. We introduce two
other scenarios.

Scenario: Audio Jitter Buffer (JB) Hysteresis The JB component absorbs
variability in network packet arrivals (i.e., jitter) to present a smooth
playout of audio frames transmitted via the network. _JBHysteresis_ is a meta-
parameter controlling the amount of inertia associated with changes in the
buffer size. The optimization metric to learn this parameter was Poor-Audio-
Rating, an objective metric estimating speech quality based on packet loss,
jitter, and conversational delay derived from user ratings ding2003speech .

Scenario: Video Encoding Bitrate Allocation for Screen Share Screen sharing
sessions in VoIP require careful selection of the bitrate. Large bit rates
lead to a higher image quality, but are also susceptible to frame freezes due
to packet losses under poor network conditions. This quality-distortion
tradeoff is characterized by a constant that weights transmission rates and
frame freezes. Similar to audio, we optimized for a video technical metric
termed _Poor-Video-Rating_.

Experiments and Results: A summary of the experimental data for all three
scenarios is presented in Table 1. The data is collected using an
$\epsilon$-greedy policy. The $\epsilon$ was set to 0.2 for the experiments.
For each experiment, we report the number of unique contexts to convey the
range of inputs going into the _FeatureBroker_. Each of these experiments
showed statistically significant improvements between the fixed-constant
(default) policy and the model-based policy with a p-value less than $0.01$.
We note the following:

  * •

These experiments show that we could successfully replace constants with CB
models for RTC scenarios. Since these experiments were done in three different
components by three different teams, it demonstrates the generality of the
_FeatureBroker_ solution.

  * •

New experiments were able to build and re-use the on-boarding of features done
by previous experiments. For example, the on-boarding of features for
_ScreenShareEncoding_ experiment simplified the integration of the
_NetworkReconnect_ model as the two models shared features. Prior to the
introduction of _FeatureBroker_ , engineering teams would have to coordinate
and align on API changes to expose features. This process would often take
multiple days, and has been reduced by an order of magnitude.

  * •

Due to the limited memory and CPU budget in RTC applications, we focused on
keeping the footprint of CB models small. The main reason was to scale this
methodology so that it can be applied to 100s of scenarios within Skype.

##  4 Discussion

This paper introduces Resonance, a system for replacing application constants
with context-aware models. We built and deployed this system for the real-time
production scenarios of Skype. We presented _FeatureBroker_ , a library for
integrating ML models relying on global context. The library presents a
solution to the problem of inference when the context is distributed across
strong component boundaries. We make this library available to practitioners
to bring down the adoption and development costs. Using three statistically
significant real-world experiments, we showed this methodology can improve
system performance. Such a methodology provides a new tool for component
owners to optimize their components.

Future Work: The problem of efficient discovery of sensitive constants in a
live system containing thousands of constants remains challenging.

In conclusion, we emphasize that the ideas presented in Resonance and
challenges solved by _FeatureBroker_ are applicable to many software systems,
particularly large codebases that ship with 1000s of system constants.

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