# Weak-Supervised Dysarthria-invariant Features for Spoken Language
Understanding using an FHVAE and Adversarial Training

###### Abstract

The scarcity of training data and the large speaker variation in dysarthric
speech lead to poor accuracy and poor speaker generalization of spoken
language understanding systems for dysarthric speech. Through work on the
speech features, we focus on improving the model generalization ability with
limited dysarthric data. Factorized Hierarchical Variational Auto-Encoders
(FHVAE) trained unsupervisedly have shown their advantage in disentangling
content and speaker representations. Earlier work showed that the dysarthria
shows in both feature vectors. Here, we add adversarial training to bridge the
gap between the control and dysarthric speech data domains. We extract
dysarthric and speaker invariant features using weak supervision. The
extracted features are evaluated on a Spoken Language Understanding task and
yield a higher accuracy on unseen speakers with more severe dysarthria
compared to features from the basic FHVAE model or plain filterbanks.

Index Terms—  Dysarthric speech, FHVAE, adversarial training, weak
supervision, end-to-end spoken language understanding.

##  1 Introduction

Dysarthric speech data, especially accurately labeled data, are scarce due to
difficulties in recruitment, collection and labeling. This data insufficiency
problem has been a constant barrier for automatic speech recognition (ASR)
model training. Recent ASR models[1, 2, 3] tend to be more complex and employ
more parameters, which increases the difficulty to train from limited
dysarthric data even further.

Current research mainly solves the dysarthric speech data deficiency problem
in three ways. One solution is taking abundant canonical (control) speech data
into account during training, e.g. pretraining on control speech and
finetuning on dysarthric speech [4, 5, 6]. A second approach is to decrease
the model size [7], or to train an inserted small module instead of finetuning
the whole model [8, 9], so the number of parameters learned on the dysarthric
data is limited. Thirdly and differently from the solutions that work on
training strategy or model structure, [10, 11, 12, 13] focus directly on the
data and do augmentation to generate more dysarthric speech for use in
training.

The features used in most methods preserve the speaker characteristics and the
dysarthria information, and thus bound the model generalization ability.
However, in practical scenarios, we would like the ASR model to have great
generalization ability and work well on unseen speakers (users), to reduce any
training effort for dysarthric users. In this work, we focus on improving the
automatic recognition model generalization ability with limited dysarthric
data. Instead of training and testing the model with speech features that
preserve variability between dysarthric speakers, we propose to use
dysarthric-invariant and speaker independent speech features for automatic
recognition use, which can both boost the model generalization ability on
unseen speakers and gives the possibility to involve abundant canonical speech
into training.

Factorized Hierarchical Variational Auto-Encoders (FHVAE) [14, 15] have shown
their ability to disentangle the content space and the speaker space in input
speech. An FHVAE models the generative process of a sequence of segments in a
hierarchical structure. It encodes a speech utterance into a segment-related
variable (short time scale) and a sequence-related variable (long time scale)
via two linked encoders. The segment-related latent variable represents the
information that only appears in a single segment in the sequence, such as the
content of the segment, and is conditioned on the sequence-related variable.
The sequence-related variable reflects the features of the whole sequence,
like the acoustic environment or speaker characteristics. The model hence
offers a separation between speaker characteristics (sequence) and content
(segment). The content variable (segment-related variable) appears to fit the
speaker-independence desideratum outlined above. However, our previous work
[16] shows that the dysarthria is reflected in both latent variables.

To achieve a greater deal of dysarthria-invariance in the content variable,
instead of using forced regularization with phoneme alignment [16], inspired
by [17], we introduce adversarial training [18] to the FHVAE model. We feed
the FHVAE model with both control and dysarthric speech, regarding the content
variable encoder as the generator. We add a discriminator that tries to
distinguish control from dysarthric content variables. With adversarial
training between the generator and the discriminator, we can force the content
variable encoder (generator) to extract content information from both control
and dysarthric speech that shares the same dysarthria-invariant latent space.
To make the extracted variable closer to the control data space and improve
disentanglement, we also try two additional loss terms called reference loss
and disentanglement loss (see section 2). Note that we only need the speaker
type label as weak supervision of the training of the complete proposed model.
The content encoder is used by both dysarthric speech and canonical speech,
which gives opportunities to involve abundant canonical speech features into
the downstream model (ASR) training in future work.

For evaluation, we use the trained content encoder to extract the content
variables from other datasets and use these variables to train and test a
mature End-to-End (E2E) Spoken Language Understanding (SLU) system [19, 20].
The system consists of a recurrent neural network (RNN) encoder, a capsule
network [21] and output layers, and can transfer input speech features into an
intent (semantic) representation without using an explicit textual
representation. E2E SLU evaluation is different from the evaluation on
speaker-independent, in the sense that we have (a limited number of) task-
specific recordings by dysarthric speakers at our disposal for training the
SLU system, whereas ASR systems would typically be trained on task-agnostic
data. Because of the domain adaptation inherent in the evaluation, it is
harder to show generalization properties on E2E SLU tasks.

Apart from SLU evaluation, we also train and test a binary speaker type
classifier (control/dysarthria) using both content variables and sequence-
related variables, and use the classification accuracy to show how the
dysarthria information distribution changes after the adversarial training.

We introduce the feature extraction methods, the FHVAE model with adversarial
training, reference loss and disentanglement loss, in section 2. Section 3
describes the data and the experimental settings. Results and analysis will be
provided in section 4, and section 5 gives conclusions.

##  2 Method

In this section, we introduce the feature extraction methods: the FHAVE model
extended with adversarial training, reference loss and disentanglement loss.

###  2.1 FHVAE with adversarial training

Suppose we have $I$ speech data sequences, each sequence $\textbf{X}^{i}$
containing $N_{i}$ segments $\textbf{x}^{i,n}$, $n=1,2,...,N_{i}$. The segment
$\textbf{x}^{i,n}$ will be represented by two latent variables
$\textbf{z}^{i,n}_{1}$ and $\textbf{z}^{i,n}_{2}$ via an FHVAE:

$q(\textbf{z}^{i,n}_{2}|\textbf{x}^{i,n})=\mathcal{N}(\mathbf{Enc}_{\mu_{\textbf{z}_{2}}}(\textbf{x}^{i,n}),\mathbf{Enc}_{\sigma^{2}_{\textbf{z}_{2}}}(\textbf{x}^{i,n}))$
(1)
$q(\textbf{z}^{i,n}_{1}|\textbf{x}^{i,n},\textbf{z}^{i,n}_{2})=\mathcal{N}(\mathbf{Enc}_{\mu_{\textbf{z}_{1}}}(\textbf{x}^{i,n},\textbf{z}^{i,n}_{2}),\mathbf{Enc}_{\sigma^{2}_{\textbf{z}_{1}}}(\textbf{x}^{i,n},\textbf{z}^{i,n}_{2})$
(2)

where $\mathbf{Enc}_{\mu_{\textbf{z}_{2}}}(\cdot)$ and
$\mathbf{Enc}_{\sigma^{2}_{\textbf{z}_{2}}}(\cdot)$ are the networks encoding
input x to mean and variance of $\textbf{z}_{2}$,
$\mathbf{Enc}_{\mu_{\textbf{z}_{1}}}(\cdot)$ and
$\mathbf{Enc}_{\sigma^{2}_{\textbf{z}_{1}}}(\cdot)$ are the encoders of
$\textbf{z}_{1}$. Following the theory of variational auto-encoders, the
latent variables should follow prior distributions:

$p(\textbf{z}^{i,n}_{1})=\mathcal{N}(\textbf{0},\sigma^{2}_{\textbf{z}_{1}}\textbf{I})$
(3)
$p(\textbf{z}^{i,n}_{2}|\boldsymbol{\mu}^{i}_{2})=\mathcal{N}(\boldsymbol{\mu}^{i}_{2},\sigma^{2}_{\textbf{z}_{2}}\textbf{I}),\quad
p(\boldsymbol{\mu}^{i}_{2})=\mathcal{N}(\textbf{0},\sigma^{2}_{\boldsymbol{\mu}_{2}}\textbf{I})$
(4)

where $\sigma^{2}_{\textbf{z}_{1}},\sigma^{2}_{\textbf{z}_{2}}$ and
$\sigma^{2}_{\boldsymbol{\mu}_{2}}$ are chosen upfront and
$\textbf{z}^{i,n}_{1},\textbf{z}^{i,n}_{2}$ are i.i.d samples. The prior
enforces variables $\textbf{z}^{i,n}_{2}$ within the same sequence $i$ to be
close.

The $\textbf{z}^{i,n}_{1}$, conditioned on $\textbf{z}^{i,n}_{2}$ (see
equation2), only contains information with the speech segment
$\textbf{x}^{i,n}$, and is regarded as the speaker-independent content
variable. The variable $\textbf{z}^{i,n}_{2}$ reflects the features of the
whole sequence $\textbf{X}^{i}$ and can be seen as speaker variable. The mean
of sequence-level variable $\boldsymbol{\mu}^{i}_{2}$ can be inferred as:

$\tilde{\mu}^{i}_{2}=\frac{\sum^{N}_{n=1}\mathbf{Enc}_{\mu_{\textbf{z}_{2}}}(\textbf{x}^{i,n})}{N+\frac{\sigma^{2}_{\textbf{z}_{2}}}{\sigma^{2}_{\boldsymbol{\mu}_{2}}}}$
(5)

Then for the whole sequence $\textbf{X}^{i}$, the inference model is written
as:

$q(\textbf{Z}^{i}_{1},\textbf{Z}^{i}_{2},\boldsymbol{\mu}^{i}_{2}|\textbf{X}^{i})=q(\boldsymbol{\mu}^{i}_{2}|\textbf{x}^{i,n})\prod^{N}_{n=1}q(\textbf{z}^{i,n}_{1}|\textbf{x}^{i,n})q(\textbf{z}^{i,n}_{2}|\textbf{x}^{i,n})$
(6)

where $\textbf{Z}^{i}_{1}=\\{\textbf{z}_{1}^{i,n}\\}^{N}_{n=1}$,
$\textbf{Z}^{i}_{2}=\\{\textbf{z}_{2}^{i,n}\\}^{N}_{n=1}$. The generative
model for the data is then:

$p(\textbf{x}^{i,n}|\textbf{z}^{i,n}_{1},\textbf{z}^{i,n}_{2})=\mathcal{N}(\mathbf{Dec}_{\mu_{x}}(\textbf{z}^{i,n}_{1},\textbf{z}^{i,n}_{2}),\mathbf{Dec}_{\sigma^{2}_{x}}(\textbf{z}^{i,n}_{1},\textbf{z}^{i,n}_{2}))$
(7)

where $\mathbf{Dec}_{\mu_{x}}(\cdot)$ and
$\mathbf{Dec}_{\sigma^{2}_{x}}(\cdot)$ form the decoder networks of the FHVAE.

The dysarthria information, as a speaker characteristic, should be extracted
to speaker variable $\textbf{z}^{i,n}_{2}$ by the FHVAE. However, our previous
work [16] found that the FHVAE does not separate the dysarthria and content
information and speech impairment is identifiable from $\textbf{z}^{i,n}_{1}$.

To obtain dysarthria-invariant features $\textbf{z}^{i,n}_{2}$, inspired by
[17], we introduce adversarial training into the FHVAE model. The data flow of
the proposed model is provided in figure 1.

Repurposing the $\mathbf{z}_{1}$ mean encoder
$\mathbf{Enc}_{\mu_{\textbf{z}_{1}}}(\cdot)$ as the generator in adversarial
training, we get $\mu_{\textbf{z}_{1}}^{i,n}$ as the generator output and feed
this into a discriminator $\mathbf{D}(\cdot)$ which is trained to yield the
probability $p^{i,n}_{D}$ that a speech segment $\textbf{x}^{i,n}$ stems from
a control speaker:

$p^{i,n}_{D}=\mathbf{D}(\mu_{\textbf{z}_{1}}^{i,n})=\mathbf{D}(\mathbf{Enc}_{\mu_{\textbf{z}_{1}}}(\textbf{x}^{i,n}))$
(8)

Training of $\mathbf{D}(\cdot)$ involves dysarthric and control data and
minimizes cross-entropy:

$L_{D}=CE_{dys}+CE_{ctrl}$ (9)

where

$\begin{split}CE_{dys/ctrl}=\mathrm{E}_{\mu_{\textbf{z}_{1}}\sim
p_{dys/ctrl}({\mu_{\textbf{z}_{1}})}}[-(l^{i,n}_{dys/ctrl}\times\log
p^{i,n}_{D})&\\\
+(1-l^{i,n}_{dys/ctrl})\times\log(1-p^{i,n}_{D})]&\end{split}$ (10)

and $l^{i,n}_{dys/ctrl}$ is the label of speech segment $\textbf{x}^{i,n}$,
i.e. $l^{i,n}_{dys/ctrl}$ is $1$ when $\textbf{x}^{i,n}$ is dysarthric and is
$0$ for control speech.

The loss function of the FHVAE model can be written as:

$L_{FHVAE}=L_{LB}+L_{z_{2},disc}*W_{z_{2},disc}+L_{Gen}*W_{Gen}$ (11)

where $L_{LB}$ is the variational lower bound loss of the FHVAE model that
guarantees the performance of the FHVAE model as an auto-encoder;
$L_{z_{2},disc}$ is the discriminative loss of the sequence-related latent
variable, which expresses that sequence identity should be inferrable from the
sequence-related embedding vector, i.e., it is used to encourage separation in
the latent space. $L_{Gen}$ is the cross-entropy between the discriminator
output and the opposite dysarthric/control label (control is $0$ and
dysarthric is $1$), similar to equation 9. Unlike a classical Generative
Adversarial Network (GAN), the $L_{Gen}$ here acts on the whole FHVAE model
instead of only on the generator part.

Fig. 1: Dataflow in the proposed model with adversarial training, reference
loss and disentanglement loss.

###  2.2 Extension with reference loss and disentanglement loss

In the model presented so far, the adversarial training will modify the
encoder output on both control and dysarthric input, as we want a uniform
feature extractor for both types of speech. Modifying the behavior for control
data is not desirable since the FHVAE is supposed to be pre-trained well in
that part of the input space. Thus we add an additional loss term
$L_{z_{1},ref}$ called reference loss into the FHVAE model training loss with
weight $W_{z_{1},ref}$. The reference loss is active on control speech only
and is the Kullback-Leibler divergence between the reference $z_{1}$
distribution $\mathcal{N}(\mu_{z_{1},ref},\sigma^{2}_{z_{1},ref})$ of the
FHVAE pre-trained on control speech and the $z_{1}$ distribution during
training. We also consider evaluating the $L_{Gen}$ term in Eq. 11 only on the
dysarthric data ($CE_{dys}$), while the other terms are unchanged. In doing
so, the content encoder emphasizes changes for dysarthric speech as well.

Finally, to further disentangle speaker and content information, we add the
disentanglement loss $L_{dstg}$ with weight $W_{dstg}$, which minimizes the
sum of squares of the entries of the correlation matrix between $\mu_{z_{1}}$
and $\mu_{z_{2}}$.

The reference loss module and disentanglement loss module are shown in figure
1.

##  3 Experiments

This section explains how we implement and evaluate the proposed model, the
datasets and the experimental details.

###  3.1 Datasets

We use four datasets for pretraining, finetuning and evaluation, respectively:

  * •

CGN dataset [22]: we use the ’k’ and ’o’ subsets from the Dutch CGN dataset
for FHVAE model pretraining. It contains 35.2-hours of broadcast news and read
speech from 326 control Dutch speakers.

  * •

Torgo dataset [23] is used for model finetuning, containing 6.3 hours of
English speech from 7 control speakers (3.8 h) and 8 speakers with dysarthria
(2.5 h).

  * •

Domotica dataset [24] is used for SLU evaluation, containing 8.8 h Dutch
speech from 15 speakers with dysarthria (2 speakers have no intelligibility
score and are excluded), uttering commands related to home automation, such as
”turn on the light in the kitchen”. The commands are encoded to 22 slot value
labels and 4 categories, and each category command is represented as 1 or 2
active labels. There are in total 3789 speech commands. The severity level
(intelligibility score) of dysarthric speakers can be found in [24].

  * •

COPAS dataset [25] is used for speaker type classification evaluation,
containing 7 hours of Dutch speech from 131 control speakers (3 h) and 230
speakers with speech disorders (4 h). 4699 utterances in total with an average
duration of 6 seconds.

###  3.2 Experimental settings

We first pre-train the FHVAE model, then finetune it using the adversarial
training module and the two additional loss terms. We also provide a
comparison where we finetune the pretrained FHVAE without adversarial
training. Then we use the trained encoders
$\mathbf{Enc}_{\mu_{\textbf{z}_{1}}}(\cdot)$ and
$\mathbf{Enc}_{\mu_{\textbf{z}_{2}}}(\cdot)$ to extract content and sequence-
related variables for each segment. The content-related variables are then
used to train and test the SLU system, while the speaker type classifier uses
both. The experimental design is illustrated in figure 2.

Fig. 2: Experimental design.

####  3.2.1 Evaluation setup and metrics

Since our SLU data contains only speakers with dysarthria, we perform
generalization tests across different severity levels rather than control
versus dysarthric speech.

The evaluation on SLU uses two train/test divisions. For the Out-of Domain
evaluation, we use speech data ($2642$ speech utterances) from 11 speakers
with mild to medium dysarthria (intelligibility score $\geq 70$) to train the
SLU model and use 4 speakers with severe dysarthria (intelligibility score
$<70$) for testing. In this experiment, we evaluate the generalization ability
of the model across different severity levels. If the content variable is more
dysarthria-invariant, the SLU model should perform better thanks to better
generalization.

In the In-Domain type evaluation, we order the speakers by their
intelligibility score from high to low, and use the data ($1901$ speech
utterances) from the 8 odd-numbered speakers for training and the rest for
testing. In this type, the SLU model sees a wide range of severity levels but
tests on unseen speakers within the data domain.

For SLU system evaluation, the micro-averaged F1-score of the detected labels
is used as the metric. For each type, we repeat the experiment 5 times and use
the average F1 score as the final result.

For the dysarthria classification, we do 6-fold cross-validation experiment
that we divide all COPAS speakers into 6 blocks, each with the same number of
control and disordered speech utterances. For each fold, we use four blocks to
train the classifier, one for validation and one for testing. We repeat the
6-fold experiment 5 times and use the averaged classification accuracy
(correct rate) as the final classification results.

####  3.2.2 Network and training setup

The speech features used in the FHVAE model are $80$ MEL filter bank energies
with a frame advance of 10 ms. During training, each FHVAE segment is 20
frames with 8 frames shift, i.e., 200 ms of speech. At inference, the shift is
one frame.

In the FHVAE network, the encoder consists of 2 layers of bidirectional LSTM
layers and 2 parallel dense layers for each latent variable. Each LSTM has 256
units, and the dimension of each latent variable is 32. The decoder consists
of 2 bidirectional LSTM layers of the same size and 2 dense layers of 80 units
for $\mu_{x}$ and $\sigma^{2}_{x}$ prediction, respectively. The discriminator
consists of two dense layers with 32 hidden nodes and 1 hidden node,
respectively. The first layer has $leaky\,ReLU$ activation and a $sigmoid$ is
used for the discriminator output.

We choose the Adam optimizer for training with loss weights
$W_{z_{2},disc}=10$, $W_{Gen}=500$, $W_{z_{1},ref}=0.1$, $W_{dstg}=1$ to match
the magnitude of other losses in the final loss function. The hierarchical
sampling size is $5000$ [26], the batch size is $500$ and the maximal number
of training epochs is $50$. Early stopping with patience of $10$ is used
during baseline training and simple finetuning. For adversarial training,
training stops when it reaches the maximal number of epochs. The default
learning rate for the FHVAE model (pre)training and finetuning is $0.001$. For
the proposed model with adversarial training, we use a learning rate of
$0.001$ for the FHVAE model and $0.0002$ for the discriminator.

We choose the SLU system of [20] with unmodified setting for evaluation. We
set the number of epochs to $50$, use Adam Optimizer with a learning rate
$0.001$ and use batch size $16$. For dysarthria classification, the classifier
consists of 2 dense layers with 100 and 2 hidden nodes, respectively. The
first layer has $ReLU$ activation and the second has $Softmax$ activation. We
use a batch size of $128$. We aggregate the input data over the utterance by
taking the mean vector as the classifier input data.

Table 1: F1-scores of SLU tasks (left of the table) and accuracy of dysarthria classifier (right part of the table) using extracted content variables, speaker variables or $Fbank$ features as training and testing data. (${}^{*}:p-value\leq 0.05$ compared to results of data extracted by FHVAE simply finetuned on Torgo, ${}^{\dagger}:p-value\leq 0.05$ compared to results of $Fbank$ features. The $p-value$s are calculated by Wilcoxon signed rank test on recognition accuracy, where the score per utterance is averaged over 5 trials.) Feature Extraction method | SLU Task Input | Speaker Type Classifier Input
---|---|---
Out-of-Domain | In-Domain | $Fbank$ | $z_{1}$ | $z_{2}$ | $z_{12}$
$Fbank$ | $z_{1}$ | $Fbank$ | $z_{1}$
- |  | 0.526 | - | 0.576* | - | 0.719 | - | - | -
FHVAE pretrained on CGN | - | 0.519 | - | 0.651† | - | 0.734† | 0.693* | 0.757†
|  +Simply Finetuned on Torgo | - | 0.522 | - | 0.653† | - | 0.742† | 0.680† | 0.756†
|  +Adversarial training on Torgo | - | 0.520 | - | 0.638† | - | 0.724* | 0.701* | 0.755†
|  +Reference loss | - | 0.537* | - | 0.641† | - | 0.714* | 0.726* | 0.754†
|  +Only use $CE_{dys}$ in $L_{Gen}$ | - | 0.571*† | - | 0.651† | - | 0.720* | 0.726* | 0.757†
|  +Disentanglement loss | - | 0.493*† | - | 0.600*† | - | 0.707* | 0.737*† | 0.764†

##  4 Results and Discussion

###  4.1 E2E SLU system evaluation

We evaluate how dysarthria and speaker invariant the extracted content
variable is.

We first compare SLU performance using filter bank ($Fbank$) versus
$\mu_{z_{1}}$-features extracted by the baselines (pretrained model and simply
finetuned model) and proposed models. In table 1, we provide the averaged slot
value F1-score for the SLU task using different training sets.

For the Out-of-Domain experiment, the $z_{1}$ extracted by the FHVAE model
with adversarial training has no advantage over the $Fbank$ features when
testing the generalization ability between different dysarthria severity
levels. The reason might be that the adversarial training strategy
successfully narrows the gap between the dysarthric and canonical speech and
it does so by finding common ground in which some of the detail is lost for
speakers with high intelligibility, leading to accuracy loss in SLU.

By adding the reference loss and including only $CE_{dys}$ in $L_{Gen}$, the
modification of the control content space is weakened and the dysarthric
speech content space moves closer to the control space. Due to the dysarthria
invariance of the content variable, the F1-score increases and we obtain the
highest F1-score $0.571$ from the model with adversarial training, reference
loss and $CE_{dys}$ only. In In-Domain SLU, the baseline FHVAE already
outperforms the $Fbank$ features by a great margin. The content variable seems
to be a powerful speaker-independent speech representation, a point also made
in [14]. For the In-Domain task, it is however less important to narrow the
gap between canonical and dysarthric data, since the SLU backend sees both
kinds of data in training. The proposed methods for narrowing the gap between
canonical and dysarthric speech are hence less relevant in this scenario.
However, for the final design with adversarial training, reference loss and
restricting the $L_{Gen}$ term to the dysarthric data, there is also no
degradation. Hence, the design provides reliable features for SLU in both
scenarios.

Finally, the disentanglement loss seems to hurt the content representation and
leads to lower accuracy for SLU.

###  4.2 Dysarthria classification

To further assess the effect of adversarial training in generating a
dysarthria-invariant latent variable, we implement a binary classifier that
classifies input utterance-level mean vectors into either dysarthric or
control type. In the right of table 1, we provide the classification accuracy
using the content variable $z_{1}$, speaker variable $z_{2}$ and $z_{12}$
which concatenates both variables.

From the table, we see that when finetuning on the dysarthric dataset Torgo,
although the F1-score increases, the classification accuracy of $z_{1}$
increases with an accuracy decrease using $z_{2}$. The dysarthria information
reassignment shows that when processing dysarthric data, the FHVAE tends to
extract dysarthria-related information into the content variable $z_{1}$. When
we include the adversarial training into the finetuning procedure, we see a
drop in accuracy when using $z_{1}$ while an increase with $z_{2}$. This shows
that the adversarial training strategy does increase the invariance of the
dysarthric information in $z_{1}$, narrows the gap between dysarthric and
control latent spaces and promotes disentanglement of the speaker and content
space. Meanwhile, when using both content and speaker variable, classification
accuracy with $z_{12}$ is higher than using only $Fbank$ features, showing the
potential of the FHVAE model in dysarthric speech diagnosis.

When we engage the disentanglement loss into model training, there is even
lower classification accuracy with $z_{1}$ and higher accuracy with $z_{2}$.
Hence, the technique is successful in the goal to push dysarthria-related
information into the sequence variable. However, this seems to happen at the
expense of the quality of the content variable (see section 4.1). Indeed, a
lower correlation can also be achieved by making the content variable more
noisy.

##  5 Conclusions

We propose a dysarthria-invariant and speaker-independent feature extraction
method using an FHVAE model and adversarial training. We work in a setting of
limited task-agnostic dysarthric speech data for building the feature
extractor and evaluate - again on limited task-specific training data - on an
SLU task. By repurposing the content encoder in the FHVAE as a generator and
adding a discriminator network, we are able to narrow the gap in the latent
space for dysarthric and control speech, using only weak supervision. Using
the extracted content variable to train and test a mature E2E SLU system, we
see significant F1-score improvement on speakers with more severe dysarthria,
compared to using filter bank features or representations from a basic FHVAE
model. The binary dysarthria/control classifier also yields lower accuracy
when using the extracted content variable, showing the effect of the proposed
model in disentangling speaker and content information in dysarthric speech.
Interestingly, the extra disentanglement loss does improve disentanglement,
but it seems to spoil the content information as well.

We test the effectiveness of the extracted dysarthria-invariant and speaker-
independent features across different dysarthria severity levels. In future
work, we intend to evaluate the proposed features of both dysarthric and
control speech in other speech recognition models. Moreover, we see obvious
improvement in In-Domain evaluation when using pretraining and finetuning from
our current work.

##  6 Acknowledgements

The research was supported by KU Leuven Special Research Fund grant
C24M/22/025 and the Flemish Government under the ”Onderzoeksprogramma
Artificiële Intelligentie (AI) Vlaanderen” programme.

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# Sales Channel Optimization via Simulations Based on Observational Data with
Delayed Rewards: A Case Study at LinkedIn

Diana M. Negoescu  Pasha Khosravi∗ Shadow Zhao  Nanyu Chen∗∗ Parvez Ahammad
Humberto Gonzalez

(∗UCLA
∗∗Gopuff
LinkedIn Corporation

)

###### Abstract

Training models on data obtained from randomized experiments is ideal for
making good decisions. However, randomized experiments are often time-
consuming, expensive, risky, infeasible or unethical to perform, leaving
decision makers little choice but to rely on observational data collected
under historical policies when training models. This opens questions regarding
not only which decision-making policies would perform best in practice, but
also regarding the impact of different data collection protocols on the
performance of various policies trained on the data, or the robustness of
policy performance with respect to changes in problem characteristics such as
action- or reward- specific delays in observing outcomes. We aim to answer
such questions for the problem of optimizing sales channel allocations at
LinkedIn, where sales accounts (leads) need to be allocated to one of three
channels, with the goal of maximizing the number of successful conversions
over a period of time. A key problem feature constitutes the presence of
stochastic delays in observing allocation outcomes, whose distribution is both
channel- and outcome- dependent. We built a discrete-time simulation that can
handle our problem features and used it to evaluate: a) a historical rule-
based policy; b) a supervised machine learning policy (XGBoost); and c) multi-
armed bandit (MAB) policies, under different scenarios involving: i) data
collection used for training (observational vs randomized); ii) lead
conversion scenarios; iii) delay distributions. Our simulation results
indicate that LinUCB, a simple MAB policy, consistently outperforms the other
policies, achieving a 18-47% lift relative to a rule-based
policy00footnotetext: Contact<EMAIL_ADDRESS>Accepted at REVEAL’22
RecSys Workshop.

##  1 Introduction

The problem known as _sales channel optimization_ is shared among all
organizations reaching out to potential customers, also defined as _leads_ ,
where they must decide the initial communication channel for customer contact
(e.g., email, phone, etc.). Given a fixed set of channels and a pool of leads,
the sales channel optimization problem aims to find a suitable balance between
cost and an unknown success likelihood of each potential communication, which
can be naturally formulated as a multi-armed bandit (MAB) [27]. Data-driven
policies for this optimization problem can only be built if previously
collected historical training data is available, including lead contextual
information, channel(s) used to communicate with them, feedback delay, and
whether the lead was converted into a client. While such a data set is best
generated through randomized experiments, in many situations these experiments
are risky, expensive or time-consuming, particularly when actions have long
delays in providing outcome feedback, or involve extensive labor or financial
resources to implement. In the absence of data from randomized experiments,
practitioners often rely on observational data, which can suffer from various
forms of bias [23], resulting in suboptimal performance of models trained on
it. In this paper, we develop a simulation framework to empirically study the
impact of randomized vs. observational training data generation, and estimate
the performance of MAB and supervised machine learning (ML) policies in a
sales channel optimization problem at LinkedIn.

LinkedIn’s go-to-market strategy includes designing and implementing policies
for the sales channel optimization of our products [7]. In particular, in a
LinkedIn Sales Solutions product, we consider three channels:

  * •

Channel  _A_ : this channel relies on online communications, is highly
automated, and thus requires little manual effort. Outcome feedback delay is
short, measured in tens of days.

  * •

Channel  _B_ : this channel relies on personal communication with
representatives, with the goal of establishing a long-term relationship.
Outcome feedback delay is long, between a few months to a year.

  * •

Channel  _C_ : this channel strikes a balance between the two above, involving
less manual effort than Channel  _B_ , but with longer feedback delays than
Channel  _A_.

Historically, channel allocations among these three options have been made
according to rule-based policies that allocate leads depending on a handful of
arbitrarily chosen features, raising concerns that the outcomes data collected
through this policy may suffer from selection bias. Moreover, reward delays in
this setting, as well as cost of allocations made via Channels  _B_ and  _C_ ,
result in prohibitive conditions for randomized experiments.

##  2 Related Work

Simulation models have been used extensively in many domains, including
business, computing, medical, and social sciences [26]. In particular, the
evaluation of optimization policies typically involves either generating
synthetic data or attempting to evaluate the policies using previously
collected (or logged) data, a problem commonly known as “off-policy learning”
[20]. When large, uniformly randomized, logged data sets are available, [21]
shows that unbiased evaluation of a new policy is possible. Other policy
evaluation techniques for non-uniform randomized logged data sets originated
in the causal analysis field [15, 9, 8, 24, 16, 29, 14], which require the
hard-to-achieve constraint that policies have coverage of the entire feature
space.

Multi-armed bandits are used to tackle exploration-exploitation trade-offs in
problems where a choice must be made between multiple actions with unknown
rewards [28, 3, 5, 11, 4, 25, 19]. Contextual bandits [13], in which the
choice is modulated by extra features, have gained popularity especially since
the introduction of _LinUCB_ [21, 18, 1]. Of particular interest to our result
are bandits with delayed feedback [17, 22, 12, 10, 30, 2]. We expand on their
work by focusing on empirical evaluation under the practical constraints of
sales channel optimization problems at LinkedIn.

##  3 Methods

###  3.1 Problem setup

We use the contextual multi-armed bandit framework to model our sales channel
optimization problem. Specifically, at time $t$ we are given an account
described by a $d$-dimensional _context_ feature vector
$x_{t}\in\mathcal{X}\subset\mathbb{R}^{d}$. We need to allocate each account
$x_{t}$ to one of the available _actions_ $a_{t}\in\mathcal{D}$, resulting in
a _reward_ $r(x_{t},a_{t})\in\\{0,1\\}$, observed after a random delay
$\tau(x_{t},a_{t})$. We note that the delay time $\tau$ depends on the action
and is not independent of the reward $r(x_{t},a_{t})$. Our goal is then to
maximize expected cumulative rewards observed over a given time horizon $T$:
$\mathbb{E}[\sum_{t=1}^{T}r(x_{t},a_{t})\mathbf{1}_{\\{t+\tau(x_{t},a_{t})\leq
T\\}}]$.

In the context of our product of interest at LinkedIn, we typically have to
allocate a few hundred leads each day to one of three channels, i.e.,
$\mathcal{D}=\\{\emph{A},\emph{B},\emph{C}\\}$. Rewards in this case are lead
conversions into clients, set to 1 when converted and 0 otherwise, which are
executed at time $t$ and observed at time $t+\tau(x_{t},a_{t})$. Thus, our
objective is to maximize the total number of lead conversions.

###  3.2 Simulation details

We build a discrete time simulation informed by a data set of leads initiated
at LinkedIn between July 1st, 2020 and April 1st, 2021, in order to evaluate
the performance of different channel allocation policies under several state-
of-the-world scenarios regarding the nature of available training sets
(randomized vs. observational), reward ground truth and feedback delay
distributions. The original data set consisted of 200,000+ accounts, each
encoded as a context vector of 13 post-processed features, together with
allocated channels, and binary conversion labels. All labels in our data set
were either 0 or 1. We used one-hot-encoding for categorical variables,
normalized z-scores for real-valued variables capped to the interval $[-5,5]$,
and imputed missing real-valued features with the population mean.

Figure 1: Distribution of outcome delays on the LinkedIn sales channel
training data set for each communication channel and conversion status. While
not converted outcome delays vary widely among different channels, the
conversion outcome delays have a more consistent behavior, with Channel  _A_
having the shortest delays on average, and Channel  _B_ the longest.

The simulation tracks daily leads that are allocated according to the user-
specified policy (Appendix Algorithm 1). Figure 1 shows delay distributions
for each channel and conversion status on this data set.

####  3.2.1 Logged data set collection scenarios

First, we simulate different logged data set collection methodologies, which
the simulated policy will consume as available data, using the following three
scenarios:

  * •

Fully Randomized: Assign channel allocations uniformly at random.

  * •

Observational: Assigning channels according to the historically used rule-
based policy based on one feature value, denoted $\hat{f}$. That is,
$a_{t}=\emph{A}$ if $\hat{f}<10$, $a_{t}=\emph{B}$ if $\hat{f}>20$, and
$a_{t}=\emph{C}$ otherwise.

  * •

Partially Randomized: Randomize by $\hat{f}$ values within a range to collect
training data, in order to avoid major deviations from the historically
applied rule-based policy. In this case, $a_{t}=\emph{A}$ if $\hat{f}<10$,
$a_{t}=\emph{B}$ if $\hat{f}>20$, and, either $a_{t}=\emph{C}$ with
probability $\frac{2}{3}$ or $a_{t}=\emph{B}$ with probability $\frac{1}{3}$
otherwise.

####  3.2.2 Lead conversion scenarios

To simulate non-trivial, yet representative, lead conversion distribution
scenarios, we first used $k$-means clustering (with $k=10$) to aggregate the
accounts in our training data set based on their feature vectors $x_{t}$ (not
including the allocations and outcomes). We then considered the following lead
conversion scenarios:

  * •

Historical: We assign each cluster-action pair a Bernoulli probability of
success, whose parameter $p$ is computed as the MLE on our LinkedIn data set.

  * •

Uniform: We assign each cluster-action-$\hat{f}$ triple a Bernoulli
probability of success, whose parameter $p$ is sampled from a uniform
distribution over an interval informed by the success probabilities of the
Historical scenario.

  * •

$\hat{f}$-adjusted uniform: Similar to the Uniform scenario, but give more
importance to $\hat{f}$ values by increasing the probability of conversion
when the action-$\hat{f}$ pair is the same as in the Observational scenario.

####  3.2.3 Delay scenarios

We simulated delays by sampling from the population distribution on our
LinkedIn data set conditional on action and reward. On top of the samples, we
enabled three scenarios where each resulting simulated delay is multiplied by
a constant factor $\lambda_{delay}\in\\{0,\frac{1}{2},1\\}$.

####  3.2.4 Policy simulation

Finally, we implemented the following LinkedIn-relevant policies using our
simulator:

  * •

Rule-based $\hat{f}$ policy: Rule-based channel assignment policy identical to
the Observational scenario in 3.2.1.

  * •

XGBoost supervised ML: We train an _XGBoost_ [6] model using simulated logged
data from 3.2.1, and choose the action with the highest predicted probability
of conversion. The simulator retrains the model every 90 days.

  * •

$\epsilon$-greedy MAB: As described in [27], $\epsilon$ percent of the time we
allocate a channel uniformly at random, the rest of the time follow the same
recommendation as the XGBoost supervised ML policy, retraining every 90 days.

  * •

LinUCB MAB: As described in [21], we choose the channel with highest upper-
confidence bound (predicted mean + confidence interval) as defined by a linear
relationship between features and conversion. Updates the upper-confidence
bound online as new data becomes available.

##  4 Results

Figure 2: Lift % with respect to $\hat{f}$ policy (100 runs, 95% CI). Data
collection was simulated for 90 days, simulation horizon was one year. Unless
varied, data collection was simulated using the Observational scenario, lead
conversion was according to the $\hat{f}$-adjusted Uniform scenario, and
delays were simulated using historical data.

Results from our simulation are shown in Figure 2 (numerical values in
Appendix Table 1). We measure cumulative reward lift (as percentage
improvement) relative to the historically used rule-based $\hat{f}$ policy as
estimated by our simulator (Appendix Algorithm 1). We found that LinUCB
outperforms XGBoost and $\epsilon$-greedy under most simulated scenarios
($18\%$ to $47\%$ lift compared to the ruled-based $\hat{f}$ policy), and
especially more so when the data used for training was observational (Figure
2a). These results were robust to changes in reward distribution (Figure 2b)
and delay distribution shifts (Figure 2c), regardless of the choice of policy.
Moreover, even partial randomization improves the performance of supervised ML
policies, but MABs performed well whether or not the training data was
collected under a randomized policy.

##  5 Acknowledgments

We particularly thank Justin Dyer, Adrian Rivera Cardoso, Jilei Yang, Ryan
Rogers, Saad Eddin Al Orjany, Wenrong Zeng, Yu Liu, Rahul Todkar, Neha Gupta,
Maggie Zhang and Jennifer Kloke for their helpful comments and feedback.

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##  Appendix A Appendix: Simulator algorithm for sales channel allocation

Algorithm 1 Channel allocation simulator

1:

2:Number of days to simulate $T$;

3:Channel allocation policy $\pi$ (except for $\hat{f}$, this policy is
trained on data collected using scenario $\pi_{0}$);

4:Lead conversion scenario $L$;

5:Feature distribution from training data $F(X)$;

6:Delay distribution from training data per action and reward
$\Delta=\\{\Delta(a,r)\\}_{a\in\mathcal{D},r\in\\{0,1\\}}$.

7:procedure Simulate($\pi$, $L$, $F(X)$, $\Delta$)

8: Initialize reward scheduler pq.

9: for each day in 1 to $T$ do:

10: today’s leads $\leftarrow$ Sample from $F(X)$.

11: for $x_{t}\in$ today’s leads do:

12: for $a_{t}\in\mathcal{D}$ do:

13: Sample reward $r(x_{t},a_{t})\sim L$.

14: Sample delay $\tau(x_{t},a_{t})\sim\Delta(a_{t},r(x_{t},a_{t}))$.

15: end for

16: Take action $a^{\pi}_{t}$ according to the policy to be evaluated $\pi$.

17: Schedule reward $r(x_{t},a^{\pi}_{t})$ to be observed after
$\tau(x_{t},a^{\pi}_{t})$ days.

18: end for

19: Observe today’s scheduled rewards and accumulate total rewards.

20: Update policy $\pi$ using today’s observations (if necessary).

21: end for

22:end procedure

A high level description of our policy simulator can be found in Algorithm 1.
We note that the same algorithm is run prior to the evaluation stage with
$\pi=\pi_{0}$ in order to generate the logged data according to the data set
collection scenarios in Section 3.2.1, where policy $\pi_{0}$ in that case is
one of the three scenarios: Fully Randomized, Observational or Partially
Randomized.

##  Appendix B Appendix: Results in tabular form

Table 1 shows the results, plotted in Figure 2, in tabular form.

| Data collection scenario
---|---
| Observational | Partially randomized | Fully Random
Policy | Rule-based $\hat{f}$ | [0.0, 0.0] | [0.0, 0.0] | [0.0, 0.0]
_LinUCB_ | [19.9, 29.6] | [18.5, 25.0] | [18.7, 24.8]
_XGBoost_ | [7.3, 16.3] | [14.1, 20.6] | [13.8, 19.8]
_$\epsilon$ -greedy_ ($\epsilon=0.05$) | [10.6, 19.8] | [14.4, 20.8] | [14.0, 19.9]
_$\epsilon$ -greedy_ ($\epsilon=0.1$) | [11.3, 20.5] | [13.7, 19.9] | [13.0, 18.7]
(a) Varying data collection: Lift % wrt rule-based $\hat{f}$ policy (100 runs,
95% CI).

| Lead conversion scenario
---|---
| Historical | Uniform |  $\hat{f}$-adjusted Uniform
Policy | Rule-based $\hat{f}$ | [0.0, 0.0] | [0.0, 0.0] | [0.0, 0.0]
_LinUCB_ | [18.5, 25.5] | [34.2, 47.0] | [20.0, 29.9]
_XGBoost_ | [-1.1, 6.9] | [20.3, 32.2] | [6.6, 16.4]
_$\epsilon$ -greedy_ ($\epsilon=0.05$) | [14.2, 21.1] | [25.5, 37.2] | [11.5, 21.2]
_$\epsilon$ -greedy_ ($\epsilon=0.1$) | [13.7, 20.4] | [26.2, 38.4] | [11.9, 21.8]
(b) Varying lead conversion: Lift % wrt rule-based $\hat{f}$ policy (100 runs,
95% CI).

| Delay distribution scenario
---|---
| No Delays | 1/2 of Usual Delays | Usual Delays
Policy | Rule-based $\hat{f}$ | [0.0, 0.0] | [0.0, 0.0] | [0.0, 0.0]
_LinUCB_ | [23.7, 29.8] | [18.6, 27.2] | [22.9, 32.7]
_XGBoost_ | [12.4, 17.0] | [6.0, 14.0] | [9.4, 19.4]
_$\epsilon$ -greedy_ ($\epsilon=0.05$) | [13.0, 17.2] | [7.8, 15.9] | [13.9, 23.4]
_$\epsilon$ -greedy_ ($\epsilon=0.1$) | [12.5, 16.7] | [7.5, 15.4] | [14.4, 23.9]
(c) Varying delay: Lift % wrt rule-based $\hat{f}$ policy (100 runs, 95% CI).

Table 1: Results after changing simulation scenarios as described in Section
4.


# Some properties of certain close-to-convex harmonic mappings

Xiao-Yuan Wang, Zhi-Gang Wang∗, Jin-Hua Fan and Zhen-Yong Hu  Xiao-Yuan Wang
School of Science, Nanjing University of Science and Technology, Nanjing
210094, Jiangsu, P. R. China  Institute of Sponge City Research, Pingxiang
University, Pingxiang 337055, Jiangxi, P. R. China<EMAIL_ADDRESS>Zhi-Gang WangSchool of Mathematics and Statistics, Hunan First Normal
University, Changsha 410205, Hunan, P. R. China<EMAIL_ADDRESS>Jin-Hua
FanSchool of Science, Nanjing University of Science and Technology, Nanjing
210094, Jiangsu, P. R. China<EMAIL_ADDRESS>Zhen-Yong HuSchool of
Science, Nanjing University of Science and Technology, Nanjing 210094,
Jiangsu, P. R. China<EMAIL_ADDRESS>
###### Abstract.

In the present paper, we determine the estimates for Toeplitz determinants of
a subclass of close-to-convex harmonic mappings. Moreover, we obtain an
improved version of Bohr’s inequalities for a subclass of close-to-convex
harmonic mappings, whose analytic parts are Ma-Minda convex functions.

###### Key words and phrases:

Univalent harmonic mappings; close-to-convex harmonic mappings; Bohr radius;
Toeplitz determinant; Ma-Minda convex function

###### 2010 Mathematics Subject Classification:

Primary 30C45, 30C50, 30C80

∗Corresponding author.

##  1\. Introduction

A complex-valued function $f$ in the unit disk $\mathbb{D}=\\{z:|z|<1\\}$ is
called a harmonic mapping if $\Delta f=4f_{z\overline{z}}=0$. Let
$\mathcal{H}$ denote the sense-preserving harmonic mappings $f=h+\overline{g}$
in $\mathbb{D}$. Such mapping has the canonical representation
$f=h+\overline{g}$, where

$\ h(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}\ \text{and}\
g(z)=\sum_{n=1}^{\infty}b_{n}z^{n}$ (1.1)

are $analytic$ functions in $\mathbb{D}$. Let $\mathcal{S}_{\mathcal{H}}$ be
the subclass of $\mathcal{H}$ consisting of univalent and sense-preserving
mappings. We observe that $\mathcal{S}_{\mathcal{H}}$ reduces to the class
$\mathcal{S}$ of normalized univalent analytic functions, if the co-analytic
part $g\equiv 0$. Denote by $\mathcal{K}_{\mathcal{H}}$ the close-to-convex
subclass in $\mathcal{S}_{\mathcal{H}}$. If $b_{1}=0$, then
$\mathcal{K}_{\mathcal{H}}$ reduces to the class
$\mathcal{K}^{0}_{\mathcal{H}}$.

Lewy [35] proved that $f=h+\overline{g}$ is locally univalent in $\mathbb{D}$
if and only if the Jacobian
$J_{f}=\left|h^{\prime}\right|^{2}-\left|g^{\prime}\right|^{2}\neq 0$ in
$\mathbb{D}$. Noting that the harmonic mapping $f$ is sense-preserving, i.e.
$J_{f}>0$ or $\left|h^{\prime}\right|>\left|g^{\prime}\right|$ in
$\mathbb{D}$. At this point, its dilatation $\omega_{f}=g^{\prime}/h^{\prime}$
has the property that $\left|\omega_{f}\right|<1$ in $\mathbb{D}$. The reader
can find much information about planar harmonic mappings from [17, 21, 43].

Let $\mathcal{P}$ denote the class of analytic functions $p$ in $\mathbb{D}$
of the form

$p(z)=1+\sum_{n=1}^{\infty}p_{n}z^{n}$ (1.2)

such that $\operatorname{Re}(p(z))>0$ in $\mathbb{D}.$

Denote by $\mathcal{A}$ the class of analytic functions in $\mathbb{D}$ with
$f(0)=f^{\prime}(0)-1=0$, and $\mathcal{K}(\alpha)$ denote the class of
functions $f\in\mathcal{A}$ such that

${\rm
Re}\left(1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right)>\alpha\quad\left(-\frac{1}{2}\leq\alpha<1;\
z\in\mathbb{D}\right).$

Particularly, the elements in $\mathcal{K}(-1/2)$ are close-to-convex but are
not necessarily starlike in $\mathbb{D}$. For $0\leq\alpha<1$, the elements in
$\mathcal{K}(\alpha)$ are known to be convex functions of order $\alpha$ in
$\mathbb{D}$. For more properties of starlike and convex functions, the reader
can refer to the books [22, 49].

By making use of the subordination in analytic functions, Ma and Minda [39]
introduced a more general class $\mathcal{C}(\phi)$, consisting of functions
in $\mathcal{S}$ for which

$1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\prec\phi(z).$

Here the function $\phi:\mathbb{D}\rightarrow\mathbb{C}$, called Ma-Minda
function, is analytic and univalent in $\mathbb{D}$ such that
$\phi(\mathbb{D})$ has positive real part, symmetric with respect to the real
axis, starlike with respect to $\phi(0)=1$ and $\phi^{\prime}(0)>0$ (for more
details, see [51, 46]). A Ma-Minda function has the form

$\phi(z)=1+\sum_{n=1}^{\infty}B_{n}z^{n}.$

The extremal function $K$ for the class $\mathcal{C}(\phi)$ is given by

$K(z)=\int_{0}^{z}\exp\left(\int_{0}^{\zeta}\frac{\phi(t)-1}{t}dt\right)d\zeta\quad(z\in\mathbb{D}),$
(1.3)

which satisfies the condition

$1+\frac{zK^{\prime\prime}(z)}{K^{\prime}(z)}=\phi(z).$

We recall the natural class of close-to-convex harmonic mappings
$\mathcal{M}(\alpha,\zeta,n)$ which belongs to $\mathcal{K}^{0}_{\mathcal{H}}$
due to Wang et al. [50] (see also [44]).

######  Definition 1.1.

A harmonic mapping $f=h+\overline{g}\in\mathcal{H}$ is said to be in the class
$\mathcal{M}(\alpha,\zeta,n)$ if $h$ and $g$ satisfy the conditions

${\rm
Re}\left(1+\frac{zh^{\prime\prime}(z)}{h^{\prime}(z)}\right)>\alpha\quad\left(-\frac{1}{2}\leq\alpha<1;\
z\in\mathbb{D}\right)$ (1.4)

and

$g^{\prime}(z)=\zeta z^{n}h^{\prime}(z)\quad\left(\zeta\in\mathbb{C}\ {\rm
with}\
\left|\zeta\right|\leq\frac{1}{2n-1};\,n\in\mbox{$\mathbb{N}$}:=\\{1,2,3\ldots\\}\right).$
(1.5)

Motivated essentially by the class $\mathcal{M}(\alpha,\zeta,n)$, we define a
new subclass of $\mathcal{H}$ as follows:

######  Definition 1.2.

A harmonic mapping $f=h+\overline{g}\in\mathcal{H}$ is said to be in the class
$\mathcal{HC}(\phi)$ if $h\in\mathcal{C}(\phi)$ and $g$ satisfies the
condition (1.5).

In recent years, the Toeplitz determinants and Hankel determinants of
functions in the class $\mathcal{S}$ or its subclasses have attracted many
researchers’ attention (see [11, 18, 31, 27, 16, 19, 33, 32, 34, 28]). Among
them, the symmetric Toeplitz determinant $|T_{q}(n)|$ estimates for subclasses
of $\mathcal{S}$ with small values of $n$ and $q$, are investigated by [10, 7,
45, 2, 52, 53].

The symmetric Toeplitz determinant $T_{q}(n)$ for analytic functions $f$ is
defined as follows:

$T_{q}(n)[f]:=\begin{vmatrix}a_{n}&a_{n+1}&\cdots&a_{n+q-1}\\\
a_{n+1}&a_{n}&\cdots&a_{n+q-2}\\\ \vdots&\vdots&\vdots&\vdots&\\\
a_{n+q-1}&a_{n+q-2}&\cdots&a_{n}\end{vmatrix},$

where $n,q\in\mbox{$\mathbb{N}$}$ and $a_{1}=1$. In particular, for functions
in starlike and convex classes, $T_{2}(2)[f],T_{3}(1)[f]$ and $T_{3}(2)[f]$
were studied by Ali $et\ al.$ [7]. Sun $et\ al.$ [47] investigated the upper
bounds of the third Hankel determinants for the subclass
$\mathcal{M}(\alpha,1,1)$ of close-to-convex harmonic mappings.

Let $\mathcal{B}$ be the class of analytic functions $f$ in $\mathbb{D}$ such
that $|f(z)|<1$ for all $z\in\mathbb{D}$, and let
$\mathcal{B}_{0}=\\{f\in\mathcal{B}:f(0)=0\\}$. In $1914$, Bohr [15] proved
that if $f\in\mathcal{B}$ is of the form $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$,
then the majorant series $M_{f}(r)=\sum_{n=0}^{\infty}|a_{n}||z|^{n}$ of $f$
satisfies

$M_{f_{0}}(r)=\sum_{n=1}^{\infty}|a_{n}||z|^{n}\leq 1-|a_{0}|=d(f(0),\partial
f(\mathbb{D}))$ (1.6)

for all $z\in\mathbb{D}$ with $|z|=r\leq 1/3$, where $f_{0}(z)=f(z)-f(0)$.
Bohr actually obtained the inequality (1.6) for $|z|\leq 1/6$. Later, Wiener,
Riesz and Schur, independently established the Bohr inequality (1.6) for
$|z|\leq 1/3$ (known as Bohr radius for the class $\mathcal{B}$) and hence
proved that $1/3$ is the best possible.

The Bohr phenomenon was reappeared in the $1990$s due to Dixon [20]. Later,
Boas and Khavinson [14] found bounds for Bohr’s radius in any complete
Reinhard domains. Other works we can see [42, 13, 41, 3, 4]. In recent years,
Bohr inequality and Bohr radius have become an active research field in the
theory of univalent functions, see [8, 5, 36, 29, 37, 26]. Furthermore, the
Bohr’s phenomenon for the complex-valued harmonic mappings have been widely
studied (see [40, 1, 6, 24, 25, 30, 38]).

In this paper, we aim at determining the estimates for Toeplitz determinants
of a subclass of close-to-convex harmonic mappings
$\mathcal{M}(\alpha,\zeta,n)$. Moreover, we will derive an improved version of
Bohr’s inequalities for a subclass $\mathcal{HC}(\phi)$ of close-to-convex
harmonic mappings, whose analytic parts are Ma-Minda convex functions.

##  2\. Preliminary results

To prove our main results, we need the following lemmas.

######  Lemma 2.1.

([22, p. 41]) For a function $p\in\mathcal{P}$ of the form (1.2), the sharp
inequality $\left|p_{n}\right|\leq 2$ holds for each $n\geq 1.$ Equality holds
for the function $p(z)=(1+z)/(1-z).$

######  Lemma 2.2.

([23, Theorem 1]) Let $p\in\mathcal{P}$ be of the form (1.2) and
$\mu\in\mathbb{C}$. Then

$\left|p_{n}-\mu p_{k}p_{n-k}\right|\leq 2\max\\{1,|2\mu-1|\\}\quad(1\leq
k\leq n-1).$

If $|2\mu-1|\geq 1$ then the inequality is sharp for the function
$p(z)=(1+z)/(1-z)$ or its rotations. If $|2\mu-1|<1$ then the inequality is
sharp for $p(z)=\left(1+z^{n}\right)/\left(1-z^{n}\right)$ or its rotations.

######  Lemma 2.3.

([50]) Let $f=h+\overline{g}\in\mathcal{M}(\alpha,\zeta,n)$. Then the
coefficients $a_{k}\ (k\in\mbox{$\mathbb{N}$}\setminus\\{1\\})$ of $h$ satisfy

$\left|a_{k}\right|\leq\frac{1}{k!}\prod_{j=2}^{k}(j-2\alpha)\
(k\in\mbox{$\mathbb{N}$}\setminus\\{1\\}).$ (2.1)

Moreover, the coefficients $b_{k}\ (k=n+1,n+2,\cdots;n\in\mbox{$\mathbb{N}$})$
of $g$ satisfy

$\left|b_{n+1}\right|\leq\frac{\left|\zeta\right|}{n+1}\ \text{ and }\
\left|b_{k+n}\right|\leq\frac{\left|\zeta\right|}{(k+n)(k-1)!}\prod_{j=2}^{k}(j-2\alpha)\
(k\in\mbox{$\mathbb{N}$}\setminus\\{1\\};\,n\in\mbox{$\mathbb{N}$}).$ (2.2)

The bounds are sharp for the extremal function given by

$f(z)=\int_{0}^{z}\frac{dt}{(1-\delta
t)^{2-2\alpha}}+\overline{\int_{0}^{z}\frac{\zeta t^{n}}{(1-\delta
t)^{2-2\alpha}}dt}\quad(\left|\delta\right|=1;\,z\in\mbox{$\mathbb{D}$}).$
(2.3)

######  Lemma 2.4.

([50]) Let $f\in\mathcal{M}(\alpha,\zeta,n)$ with $0\leq\alpha<1$ and
$0\leq\zeta<\frac{1}{2n-1}\ (n\in\mbox{$\mathbb{N}$})$. Then

$\Phi(r;\alpha,\zeta,n)\leq\left|f(z)\right|\leq\Psi(r;\alpha,\zeta,n)\quad(r=|z|<1),$
(2.4)

where

$\Phi(r;\alpha,\zeta,n)=\left\\{\begin{array}[]{ll}\log(1+r)-\displaystyle\frac{\zeta\,r^{n+1}\,_{2}F_{1}(1,\,n+1;\,n+2;\,-r)}{n+1}&(\alpha=1/2),\\\
\\\
\displaystyle\frac{(1+r)^{2\alpha-1}-1}{2\alpha-1}-\frac{\zeta\,r^{n+1}\,_{2}F_{1}(n+1,\,2-2\alpha;\,n+2;\,-r)}{n+1}&(\alpha\neq
1/2),\end{array}\right.$

and

$\Psi(r;\alpha,\zeta,n)=\left\\{\begin{array}[]{ll}-\log(1-r)+\displaystyle\frac{\zeta\,r^{n+1}\,_{2}F_{1}(1,\,n+1;\,n+2;\,r)}{n+1}&(\alpha=1/2),\\\
\\\
\displaystyle\frac{1-(1-r)^{2\alpha-1}}{2\alpha-1}+\frac{\zeta\,r^{n+1}\,_{2}F_{1}(n+1,\,2-2\alpha;\,n+2;\,r)}{n+1}&(\alpha\neq
1/2).\end{array}\right.$

All these bounds are sharp, the extremal function is
$f_{\alpha,\zeta,n}=h_{\alpha}+\overline{g_{\alpha,\zeta,n}}$ or its
rotations, where

$f_{\alpha,\zeta,n}(z)=\left\\{\begin{array}[]{ll}-\log(1-z)+\displaystyle\overline{\frac{\zeta\,z^{n+1}\,_{2}F_{1}(1,\,n+1;\,n+2;\,z)}{n+1}}&(\alpha=1/2),\\\
\\\
\displaystyle\frac{1-(1-z)^{2\alpha-1}}{2\alpha-1}+\overline{\frac{\zeta\,z^{n+1}\,_{2}F_{1}(n+1,\,2-2\alpha;\,n+2;\,z)}{n+1}}&(\alpha\neq
1/2).\end{array}\right.$ (2.5)

The following two results are due to Ma-Minda [39].

######  Lemma 2.5.

Let $f\in\mathcal{C}(\phi)$. Then $zf^{\prime\prime}(z)/f^{\prime}(z)\prec
zK^{\prime\prime}(z)/K^{\prime}(z)$ and $f^{\prime}(z)\prec K^{\prime}(z)$,
where $K$ is given by (1.3).

######  Lemma 2.6.

Assume that $f\in\mathcal{C}(\phi)$ and $|z|=r<1$. Then

$K^{\prime}(-r)\leq|f^{\prime}(z)|\leq K^{\prime}(r).$ (2.6)

where $K$ is given by (1.3). Equality holds for some $z\neq 0$ if and only if
$f$ is a rotation of $K$.

######  Lemma 2.7.

([12]) Let $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ and
$g(z)=\sum_{n=0}^{\infty}b_{n}z^{n}$ be two analytic functions in $\mathbb{D}$
and $g\prec f$. Then

$\sum_{n=0}^{\infty}|b_{n}|r^{n}\leq\sum_{n=0}^{\infty}|a_{n}|r^{n}$ (2.7)

for $z|=r\leq 1/3.$

##  3\. Toeplitz determinant for the class $\mathcal{M}(\alpha,\zeta,n)$

In this section, we will give estimates for Toeplitz determinants
$|T_{q}(n)[\cdot]|$ of functions in $\mathcal{M}(\alpha,\zeta,n)$.

######  Theorem 3.1.

Let $f\in\mathcal{M}(\alpha,\zeta,n)$. Then

$|T_{2}(n)[h]|\leq\left(\frac{1}{n!}\prod_{j=2}^{n}(j-2\alpha)\right)^{2}+\left(\frac{1}{(n+1)!}\prod_{j=2}^{n+1}(j-2\alpha)\right)^{2}\quad(n\in\mathbb{N}\backslash\\{1\\})$
(3.1)

and

$|T_{2}(n)[g]|\leq\frac{|\zeta|^{2}}{(n+1)^{2}}.$ (3.2)

The inequalities in (3.1) and (3.2) are sharp.

###### Proof.

Suppose that $f\in\mathcal{M}(\alpha,\zeta,n)$. Then by Lemma 2.3, we see that

$|T_{2}(n)[h]|=|a_{n}^{2}-a_{n+1}^{2}|\leq|a_{n}^{2}|+|a_{n+1}^{2}|$ (3.3)

yields (3.1). Equality holds in (3.3) for the function defined by

$\ h(z)=\int_{0}^{z}\frac{dt}{(1-\delta
t)^{2-2\alpha}}\quad(\left|\delta\right|=1;\,z\in\mbox{$\mathbb{D}$}).$

By the coefficients $b_{k}\ (k=n+1,n+2,\cdots;n\in\mbox{$\mathbb{N}$})$ of
$g$, we get the assertion (3.2) by (2.2). The equalities in (3.1) and (3.2)
are sharp for the extremal function given by (2.3). ∎

######  Corollary 3.1.

Let $f\in\mathcal{M}(\alpha,\zeta,n)$. Then

$|T_{2}(2)[h]|\leq\frac{2}{9}(1-\alpha)^{2}\left(2\alpha^{2}-6\alpha+9\right),$
(3.4)

and

$|T_{2}(2)[g]|\leq\frac{|\zeta|^{2}}{9}.$ (3.5)

The inequalities in (3.4) and (3.5) are sharp.

######  Theorem 3.2.

Let $f\in\mathcal{M}(\alpha,\zeta,n)$. Then

$|T_{3}(1)[h]|\leq\begin{cases}{\frac{1}{9}\big{(}8\alpha^{4}-34\alpha^{3}+71\alpha^{2}-72\alpha+36\big{)}\quad(-\frac{1}{2}\leq\alpha\leq\frac{1}{2}}),\\\
\\\\[-12.0pt]
{\frac{1}{9}\left(-2\alpha^{3}+25\alpha^{2}-44\alpha+30\right)\quad\quad\quad\quad\
(\frac{1}{2}\leq\alpha<1}),\end{cases}$ (3.6)

and

$|T_{3}(1)[g]|\leq\frac{|\zeta|^{3}}{6}(1-\alpha).$ (3.7)

###### Proof.

For $f\in\mathcal{M}(\alpha,\zeta,n)$, we see that

$p(z)=\frac{1}{1-\alpha}\left(1+\frac{zh^{\prime\prime}(z)}{h^{\prime}(z)}-\alpha\right)\in\mathcal{P}\quad\left(-\frac{1}{2}\leq\alpha<1;z\in\mathbb{D}\right).$

It follows that

$n(n-1)a_{n}=(1-\alpha)\sum_{k=1}^{n-1}ka_{k}p_{n-k}\quad(n\geq 2).$ (3.8)

From (3.8), we obtain

$\begin{cases}a_{2}=\frac{1}{2}(1-\alpha)p_{1},\\\ \\\\[-12.0pt]
a_{3}=\frac{1}{6}(1-\alpha)\left[(1-\alpha)p_{1}^{2}+p_{2}\right],\\\
\\\\[-12.0pt]
a_{4}=\frac{1}{24}(1-\alpha)\left[(1-\alpha)^{2}p_{1}^{3}+3(1-\alpha)p_{1}p_{2}+2p_{3}\right].\\\
\end{cases}$ (3.9)

By virtue of Lemma 2.2 and (3.9), we get

$\displaystyle\left|T_{3}(1)[h]\right|$
$\displaystyle=\left|1-2a_{2}^{2}+2a_{2}^{2}a_{3}-a_{3}^{2}\right|$ (3.10)
$\displaystyle\leq
1+2\left|a_{2}^{2}\right|+\left|a_{3}\right|\left|a_{3}-2a_{2}^{2}\right|$
$\displaystyle\leq
1+\frac{1}{2}(1-\alpha)^{2}p_{1}^{2}+\frac{1}{36}(1-\alpha)^{2}|(1-\alpha)p_{1}^{2}+p_{2}||p_{2}-2(1-\alpha)p_{1}^{2}|$
$\displaystyle\leq\begin{cases}{\frac{1}{9}\big{(}8\alpha^{4}-34\alpha^{3}+71\alpha^{2}-72\alpha+36\big{)}\quad(-\frac{1}{2}\leq\alpha\leq\frac{1}{2}}),\\\
\\\\[-12.0pt]
{\frac{1}{9}\big{(}-2\alpha^{3}+25\alpha^{2}-44\alpha+30\big{)}\qquad\quad\ \
(\frac{1}{2}\leq\alpha<1}).\end{cases}$

By the power series representations of $h$ and $g$ for
$f=h+\bar{g}\in\mathcal{M}(\alpha,\zeta,n)$, we see that

$b_{k+n-1}=0,\quad(k+n)b_{k+n}=\zeta ka_{k}\quad(k,n\in\mathbb{N};\,a_{1}=1).$

For $n=1$, we know that

$\begin{cases}b_{2}=\frac{\zeta}{2}a_{1}=\frac{\zeta}{2},\\\ \\\\[-12.0pt]
b_{3}=\frac{2\zeta}{3}a_{2}=\frac{\zeta}{3}\big{(}1-\alpha\big{)}p_{1},\\\
\\\\[-12.0pt]
b_{4}=\frac{3\zeta}{4}a_{3}=\frac{\zeta}{8}(1-\alpha)\left[(1-\alpha)p_{1}^{2}+p_{2}\right].\\\
\end{cases}$ (3.11)

For $n=2$, we see that

$\begin{cases}b_{3}=\frac{\zeta}{3}a_{1}=\frac{\zeta}{3},\\\ \\\\[-12.0pt]
b_{4}=\frac{\zeta}{2}a_{2}=\frac{\zeta}{4}\big{(}1-\alpha\big{)}p_{1}.\\\
\end{cases}$ (3.12)

Thus, by Lemma 2.1, we deduce that the assertion (3.7) of Theorem 3.2 holds. ∎

######  Theorem 3.3.

Let $f\in\mathcal{M}(\alpha,\zeta,n)$. Then

$|T_{3}(2)[h]|\leq\begin{cases}{\frac{1}{108}\left(1-\alpha)^{3}(2\alpha^{2}-7\alpha+12)(10\alpha^{2}-27\alpha+36\right)\quad(-\frac{1}{2}\leq\alpha\leq\frac{1}{7}}),\\\
\\\\[-12.0pt]
{\frac{5}{108}\left(1-\alpha)^{3}(2\alpha^{2}-7\alpha+12)(2\alpha^{2}-4\alpha+7\right)\qquad\
\,(\frac{1}{7}\leq\alpha<1}),\end{cases}$ (3.13)

and

$|T_{3}(2)[g]|=|2b_{3}^{2}b_{4}|\leq\frac{|\zeta|^{3}}{9}\left(1-\alpha\right).$
(3.14)

###### Proof.

Let $f\in\mathcal{M}(\alpha,\zeta,n)$. By noting that

$T_{3}(2)[h]=\left(a_{2}-a_{4}\right)\left(a_{2}^{2}-2a_{3}^{2}+a_{2}a_{4}\right),$

by (3.9) and Lemma 2.1, it clearly that

$\displaystyle\left|a_{2}-a_{4}\right|$
$\displaystyle\leq\left|a_{2}\right|+\left|a_{4}\right|$ (3.15)
$\displaystyle\leq\left|\frac{1}{2}(1-\alpha)p_{1}\right|+\left|\frac{1}{24}(1-\alpha)\left[(1-\alpha)^{2}p_{1}^{3}+3(1-\alpha)p_{1}p_{2}+2p_{3}\right]\right|$
$\displaystyle\leq\frac{1}{6}\left(1-\alpha)(2\alpha^{2}-7\alpha+12\right).$

Next, we shall maximize $\left|a_{2}^{2}-2a_{3}^{2}+a_{2}a_{4}\right|$. With
the help of (3.9), Lemma 2.1 and Lemma 2.2, we get

$\displaystyle|a_{2}^{2}-2a_{3}^{2}+a_{2}a_{4}|$
$\displaystyle=\frac{(1-\alpha)^{2}}{144}|-5(1-\alpha)^{2}p_{1}^{4}+36p_{1}^{2}-7(1-\alpha)p_{1}^{2}p_{2}-8p_{2}^{2}+6p_{1}p_{3}|$
(3.16)
$\displaystyle\leq\frac{(1-\alpha)^{2}}{144}\left[5(1-\alpha)^{2}|p_{1}|^{4}+36|p_{1}|^{2}+8|p_{2}|^{2}+6|p_{1}||p_{3}-\frac{7}{6}(1-\alpha)p_{1}p_{2}|\right]$
$\displaystyle\leq\begin{cases}{\frac{1}{18}\big{(}1-\alpha)^{2}\big{(}10\alpha^{2}-27\alpha+36\big{)}\quad\
\,(-\frac{1}{2}\leq\alpha\leq\frac{1}{7}}),\\\ \\\\[-12.0pt]
{\frac{5}{18}\big{(}1-\alpha)^{2}\big{(}2\alpha^{2}-4\alpha+7\big{)}\quad\qquad(\frac{1}{7}\leq\alpha<1}).\end{cases}$

Therefore, combining (3.15) with (3.16), we obtain the inequality (3.13). From
(3.12) and Lemma 2.1, we get the assertion (3.14) of Theorem 3.3. ∎

######  Remark 3.1.

By setting $\alpha=0$ in Corollary 3.1, Theorem 3.2 and Theorem 3.3, we get
$|T_{2}(2)[h]|\leq 2,|T_{3,1}[h]|\leq 4,|T_{3,2}[h]|\leq 4$. The bounds for
convex functions were recently obtained by Ali $et\ al.$ [7].

##  4\. Bohr inequality for the class $\mathcal{HC}(\phi)$

In this section, we firstly give the sharp growth estimate for the class
$\mathcal{HC}(\phi)$.

######  Proposition 4.1.

Let $f\in\mathcal{HC}(\phi)$. Then

$L(\zeta,n,r)\leq|f(z)|\leq R(\zeta,n,r),$ (4.1)

where

$L(\zeta,n,r)=-K(-r)-|\zeta|\int_{0}^{r}t^{n}K^{\prime}(-t)dt$

and

$\ R(\zeta,n,r)=K(r)+|\zeta|\int_{0}^{r}t^{n}K^{\prime}(t)dt.$

The bounds are sharp for the extremal function
$f_{\zeta}=h_{\zeta}+\overline{g_{\zeta}}$ with $h_{\zeta}=K$, where $K$
satisfies (1.3) or its rotations and $g_{\zeta}$ satisfies
$g^{\prime}_{\zeta}=\zeta z^{n}h^{\prime}_{\zeta}$.

###### Proof.

Let $f=h+\overline{g}\in\mathcal{HC}(\phi)$. By Lemma 2.6, we know that

$K^{\prime}(-r)\leq|h^{\prime}(z)|\leq K^{\prime}(r)\quad(|z|=r).$ (4.2)

Let $\gamma$ be the linear segment joining $0$ to $z$ in $\mathbb{D}$. Then we
see that

$\displaystyle|f(z)|=\left|\int_{\gamma}\frac{\partial
f}{\partial\theta}\,\,d\theta+\frac{\partial
f}{\partial\overline{\theta}}\,\,d\overline{\theta}\right|$
$\displaystyle\leq\int_{\gamma}\left(|h^{\prime}(\theta)|+|g^{\prime}(\theta)|\right)\,|d\theta|=\int_{\gamma}\left(1+|\zeta||\theta|^{n}\right)|h^{\prime}(\theta)|\,|d\theta|.$
(4.3)

Combining (4.2) and (4.3), we obtain

$|f(z)|\leq\int_{0}^{r}\left(1+|\zeta|t^{n}\right)K^{\prime}(t)\,dt=K(r)+|\zeta|\int_{0}^{r}t^{n}K^{\prime}(t)\,\,dt=R(\zeta,n,r).$
(4.4)

Let $\Gamma$ be the preimage of the line segment joining $0$ to $f(z)$ under
the function $f$, it follows that

$\displaystyle|f(z)|=\left|\int_{\Gamma}\frac{\partial
f}{\partial\theta}\,d\theta+\frac{\partial
f}{\partial\overline{\theta}}\,\,d\overline{\theta}\right|$
$\displaystyle\geq\int_{\Gamma}\left(|h^{\prime}(\theta)|-|g^{\prime}(\theta)|\right)\,|d\theta|$
(4.5)
$\displaystyle=\int_{\Gamma}\left(1-|\zeta||\theta|^{n}\right)|h^{\prime}(\theta)|\,|d\theta|.$

From (4.2) and (4.5), we have

$|f(z)|\geq\int_{0}^{r}\left(1-|\zeta|t^{n}\right)K^{\prime}(-t)\,dt=-K(-r)-|\zeta|\int_{0}^{r}t^{n}K^{\prime}(-t)\,dt=L(\zeta,n,r).$
(4.6)

In view of (4.4) and (4.6), we deduce that

$L(\zeta,n,r)\leq|f(z)|\leq R(\zeta,n,r).$ (4.7)

To show the sharpness, we consider the function
$f_{\zeta}=h_{\zeta}+\overline{g_{\zeta}}$ with $h_{\zeta}=K$ or its
rotations. It is easy to see that $h_{\zeta}=K\in\mathcal{C}(\phi)$ and
$g_{\zeta}$ satisfies $g^{\prime}_{\zeta}(z)=\zeta
z^{n}h^{\prime}_{\zeta}(z)$, which shows that
$f_{\zeta}\in\mathcal{HC}(\phi)$. The equality holds on both sides of (4.2)
for suitable rotations of $K$. For $0\leq\zeta<1/{(2n-1)}$, we see that
$f_{\zeta}(r)=R(\zeta,n,r)$ and $f_{\zeta}(-r)=-L(\zeta,n,r)$. Hence
$|f_{\zeta}(r)|=R(\zeta,n,r)$ and $|f_{\zeta}(-r)|=L(\zeta,n,r)$. This
completes the proof of Proposition 4.1. ∎

######  Proposition 4.2.

Let $f\in\mathcal{HC}(\phi)$ and $S_{r}$ be the area of the image
$f(\mathbb{D}_{r})$ ($\mathbb{D}_{r}:=\\{z\in\mathbb{D}:|z|<r<1\\}$ ). Then

$2\pi\int_{0}^{r}t\left(1-|\zeta|^{2}t^{2n}\right)(K^{\prime}(-t))^{2}\,dt\leq
S_{r}\leq
2\pi\int_{0}^{r}t\left(1-|\zeta|^{2}t^{2n}\right)(K^{\prime}(t))^{2}\,dt.$
(4.8)

###### Proof.

Let $f=h+\overline{g}\in\mathcal{HC}(\phi)$. Then the area of image of
$\mathbb{D}_{r}$ under a harmonic mapping $f$ is given by

$S_{r}=\iint_{\mathbb{D}_{r}}\left(|h^{\prime}(z)|^{2}-|g^{\prime}(z)|^{2}\right)\,dxdy=\iint_{\mathbb{D}_{r}}\left(1-|\zeta|^{2}|z|^{2n}\right)|h^{\prime}(z)|^{2}dxdy.$
(4.9)

Since $h\in\mathcal{C}(\phi)$, in view of (4.2) and (4.9), we have

$\int_{0}^{r}\int_{0}^{2\pi}t\left(1-|\alpha|^{2}t^{2}\right)(K^{\prime}(-t))^{2}d\theta
dt\leq
S_{r}\leq\int_{0}^{r}\int_{0}^{2\pi}t\left(1-|\alpha|^{2}t^{2}\right)(K^{\prime}(t))^{2}d\theta
dt.$ (4.10)

Therefore, the assertion (4.8) of Proposition 4.2 follows directly from
(4.10). ∎

Next, we derive the Bohr inequality for the class $\mathcal{HC}(\phi)$.

######  Theorem 4.1.

Let $f\in\mathcal{HC}(\phi)$. Then the majorant series of $f$ satisfies the
inequality

$|z|+\sum_{n=2}^{\infty}(|a_{n}|+|b_{n}|)|z|^{n}\leq d(f(0),\partial
f(\mathbb{D}))$ (4.11)

for $|z|=r\leq\min\\{1/3,r_{f}\\}$, where $r_{f}$ is the smallest positive
root in $(0,1)$ of

$L(\zeta,n,1)=M_{K}(r)+|\zeta|\int_{0}^{r}t^{n}M_{K^{\prime}}(t)\,dt,$

and $L(\zeta,n,1)$ is given in Proposition 4.1.

###### Proof.

Let $f=h+\overline{g}\in\mathcal{HC}(\phi)$. Since $h\in\mathcal{C}(\phi)$,
from Lemma 2.5, we know that

$h^{\prime}\prec K^{\prime}.$ (4.12)

Let $K(z)=z+\sum\limits_{n=2}^{\infty}k_{n}z^{n}$. In view of Lemma 2.7 and
(4.12), we have

$1+\sum\limits_{n=2}^{\infty}n|a_{n}|r^{n-1}=M_{h^{\prime}}(r)\leq
M_{K^{\prime}}(r)=1+\sum\limits_{n=2}^{\infty}n|k_{n}|r^{n-1}$ (4.13)

for $|z|=r\leq 1/3$. Integrating (4.13) with respect to $r$ from $0$ to $r$,
we get

$M_{h}(r)=r+\sum\limits_{n=2}^{\infty}|a_{n}|r^{n}\leq
r+\sum\limits_{n=2}^{\infty}|k_{n}|r^{n}=M_{K}(r)\quad(r\leq 1/3).$ (4.14)

From the definition of $\mathcal{HC}(\phi)$, we have $g^{\prime}(z)=\zeta
z^{n}h^{\prime}(z)$. This relation along with (4.13) provides that

$\sum\limits_{n=2}^{\infty}n|b_{n}|r^{n-1}=M_{g^{\prime}}(r)=|\zeta|r^{n}M_{h^{\prime}}(r)\leq|\zeta|r^{n}M_{K^{\prime}}(r)\quad(r\leq
1/3).$ (4.15)

By integrating (4.15) with respect to $r$ from $0$ to $r$, it follows that

$M_{g}(r)=\sum\limits_{n=2}^{\infty}|b_{n}|r^{n}\leq|\zeta|\int_{0}^{r}t^{n}M_{K^{\prime}}(t)dt\quad(r\leq
1/3).$ (4.16)

Therefore, for $|z|=r\leq 1/3$, from (4.14) and (4.16), we obtain

$M_{f}(r)=|z|+\sum_{n=2}^{\infty}(|a_{n}|+|b_{n}|)r^{n}\leq
M_{K}(r)+|\zeta|\int_{0}^{r}t^{n}M_{K^{\prime}}(t)dt=R_{\mathcal{C}}(n,r).$
(4.17)

In view of (4.1), it is evident that the Euclidean distance between $f(0)$ and
the boundary of $f(\mathbb{D})$ is given by

$d(f(0),\partial f(\mathbb{D}))=\liminf\limits_{|z|\rightarrow
1}|f(z)-f(0)|\geq L(\zeta,n,1).$ (4.18)

We note that $R_{\mathcal{C}}(n,r)\leq L(\zeta,n,1)$ whenever $r\leq r_{f}$,
where $r_{f}$ is the smallest positive root of
$R_{\mathcal{C}}(n,r)=L(\zeta,n,1)$ in $(0,1)$. Let
$H_{1}(n,r)=R_{\mathcal{C}}(n,r)-L(\zeta,n,1)$, then $H_{1}(n,r)$ is a
continuous function in $[0,1]$. Since $M_{K}(r)\geq K(r)>-K(-r)$, it follows
that

$\displaystyle H_{1}(n,1)$ $\displaystyle=R_{\mathcal{C}}(n,1)-L(\zeta,n,1)$
(4.19)
$\displaystyle=M_{K}(1)+K(-1)+|\zeta|\int_{0}^{r}t^{n}\left(M_{K^{\prime}}(t)+K^{\prime}(t)\right)dt$
$\displaystyle\geq
K(1)+K(-1)+|\zeta|\int_{0}^{r}t^{n}\left(M_{K^{\prime}}(t)+K^{\prime}(t)\right)dt>0.$

On the other hand,

$H_{1}(n,0)=-L(\zeta,n,1)=K(-1)(1-|\zeta|)+n|\zeta|\int_{0}^{1}t^{n-1}K(-t)\,dt<0.$
(4.20)

Therefore, $H_{1}$ has a root in $(0,1)$. Let $r_{f}$ be the smallest root of
$H_{1}$ in $(0,1)$. Then $R_{\mathcal{C}}(n,r)\leq L(\zeta,n,1)$ for $r\leq
r_{f}$. Now in view of the inequalities (4.17) and (4.18) with the relation
$R_{\mathcal{C}}(n,r)\leq L(\zeta,n,1)$ for $r\leq r_{f}$, we obtain

$|z|+\sum_{n=2}^{\infty}(|a_{n}|+|b_{n}|)r^{n}\leq d(f(0),\partial
f(\mathbb{D}))$

for $|z|=r\leq\min\\{1/3,r_{f}\\}$. ∎

######  Remark 4.1.

The Bohr inequality holds for $|z|=r\leq r_{f}$, which has been extensively
studied by Allu and Halder [8, 9] and for particular values of $\phi$, they
obtained the Bohr radius $r_{f}$.

######  Corollary 4.1.

Let $f\in\mathcal{M}(\alpha,\zeta,n)$ with $0\leq\alpha<1$ and
$0\leq\zeta<1/(2n-1)$. Then the inequality (4.11) holds for $|z|=r\leq r_{f}$,
where $r_{f}$ is the smallest root in $(0,1)$ of

$F_{n}(r):=R(\alpha,\zeta,n,r)-L(\alpha,\zeta,n,1)=0.$

The radius $r_{f}$ is sharp.

###### Proof.

From Lemma 2.4, the Euclidean distance between $f(0)$ and the boundary of
$f(\mathbb{D})$ shows that

$d(f(0),\partial f(\mathbb{D}))=\liminf\limits_{|z|\rightarrow
1}|f(z)-f(0)|\geq L(\alpha,\zeta,n,1).$ (4.21)

We note that $r_{f}$ is the root of the equation
$R(\alpha,\zeta,n,r)=L(\alpha,\zeta,n,1)$ in $(0,1)$. The existence of the
root is ensured by the relation $R(\alpha,\zeta,n,1)>L(\alpha,\zeta,n,1)$ with
(2.4). For $0<r\leq r_{f}$, it is evident that $R(\alpha,\zeta,n,r)\leq
L(\alpha,\zeta,n,1)$. In view of Lemma 2.3 and (4.21), for $|z|=r\leq r_{f}$,
we have

$\displaystyle|z|+\sum\limits_{n=2}^{\infty}(|a_{n}|+|b_{n}|)|z|^{n}$
$\displaystyle\leq
r_{f}+(|a_{2}|+|b_{2}|)r_{f}^{2}+\sum\limits_{n=3}^{\infty}(|a_{n}|+|b_{n}|)r_{f}^{n}$
$\displaystyle=R(\alpha,\zeta,n,r_{f})\leq L(\alpha,\zeta,n,1)\leq
d(f(0),\partial f(\mathbb{D})).$

To show the sharpness of the radius $r_{f}$, we consider the function
$f=f_{\alpha,\zeta,n}$, which is defined in Lemma 2.4. We see that
$f_{\alpha,\zeta,n}$ belongs to $\mathcal{M}(\alpha,\zeta,n)$. Since the left
side of the growth inequality in Lemma 2.4 holds for $f=f_{\alpha,\zeta,n}$ or
its rotations, we have $d(f(0),\partial f(\mathbb{D}))=L(\alpha,\zeta,n,1)$.
Therefore, the function $f=f_{\alpha,\zeta,n}$ for $|z|=r_{f}$ gives

$\displaystyle|z|+\sum\limits_{n=2}^{\infty}(|a_{n}|+|b_{n}|)|z|^{n}$
$\displaystyle=r_{f}+(|a_{2}|+|b_{2}|)r_{f}^{2}+\sum\limits_{n=3}^{\infty}(|a_{n}|+|b_{n}|)r_{f}^{n}$
$\displaystyle=R(\alpha,\zeta,n,r_{f})=L(\alpha,\zeta,n,1)=d(f(0),\partial
f(\mathbb{D})),$

which reveals that the radius $r_{f}$ is the best possible. ∎

The roots $r_{f}$ of $F_{n}(r)=0$ for different values of $\alpha$, $\zeta$
and $n$ have been shown in Table 1, Table 2 and Figure 1.

Figure 1. The graphs of $F_{n}(r)$ respectively for $\alpha=0.5,\alpha=0.9$ when $n=1,2,3,4,5$. $n$ | $1$ | $2$ | $3$ | $4$ | $5$ | $10$ | $100$ | $1000$
---|---|---|---|---|---|---|---|---
$\zeta$ | $1/2$ | $1/4$ | $1/6$ | $1/8$ | $1/10$ | $1/20$ | $1/200$ | $1/2000$
$r_{f}$ | $0.386555$ | $0.468176$ | $0.486196$ | $0.492459$ | $0.495252$ | $0.498809$ | $0.499988$ | $0.500000$
Table 1. The roots $r_{f}$ of $F_{n}(r)=0$ for different values of $\zeta$ when $\alpha=0.5$. $n$ | $1$ | $2$ | $3$ | $4$ | $5$ | $10$ | $100$ | $1000$
---|---|---|---|---|---|---|---|---
$\zeta$ | $1/2$ | $1/4$ | $1/6$ | $1/8$ | $1/10$ | $1/20$ | $1/200$ | $1/2000$
$r_{f}$ | $0.567721$ | $0.731273$ | $0.774894$ | $0.792253$ | $0.800709$ | $0.812036$ | $0.815292$ | $0.815323$
Table 2. The roots $r_{f}$ of $F_{n}(r)=0$ for different values of $\zeta$
when $\alpha=0.9$.

######  Remark 4.2.

For $\alpha=0.5$, as $n\rightarrow\infty$, the sharp radius is 0.500000. For
$\alpha\rightarrow 1$ when $n=1$, the sharp radius is 0.645750. These bounds
are generalize the corresponding results obtained in [48, 9].

Now, we give an improved version of Bohr inequality for the class
$\mathcal{HC}(\phi)$. Note that by adding area quantity $S_{r}/2\pi$ with the
Majorant series of $f\in\mathcal{HC}(\phi)$, the sum is still less than
$d(f(0),\partial f(\mathbb{D}))$ for some radius
$r\leq\min\\{1/3,\widetilde{r}_{f}\\}<1$.

######  Theorem 4.2.

Let $f\in\mathcal{HC}(\phi)$ and $S_{r}$ be the area of the image
$f(\mathbb{D}_{r})$. Then the inequality

$M_{f}(r)+\frac{S_{r}}{2\pi}\leq d(f(0),\partial f(\mathbb{D}))$

holds for $|z|=r\leq\min\\{1/3,\widetilde{r}_{f}\\}$, where
$\widetilde{r}_{f}$ is the smallest positive root in $(0,1)$ of

$L(\zeta,n,1)=M_{K}(r)+|\zeta|\int_{0}^{r}t^{n}M_{K^{\prime}}(t)\,dt+\int_{0}^{r}t\left(1-|\zeta|^{2}t^{2n}\right)(K^{\prime}(t))^{2}dt,$

and $L(\zeta,n,1)$ is defined as in Proposition 4.1.

###### Proof.

Let $f\in\mathcal{HC}(\phi)$ be of the form (1.1). Then, from the right hand
inequality in (4.8) and (4.17), we obtain

$\displaystyle M_{f}(r)+\frac{S_{r}}{2\pi}$ $\displaystyle\leq
M_{K}(r)+|\zeta|\int_{0}^{r}t^{n}M_{K^{\prime}}(t)\,dt+\int_{0}^{r}t\left(1-|\zeta|^{2}t^{2n}\right)(K^{\prime}(t))^{2}\,dt$
(4.22)
$\displaystyle=R_{\mathcal{C}}(n,r)+\int_{0}^{r}t\left(1-|\zeta|^{2n}t^{2}\right)(K^{\prime}(t))^{2}\,dt=\widetilde{R}_{f}(n,r)$

for $r\leq 1/3$. Let $H_{2}(n,r)=\widetilde{R}_{f}(n,r)-L(\zeta,n,1)$, then
$H_{2}(n,r)$ is a continuous function in $[0,1]$. The inequality (4.20) yields
that $H_{2}(n,0)=-L(\zeta,n,1)<0$. In view of (4.19), we get

$R_{\mathcal{C}}(n,1)-L(\zeta,n,1)>0.$ (4.23)

For $|\zeta|<1/(2n-1)$, we observe that

$t\left(1-|\zeta|^{2}t^{2n}\right)(K^{\prime}(t))^{2}\geq 0,$

and hence

$\int_{0}^{r}t\left(1-|\zeta|^{2}t^{2n}\right)(K^{\prime}(t))^{2}dt>0.$ (4.24)

From (4.22) and (4.23), we obtain

$H_{2}(n,1)=R_{\mathcal{C}}(n,1)-L(\zeta,n,1)+\int_{0}^{1}t\left(1-|\zeta|^{2}t^{2n}\right)(K^{\prime}(t))^{2}\,dt>0.$

Since $H_{2}(n,0)<0$ and $H_{2}(n,1)>0$, $H_{2}$ has a root in $(0,1)$ and
choose $\widetilde{r}_{f}$ to be the smallest root in $(0,1)$. Therefore,
$\widetilde{R}_{f}(n,r)\leq L(\zeta,n,1)$ for $r\leq\widetilde{r}_{f}$. Hence,
from (4.18) and (4.22), we conclude that

$M_{f}(r)+\frac{S_{r}}{2\pi}\leq d(f(0),\partial f(\mathbb{D}))$

for $r\leq\min\\{1/3,\widetilde{r}_{f}\\}$. ∎

Acknowledgments

The present investigation was supported by the Key Project of Education
Department of Hunan Province under Grant no. 19A097 of the P. R. China.

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