# DH-PTAM: A Deep Hybrid Stereo Events-Frames Parallel Tracking And Mapping
System

Abanob Soliman$\,{}^{\href https://orcid.org/0000-0003-4956-8580}$, Fabien
Bonardi$\,{}^{\href https://orcid.org/0000-0002-3555-7306}$, Désiré
Sidibé$\,{}^{\href https://orcid.org/0000-0002-5843-7139}$, and Samia
Bouchafa$\,{}^{\href https://orcid.org/0000-0002-2860-8128}$

###### Abstract

This paper presents a robust approach for a visual parallel tracking and
mapping (PTAM) system that excels in challenging environments. Our proposed
method combines the strengths of heterogeneous multi-modal visual sensors,
including stereo event-based and frame-based sensors, in a unified reference
frame through a novel spatio-temporal synchronization of stereo visual frames
and stereo event streams. We employ deep learning-based feature extraction and
description for estimation to enhance robustness further. We also introduce an
end-to-end parallel tracking and mapping optimization layer complemented by a
simple loop-closure algorithm for efficient SLAM behavior. Through
comprehensive experiments on both small-scale and large-scale real-world
sequences of VECtor and TUM-VIE benchmarks, our proposed method (DH-PTAM)
demonstrates superior performance in terms of robustness and accuracy in
adverse conditions, especially in large-scale HDR scenarios. Our
implementation’s research-based Python API is publicly available on GitHub for
further research and development: https://github.com/AbanobSoliman/DH-PTAM.

###### Index Terms:

Stereo, events, SuperPoint, R2D2, SLAM.

##  I Introduction

Simultaneous Localization and Mapping (SLAM) is pivotal in robotics and
computer vision, aiming to chart unknown terrains while discerning an agent’s
location. Among the key SLAM contributions is the Parallel Tracking and
Mapping (PTAM) method [1], which uniquely separates tracking and mapping into
parallel threads for augmented efficiency and real-time performance in
monocular systems. However, PTAM faced scale ambiguity challenges inherent to
monocular SLAM. Addressing this issue, its successor, Stereo Parallel Tracking
and Mapping (S-PTAM) [2], leveraged stereo vision to extract depth information
directly, eliminating scale ambiguity and fortifying robustness. In recent
years, learning-based features extraction and description methods [3, 4], and
deep learning based approaches [5] have been applied to improve robustness.

Visual Odometry (VO), an integral component of SLAM, has predominantly
depended on conventional cameras to determine motion. However, such methods
often fail in high dynamic range (HDR) scenarios where lighting conditions
fluctuate [6] (see Fig. 1). Fortunately, the innovation of event cameras, also
termed asynchronous or dynamic vision sensors (DVS) [7], offers a
groundbreaking solution. Unlike their traditional counterparts that capture
frames at fixed intervals, event cameras relay a continuous stream of ”events”
showcasing pixel-wise brightness alterations. This not only empowers them to
operate at unparalleled speeds and in dimly lit conditions but also markedly
diminishes motion blur [8].

Figure 1: Experiments on school-scooter and corner-slow sequences from the
VECtor dataset show the estimated trajectory with the constructed scene map
(green dotted rectangle). The red dotted rectangle highlights an HDR use-case
where DH-PTAM estimates the trajectory continuously based on the two fusion
modes (Dynamic Vision Sensor (DVS) or Active Pixel Sensor (APS) biased). APS:
denotes the standard camera’s global shutter frames.

With the ability to adeptly handle swift motions, scenes with significant HDR,
and challenging lighting, event cameras emerge as superior alternatives,
increasing both SLAM and VO’s robustness and precision in environments that
would confound traditional cameras. The distinctive event-based design makes
these sensors incredibly adapted at tracking fast-moving objects, setting them
apart as an invaluable asset to visual odometry, especially under adverse
conditions.

Deep learning-based features are more robust than traditional methods [3, 4],
as they can learn from large amounts of data and generalize well to unseen
data. They are also more invariant to changes in viewpoint and lighting,
making them suitable for real-world applications. Recently, pre-trained models
have been widely adopted in computer vision and have achieved state-of-the-art
performance in object detection, semantic segmentation, and image
classification tasks.

This paper proposes a deep hybrid stereo events-frames parallel tracking and
mapping system (see Fig. 2) that significantly improves SLAM accuracy and
robustness in dynamic environments. This system combines the advantages of
stereo RGB and event cameras, which can capture visual information at high
temporal resolution. The use of deep learning techniques in this system allows
for the extraction of robust features from the stereo hybrid image and event
frames, which improves the accuracy of the feature-matching process and the
estimation of the camera pose.

The main contributions can be summarized as follows:

  * •

We propose an end-to-end parallel tracking and mapping (PTAM) approach based
on a novel spatio-temporal synchronization of stereo visual frames with event
streams.

  * •

We propose a simple mid-level feature loop-closure algorithm for prompt SLAM
behavior based on a learning-based feature description method to maximize
robustness.

  * •

DH-PTAM’s effectiveness is evaluated in both stereo event-aided and image-
based visual SLAM modes, achieving improved accuracy when incorporating event
information, shown in an ablation study on the CPU versus the GPU of a
consumer-grade laptop.

This paper is organized as follows: Section II gives a brief overview of the
state-of-the-art SLAM methods. Section III provides an in-detail overview of
the proposed method and offers insights into the novel parts of the algorithm.
Section IV comprehensively evaluates the algorithm on the most recent VECtor
[9] and TUM-VIE [10] benchmarks, along with defining the limitations. Section
V summarizes the experiments’ main observations, the proposed method’s
behavioral aspects, and the start points for future works.

##  II Related Work

Figure 2: Block diagram of the proposed hybrid event-aided stereo visual
odometry approach (DH-PTAM). $f$ denotes the fusion function defined in (6).
$\Delta{t}^{k}$ is the event volume $\mathcal{V}_{0}(x,y,t)$ accumulation time
defined in (2). E3CT denotes the Event 3-Channel Tensor [11], an image-like
event representation.

Events-Frames hybridization. Event-aided systems leverage the high-quality
representations that events can produce after processing, especially in
dynamic and dimmed environments where RGB camera frames fail. Some of the
well-known event representations are event image (EI) [12], Time Surfaces (TS)
[13], Event Spike Tensor (EST) [14], and recently Event 3-Channel Tensor
(E3CT) [11]. Others [7] build the front-end on an Event Generation Model (EGM)
[15] or construct motion-compensated event frames (MEF) [16] aided by a
gyroscope. Towards a traditional frame reconstruction from events, [17]
proposes a Log Intensity Reconstruction (LIR), a model-based method, and [18]
proposes Spade-e2vid, a learning-based method.

Indirect methods [19], such as frame-based approaches, extract keypoints from
the input data in the front-end. This front-end stage typically involves
detecting and matching salient features in the sensory data, such as images or
event streams. These keypoints are then passed to the back-end, where state
estimation algorithms are used to estimate the robot’s pose and build a
consistent map of the environment.

Conversely, direct methods [7] attempt to process all available sensor data,
such as individual pixel intensity changes in images (events) or all RGB frame
pixels, without any intermediate filtering or feature extraction in the front-
end, relying on the back-end to handle the entire data. The proposed method
adopts a hybrid approach where all events are directly processed during the
events-frames fusion in the front-end, while only the reliable learning-based
features from the fusion frames are fed to the back-end (see Fig. 2).

Table I compares the latest event-based and event-aided VO solutions
concerning the sensor setup, events pre-processing layer (EPL), direct or
indirect event processing, and the loop-closure capability to minimize visual
drifts.

Event-aided visual-SLAM. DH-PTAM builds upon the pioneering work of [7], which
introduces a monocular 6-DoF visual odometry model that synergistically
integrates events and grayscale frames using a direct approach. In this model,
the fusion of events and frames is achieved by applying EGM to the intensity
frame, transforming it into a unified reference that aligns with the event
stream, termed as brightness increments (event-like). This process is followed
by an error minimization cost function, which estimates the extrinsics in
real-time for accurate mapping and is augmented by a photometric bundle
adjustment for tracking. However, DH-PTAM distinguishes itself from [7] by
employing a image-like unified reference in its front-end, harmoniously
combining stereo event streams with stereo RGB frames.

Loop-closure detection is paramount in visual-SLAM, effectively minimizing
drifts by allowing a system to recognize previously traversed locations. The
realm of loop-closure detection predominantly features two methodologies: mid-
level features [20] and the bag-of-words model [21]. While mid-level features
offer a more nuanced representation than low-level features such as edges and
corners, they don’t consider the specificity of high-level features like
object recognition. Deep learning descriptors, as referenced in [22], typify
mid-level features. They extract more sophisticated information from raw data
than low-level features like pixel values, yet they don’t reach the
specificity of direct task-related features, for instance, object labels.

TABLE I: Direct and Indirect (D/I) Visual Odometry methods based (B) on events
and/or aided (A) by events

Method | B/A | D/I | EPLa | LCb | More Information
---|---|---|---|---|---
Kim [17] | B | D | LIR | $\times$ | 3 EKFs + Image reconst.
Rebecq [12] | B | D | EI | $\times$ | Monocular (PTAM)
Zhou [23] | B | D | TS | $\times$ | Stereo (PTAM)
Kueng [24] | A | I | $\times$ | $\times$ | Event-aided Tracking
Rosinol [16] | A | I | MEF | $\times$ | Mono + IMU (front-end)
Hidalgo-Carrió [7] | A | D | EGM | $\times$ | Monocular Odometry
Proposed | A | D/I | E3CT | ✓ | Stereo (PTAM) + DLc

a denotes an Event Pre-processing Layer. b denotes Loop-Closure capability. c
denotes the only method incorporating Deep Learning-aided features.

##  III Methodology

###  III-A System Overview

Fig. 2 illustrates the main components and the process of DH-PTAM. The system
establishes a global reference frame based on the camera position in the
initial frame. A preliminary map is created by identifying and triangulating
distinctive points in the first stereo pair of images. For subsequent frames,
the tracking thread calculates the 6D pose of each stereo frame by minimizing
the discrepancy between the projected map points and their matches. The system
chooses a subset of keyframes used in another thread to update the map at a
slower pace.

Map points are derived from the stereo matches of each keyframe and added to
the map. The mapping thread constantly improves the local discrepancy by
adjusting all map points, and stereo poses using Bundle Adjustment. A pose
graph is utilized to preserve the global consistency of the map which is a
shared resource among the tracking, mapping, and loop-closing threads. Point
correspondences are actively searched between keyframes to strengthen the
constraints of the pose graph optimization smoothing process.

Notations. The odometry state representation comprises the 3D points
$X_{w}^{k}$ and a 7-increment vector $\mu\in\mathfrak{se}(3)$, which is the
current pose of the left fusion frame at time $k$:

$\mu^{k}=\left[\delta x\;\delta y\;\delta z\;\delta q_{x}\;\delta
q_{y}\;\delta q_{z}\;\delta q_{w}\right]^{\top}\;,$ (1)

where $[\delta x\;\delta y\;\delta z]^{\top}$ is the incremental translation
vector and $[\delta q_{x}\;\delta q_{y}\;\delta q_{z}\;\delta q_{w}]^{\top}$
is the incremental quaternion vector.

###  III-B Temporal Synchronization Approach

Figure 3: Temporal synchronization scheme. $t_{exp}$ is the global shutter
camera exposure time. $\Delta{t}$ is the event representation (E3CT) volume
accumulation window. $t_{f}$ is the fusion frame calculated timestamp.
$t_{\text{DVS},\text{CAM}}$ are the DVS events, and RGB camera frames
timestamps, respectively.

Our temporal synchronization approach (see Fig. 3) considers the general case
of global shutter cameras where the exposure time $t_{exp}$ is known. We adopt
the constant-time $\Delta{t}^{k}$ events accumulation window $k$ approach
where the number of accumulated events during this temporal window is ablated
in the qualitative analysis in Fig. 4.

Figure 4: Ablation study on reducing the temporal window width versus
controlling the number of events in the designed window. All event frames are
post-processed E3CTs by median filtering followed by a binary threshold.

As soon as stereo RGB camera frames are received at timestamps
$t_{\text{CAM}}$, we calculate the fusion frames timestamps assuming the
hardware synchronization of stereo RGB images and stereo event streams, using:

$t_{f}=t_{\text{CAM}}+\frac{t_{exp}}{2}\;,\;\Delta{t}^{k}=t_{f}^{k}-t_{f}^{k-1}\;,$
(2)

where the left $t_{\text{CAM}}$ is the selected stereo keyframe timestamp.

###  III-C Spatial Hybridization Approach

Leveraging the findings of [25], where Scale-Consistent monocular depth
learning is explored, our method is firmly rooted in the depth scale
equivalence principle. Rather than just an assumption, this principle is an
inference drawn from the inherent properties of closely-spaced APS-DVS
sensors. It asserts that the depth scales between frames maintain a consistent
relationship. Diverging from the approach in [25], we incorporate the front-
end’s depth uncertainty into the back-end’s stereo bundle adjustment process.
Here, it acts as an adaptive mechanism, progressively refining depth
estimates, thereby boosting system performance. The E3CT events pre-processing
layer is adopted and modeled as two consecutive filtering kernel convolutions
on the event volume $\mathcal{V}_{0}(x,y,t)$ of temporal width $\Delta{t^{k}}$
(see Fig. 2). The first kernel to filter the time decaying events in the
volume, is the $\alpha$-exponential time decay kernel and is modeled as:

$\mathcal{V}_{1}(x,y,t)\doteq\exp\left(-\alpha\left(\frac{\mathcal{V}_{0}(x,y,t)-\eta/2}{\eta/6}\right)^{2}\right)\;,$
(3)

where $\alpha=0.5$ and the decay rate $\eta=30$ [ms] for our model. Followed
by a trilinear voting kernel to stack the events in the three channels tensor
so that each event contributes to two consecutive channels depending on their
location from a vertex of this trilinear kernel. An event near the top
contributes a higher weight to the current channel and a lower weight to the
neighboring ones. These contribution weights of the three channels can
represent a percentage of an R-G-B color map; hence, the E3CT can be
considered a synthetic RGB frame of events. The trilinear voting kernel can be
modeled as follows:

$\mathcal{V}_{2}(x,y,t_{i})\doteq\max\left(0,\,1-\left|\frac{\mathcal{V}_{1}(x,y,t_{i})}{\delta{t}}\right|\right)\;,$
(4)

where $\delta{t}$ is the temporal bin $i$ size as discussed in [14].

After applying the trilinear temporal voting kernel on the exponential-decay
time surface, we stack the 3-channel tensor temporal bins together, resulting
in a synthetically colored 2D frame called the Event 3-Channel Tensor (E3CT).
In Fig. 2, we can observe that the constructed synthetic colors are always
consistent, meaning that the stereo left and right constructed E3CTs have
identical colors for the same scene.

Conventional frame-based post-processing operations can be applied to the
constructed E3CTs, such as adaptive threshold, contrast stretching, color
correction and balance, and denoising functions. We consider a fully
calibrated stereo RGB and event cameras stack as represented in Fig. 5, so
that the rigid-body transformations $\mathcal{T}_{cd}$ and the cameras
intrinsic parameters $\mathcal{K}_{c},\mathcal{K}_{d}$ are known.

Figure 5: Geometry of the stereo hybrid event-RGB cameras stack.
$\mathcal{T}_{cd}$ denotes the rigid-body transformations. $P^{h}_{d\in
c}\;\text{and}\;P^{h}_{d}$ denote pixels locations.

Given that the same post-processing operations are applied on the current
stereo E3CT frames (see Fig. 4), the 2D-to-3D-to-2D consecutive inverse-
forward projections of the pixels on the E3CT frames $P^{h}_{d}$ to the RGB
camera frames $P^{h}_{d\in c}$ can be performed as follows (comparable to
Equation (2) in [25]):

$P^{h}_{d\in
c}\doteq\frac{1}{\lambda{{}^{\prime}}}\,\mathcal{K}_{c}\,[R|t]_{cd}\,\lambda\,\mathcal{K}_{d}^{-1}\,P^{h}_{d}+\delta{P^{h}_{align}}\;,$
(5)

where $(.)^{h}$ denotes the pixel location in homogeneous coordinates. The
term $\delta{P^{h}_{align}}$ denotes the pixel location alignment correction
factor for the RGB and event frames so that the same 3D world point
$X^{h}_{w}$ should correspond exactly to the pixel locations $P^{h}_{d\in
c},\,P^{h}_{d}$. This alignment term is observed to be constant for the same
sensor rig with non-varying intrinsic and extrinsic parameters.
$\delta{P^{h}_{align}}$ value can be estimated using an offline optimization
process only once on a selected number of frames (the more the accurate) with
high confidence feature matches, and this value is given in Section IV for
both VECtor and TUM-VIE sequences.

Finally, the fusion function (and frame) $f(.)$ performs a temporal cross-
dissolve (linear blending) between both the left ($D_{0},C_{0}$) and right
($D_{1},C_{1}$) E3CTs and RGB camera frames, respectively, and is formulated
as:

$f(C,D)\,=\,(1-\beta)*C+\beta*D\;,$ (6)

where $\beta\in[0,1]$ is the E3CT contribution weight in the current fusion
frame. $\beta$ value is calculated online and depends on the scene lighting
and texture conditions. It is set to chirp-shaped values
$\beta=\max(\bar{C}/{C^{\text{max}}},\,1-\bar{C}/{C^{\text{max}}})$ when the
RGB camera frame fails to detect features due to adverse HDR conditions and
low-textured scenes, and this is the DVS-biased fusion mode. For situations
where RGB camera frames can detect reliable scene features with good lighting
and enough texture, the $\beta$ value is harmonic with the scene lighting
conditions according to
$\beta=\min(\bar{C}/{C^{\text{max}}},\,1-\bar{C}/{C^{\text{max}}})$, and this
is the APS-biased fusion mode. To reduce the amount of extracted features and
maintain the back-end processing complexity and latency in reasonable ranges,
$\beta$ value can be capped at a certain value, as shown in Fig. 6 by setting
$\beta=0.3$ as an example.

Dynamic scenes with challenging and adverse conditions can easily trigger
rapid switching between these two fusion modes during long-term navigation.
This causes a critical problem during the feature tracking process using
conventional low-level feature detectors, such as ORB, SIFT, SURF, BRIEF, and
FAST. Accordingly, applying mid-level feature detectors that depend mainly on
learning-based architectures could solve this fusion frame modes alternation
problem. We employ the learning-based feature extractors and descriptors [3,
4] for their high robustness and feature detection speed. Fig. 6 elaborates in
a real-world case-study, the fusion mode automatic switch using the $\beta$
parameter in response to the rapid intensity fluctuations of the RGB camera
frame $C$ in HDR scenarios represented by the parameter
$\bar{C}/C^{\text{max}}$.

Figure 6: Experiments on mocap-desk2 TUM-VIE dataset that show the capability
of the proposed events-frames fusion method to maintain and track features in
bight and dimmed scenes where grayscale-only frames may fail. For each
scenario in top: grayscale frame (left) and fusion frame (right).

###  III-D Optimization-based State Estimation

As our work is based on the original S-PTAM system, all the optimization
Jacobians mentioned in this section can be found with detailed proofs in [2].
All objective functions are minimized with the Levenberg-Marquardt algorithm
implemented in the $g^{2}o$ optimization library. We employ the Huber loss
function for outliers rejection $\rho(.)$.

System bootstrapping. The first stereo fusion frames are considered a
keyframe. Then, a triangulation for the collected feature matches on the left
and right fusion frames is performed to initialize the map.

Figure 7: Normal versus HDR semi-circular spatio-temporal matching for two
stereo consecutive fusion frames. Green and blue lines denote the spatial
stereo SuperPoints matching at frames $k$ and $k-1$, respectively. Red lines
denote the temporal matching for keypoints of two consecutive keyframes
(left).

Pose tracking thread. Each map point is projected into the viewing frustum of
the anticipated stereo position, and we then look nearby for the match. A
valid prediction of the current pose is required for such a projection. By
contrasting the descriptions, map points, and features are matched. The
$L_{2}$ norm is computed using the binary descriptors of SuperPoint and R2D2
(see Fig. 7). The match is valid if the distance falls below a certain
threshold; otherwise, it is ignored. The pose refinement is then applied to
recover the current pose knowing the previous one using the following
objective function:

$L^{\text{refine}}=\operatorname*{arg\,min}_{\mu}\sum_{i\in
N}\rho(\lvert\lvert
J_{i}^{k}\mu_{k}-\Delta{z}_{i}(\mu_{k-1},X_{w}^{i})\rvert\rvert^{2})\;,$ (7)

where $N=\\{z_{1},\,\dots\,,\,z_{M}\\}$ and $M$ is the number of matched
measurements. The measurement $z=[u,v]^{\top}$ is a pixel 2D location of the
forward projection of a 3D map point $X_{w}$ using the pinhole model
projection function
$\pi(X_{w}^{i})=\mathcal{K}_{c}\mathcal{T}^{i}_{f_{0}w}X_{w}^{i}$.
$J_{i}^{k}=\frac{\partial\Delta{z}_{i}(\mu)}{\partial\mu_{k}}$ is the re-
projection error’s Jacobian with respect to the current odometry state vector.
$\Delta{z}$ is the re-projection error of a matched set of measurements on the
current $k$ stereo fusion frames and is defined as:

$\Delta{z}_{i}(\mu,X_{w})=z_{i}-\pi(\exp{(\mu)}\mathcal{T}^{k-1}_{f_{0}w}X_{w}^{i})\;,$
(8)

where the 3D point cloud $X_{w}$ is considered a constant optimization
parameter and not updated in the tracking thread and
$\mathcal{T}^{k-1}_{f_{0}w}=exp(\mu)\in SE(3)$ with $\exp{(.)}$ the
exponential map in the $\mathfrak{Lie}$ group for the previous increment state
vactor. If the number of observed points is less than 90% of the points
recorded in the previous keyframe, a frame is chosen to be a keyframe after
the current pose has been evaluated. Then, new map points are created by
triangulating the stereo pair’s remaining mismatched features. The keyframe is
then placed in the local mapping thread for processing.

Mapping thread. We apply Bundle Adjustment (BA) to fine-tune the camera poses
(keyframe map) and the 3D points (point cloud map). Local Bundle Adjustment
minimizes the re-projection error of every point in every keyframe
$f^{k}_{0}$. Given an initial set of $N$ keyframe poses
$\\{\mathcal{T}_{f_{0}w}^{1},\,\dots\,,\,\mathcal{T}_{f_{0}w}^{N}\\}$, an
initial set of $M$ 3D points $X_{w}^{i}$, and measurement sets
$S\in\\{S_{1},\,\dots\,,\,S_{N}\\}$, where each set comprises the measurement
$z_{i}^{k}$ of the $i^{th}$ point in the $k^{th}$ keyframe, the local BA is
performed using the following objective function on all keyframes in a pre-
defined sliding-window size $N$:

$L^{\text{BA}}=\operatorname*{arg\,min}_{\mu,\,X_{w}}\sum_{k=1}^{N}\sum_{i\in
S_{k}}\rho(\lvert\lvert J_{i}^{k}\begin{bmatrix}\mu_{k}\\\
X_{w}^{i}\end{bmatrix}-\Delta{z}_{i}(\mu_{k},X_{w}^{i})\rvert\rvert^{2})\;,$
(9)

where the 3D point cloud $X_{w}$ is considered a variable optimization
parameter and is updated in the mapping thread. Hence, the
$J_{i}^{k}=[\frac{\partial\Delta{z}_{i}(\mu_{k},X_{w}^{i})}{\partial\mu_{k}},\,\frac{\partial\Delta{z}_{i}(\mu_{k},X_{w}^{i})}{\partial
X_{w}^{i}}]$ is the re-projection error’s Jacobian with respect to the current
odometry state vector and the 3D point as well.

Loop-closure thread. Instead of the conventional way of keyframe embedding
assignments using a bag-of-words, we adopt a simple loop-closure detection
method based on the mean of the mid-level learning-based feature descriptors
(SuperPoint and R2D2) for each keyframe and assign this mean value as the
embedding identity of each keyframe. Once a potential loop closure is
detected, the system performs geometric verification through RANSAC-based pose
estimation to validate the candidate. If the verification is successful, a
loop closure constraint is added to the pose graph, and a graph optimization
is performed to distribute the error and update the global map, thus
correcting the accumulated drift.

##  IV Evaluation

We perform a thorough, comprehensive evaluation during navigation in real-
world, large-scale, and small-scale areas in challenging settings. In
subsection IV-A, we compare DH-PTAM with other RGB image-based and event-
based/-aided methods on the HDR large-scale sequences of the publicly
available dataset VECtor [9] due to its high-quality ground truth values and
sensors calibration parameters. In subsection IV-B, we evaluate the small-
scale (mocap-) sequences of TUM-VIE [10] to test the quality of the DH-PTAM
spatio-temporal synchronization method with degraded event camera calibration
parameters. Moreover, the first 45 frames of TUM-VIE sequences suffer a high
over-/under-exposure global shutter alternation, which tests the DH-PTAM’s
pose estimation stability. We perform a comparative quantitative analysis to
evaluate the accuracy of our system in Table II and a qualitative analysis in
Fig. 8. The accuracy of DH-PTAM is measured with absolute trajectory error
(ATE), and relative pose error (RPE) metrics calculated using the baseline
SLAM evaluation tool [26].

TABLE II: DH-PTAM Quantitative Comparison Against the best performing open-
source State-of-the-art SLAM Systems using ATE [m] metric. The upper sub-table
is for Standard Stereo VIO Methods, the middle is for event-based VO/VIO
Methods, and the lower is for DH-PTAM with RPE [m] metric. Bold denotes best
performing, Underline for second best performing, and ($\times$) denotes
failure

Method | VECtor sequences [9] | TUM-VIE sequences [10] | Mean | Mean
---|---|---|---|---
corridors | corridors | units | units | school | school | mocap | mocap | mocap | mocap | mocap | VECtor | TUM-VIE
dolly | walk | dolly | scooter | dolly | scooter | 1d-trans | 3d-trans | 6dof | desk | desk2 | large-scale | small-scale
ORB-SLAM3 (SVIO) [19] | 0.802 | 1.031 | 18.063 | 14.504 | 0.921 | 0.752 | 0.007 | 0.012 | 0.018 | 0.007 | 0.025 | 6.012 | 0.013
BASALT (SVIO) [27] | 1.625 | 2.152 | 11.151 | 13.256 | 1.852 | 1.482 | 0.003 | 0.009 | 0.014 | 0.016 | 0.011 | 5.253 | 0.011
VINS-Fusion (SVIO) [28] | 1.464 | 0.392 | 10.391 | 11.471 | 1.791 | 0.562 | 0.011 | 0.011 | 0.017 | 0.058 | 0.013 | 4.345 | 0.022
EVO (Mono Events) [12] | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 0.075 | 0.125 | 0.855 | 0.541 | 0.752 | $\times$ | 0.470
ESVO (Stereo Events) [23] | $\times$ | $\times$ | $\times$ | $\times$ | 13.710 | 9.830 | 0.009 | 0.028 | 0.058 | 0.033 | 0.032 | 11.77 | 0.032
Ultimate SLAM (EVIO)+ [16] | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | 6.830 | 0.039 | 0.047 | 0.353 | 0.195 | 0.341 | 6.830 | 0.195
DH-PTAM (Stereo Fusion) | 1.884 | 1.299 | 5.274 | 8.433 | 1.093 | 0.796 | 0.103 | 0.007 | 0.024 | 0.016 | 0.015 | 3.130 | 0.033
(SuperPoint on CPU) - RPE ($\sigma$) | 0.073 | 0.038 | 0.055 | 0.149 | 0.178 | 0.074 | 0.006 | 0.007 | 0.009 | 0.009 | 0.007 | 0.095 | 0.008
DH-PTAM∗ (Stereo Image) | 1.841 | 1.543 | 5.738 | 5.010 | 1.559 | 0.877 | 0.099 | 0.004 | 0.045 | 0.011 | 0.008 | 2.761 | 0.033
(R2D2 on GPU) - RPE ($\sigma$) | 0.116 | 0.141 | 0.134 | 0.308 | 0.202 | 0.331 | 0.014 | 0.020 | 0.022 | 0.023 | 0.021 | 0.205 | 0.020

+ IMU sensor is included since it is integrated into the front-end and cannot be separated for a fair comparison with EVO, ESVO, and DH-PTAM (ours).
∗ in this ablation case study, the SuperPoint detector is replaced with the
R2D2 detector (trained for SLAM tasks), leveraging the GPU performance.

Figure 8: DH-PTAM (GPU (no events) vs. CPU (event-aided)) qualitative
analysis. All trajectories are transformed to a reference frame as the ground
truth poses using the extrinsic parameters, followed by an alignment with all
poses by Umeyama’s SE(3) method implemented in [26]. Large-scale trajectories
show high-quality loop closure detection in the case of R2D2 on GPU. The
small-scale trajectory shows the high accuracy of the event-aided version of
DH-PTAM.

| EVO [12] | EDS [7] | DH-PTAM (Ours) | EDS (point cloud) [7] | DH-PTAM (point cloud)
---|---|---|---|---|---
monitor |  |  |  |  |
boxes |  |  |  |  |
desk |  |  |  |  |
bin |  |  |  |  |

Figure 9: Stereo DAVIS sequences evaluation from [29] with
$\delta{P^{h}_{align}}=[0,0]^{\top}$ since both sensors (event-based and
frame-based) share the same photoreceptor pixel array. Initial three columns
present pseudo-colored inverse depth maps: EVO (yellow-near, blue-far), EDS
and DH-PTAM (red-near, blue-far). All sequences span a 1-7 m depth range. 3D
point clouds from EDS and DH-PTAM, utilizing grayscale keyframe values, are
presented in the last two columns.

To prevail the advantages of complementing the sensor stack with events
information, we compare our event-aided stereo visual odometry solution (DH-
PTAM) to the latest best-performing open-source visual-inertial systems in
literature in Table II. Table III gives the system parameters configuration
for large-scale and small-scale sequences. We keep these parameters constant
for all sequences of the same scale group without an online fine-tuning
process.

TABLE III: DH-PTAM Parameters Configuration

Parameter | VECtor sequences | TUM-VIE sequences
---|---|---
$\delta{P^{h}_{align}}$ \- Left | (-160, -235) [px] | (355, 40) [px]
$\delta{P^{h}_{align}}$ \- Right | (-160, -235) [px] | (375, 45) [px]
frustum_near | 0.1 [m] | 0.1 [m]
frustum_far | 30.0 [m] | 5.0 [m]
matching_cell_size | 15 [px] | 15 [px]
matching_neighborhood | 2 [px] | 1.8 [px]
matching_distance | 25 [px] | 15 [px]

All experiments are performed on the CPU and the GPU of a 16 GB RAM laptop
computer running 64-bit Ubuntu 20.04.3 LTS with AMD(R) Ryzen 7 4800h ×16 cores
2.9 GHz processor and a Radeon RTX NV166 Renoir graphics card. Table IV
reports a detailed computational complexity analysis for our DH-PTAM system
with minimal and maximal system requirements. The high CPU load observed when
detecting SuperPoint and R2D2 features can be attributed to the algorithms’
design, which prioritizes feature quality and robustness over computational
efficiency. This trade-off is often necessary for computer vision research,
where high-quality results are crucial for many applications but come at the
cost of increased computational complexity. The back-end runs with real-time
performance, and it is recommended to run the front-end on a GPU to achieve a
memory efficient, faster, and more stable performance.

TABLE IV: Computational Complexity Analysis on CPU vs. GPU

Thread | #Tasks | Operation | CPU [ms] | GPU [ms]
---|---|---|---|---
Front-end | 3 | Stereo E3CT Construction | 25-30 | 76-172
|  | Events-Frames Fusion | 439-521 | 191-352
| ($\approx$2$\times$10K) | SuperPoint Detection | 2478-3256 | 521-1752
| ($\approx$2$\times$4K) | R2D2 Detection | 8532-8752 | 1067-2254
Bootstrapping | 1 | Initialize the Map | 106-143 | 53-120
Tracking | 2 | Spatio-temporal Matching | 161-215 | 142-172
| (10 iter.) | Pose Refinement | 11-15 | 10-12
Mapping | 2 | Update Map | 1-3 | 0.452-1
| (30 iter.) | Local Bundle-Adjustment | 1-4 | 1-2
Loop-closing | 3 | Loop Detection | 14-20 | 10-15
|  | Compute and Validate | 2-5 | 1-3
| (30 iter.) | Pose Graph Optimization | 1-3 | 0.524-2
End-to-End | 11 | SuperPoint Detector | 3153-4208 | 356-1962
|  | R2D2 Detector | 8560-9125 | 1226-2486

###  IV-A VECtor large-scale experiments

We notice a prominent estimation failure in Table II while evaluating the
event-based methods EVO, ESVO and Ultimate SLAM on the large-scale sequences.
Numerous factors may contribute to the failure of these systems, including
stringent initialization requirements. For instance, the system EVO
necessitates running in a sensor-planar scene for several seconds to bootstrap
the system. Additionally, these systems are susceptible to parameter tuning,
as demonstrated by using different parameters for different sequences in the
same scenarios, even within their open-source projects.

Table II shows a good performance for DH-PTAM compared to the competing VI-
SLAM systems. Although Fig. 8 shows high visual drifts for our vision-only
system in the case of units sequences, DH-PTAM could outperform the VI-SLAM
systems based on the ATE metric. Fig. 8 gives an overview of the high-quality
loop detection of DH-PTAM in the case of corridors sequences. Loop detection
failure can be noticed only when the RAM overflows while running the system
with enormous point clouds, as in the case of units sequences. We provide
trajectory smoothing and post-processing script with our open-source
implementation to join estimated trajectory increments in case of RAM overflow
failures.

###  IV-B TUM-VIE small-scale experiments

As noticed in [30], the calibrationA (mocap-desk, mocap-desk2) sequences have
more accurate depth estimation results than calibrationB (rest of mocap and
TUM-VIE large-scale) sequences due to the significant calibration errors in
the latter. Hence, we perform our comparative evaluation on TUM-VIE small-
scale (mocap-) sequences using calibrationA parameters. Although the same
high-quality calibrationA parameters apply to both desk2 and desk sequences
with the same spiral motion, DH-PTAM performs the best with desk2 sequence but
the worst with desk sequence. This occurs since the scene of the desk sequence
is bounded by a close-by white wall that strict the depth, and hence DH-PTAM
front-end detects low quality and fewer features for desk than desk2. Table II
shows that the more DoF excited (6dof, desk2) and the consistent loops
detection (1d-trans), the better the pose estimation quality.

###  IV-C Ablation experiments

No event streams ($\beta=0$). In Table II, we show an ablation study where we
run DH-PTAM on stereo images. We notice estimation failure with all the
conventional and learning-based feature detectors except R2D2. Although the
ATE metric shows slightly better results without using events, the RPE metric
shows much more accurate values when using events. These better ATE values are
due to the high performance of the GPU in loop-closures using R2D2 features
(see Fig. 8).

Front-end depth maps. Qualitatively in Fig. 9, we rigorously assess the front-
end of our events-frames fusion technique by comparing its inverse depth maps
against established methods such as EVO and EDS. By channeling the stereo
fusion frames through the SGM matching method [31], our method capitalizes on
direct front-end fusion where the inverse depth maps are derived from events
projected onto the RGB frames. Remarkably, the point clouds generated through
our method exhibited richer details compared to those of EDS. This affirms the
potential of our method to set a new benchmark in event-frame fusion
paradigms.

##  V Conclusion

In this paper, we presented the DH-PTAM system for robust parallel tracking
and mapping in dynamic environments using stereo images and event streams. The
proposed system builds upon the principles of S-PTAM and extends it with a
learning-based approach to handle the sparse and noisy nature of event-based
sensors while leveraging the rich information provided by fusion frames. Our
experiments demonstrate that DH-PTAM outperforms state-of-the-art visual-SLAM
methods, particularly in challenging scenarios such as HDR and occlusions. The
proposed system can achieve better performance on a GPU and provides a
scalable and accurate solution for 3D reconstruction and pose estimation.
Future work includes investigating the potential of improving the spatial
synchronization through online optimization and exploring the integration of
an adaptive paradigm for choosing the temporal window width to further boost
robustness and system latency. DH-PTAM proves to have the potential to provide
robust and accurate 3D mapping and localization, which are crucial for the
successful operation of long-term navigation systems.

## References

  * [1] G. Klein and D. Murray, “Parallel tracking and mapping for small ar workspaces,” in _2007 6th IEEE and ACM International Symposium on Mixed and Augmented Reality_ , 2007, pp. 225–234.
  * [2] T. Pire, T. Fischer, G. Castro, P. De Cristóforis, J. Civera, and J. J. Berlles, “S-ptam: Stereo parallel tracking and mapping,” _Robotics and Autonomous Systems_ , vol. 93, pp. 27–42, 2017.
  * [3] D. DeTone, T. Malisiewicz, and A. Rabinovich, “Superpoint: Self-supervised interest point detection and description,” in _Proceedings of the IEEE conference on computer vision and pattern recognition workshops_ , 2018, pp. 224–236.
  * [4] J. Revaud, C. De Souza, M. Humenberger, and P. Weinzaepfel, “R2d2: Reliable and repeatable detector and descriptor,” _Advances in neural information processing systems_ , vol. 32, 2019.
  * [5] Z. Teed and J. Deng, “Droid-slam: Deep visual slam for monocular, stereo, and rgb-d cameras,” _Advances in Neural Information Processing Systems_ , vol. 34, pp. 16 558–16 569, 2021.
  * [6] Z. Zhang, C. Forster, and D. Scaramuzza, “Active exposure control for robust visual odometry in hdr environments,” in _2017 IEEE International Conference on Robotics and Automation (ICRA)_. IEEE, 2017, pp. 3894–3901.
  * [7] J. Hidalgo-Carrió, G. Gallego, and D. Scaramuzza, “Event-aided direct sparse odometry,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2022, pp. 5781–5790.
  * [8] S. Sun, G. Cioffi, C. De Visser, and D. Scaramuzza, “Autonomous quadrotor flight despite rotor failure with onboard vision sensors: Frames vs. events,” _IEEE Robotics and Automation Letters_ , vol. 6, no. 2, pp. 580–587, 2021.
  * [9] L. Gao, Y. Liang, J. Yang, S. Wu, C. Wang, J. Chen, and L. Kneip, “VECtor: A versatile event-centric benchmark for multi-sensor slam,” _IEEE Robotics and Automation Letters_ , vol. 7, no. 3, pp. 8217–8224, 2022.
  * [10] S. Klenk, J. Chui, N. Demmel, and D. Cremers, “Tum-vie: The tum stereo visual-inertial event dataset,” in _2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2021, pp. 8601–8608.
  * [11] A. Soliman, F. Bonardi, D. Sidibé, and S. Bouchafa, “IBISCape: A simulated benchmark for multi-modal SLAM systems evaluation in large-scale dynamic environments,” _Journal of Intelligent & Robotic Systems_, vol. 106, no. 3, p. 53, Oct 2022.
  * [12] H. Rebecq, T. Horstschäfer, G. Gallego, and D. Scaramuzza, “Evo: A geometric approach to event-based 6-dof parallel tracking and mapping in real time,” _IEEE Robotics and Automation Letters_ , vol. 2, no. 2, pp. 593–600, 2016.
  * [13] A. Sironi, M. Brambilla, N. Bourdis, X. Lagorce, and R. Benosman, “Hats: Histograms of averaged time surfaces for robust event-based object classification,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 2018, pp. 1731–1740.
  * [14] D. Gehrig, A. Loquercio, K. G. Derpanis, and D. Scaramuzza, “End-to-end learning of representations for asynchronous event-based data,” in _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , 2019, pp. 5633–5643.
  * [15] D. Gehrig, H. Rebecq, G. Gallego, and D. Scaramuzza, “Eklt: Asynchronous photometric feature tracking using events and frames,” _International Journal of Computer Vision_ , vol. 128, no. 3, pp. 601–618, 2020.
  * [16] A. R. Vidal, H. Rebecq, T. Horstschaefer, and D. Scaramuzza, “Ultimate slam? combining events, images, and imu for robust visual slam in hdr and high-speed scenarios,” _IEEE Robotics and Automation Letters_ , vol. 3, no. 2, pp. 994–1001, 2018.
  * [17] H. Kim, S. Leutenegger, and A. J. Davison, “Real-time 3d reconstruction and 6-dof tracking with an event camera,” in _European conference on computer vision_. Springer, 2016, pp. 349–364.
  * [18] P. R. G. Cadena, Y. Qian, C. Wang, and M. Yang, “Spade-e2vid: Spatially-adaptive denormalization for event-based video reconstruction,” _IEEE Transactions on Image Processing_ , vol. 30, pp. 2488–2500, 2021.
  * [19] C. Campos, R. Elvira, J. J. G. Rodríguez, J. M. Montiel, and J. D. Tardós, “OrbSLAM3: An accurate open-source library for visual, visual–inertial, and multimap slam,” _IEEE Transactions on Robotics_ , vol. 37, no. 6, pp. 1874–1890, 2021.
  * [20] P. Koniusz, F. Yan, and K. Mikolajczyk, “Comparison of mid-level feature coding approaches and pooling strategies in visual concept detection,” _Computer vision and image understanding_ , vol. 117, no. 5, pp. 479–492, 2013.
  * [21] D. Gálvez-López and J. D. Tardós, “Bags of binary words for fast place recognition in image sequences,” _IEEE Transactions on Robotics_ , vol. 28, no. 5, pp. 1188–1197, October 2012.
  * [22] Z. Ji, F. Wang, X. Gao, L. Xu, and X. Hu, “Ssnet: Learning mid-level image representation using salient superpixel network,” _Applied Sciences_ , vol. 10, no. 1, 2020.
  * [23] Y. Zhou, G. Gallego, and S. Shen, “Event-based stereo visual odometry,” _IEEE Transactions on Robotics_ , vol. 37, no. 5, pp. 1433–1450, 2021.
  * [24] B. Kueng, E. Mueggler, G. Gallego, and D. Scaramuzza, “Low-latency visual odometry using event-based feature tracks,” in _2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2016, pp. 16–23.
  * [25] L. Wang, Y. Wang, L. Wang, Y. Zhan, Y. Wang, and H. Lu, “Can scale-consistent monocular depth be learned in a self-supervised scale-invariant manner?” in _2021 IEEE/CVF International Conference on Computer Vision (ICCV)_ , 2021, pp. 12 707–12 716.
  * [26] M. Grupp, “evo: Python package for the evaluation of odometry and slam,” Note: https://github.com/MichaelGrupp/evo, 2017.
  * [27] V. Usenko, N. Demmel, D. Schubert, J. Stückler, and D. Cremers, “Visual-inertial mapping with non-linear factor recovery,” _IEEE Robotics and Automation Letters_ , vol. 5, no. 2, pp. 422–429, 2019.
  * [28] T. Qin, J. Pan, S. Cao, and S. Shen, “A general optimization-based framework for local odometry estimation with multiple sensors,” _arXiv preprint arXiv:1901.03638_ , 2019.
  * [29] Y. Zhou, G. Gallego, H. Rebecq, L. Kneip, H. Li, and D. Scaramuzza, “Semi-dense 3d reconstruction with a stereo event camera,” in _Proceedings of the European conference on computer vision (ECCV)_ , 2018, pp. 235–251.
  * [30] S. Ghosh and G. Gallego, “Multi-event-camera depth estimation and outlier rejection by refocused events fusion,” _Advanced Intelligent Systems_ , vol. 4, no. 12, p. 2200221, 2022.
  * [31] H. Hirschmuller, “Stereo processing by semiglobal matching and mutual information,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , vol. 30, no. 2, pp. 328–341, 2008.


#  Some coefficient estimates on complex-valued kernel $\alpha-$harmonic
mappings∗

Bo-Yong Long

School of Mathematical Sciences, Anhui University, Hefei 230601, China

††footnotetext:  ∗ Supported by NSFC (No.12271001) and Natural Science
Foundation of Anhui Province (2308085MA03), China. E-mail<EMAIL_ADDRESS>
> Abstract: We call a kind of mappings induced by a kind of weighted Laplace
> operator as complex-valued kernel $\alpha-$harmonic mappings. In this
> article, for this class of mappings, the Heinz type lemma is established,
> and the best Heinz type inequality is obtained. Next, the extremal function
> of Schwartz’s Lemma is discussed. Finally, the coefficients are estimated
> for the subclass of complex-valued kernel $\alpha-$harmonic mappings whose
> coefficients are real numbers.
> Keywords: Weighted harmonic mapping; Heinz’s type lemma; Schwarz lemma;
> typically real; coefficient estimates
> 2010 Mathematics Subject Classification: Primary 30C50, Secondary ; 31A05,
> 31A30

##  1 Introductions and main results

Let $\mathbb{C}$ be the complex plane, $\mathbb{D}$ the unit disk and
$\mathbb{T}=\partial\mathbb{D}$ the boundary of unit disk.

A kind of weighted Laplace operator be given by

$\Delta_{\alpha,z}=\frac{\partial}{\partial
z}(\omega_{\alpha})^{-1}\frac{\partial}{\partial\bar{z}}=\frac{\partial}{\partial
z}(1-|z|^{2})^{-\alpha}\frac{\partial}{\partial\bar{z}},$

where $\alpha>-1$, $\omega_{\alpha}=(1-|z|^{2})^{\alpha}$ is the so-called
standard weight. The corresponding homogeneous partial differential equation
is

$\displaystyle\Delta_{\alpha,z}(u)=0,\quad\mbox{in}\quad\mathbb{D}.$ (1.1)

Denote the associated Dirichlet boundary value problem as follows

$\;\left\\{\begin{array}[]{rr}\Delta_{\alpha,z}(u)=0&\quad\mbox{in}\,\mathbb{D},\\\
u=u^{*}&\quad\mbox{on}\,\mathbb{T}.\end{array}\right.$ (1.2)

Here, the boundary data $u^{*}\in\mathfrak{D}^{\prime}(\mathbb{T})$ is a
distribution on $\mathbb{T}$, and the boundary condition in (1.2) is
interpreted in the distributional sense that $u_{r}\rightarrow u^{*}$ in
$\mathfrak{D}^{\prime}(\mathbb{T})$ as $r\rightarrow 1^{-}$, where

$u_{r}(e^{i\theta})=u(re^{i\theta}),\quad e^{i\theta}\in\mathbb{T},$

for $r\in[0,1)$.

In [22], the authors proved that, for parameter $\alpha>-1$, if a function
$u\in\mathcal{C}^{2}(\mathbb{D})$ satisfies (1.1) with $\lim_{r\rightarrow
1^{-}}u_{r}=u^{*}\in\mathfrak{D}^{\prime}(\mathbb{T})$ , then it has the form
of Poisson type integral

$u(z)=\frac{1}{2\pi}\int_{0}^{2\pi}\mathcal{P}_{\alpha}(ze^{-i\tau})u^{*}(e^{i\tau})d\tau,\quad\,z\in\mathbb{D},$
(1.3)

where

$\mathcal{P}_{\alpha}(z)=\frac{(1-|z|^{2})^{\alpha+1}}{(1-z)(1-\bar{z})^{\alpha+1}}.$
(1.4)

Observe that the kernel $\mathcal{P}_{\alpha}$ in (1.4) is complex-valued. We
call the solution $u$ of the equation (1.1) with $\alpha>-1$ complex-valued
kernel $\alpha-$harmonic mappings (or functions).

######  Theorem 1.1.

[22] A function $u$ in $\mathbb{D}$ is a complex-valued kernel
$\alpha$-harmonic mapping if and only if it is given by a convergent power
series expansion of the form

$u(z)=\sum_{k=0}^{\infty}c_{k}z^{k}+\sum_{k=1}^{\infty}c_{-k}P_{\alpha,k}(|z|^{2})\bar{z}^{k},\quad
z\in\mathbb{D},$ (1.5)

for some sequence $\\{c_{k}\\}_{-\infty}^{\infty}$ of complex number
satisfying

$\lim_{|k|\rightarrow\infty}\sup|c_{k}|^{\frac{1}{|k|}}\leq 1,$

where

$P_{\alpha,k}(x)=\int^{1}_{0}t^{k-1}(1-tx)^{\alpha}dt,\quad-1<x<1.$ (1.6)

In the Remark 1.3 of [22], the authors pointed out that the complex-valued
kernel $\alpha-$harmonic mapping $u$ with power series expansion (1.5)
actually converges in $\mathcal{C}^{\infty}(\mathbb{D})$.

If we take $\alpha=p-1$ is a nonnegative integer, then from Theorem 1.1 of
[9], we find that the complex-valued kernel $\alpha-$harmonic mapping is
polyharmonic (exactly, $p-$harmonic). As to the general polyharmonic mappings,
we refer to [1, 7, 23, 6, 24, 2, 14]. In particular, if $\alpha=0$, then the
complex-valued kernel $\alpha-$harmonic mapping is harmonic. As to the general
harmonic mappings, we refer to [11]. Thus, the complex-valued kernel
$\alpha-$harmonic mappings is a kind of generalization of classical harmonic
mappings. Furthermore, by [22, 20], we know that complex-valued kernel
$\alpha-$harmonic mappings are related to the real kernel $\alpha-$harmonic
mappings. For the related study on real kernel $\alpha-$harmonic mappings, see
[8, 13, 19].

For the complex-valued kernel $\alpha-$harmonic mappings, the representation
theorem and Lipschitz continuity are studied in [9] when $\alpha$ is a
nonnegative integer; the Lipschitz continuity for general $\alpha>-1$ are
considered in [15] and [21]; the Schwarz-Pick type inequality and coefficient
estimates are obtained in [16]; the starlikeness, convexity and Radó-Kneser-
Choquet type theorem are explored in [18]. The corresponding case of complex-
valued kernel $\alpha-$harmonic mappings defined on the upper half plane are
researched in [5].

Similar to the definition of sense-preserving harmonic mappings [12], we
define the sense-preserving complex-valued kernel $\alpha-$harmonic mappings.
A complex-valued kernel $\alpha-$harmonic mapping $u$, not identically
constant, is called sense-preserving at a point $z_{0}$ if $u_{z}\not\equiv
0$, $\omega:=\overline{u}_{z}/u_{z}$ is well defined at $z_{0}$ (possibly with
a removable singularity), and $|\omega(z_{0})|<1$. If $u$ is sense-preserving
at $z_{0}$, then either the Jacobian
$J_{u}(z_{0})=|u_{z}(z_{0})|^{2}-|u_{\bar{z}}(z_{0})|^{2}=|u_{z}(z_{0})|^{2}-|\overline{u}_{z}(z_{0})|^{2}>0$
or $u_{z}(z_{0})=0$ for an isolated point $z_{0}$. We say $u$ is sense-
preserving in $\mathbb{D}$ if $u$ is sense-preserving at all $z\in\mathbb{D}$.

In the harmonic mapping theory, Heinz’s lemma is an important and fundamental
result. It has applications in many aspects including minimal surface theory.
For complex-valued kernel $\alpha-$harmonic self-mappings of the unit disk, we
establish a Heinz’s type lemma which is as follows:

######  Theorem 1.2.

(Heinz’s type lemma) Let $u(z)$ be a complex-valued kernel $\alpha-$harmonic
mapping of the form (1.5). Suppose $u(z)$ maps the unit disk $\mathbb{D}$ onto
itself. Then

$\displaystyle|c_{1}|^{2}+\frac{3\sqrt{3}}{\pi}|c_{0}|^{2}+\frac{1}{(\alpha+1)^{2}}|c_{-1}|^{2}\geq\frac{27}{4\pi^{2}}.$
(1.7)

The lower bound $\frac{27}{4\pi^{2}}$ is sharp.

It is worth pointing out that if $\alpha\in(-1,0)$, the image domain of the
extremal function in the Theorem 1.2 is a relatively compact subset of the
unit disc $\mathbb{D}$. See Remark 1 in Section 2 for the detail. If
$\alpha=0$, then Theorem 1.2 reduces to the case of harmonic mappings, see
[11].

Define $\Lambda_{u}:=|u_{z}|+|u_{\bar{z}}|$ and
$\lambda_{u}:=||u_{z}|-|u_{\bar{z}}||$. We get the following conclusion that
involves the extremal function of Schwarz type lemma of the class of complex-
valued kernel $\alpha-$harmonic mappings.

######  Theorem 1.3.

Suppose $u(z)$ is a sense-preserving complex-valued kernel $\alpha-$harmonic
mapping of the unit disk $\mathbb{D}$, $u(0)=0$ and $|u(z)|<1$. If
$J_{u}(0)=1$ or $\lambda_{u}(0)=1$, then $u(z)=\beta z$ with $|\beta|=1$.

If $\alpha=0$, then Theorem 1.3 reduces to the case of harmonic functions, see
[17].

According to Theorem 1.3 and its proof, we obtain the following proposition.

######  Proposition 1.4.

There does not exist a sense-preserving complex-valued kernel
$\alpha-$harmonic mapping $u(z)$ on $\mathbb{D}$ with $u(0)=0$ and $|u(z)|<1$,
such that $J_{u}(0)>1$.

Let $H_{\alpha}$ be the family of all complex-valued kernel $\alpha-$harmonic
mappings with the form (1.5), $S_{H_{\alpha}}$ be the subclass of $H_{\alpha}$
consisting of $u$ that $u$ are univalent and sense-preserving in $\mathbb{D}$
and satisfy the normalization condition $u(0)=0$ and $c_{1}=1$. Observe that
if $u\in S_{H_{\alpha}}$, then $J_{u}(0)>0$ implies that $|c_{-1}|<1$, where
$J_{u}$ is the Jacobian of $u$. We denote by $S^{o}_{H_{\alpha}}$ the subclass
of $S_{H_{\alpha}}$ consisting of $u$ such that $c_{-1}=0$.

In [10, 25, 4], the typically real harmonic mappings are studied. Similar to
the typically real functions of analytic functions or harmonic functions, we
want to study the complex-valued kernel $\alpha-$harmonic functions with real
coefficients. Let $TS_{H_{\alpha}}$ denote the subclass of $S_{H_{\alpha}}$
consisting of $u$ such that all the coefficients $c_{k}$ and $c_{-k}$ are real
numbers, $TS^{o}_{H_{\alpha}}$ be the subclass of $TS_{H_{\alpha}}$ with
$c_{-1}=0$. If $u\in TS_{H_{\alpha}}$, then we call $u$ a typically real
complex-valued kernel $\alpha-$harmonic function.

We can get the coefficient estimation for $TS_{H_{\alpha}}$ as follows.

######  Theorem 1.5.

Let $u(z)$ be in $TS_{H_{\alpha}}$ with the form (1.5) and $k=2,3,4,...$. Then
it holds that

$\left|\frac{c_{k}-c_{-k}B(k,\alpha+1)}{1-c_{-1}B(1,\alpha+1)}\right|\leq k$
(1.8)

if one of the following two conditions is satisfied:

(1) $\alpha\in[0,+\infty)$;

(2) $\alpha\in(-1,0)$ and $|c_{-1}|<1+\alpha$.

Here $B(\cdot,\cdot)$ is Beta function.

##  2 Proofs and Remarks

Let $k=1,2,...$, $x=|z|^{2}=r^{2}$, $P_{\alpha,k}=P_{\alpha,k}(x)$ defined as
in (1.6) and $P_{\alpha,k}(1)=\lim_{x\rightarrow 1^{-}}P_{\alpha,k}(x)$. Then

$\displaystyle
P_{\alpha,k}(1)=B(k,\alpha+1)=\frac{\Gamma(\alpha+1)\Gamma(k)}{\Gamma(k+\alpha+1)},$
(2.1)

where $B(\cdot,\cdot)$ and $\Gamma(\cdot)$ are Beta function and Gamma
function, respectively. In the rest of this paper, we will use the notations
of $P_{\alpha,k}$ and $P_{\alpha,k}(1)$.

######  Lemma 2.1.

(1) If $\alpha\in[0,+\infty)$, then $1\geq P_{\alpha,1}>\frac{1}{1+\alpha}$;

(2) If $\alpha\in(-1,0)$, then $1\leq P_{\alpha,1}<\frac{1}{1+\alpha}$;

###### Proof.

By (1.6), we have that
$P_{\alpha,1}^{\prime}(x)=-\alpha\int^{1}_{0}t(1-tx)^{\alpha-1}dt$. Thus,
$P_{\alpha,1}^{\prime}(x)\leq 0$ for given $\alpha\in[0,+\infty)$ and
$P_{\alpha,1}^{\prime}(x)>0$ for given $\alpha\in(-1,0)$. Therefore, Lemma 2.1
follows from the monotonicity of $P_{\alpha,1}(x)$ and the fact that
$P_{\alpha,1}(0)=1$ and $P_{\alpha,1}(1)=\frac{1}{1+\alpha}$. ∎

######  Lemma 2.2.

Let $g(\alpha)=\frac{\Gamma(\alpha+1)}{(\Gamma(\frac{\alpha}{2}+1))^{2}}$.
Then $g(\alpha)<1$ for $\alpha\in(-1,0)$.

###### Proof.

Let $\psi(x)=\Gamma^{\prime}(x)/\Gamma(x)$ be the digamma function. It is well
known that $\psi(x)$ is strictly increasing on $(0,+\infty)$, cf. [3]. Then we
have

$\displaystyle\frac{d}{d\alpha}(\log
g(\alpha))=\psi(\alpha+1)-\psi(\frac{\alpha}{2}+1))<0$

for $\alpha\in(-1,0)$. Observe $\log g(\alpha)$ and $g(\alpha)$ have the same
monotonicity. Therefore, Lemma 2.2 follows from the monotonicity of
$g(\alpha)$ and the fact that $g(0)=1$.

∎

###### Proof of Theorem 1.2.

Without loss generality, we assume that $u(z)$ is sense-preserving. Let
$z=re^{i\varphi}$. By (1.5) and (2.1), we have

$\displaystyle e^{i\theta(\varphi)}$ $\displaystyle:=\lim_{r\rightarrow
1^{-}}u(z)$
$\displaystyle=\sum_{k=0}^{\infty}c_{k}e^{ik\varphi}+\sum_{k=1}^{\infty}c_{-k}B(k,\alpha+1)e^{-ik\varphi}.$

Then $c_{k}$ and $c_{-k}B(k,\alpha+1)$ may be regarded as Fourier coefficients
of the boundary function $e^{i\theta(\varphi)}$, where $\theta(\varphi)$ is a
continuous nondecreasing function with
$\theta(\varphi+2\pi)=\theta(\varphi)+2\pi$. It follows that

$\displaystyle
e^{i\theta(\phi+\varphi)}=\sum_{k=0}^{\infty}c_{k}e^{ik(\phi+\varphi)}+\sum_{k=1}^{\infty}c_{-k}B(k,\alpha+1)e^{-ik(\phi+\varphi)},$
$\displaystyle
e^{-i\theta(\phi-\varphi)}=\sum_{k=0}^{\infty}\overline{c_{k}}e^{-ik(\phi-\varphi)}+\sum_{k=1}^{\infty}\overline{c_{-k}}B(k,\alpha+1)e^{ik(\phi-\varphi)}.$

Using Parseval’s relation, direction computation leads to

$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}e^{i[\theta(\phi+\varphi)-\theta(\phi-\varphi)]}d\phi=|c_{0}|^{2}+\sum_{k=1}^{\infty}|c_{k}|^{2}e^{2ik\varphi}+\sum_{k=1}^{\infty}|c_{-k}|^{2}(B(k,\alpha+1))^{2}e^{-2ik\varphi}$

for arbitrary $\varphi\in\mathbb{R}$. Taking real parts, we have that

$\displaystyle
1-2J(\varphi)=|c_{0}|^{2}+\sum_{k=1}^{\infty}\left(|c_{k}|^{2}+|c_{-k}|^{2}(B(k,\alpha+1))^{2}\right)\cos(2k\varphi),$
(2.2)

where

$\displaystyle
J(\varphi)=\frac{1}{2\pi}\int_{0}^{2\pi}\sin^{2}\left(\frac{\theta(\phi+\varphi)-\theta(\phi-\varphi)}{2}\right)d\phi.$

Let $N(\varphi)$ be the even function with period $\pi/3$ that has the values
$\cos^{2}(\pi/3+\varphi)$ in the interval $0\leq\varphi\leq\pi/6$.
Furthermore, let

$\displaystyle M(\varphi)=\cos^{2}\varphi-N(\varphi),\quad
0\leq\varphi\leq\pi/2.$

Then

$\displaystyle\;M(\varphi)=\left\\{\begin{array}[]{rr}\cos^{2}\varphi-\cos^{2}(\frac{\pi}{3}+\varphi),&\quad
0\leq\varphi\leq\frac{\pi}{6},\\\
\cos^{2}\varphi-\cos^{2}(\frac{2\pi}{3}-\varphi),&\quad\frac{\pi}{6}\leq\varphi\leq\frac{\pi}{3},\\\
0,&\quad\frac{\pi}{3}\leq\varphi\leq\frac{\pi}{2}.\end{array}\right.$

By the Fourier expansion

$\displaystyle
N(\varphi)=\frac{1}{2}-\frac{3\sqrt{3}}{4\pi}+\frac{3\sqrt{3}}{2\pi}\sum_{k=1}^{\infty}\frac{1}{9k^{2}-1}\cos
6k\varphi$

and equation (2.2), one has

$\displaystyle\frac{8}{\pi}\int_{0}^{\frac{\pi}{2}}M(\varphi)(1-2J(\varphi))d\varphi$
$\displaystyle=$
$\displaystyle|c_{1}|^{2}+\frac{3\sqrt{3}}{\pi}|c_{0}|^{2}+|c_{-1}|^{2}(B(1,\alpha+1))^{2}-\frac{3\sqrt{3}}{\pi}\sum_{k=1}^{\infty}\frac{1}{9k^{2}-1}\left(|c_{3k}|^{2}+|c_{-3k}|^{2}(B(3k,\alpha+1))^{2}\right)$
$\displaystyle\leq$
$\displaystyle|c_{1}|^{2}+\frac{3\sqrt{3}}{\pi}|c_{0}|^{2}+|c_{-1}|^{2}(B(1,\alpha+1))^{2}.$
(2.3)

Direct computation leads to

$\displaystyle\frac{8}{\pi}\int_{0}^{\frac{\pi}{2}}M(\varphi)d\varphi=\frac{3\sqrt{3}}{\pi}.$
(2.4)

Furthermore, inequality

$\displaystyle\frac{16}{\pi}\int_{0}^{\frac{\pi}{2}}M(\varphi)J(\varphi)d\varphi\leq\frac{3\sqrt{3}}{\pi}-\frac{27}{4\pi^{2}}.$
(2.5)

has been proved in $\S$ 4.4 of [11]. Therefore, inequality (1.7) follows from
inequalities (2), (2.4), and (2.5).

Next, let us to show that inequality (1.7) is sharp. We rewrite the Poisson
type representation (1.3) as

$\displaystyle
u(z)=\frac{1}{2\pi}\int_{0}^{2\pi}\mathcal{P}_{\alpha}(ze^{-i\varphi})e^{i\theta(\varphi)}d\varphi,$
(2.6)

where $e^{i\theta(\varphi)}$ is the boundary function. By the Definition 2.1
and Theorem 2.5 of [22], we have

$\displaystyle\mathcal{P}_{\alpha}(ze^{-i\varphi})$
$\displaystyle=\sum_{k=0}^{\infty}e^{-ik\varphi}z^{k}+\sum_{k=1}^{\infty}\frac{1}{B(k,\alpha+1)}P_{\alpha,k}e^{ik\varphi}\bar{z}^{k}.$

Thus, the formulas

$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}e^{-ik\varphi}z^{k}e^{i\theta(\varphi)}d\varphi$
$\displaystyle=c_{k}z^{k}$

and

$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{B(k,\alpha+1)}P_{\alpha,k}e^{ik\varphi}\bar{z}^{k}e^{i\theta(\varphi)}d\varphi$
$\displaystyle=c_{-k}P_{\alpha,k}\bar{z}^{k}$

follow from the Poisson type integral representation (2.6) and the series
expansion (1.5). It follows that

$\displaystyle c_{k}$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}e^{-ik\varphi}e^{i\theta(\varphi)}d\varphi$
(2.7)

and

$\displaystyle c_{-k}$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{B(k,\alpha+1)}e^{ik\varphi}e^{i\theta(\varphi)}d\varphi.$
(2.8)

Now taking

$\displaystyle\theta(\varphi)=\left\\{\begin{array}[]{rr}0,&\quad
0\leq\varphi<\frac{\pi}{3},\\\
\frac{2\pi}{3},&\quad\frac{\pi}{3}<\varphi<\pi,\\\
\frac{4\pi}{3},&\quad\pi<\varphi<\frac{5\pi}{3},\\\
0,&\quad\frac{5\pi}{3}<\varphi<2\pi.\end{array}\right.$

Denote by $U(z)$ the corresponding complex-valued kernel $\alpha$-harmonic
mapping produced by (2.6). Then formulas (2.7) and (2.8) imply that

$c_{1}=\frac{3\sqrt{3}}{2\pi},\quad c_{0}=c_{-1}=0.$

Thus the value $27/4\pi^{2}$ is attained in inequality (1.7) by the function
$U(z)$.

∎

######  Remark 1.

(1). We don’t know how to write down the explicit expression of the extremal
function $U(z)$.

(2). Formula (2.6) implies that

$\displaystyle|U(z)|\leq||e^{i\theta(\varphi)}||_{\infty}\frac{1}{2\pi}\int_{0}^{2\pi}|\mathcal{P}_{\alpha}(ze^{-i\varphi})|d\varphi=\frac{1}{2\pi}\int_{0}^{2\pi}|\mathcal{P}_{\alpha}(ze^{-i\varphi})|d\varphi\leq\frac{\Gamma(\alpha+1)}{(\Gamma(\frac{\alpha}{2}+1))^{2}}.$

The last inequality holds because of formula (0.6) of [22]. If
$\alpha\in(-1,0)$, then by Lemma 2.2, we have that
$|U(z)|\leq\frac{\Gamma(\alpha+1)}{(\Gamma(\frac{\alpha}{2}+1))^{2}}<1$ for
all $z\in\mathbb{D}$. That is to say that $U(\mathbb{D})$ is a relatively
compact subset of $\mathbb{D}$. If $\alpha=0$, in the other words, if it is
the classical harmonic case, then $U(\mathbb{D})$ is an equilateral triangle
embedded in the unit circle. See $\S$ 4.2 and $\S$ 4.4 of [11] for the detail.

###### Proof of Theorem 1.3.

Let $z=re^{i\theta}$. Then formula (1.5) can be rewritten as

$\displaystyle
u(re^{i\theta})=\sum_{k=1}^{\infty}c_{k}r^{k}e^{ik\theta}+\sum_{k=1}^{\infty}c_{-k}P_{\alpha,k}r^{k}e^{-ik\theta}$

for $\theta\in[0,2\pi)$. By Parseval’s identity and the assumption of
$|u(z)|<1$, we obtain

$\displaystyle\sum_{k=1}^{\infty}(|c_{k}|^{2}+|c_{-k}|^{2}(P_{\alpha,k})^{2})r^{2k}=\frac{1}{2\pi}\int^{2\pi}_{0}|u(re^{i\theta})|^{2}d\theta<1.$

Letting $r\rightarrow 1^{-}$, we obtain

$\displaystyle\sum_{k=1}^{\infty}\left(|c_{k}|^{2}+|c_{-k}|^{2}(B(k,\alpha+1))^{2}\right)\leq
1.$ (2.9)

Next, we divide it into two cases to discuss.

(1) If $J_{u}(0)=1$, then we have

$\displaystyle|c_{1}|^{2}+|c_{-1}|^{2}(B(1,\alpha+1))^{2}\geq|c_{1}|^{2}-|c_{-1}|^{2}=J_{u}(0)=1.$

Thus inequality (2.9) implies that

$\displaystyle|c_{1}|=1,\,\,|c_{-1}|=0\,\,\,\mbox{and}\,\,\,|c_{k}|=|c_{-k}|=0\,\,\mbox{
for}\,\,k=2,3,....$ (2.10)

It follows that $u(z)=\beta z$, where $|\beta|=1$.

(2) If $\lambda_{u}(0)=1$, then we have

$\displaystyle|c_{1}|^{2}+|c_{-1}|^{2}(B(1,\alpha+1))^{2}\geq|c_{1}|^{2}-|c_{-1}|^{2}=J_{u}(0)=\Lambda_{u}(0)\lambda_{u}(0)\geq(\lambda_{u}(0))^{2}=1.$

Thus inequality (2.9) still implies that (2.10) holds. ∎

###### Proof of Proposition 1.4.

Suppose that there exists a sense-preserving complex-valued kernel
$\alpha-$harmonic mapping with $u(0)=0$ and $|u(z)|<1$ on $\mathbb{D}$. Then
inequality (2.9) holds. Specially, it holds that

$\displaystyle|c_{1}|^{2}+|c_{-1}|^{2}(B(1,\alpha+1))^{2}\leq 1.$ (2.11)

If $J_{u}(0)>1$, then it holds that

$\displaystyle|c_{1}|^{2}-|c_{-1}|^{2}=J_{u}(0)>1.$

It contradicts the inequality (2.11).

∎

###### Proof of Theorem 1.5.

Let $z=re^{i\theta}$ and define the difference quotient $\mathcal{D}(z)$ as

$\displaystyle\;\mathcal{D}(re^{i\theta})=\left\\{\begin{array}[]{rr}u^{\prime}(r),&\quad
0<r<1\,\,\mbox{and}\,\,\theta=0,\\\
\frac{u(z)-u(\overline{z})}{z-\overline{z}},&\quad
0<r<1\,\,\mbox{and}\,\,0<\theta<\pi,\\\ u^{\prime}(-r),&\quad
0<r<1\,\,\mbox{and}\,\,\theta=\pi.\end{array}\right.$

Then direct computation leads to

$\displaystyle\mathcal{D}(re^{i\theta})$
$\displaystyle=\frac{\sum_{k=1}^{\infty}c_{k}r^{k}e^{ik\theta}+\sum_{k=1}^{\infty}c_{-k}P_{\alpha,k}r^{k}e^{-ik\theta}-\sum_{k=1}^{\infty}c_{k}r^{k}e^{-ik\theta}-\sum_{k=1}^{\infty}c_{-k}P_{\alpha,k}r^{k}e^{ik\theta}}{re^{i\theta}-re^{-i\theta}}$
$\displaystyle=\frac{\sum_{k=1}^{\infty}(c_{k}-c_{-k}P_{\alpha,k})r^{k}(e^{ik\theta}-e^{-ik\theta})}{r(e^{i\theta}-e^{-i\theta})}$
$\displaystyle=\sum_{k=1}^{\infty}\left((c_{k}-c_{-k}P_{\alpha,k})r^{k-1}\frac{\sin
k\theta}{\sin\theta}\right)$ (2.12)

for $0<r<1$ and $0<\theta<\pi$.

If $u(z)$ is univalent, then $\mathcal{D}(re^{i\theta})\neq 0$ for the region
$0\leq r<1$, and $0\leq\theta<2\pi$. Since all the coefficients are real,
$\mathcal{D}(z)$ is real. Furthermore, $\mathcal{D}(0):=\lim_{z\rightarrow
0}\mathcal{D}(z)=1-c_{-1}>0$. Hence, the continuity of $\mathcal{D}(z)$ and
the univalence of $u$ imply that $\mathcal{D}(z)>0$ in
$\mathbb{D}\backslash(-1,1)$ and $\mathcal{D}(z)\geq 0$ in $(-1,1)$.

Let

$\displaystyle I_{k}:=\int^{\pi}_{0}\mathcal{D}(re^{i\theta})\sin\theta\sin
k\theta d\theta,\quad k=2,3,4....$

On the one hand, by the orthogonality of $\\{\sin k\theta\\}$ on $[0,\pi]$,

$\displaystyle I_{k}$
$\displaystyle=\int^{\pi}_{0}\left((1-c_{-1}P_{\alpha,1}+\sum_{n=2}^{\infty}(c_{n}-c_{-n}P_{\alpha,n})r^{n-1}\frac{\sin
n\theta}{\sin\theta}\right)\sin\theta\sin k\theta d\theta$
$\displaystyle=(c_{k}-c_{-k}P_{\alpha,k})r^{k-1}\int^{\pi}_{0}\sin^{2}k\theta
d\theta.$ (2.13)

On the other hand

$\displaystyle I_{k}=\int^{\pi}_{0}\mathcal{D}(re^{i\theta})\sin\theta\sin
k\theta
d\theta=\int^{\pi}_{0}\mathcal{D}(re^{i\theta})\sin^{2}\theta\frac{\sin
k\theta}{\sin\theta}d\theta.$

If $m_{k}$ and $M_{k}$ denote the minimum and maximum values of $\sin
k\theta/\sin\theta$ in $[0,\pi]$, respectively, then

$\displaystyle m_{k}\int^{\pi}_{0}\mathcal{D}(re^{i\theta})\sin^{2}\theta
d\theta\leq I_{k}\leq
M_{k}\int^{\pi}_{0}\mathcal{D}(re^{i\theta})\sin^{2}\theta d\theta.$ (2.14)

It follows from (2) and (2.14) that

$\displaystyle m_{k}\int^{\pi}_{0}\mathcal{D}(re^{i\theta})\sin^{2}\theta
d\theta\leq(c_{k}-c_{-k}P_{\alpha,k})r^{k-1}\int^{\pi}_{0}\sin^{2}k\theta
d\theta\leq M_{k}\int^{\pi}_{0}\mathcal{D}(re^{i\theta})\sin^{2}\theta
d\theta.$

Using the series expansion (2) for $\mathcal{D}(re^{i\theta})$ and the
orthogonality of $\\{\sin k\theta\\}$, we obtain

$\displaystyle m_{k}(1-c_{-1}P_{\alpha,1})\int^{\pi}_{0}\sin^{2}\theta
d\theta\leq(c_{k}-c_{-k}P_{\alpha,k})r^{k-1}\int^{\pi}_{0}\sin^{2}k\theta
d\theta\leq M_{k}(1-c_{-1}P_{\alpha,1})\int^{\pi}_{0}\sin^{2}\theta d\theta.$

Now we divide it into two cases to discuss.

Case (1) $\alpha\geq 0$. Noting the sense-preservity implies that
$|c_{-1}|<1$. Then by Lemma 2.1 (1), we have that $1-c_{-1}P_{\alpha,1}>0$.
Since $\int^{\pi}_{0}\sin^{2}\theta d\theta=\int^{\pi}_{0}\sin^{2}k\theta
d\theta=\pi/2$, we have

$\displaystyle
m_{k}\leq\frac{c_{k}-c_{-k}P_{\alpha,k}}{1-c_{-1}P_{\alpha,1}}r^{k-1}\leq
M_{k}.$

Observe that

$\displaystyle\lim_{\theta\rightarrow 0^{+}}\frac{\sin
k\theta}{\sin\theta}=k,\quad\lim_{\theta\rightarrow\pi^{-}}\frac{\sin
k\theta}{\sin\theta}=(-1)^{k+1}k.$

We know that $M_{k}=k$, and $m_{k}\geq-k$. But $\min(\sin
k\theta/\sin\theta)=-k$ only if $k$ is even. If $k$ is odd, then $m_{k}>-k$.
Thus,

$\displaystyle-k\leq\frac{c_{k}-c_{-k}P_{\alpha,k}}{1-c_{-1}P_{\alpha,1}}r^{k-1}\leq
k$

for all $r\in(0,1)$. Taking $r=1^{-}$, we have

$\displaystyle\left|\frac{c_{k}-c_{-k}B(k,\alpha+1)}{1-c_{-1}B(1,\alpha+1)}\right|\leq
k.$

Case (2) $\alpha\in(-1,0)$ and $|c_{-1}|<1+\alpha$. Then Lemma 2.1 (2) still
implies that $1-c_{-1}P_{\alpha,1}>0$. The rest of the proof is same as that
of Case (1). We omit it. ∎

Data availability statement: Data sharing not applicable to this article as no
datasets were generated or analysed during the current study.

Competing interests declaration: The authors declare that he have no known
competing financial interests or personal relationships that could have
appeared to influence the work reported in this paper.

## References

  * [1] Z. Abdulhadi and Y. Abu Muhanna.  Landau’s theorem for biharmonic mappings.  J. Math. Anal. Appl., 338(1):705–709, 2008.
  * [2] K. F. Amozova, E. G. Ganenkova, and S. Ponnusamy.  Criteria of univalence and fully $\alpha$-accessibility for $p$-harmonic and $p$-analytic functions.  Complex Var. Elliptic Equ., 62(8):1165–1183, 2017.
  * [3] G. E. Andrews, R. Askey, and R. Roy.  Special functions, volume 71 of Encyclopedia of Mathematics and its Applications.  Cambridge University Press, Cambridge, 1999.
  * [4] D. Bshouty, W. Hengartner, and O. Hossian.  Harmonic typically real mappings.  Math. Proc. Cambridge Philos. Soc., 119(4):673–680, 1996.
  * [5] M. Carlsson and J. Wittsten.  The Dirichlet problem for standard weighted Laplacians in the upper half plane.  J. Math. Anal. Appl., 436(2):868–889, 2016.
  * [6] J. Chen, A. Rasila, and X. Wang.  Coefficient estimates and radii problems for certain classes of polyharmonic mappings.  Complex Var. Elliptic Equ., 60(3):354–371, 2015.
  * [7] S. Chen, S. Ponnusamy, and X. Wang.  Bloch constant and Landau’s theorem for planar $p$-harmonic mappings.  J. Math. Anal. Appl., 373(1):102–110, 2011.
  * [8] S. Chen and M. Vuorinen.  Some properties of a class of elliptic partial differential operators.  J. Math. Anal. Appl., 431(2):1124–1137, 2015.
  * [9] X. Chen and D. Kalaj.  A representation theorem for standard weighted harmonic mappings with an integer exponent and its applications.  J. Math. Anal. Appl., 444(2):1233–1241, 2016.
  * [10] M. Dorff, M. Nowak, and W. Szapiel.  Typically real harmonic functions.  Rocky Mountain J. Math., 42(2):567–581, 2012.
  * [11] P. Duren.  Harmonic mappings in the plane, volume 156 of Cambridge Tracts in Mathematics.  Cambridge University Press, Cambridge, 2004.
  * [12] P. Duren, W. Hengartner, and R. S. Laugesen.  The argument principle for harmonic functions.  Amer. Math. Monthly, 103(5):411–415, 1996.
  * [13] P. Li and S. Ponnusamy.  Lipschitz continuity of quasiconformal mappings and of the solutions to second order elliptic PDE with respect to the distance ratio metric.  Complex Anal. Oper. Theory, 12(8):1991–2001, 2018.
  * [14] P. Li, S. Ponnusamy, and X. Wang.  Some properties of planar $p$-harmonic and log-$p$-harmonic mappings.  Bull. Malays. Math. Sci. Soc. (2), 36(3):595–609, 2013.
  * [15] P. Li and X. Wang.  Lipschitz continuity of $\alpha$-harmonic functions.  Hokkaido Math. J., 48(1):85–97, 2019.
  * [16] P. Li, X. Wang, and Q. Xiao.  Several properties of $\alpha$-harmonic functions in the unit disk.  Monatsh. Math., 184(4):627–640, 2017.
  * [17] M. Liu.  Landau’s theorem for planar harmonic mappings.  Comput. Math. Appl., 57(7):1142–1146, 2009.
  * [18] B. Long and Q. Wang.  Some geometric properties of complex-valued kernel $\alpha$-harmonic mappings.  Bull. Malays. Math. Sci. Soc., 44(4):2381–2399, 2021.
  * [19] B. Long and Q. Wang.  Starlikeness, convexity and Landau type theorem of the real kernel $\alpha$-harmonic mappings.  Filomat, 35(8):2629–2644, 2021.
  * [20] A. Olofsson.  Differential operators for a scale of Poisson type kernels in the unit disc.  J. Anal. Math., 123:227–249, 2014.
  * [21] A. Olofsson.  Lipschitz continuity for weighted harmonic functions in the unit disc.  Complex Var. Elliptic Equ., 65(10):1630–1660, 2020.
  * [22] A. Olofsson and J. Wittsten.  Poisson integrals for standard weighted Laplacians in the unit disc.  J. Math. Soc. Japan, 65(2):447–486, 2013.
  * [23] M. Pavlović.  Decompositions of $L^{p}$ and Hardy spaces of polyharmonic functions.  J. Math. Anal. Appl., 216(2):499–509, 1997.
  * [24] J. Qiao.  Univalent harmonic and biharmonic mappings with integer coefficients in complex quadratic fields.  Bull. Malays. Math. Sci. Soc., 39(4):1637–1646, 2016.
  * [25] X. Wang, X. Liang, and Y. Zhang.  On harmonic typically real mappings.  J. Math. Anal. Appl., 277(2):533–554, 2003.