:
193 1 2022 Machine Learning for Health (ML4H) 2022

# Neurodevelopmental Phenotype Prediction:A State-of-the-Art Deep Learning
Model

Dániel Unyi<EMAIL_ADDRESS>
Bálint Gyires-Tóth<EMAIL_ADDRESS>
Department of Telecommunications and Media Informatics
Budapest University of Technology and Economics
Műegyetem rkp. 3  H-1111 Budapest  Hungary

###### Abstract

A major challenge in medical image analysis is the automated detection of
biomarkers from neuroimaging data. Traditional approaches, often based on
image registration, are limited in capturing the high variability of cortical
organisation across individuals. Deep learning methods have been shown to be
successful in overcoming this difficulty, and some of them have even
outperformed medical professionals on certain datasets. In this paper, we
apply a deep neural network to analyse the cortical surface data of neonates,
derived from the publicly available Developing Human Connectome Project
(dHCP). Our goal is to identify neurodevelopmental biomarkers and to predict
gestational age at birth based on these biomarkers. Using scans of preterm
neonates acquired around the term-equivalent age, we were able to investigate
the impact of preterm birth on cortical growth and maturation during late
gestation. Besides reaching state-of-the-art prediction accuracy, the proposed
model has much fewer parameters than the baselines, and its error stays low on
both unregistered and registered cortical surfaces.

###### keywords:

Biomarker Detection, Neuroimaging, Deep Neural Network, Cortical Surface,
Developing Human Connectome Project

##  1 Introduction

The goal of neuroimaging is collecting visual data of the nervous system, and
analysing this data in order to better understand its structure and function.
High-resolution images are primarily acquired through magnetic resonance
imaging (MRI), which allows non-invasive and non-radioactive measurements on
the patient. Large, publicly available MRI datasets and the corresponding
processing pipelines (Elam et al., 2021; Glasser et al., 2013) enable the
research community to work on novel machine learning models, aimed to improve
our understanding of pathological conditions, and to assist medical experts in
diagnosis, prognosis and therapeutic suggestions.

Deep learning models have two crucial advantages over other machine learning
models: 1) they can automatically identify the most important biomarkers,
automating the meticulous work known as feature selection 2) they can exploit
geometric priors, like the spatial correlation of 2D/3D MRI images or
surfaces. With state-of-the-art accuracy, novel deep learning methods provide
valuable contributions to the field of neuroimaging, for instance the
detection of neurodegenerative diseases (Farooq et al., 2017; Barbaroux et
al., 2020), segmentation of brain tumors (Pereira et al., 2016), or
parcellation of the cerebral cortex (Fawaz et al., 2021). Some of the methods
are even able to outperform medical professionals on certain datasets (Liu et
al., 2019).

The cerebral cortex is a thin, highly convoluted neural tissue, constituting
the outer layer of the brain. Billions of neurons live within its folds and
grooves, facilitating complex cognitive processes which ultimately make us
humans and ourselves. Due to its vital role and complexity, it is intensively
studied by the machine learning research community. The most common approaches
for processing cortical MRI images are convolutional neural networks (He et
al., 2016), and more recently, vision transformers (Dosovitskiy et al., 2020).
Other approaches involve the extension of convolutions to manifolds (Bronstein
et al., 2021), based on the reconstruction of the cortical surface as a
polygon mesh.

In this paper, we extend a previous line of work, organised around the
Developing Human Connectome Project (dHCP). We apply a deep neural network to
identify neurodevelopmental biomarkers, and based on these biomarkers, to
predict scan age and birth age of preterm and term neonates. During late
gestation, rapid cortical growth and maturation occurs. Since preterm birth
interrupts essential cellular and molecular processes, it increases the risk
of developing neuropsychiatric conditions in later life. The main
contributions of this work are the following:

  * •

Our deep learning model achieves state-of-the-art accuracy in scan age
prediction and birth age prediction.

  * •

Using SHAP values, we investigate which myelinated cortical areas has the most
significant effect onto the model’s final predictions.

  * •

We describe the slight modifications we made to our model, to win an
international machine learning challenge in this topic.

##  2 Related Work

The cerebral cortex is the highly convoluted outer layer of the brain, often
reconstructed from structural MRI scans as a polygon mesh. The deep learning
based analysis of polygon meshes requires the generalisation of convolution
from Euclidean grids to manifolds, which led to the development of geometric
deep learning (gDL) and intrinsic mesh CNNs (Bronstein et al., 2021).

Many previous works applied gDL techniques directly on the mesh-based
representation of the cortical surface. Cucurull et al. (2018) framed cerebral
cortex parcellation as a mesh segmentation task. By utilizing graph neural
networks (Defferrard et al., 2016; Veličković et al., 2017), they segmented
the Broca area of 100 participants from the Human Connectome Project, and
significantly improved the Dice score over algorithmic baselines and
multilayer perceptrons (which processed the cortical features independently,
just as we did in this paper). A similar approach was taken by Gopinath et al.
(2019) to parcellation: the aligned spectral embedding of the cortical meshes
were concatenated to the other cortical features, before feeding them into a
graph neural network. They segmented the whole cortical surface of 101
subjects from the Mindboggle dataset, and achieved better Dice score than the
FreeSurfer algorithm (Fischl et al., 2004), one of the most popular automatic
parcellation tools.

Others prefer to inflate the cortical meshes and project them onto a sphere,
which is commonly referred to as ”sphericalisation” (Makropoulos et al.,
2018). There are various gDL approaches explicitly designed for spherical
manifolds (Cohen et al., 2018; Defferrard et al., 2020; Zhao et al., 2019).
According to Barbaroux et al. (2020), spherical CNNs show superior performance
to traditional CNNs in classifying Alzheimer’s disease versus cognitively
normal, on 589 patients from the ADNI-1 cohort. Zhao et al. (2019) proposed
Spherical U-Net, by replacing all operations in the standard U-Net
(Ronneberger et al., 2015) with equivalent spherical operations. They
demonstrated the capability of their model not only in parcellation, but in
predicting the cortical feature development of 370 infants in their first year
after birth.

Recently, a large body of work was organised around the Developing Human
Connectome Project (dHCP) - a publicly available dataset of cortical surfaces,
acquired from preterm and term neonates. Fawaz et al. (2021) benchmarked a
collection of gDL methods, comparing them with each other and ResNet (He et
al., 2016) on sphericalised cortical surfaces. They aimed to predict the
postmenstrual age (PMA) at scan and gestational age (GA) at birth. On
registered surfaces, the encoder of Spherical U-Net (Zhao et al., 2019)
performed the best, but this performance drastically dropped on unregistered
surfaces. On unregistered surfaces, the rotationally equivariant Spherical CNN
(Cohen et al., 2018) and MoNet (Monti et al., 2017) were the best performing
models. The authors briefly investigated whether sphericalisation holds any
disadvantage, but the corresponding results were not conclusive. In the
follow-up work by Dahan et al. (2022), they adapted vision transformers to
sphericalised cortical surfaces (SiTs), representing data as a sequence of
triangular patches. SiTs scored similarly or better than gDL methods in both
prediction tasks, and could gain benefit from pretraining on ImageNet or on
the same dataset using self-supervised learning (e.g. masked patch
prediction). To encourage the endeavours of developing novel methods regarding
GA regression at birth, the authors organised a MICCAI 2022 Challenge with
deadline of 8th July, 2022, leading up to our current paper.

Table 1: Dataset description. Dataset | Total | Maturity | Sex
---|---|---|---
Preterm | Term | Male | Female
PMA at scan | 530 | 111 | 419 | 288 | 242
GA at birth | 514 | 95 | 419 | 283 | 231

##  3 Methods

###  3.1 Data

Our initial dataset was derived from the publicly available third release of
the Developing Human Connectome Project (dHCP). The reconstructed T1 and
T2-weighted magnetic resonance images (Kuklisova-Murgasova et al., 2012;
Hughes et al., 2017; Cordero-Grande et al., 2018) were processed according to
the dHCP structural pipeline111https://github.com/BioMedIA/dhcp-structural-
pipeline (Makropoulos et al., 2018), which performed tissue segmentation
(Makropoulos et al., 2012, 2014, 2016), surface extraction (Schuh et al.,
2017), and inflation and projection to a sphere. This process also generates
cortical feature descriptors, including three descriptors of cortical geometry
(sulcal depth, curvature, thickness) and the T1w/T2w ratio, highly correlated
to cortical myelination (Glasser et al., 2013).

The dataset contains the cortical surface of 588 neonatal subjects, with
accompanying metadata such as gestational age (GA) at birth, postmenstrual age
(PMA) at scan, sex, birthweight, head circumference, and radiology score.
Table 1 and Figure 1 show insight into the dataset for both tasks. Some of the
preterm neonates were scanned twice: once around birth and once at term-
equivalent age. Since our goal is to predict scan age as a marker of healthy
development, and to explore the impact of preterm birth on term-equivalent
development, we excluded the later preterm scans from scan age regression and
the earlier preterm scans from birth age regression (also important for the
direct comparison with previous methods).

We preprocessed the initial dataset as follows. For experiments in native
space (i.e. unregistered data), we used each subject’s native sphere, while in
template space (i.e. registered data), we used the transform from each
subject’s native surface to the dHCP symmetric 40-week surface template. The 4
cortical features (sulcal depth, curvature, thickness, and T1w/T2w ratio) were
resampled from the native/template sphere to the 6-th order icosphere with
barycentric interpolation, implemented by the Human Connectome Project
workbench software (Marcus et al., 2011). This pipeline results 40962 equally
spaced vertices per subject, with 4 cortical features per vertex. Right
hemispheres were mirrored onto the sagittal plane, such that their orientation
matched with left hemispheres. We computed the mean and standard deviation of
the 4 features in the train set, and standardized the input data every time
before training or inference.

###  3.2 Model

We used a multilayer perceptron (MLP) in each of our experiments. The
architecture is shown on Fig. 2. We applied the following layer sequence four
times: $\\{Linear()\rightarrow Tanh()\rightarrow BatchNorm()\\}$. After that,
we performed mean pooling along the vertex dimension, and added a regression
head $Linear()\rightarrow Tanh()\rightarrow Linear()$ to produce the final
predictions. Each hidden layer had 16 units, resulting fewer than 1400
trainable parameters. We had 4 input features in scan age prediction, but 5 in
birth age prediction, since in this case we also incorporated scan age as a
confound. The model was trained using mean squared error (MSE) as loss
function, and Adam as optimizer with learning rate 0.001. Batch size was set
to 32. Training was halted when the validation loss had not decreased in the
last 200 epochs, and the model with the lowest validation loss was restored.

##  4 Results and Discussion

###  4.1 State-of-the-Art Results

Figure 1: Distribution of the scan age and birth age datasets. Figure 2: Our
proposed architecture.

For each of the four tasks (scan age prediction and birth age prediction in
both native and template space), we trained an MLP model four times with
different initial weights, and monitored the mean absolute error (MAE) in
weeks222Code to reproduce the results is publicly available at
https://github.com/daniel-unyi-42/Neurodevelopmental-Phenotype-Prediction. For
direct comparison with previous models (Fawaz et al., 2021; Dahan et al.,
2022), we report the best MAE score achieved on the test set, alongside the
standard deviation across all four trainings, in Table 2. We compared our MLP
to the following geometric deep learning methods: S2CNN (Cohen et al., 2018),
ChebNet (Defferrard et al., 2016), GConvNet (Kipf and Welling, 2016),
Spherical U-Net (Zhao et al., 2019), MoNet (Monti et al., 2017), and the
following surface vision transformers: SiT-0 was not pretrained, SiT-1 was
pretrained on ImageNet, and SiT-2 was pretrained via masked patch prediction
on the dHCP dataset (Dahan et al., 2022). Overall, our MLP model consistently
outperformed the other models in native space, with MAE of 0.54 weeks in scan
age prediction and 1.08 weeks in birth age prediction. It was also competitive
compared to the best performing models in template space, with the exception
of Spherical U-Net in birth age prediction, which still leads the race by a
large margin. Incorporating scan age as a confound in birth age prediction
yielded lower error, in agreement with the findings of Dahan et al. (2022). We
also performed 10-fold cross validation, and report the mean and standard
deviation across the 10 different test folds in Table 2. We think this is a
fairer way to evaluate and compare future models trained on this dataset.

It is important to note that our model has less than 1400 trainable
parameters, a fraction of the several million parameters in previous models.
We attributed the success of our lightweight MLP model to two reasons: 1) it
can generalise better on unseen data 2) it was easier to optimise its
hyperparameters (for instance, it could not train as well with asymmetric
activation functions as with tanh). However, it is possible that message
passing models inherently overfit this dataset: changing the first 4 linear
layers to the convolutional layers of ChebNet, GConv-Net, or MoNet results an
immediate drop in performance, regardless of the small number of parameters.
Using an NVIDIA TITAN V GPU, our model requires only an average of 834 ms
training time per epoch and 76 ms inference time.

Table 2: Best Mean Absolute Error and standard deviation across 4 runs are reported (in weeks). Our MLP model achieves better results than surface vision transformers (Dahan et al., 2022) and geometric deep learning methods (Fawaz et al., 2021), with only a fraction of trainable parameters. We also report the result of 10-fold cross validation in the last row. Method | Scan age | Birth age
---|---|---
Template | Native | Avg | Template | Native | Avg
S2CNN | 0.63±0.02 | 0.73±0.25 | 0.68 | 1.35±0.68 | 1.52±0.60 | 1.44
ChebNet | 0.59±0.37 | 0.77±0.49 | 0.68 | 1.57±0.15 | 1.70±0.36 | 1.64
GConvNet | 0.75±0.13 | 0.75±0.26 | 0.75 | 1.77±0.26 | 2.30±0.74 | 2.04
Spherical UNet | 0.57±0.18 | 0.87±0.50 | 0.72 | 0.85±0.17 | 2.16±0.57 | 1.51
MoNet | 0.57±0.02 | 0.61±0.05 | 0.59 | 1.44±0.08 | 1.58±0.06 | 1.51
SiT-0 | 0.60±0.02 | 0.76±0.03 | 0.68 | 1.14±0.12 | 1.44±0.03 | 1.29
SiT-1 | 0.59±0.03 | 0.71±0.02 | 0.65 | 1.15±0.05 | 1.69±0.03 | 1.42
SiT-2 | 0.55±0.04 | 0.63±0.06 | 0.59 | 1.13±0.02 | 1.47±0.08 | 1.30
MLP(ours) | 0.50±0.05 | 0.54±0.01 | 0.52 | 1.08±0.05 | 1.08±0.17 | 1.08
MLP-cv(ours) | 0.55±0.05 | 0.55±0.03 | 0.55 | 1.00±0.12 | 1.12±0.22 | 1.06
Figure 3: a) and b) show the average cortical myelination of preterm and term
neonates. The increase of myelin levels can be seen in the visual (red box),
auditory (green box), and somatosensory (blue box) areas. c) and d) highlight
the SHAP values in scan age and birth prediction, with red and blue colours
indicating that the given area is important in the model’s prediction making.

###  4.2 Identifying Neurodevelopmental Biomarkers

Since our model was successful at detecting biomarkers, it is worth to
investigate how it concluded the predicted values of scan age and birth age.
For this purpose, we used the DeepExplainer method (Shrikumar et al., 2017) of
the SHapley Additive ExPlanations (SHAP)
library333https://github.com/slundberg/shap (Lundberg and Lee, 2017). SHAP
assigns a so-called SHAP value to each feature, indicating its importance in
the model’s prediction making.

We averaged the T1w/T2w ratio of preterm (GA<37 weeks) and term (GA>37 weeks)
neonates separately over the validation and test sets, to visualise the
dependence of cortical myelination on maturity. Figure 3 highlights the
difference of myelin levels in the visual, auditory and somatosensory areas
between the two gestational periods. SHAP values indicate that the latter two
areas were indeed important indicators in scan age prediction, marking the
process of healthy development. In birth age prediction, other parts of the
cortex become more important, including the superior frontal gyrus, the
praecuneus, or the temporal pole. These areas are responsible for high-level
cognitive functions, like certain aspects of consciousness, memory, or
visuospatial processing, and are likely impacted by preterm birth.

###  4.3 SLCN: A MICCAI 2022 Challenge

Our solution ranked first on the MICCAI 2022 Challenge called Surface Learning
for Clinical Neuroimaging. The goal of this challenge was to elicit
submissions of explainable machine learning methods, which are able to
identify neurodevelopmental biomarkers from cortical surface data. Thereby,
submissions were evaluated on the task of birth age regression. The same train
and validation sets (acquired from the dHCP) were provided to the participants
that we used throughout this paper, but the test set was kept hidden during
the challenge. Metadata was also provided for subjects in the train and
validation sets: the PMA at scan, sex, birthweight, and head circumference.
Since the PMA at scan was not available for test set subjects, we had to work
out a less straightforward deconfounding strategy. The key observation was
that predicting the confound and incorporating it into the loss function
consistently decreased the error on the test set. Predicting the birthweight
variable and incorporating it into the loss also yielded better results: the
standard deviation decreased among validation errors (i.e. evaluations of the
same model with different initial weights). This is understandable, since it
is strongly correlated with our target variable. In conclusion, the only
modification we made to our MLP model was adding 3 output units instead of 1.
These 3 output units correspond to the GA at birth, the PMA at scan, and the
birthweight of the subject; and since we incorporated all 3 variables into the
loss function, we required the MLP model to predict all of them as accurately
as possible. Our best test MAE was 1.226 in template space and 1.386 in native
space.

##  5 Conclusion

In this work, we applied a deep neural network to predict neurodevelopmental
phenotypes, based on the cortical surface data of preterm and term neonates.
We achieved state-of-the-art results compared to previous baselines,
demonstrating that models with fewer priors and fewer parameters can be more
effective on certain datasets. We also identified some myelinated cortical
areas that are biomarkers of healthy development, and others that are possible
biomarkers of preterm birth.

Regarding future research, we plan to dissipate the debate around whether or
not sphericalisation leads to the loss of useful information (as we mentioned,
results in Fawaz et al. (2021) are not conclusive). We are going to apply
intrinsic mesh CNNs on the high-resolution, native cortical meshes, and
identify the cortical areas whose geometry may affect the predictions. We also
plan to explore further metadata like biological sex, utilizing it both as
confound and as target variable.

The work reported in this paper, carried out at BME, has been supported by the
European Union project RRF-2.3.1-21-2022-00004 within the framework of the
Artificial Intelligence National Laboratory.

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# ACE: Zero-Shot Image to Image Translation via Pretrained Auto-Contrastive-
Encoder

Sihan XU∗
University of Michigan
Ann Arbor  Zelong Jiang∗
University of Michigan
Ann Arbor  Ruisi Liu∗
University of Illinois
Urbana-Champaign  Kaikai Yang
YanShan University
Qinhuangdao, China  Zhijie Huang
ShanghaiTech University
Shanghai, China

###### Abstract

Image-to-image translation is a fundamental task in computer vision. It
transforms images from one domain to images in another domain so that they
have particular domain-specific characteristics. Most prior works train a
generative model to learn the mapping from a source domain to a target domain.
However, learning such mapping between domains is challenging because data
from different domains can be highly unbalanced in terms of both quality and
quantity. To address this problem, we propose a new approach to extract image
features by learning the similarities and differences of samples within the
same data distribution via a novel contrastive learning framework, which we
call Auto-Contrastive-Encoder (ACE). ACE learns the content code as the
similarity between samples with the same content information and different
style perturbations. The design of ACE enables us to achieve zero-shot image-
to-image translation with no training on image translation tasks for the first
time.

Moreover, our learning method can learn the style features of images on
different domains effectively. Consequently, our model achieves competitive
results on multimodal image translation tasks with zero-shot learning as well.
Additionally, we demonstrate the potential of our method in transfer learning.
With fine-tuning, the quality of translated images improves in unseen domains.
Even though we use contrastive learning, all of our training can be performed
on a single GPU with the batch size of 8. Our code is available at
github.com/SihanXU/ACE.

##  1 Introduction

(a) training

(b) inference

Figure 1: The Main Idea of Auto-Contrastive-Encoder. When training the ACE
model, the encoder learns how to get the similarity and difference from the
same domain by contrastive learning. And the decoder uses the similarity
features as style and the difference as content to reconstruct the image in
both training and inference stage.

(a) supervised

(b) unsupervised

(c) few-shot/truly unsupervised

(d) zero-shot (ours)

Figure 2: Different kinds of image to image translation. (a) Supervised image
to image translation like pix2pix [20] requires paired datas and (b)
unsupervised image to image translation like cycleGAN [40] requires paired
dataset for training. (c) Few-shot [24] or Truly unsupervised [1] image to
image translation can only translate images in the training domain. Our
method, (d) zero-shot image translation is capable to translate images from
unseen domains to both of training domain and unseen domains.

In the field of computer vision, image-to-image translation has been well-
established and achieved promising results on various related tasks such as
image colorization, style transfer. These existing works are usually achieved
by learning the mapping between the source and target domain [20, 40, 23, 18].
Some new training methods have emerged afterwards [7, 30, 2, 24, 1, 31], but
they are still trained for learning the mapping relationship, which keeps them
from focusing on the distribution of samples. However, such works are
inherently fastidious in data distributions. For example, pix2pix requires the
data of two domains to come in pairs [20], while cycleGAN-like methods rely on
the joint distribution of two domains[40, 23, 18]. Some new methods, such as
[24, 7, 1], still have strict requirements for data. Meanwhile, one-shot image
translation (OST) [2] has only achieved limited breakthroughs with the same
idea of learning the mapping. On the other hand, most other works have focused
more on how to improve the quality of generated images [30, 31]. As a
consequence, they still fail to move the eyes off the mapping relations to
other approaches. Conversely, we believe that we can solve the problem of
strict data requirements if there is a training method that can achieve the
image translation task without learning the mapping between different
distributions.

In this paper, we propose a novel learning task (Section 3) that implements
image translation by learning similar and different features of images within
a data distribution (Fig. 1). We note that the features that are similar
before and after image translation are precisely the features that need to be
retained, while the features that are different under the same distribution
are the features that need to be translated. In this way, our model can
recognize the features to be retained or transformed by learning the
similarities and differences within the distribution. Moreover, such a
training method without learning the mapping relation can also translate the
samples in unseen domains without being trained on image translation tasks,
thereby achieving zero-shot learning as Fig. 1(b), Fig. 2(d) . Despite several
previous works claiming to attain zero-shot image translation [6, 22], they
merely perform style transformation within the features in the specific
domain.

Based on these ideas, we propose the Auto-Contrastive-Encoder (ACE), an Auto-
Encoder structure that incorporates contrastive learning (Section 3).
Contrastive learning provides us with the effectiveness of learning
similarities between positive samples and differences between negative
samples, which encourages our model to learn the similar and different
features in the same distribution. In this paper, we use a structure similar
to Simple Siamese Representation Learning (SimSiam) [5] to keep the model
simple while allowing the model to be trained with a small batch size (Section
4). For contrastive learning, we propose Adaptive Instance Augmentation and
perform contrastive learning directly on the encoded image features. Proven by
experiments, our method is able to capture the similarity and difference in
image features effectively.

It is worth noting that ACE is a framework applicable to any model instead of
a specific model. Our model in this article uses the VGG [32] model for the
encoder, while our decoder is a convolutional network (CNN) [29] using ResNet
[16]. We perform zero-shot image translation and achieve satisfactory results
on the Summer$\Leftrightarrow$Winter [18], Orange$\Leftrightarrow$Apple [8]
and Animal Face [24] datasets. Furthermore, our experiment on Animal Face [24]
shows that our method also has the potential for transfer learning by pre-
training on large datasets. The experiments in this paper can all be completed
on a single GPU with batchsize of 8.

##  2 Related Work

Image-to-image translation. Image-to-image translation, a prevalent problem
for computer vision, aims to convert an input image into another output image.
It has been applied to style transfer [21, 11, 12], image denoising [37, 34,
10], and colorization [38, 39, 26]. Many methods for image translation tasks
have been proposed since [20]. However, these methods are largely limited to
learning the mapping between images and rely on the pairs of data in datasets
for training [40, 23, 18, 7, 24, 1, 30], and thus can never realize zero-shot
learning. Although there is similar research [6, 22] about zero-shot learning
methods before, they only do translations in the same domain rather than
implement real image translation.

This paper presents a new method of image translation by learning the
similarities and differences of samples within the same distribution. Our ACE
approach demonstrates the feasibility of this idea. Our method is not only
able to achieve competitive results on zero-shot image-to-image translation
but also applicable to various image translation tasks like multimodal
translation task. Furthermore, our model has the potential for transfer
learning to improve the quality of image translation tasks by pretrain and
fine-tune.

Contrastive learning Contrastive learning is an efficient method for
unsupervised learning. Its key idea is to learn similar features between
positive samples and different features between negative samples[36]. Based on
this idea, several subsequent works with great influence have come into being,
such as [15, 4, 14, 5, 3]. There are some methods like [14, 5, 3] can learn
similar features between positive samples even without negative samples.

We use a similar structure to SimSiam [5] in this paper, but the difference is
that we augment the features of images instead of augmenting the image itself.
We first use adaptive instance augmentation to augment the features of the
images and implement contrastive learning subsequently. Unlike SimSiam [5], we
also include a predictor in the process of using the encoder. Experiments
demonstrate that our method is effective in learning similarities and
difference within a distribution.

##  3 Method

###  3.1 Assumption

(a) In-domain contrast

(b) Cross-domain contrast

Figure 3: Assumption of zero-shot translation. (a) For the examples in the
same domain, they should have same style but different content. The similarity
should be style and the difference should be the content. (b) For the paired
examples before and after translated, they should have same content with
different style. Thus, the difference is their style and the similarity is
their content. Figure 4: Image translation task. In order to realize the image
translation, content code need to be obtained from the pixel space of domain
A. And then the output, which is in the pixel space of domain B, should be
generated from the content. Additionally, style code is required for
multimodal image translation tasks.

Fig. 3 shows the fundamental assumption of our method. Based on the effect of
in-domain contrast and cross-domain contrast, we can model the content
information as the in-domain difference and cross-domain similarity, and the
style information as the in-domain similarity and cross-domain difference.
Then, we can follow the assumption in MUNIT [18] like Fig. 4 that each image
$x_{i}\in\mathcal{X}_{i}$ is generated from a content latent code
$c\in\mathcal{C}$ that is shared by both domains, and a style latent code
$s_{i}\in S_{i}$ that is specific to the individual domain. For each image,
our objective is to find a pair of underlying encoders $E_{s}$ and $E_{c}$ to
disentangle the two latent codes and a generator $G$ to reconstruct images
with these two types of codes. Suppose we have a pair of image
$(x_{1},x_{2})$. We are able to generate a translated image $x_{1\to 2}$ by
applying the encoders and the generator, namely $x_{1\to
2}=G\left(E_{c}(x_{1}),E_{s}(x_{2})\right)$. Note that now $x_{1\to 2}$ is
also a sample in domain $\mathcal{X}_{2}$ with the same content as $x_{1}$.
Then, since the content encoder generates domain-invariant representations,
ideally we have $E_{c}(x_{1})=E_{c}(x_{1\to 2})$. Similarly, considering
$x_{1\to 2}$ and $x_{2}$ are from the same domain, the constraint
$E_{s}(x_{2})=E_{s}(x_{1\to 2})$ should also hold.

###  3.2 Model

(a) training

(b) inference

(c) contrastive learning

Figure 6: Model overview. In training, we use (a) auto-encoder and (c)
contrastive learning to reconstruct images from the training domain. In
inference (b), we use two-stream like architecture to encode content and
style, and AdaIn decoder to reconstruct the image.

Fig. 6 is an overview of our model, which is similar to MUNIT [18]. Our model
consists of the encoder, content encoder, style encoder, and decoder. The
content encoder contains residual blocks [16] and a predictor, forming a
contrastive learning framework. In the process of training (Fig. 6(a)), first,
we obtain the content and style features respectively through content encoder
and style encoder. Finally, the image can be restored by the decoder. In the
fine-tuning and inference process (Fig. 6(b)), the encoder obtains features of
style and content images so that the style and content images can be learned
by the content encoder and style encoder. After these steps, we use the
decoder to restore the desired image.

We use VGG [32] as the encoder of the model as in the previous works [17, 18,
24]. Fig. 6(c) presents the learning process of content code. After acquiring
the features of input images, we augment the image features with the Adaptive
Instance Augmentation. Subsequently, we use a SimSiam similar structure [5] to
implement contrastive learning and obtain the images’ content features. Our
style encoder is a single layer CNN with Adaptive Pooling [25], which is able
to preserve the global feature information of the images. Following MUNIT
[18], we use an MLP to learn the AdaIN parameters from the style codes.

The content encoder includes two parts as shown in Fig. 6(c). The residual
blocks are CNN with skip connection[16] and BatchNorm[19]. And the predictor
is a 2-layer-MLP with a bottleneck, and it has BatchNorm between hidden
layers. We don’t use the BatchNorm at the output layer.

Our decoder is composed of residual networks and a convolutional network [29]
with adaptive instance normalization [17]. As stated in [18], instance
normalization [35] and batch normalization would destroy the style features of
the image. Therefore, we exclude these two types of normalization in our
decoder.

###  3.3 Adaptive Instance Augmentation

According to the experimental results of using the same adaptive instance
normalization (AdaIN) [17], the instance norm would affect the style of
images. Following this, [18] proposes to use MLP to dynamically produce the
parameters for Instance Normalization layers from style codes. Inspired by
these practices, we propose Adaptive Instance Augmentation, where we replace
the parameters of AdaIN with Gaussian noises:

$\displaystyle AdaIN(z,\gamma,\beta)$
$\displaystyle=\gamma\left(\frac{z-\mu(z)}{\sigma(z)}\right)+\beta,$ (1)
$\displaystyle\gamma,\beta$ $\displaystyle\sim\mathcal{N}(0,1)$ (2)

Note that this is an augmentation in the latent space. The procedure to
augment a sample $x$ is

$Aug(x)=G_{\gamma,\beta}(E_{c}(x)),$ (3)

where $G_{\gamma,\beta}$ means that instead of using style encoder, we use the
variables $\gamma$ and $\beta$ for the AdaIN layers. We first map $x$ into its
content code and then use the randomized AdaIN decoder to reconstruct the
augmented sample. This method enables us to modify the style of images while
ensuring the same image content. Based on this feature augmentation, our
contrastive learning method can make the content encoder insensitive to style
features, so that the content feature can be effectively preserved.

###  3.4 Loss Function

Originating from our assumptions, we first design our loss function to capture
the similarity between the content codes with different style features. Let
$c_{1}$ and $c_{2}$ be two latent codes extracted by content encoder and
augmented with Adaptive Instance Augmentation. We define the loss for
contrastive learning as the SimSiam loss from [5]:

$\displaystyle\mathcal{L}_{contrast}(c_{1},c_{2})=\mathcal{C}ontrastiveLoss(p(c_{1}),stopgrad(c_{2}))$
(4)

where $p$ is the predictor layer and $\mathcal{C}ontrastiveLoss$ can be any
distance measurement such as negative consine similarity. The content
consistency is also forced by minimizing the distance between content codes
extracted from each pair of original sample $x$ and reconstructed sample
$x^{\prime}$:

$\mathcal{L}^{c}_{consist}(x)=\|E_{c}(x)-E_{c}(x^{\prime})\|_{1}.$ (5)

Similarly, for style codes we have

$\mathcal{L}^{s}_{consist}(x)=\|E_{s}(x)-E_{s}(x^{\prime})\|_{1}.$ (6)

Next, to train the auto-encoder, we adopt a reconstruction loss and a GAN loss
to ensure that the reconstructed images follow the distribution of target
domain.

$\displaystyle\mathcal{L}_{recon}(x)$ $\displaystyle=\|x-x^{\prime}\|_{1}$ (7)
$\displaystyle\mathcal{L}_{GAN}=\|D(x)$
$\displaystyle-1\|_{2}^{2}+\|D(x^{\prime})\|_{2}^{2}$ (8)

We train our model with the total objective as the weighted sum of all loss
functions mentioned above.

###  3.5 Stop gradient

Since Auto-Encoder updates encoder in the training, it will greatly influence
the effect of contrastive learning. Therefore, when we are training the Auto-
Encoder, we freeze the content encoder. Which means

$x^{\prime}=G(stopgrad(E_{c}(x)),E_{s}(x))$ (9)

###  3.6 Discriminator

We use an approach similar to Generative Adversarial Network (GAN) [13] to
train our ACE to improve the quality of the images. In our experiments, we use
a loss function similar to [27] to make the training more stable; and use
SpectralNorm [28] to enable the model to generate images with higher quality.

##  4 Experiments

###  4.1 Implementation Details

Figure 7: Visualization of content code on two domains. The content code only
contains the information of the location of animal’s eyes, which is the
difference between the in-domain examples. And the content encoder is also
capable to encode the content from the unseen domain.

Our framework is comprised of a VGG encoder, a content encoder, a style
encoder and a decoder. The content encoder consists of four residual blocks
and the style encoder contains a global pooling layer and a fully connected
layer. For the decoder, we have several residual blocks, each followed by up-
sampling layers. We also use Adaptive Instance Normalization layers to
dynamically generate parameters of Instance Normalization. However, to
accelerate the convergence of style encoder, we propose to use two different
style codes to respectively represent the global style in the domain and the
individual style of each sample. The domain style code is a learnable tensor
which is shared by all data in the pretrain domain, while the individual style
code is output by the style encoder. Then we sum these two style codes up
before applying them to the AdaIN layers.

###  4.2 Datasets

We conduct the evaluation on the same datasets as [18, 40]. Our method
achieves satisfying results on Yosemite summer$\Leftrightarrow$winter,
apple$\Leftrightarrow$orange and Animal face translation (including data of
bigcats, cats and dogs).

Figure 8: Translation results on Yosemite summer$\Leftrightarrow$winter and
Apple$\Leftrightarrow$Orange.

Figure 9: Multimodal Translation results on bigcat$\Leftrightarrow$cat and
dog$\Leftrightarrow$cat.

###  4.3 Visualization

Figure 10: Comparison of zero-shot translation results with one-shot[2], few-
shot[24] and unsupervised[18] methods.

To better understand whether our designed models work as we expect, we adopt
some tools of visualization for our extracted content codes. Fig. 7 shows the
information in the content codes from the pretrain domain and the unseen
domain. Both representations indicate the animal’s eyes. It verifies that our
contrastive learning based encoder is able to extract similar content codes
regardless of the domains.

###  4.4 Effectiveness of Zero-shot Learning

Our experiment covers the zero-shot learning on datasets
Summer$\Leftrightarrow$Winter [18] and Orange$\Leftrightarrow$Apple [8]. The
final results are shown in Fig. 9.

At the same time, we conduct multimodal translation on the Animal Image
Translation Dataset [24]. We trained our model on cat dataset and applied to
bigcat2cat task and dog2cat task. With the images in Fig. 9, we can see that
our method translates images to different styles while maintaining the
original content features.

The focus of our work is not about producing high quality images. For this
reason, we only compare our method with OST [2], FUNIT [24], MUNIT [18].
During training, our model shares the same settings as MUNIT, while OST and
FUNIT keep their settings as mentioned in OST and FUNIT. The comparison is
presented in Fig. 10. It is apparent that our framework obtains outstanding
translating results in a zero-shot manner.

###  4.5 Transfer Learning

Figure 11: Fine-tune result. Even though ACE is capable of doing the zero-shot
translation to an unseen target domain, but the translation result contains
some features from the pretrain domain and thus affects the translation
image’s quality. After fine-tuning on the target domain, ACE can achieve
better translation results, which shows the potential of ACE model in transfer
learning.

Here we discuss whether our model is suitable for transfer learning. We pre-
train our model on a specific domain and test on one or two unseen domains. If
we are able to get the training data of the test domains, we can conduct fine-
tuning to improve the generated image quality. The process of fine-tuning is
quite similar to the pre-training. We use the images in the source domain to
generate the content codes and use target domain to get style codes. The fine-
tuning loss will be consisted of the latent consistency losses
$\mathcal{L}^{c}_{consist},\mathcal{L}^{s}_{consist}$ and the GAN loss
$\mathcal{L}_{GAN}$. We don’t use the contrastive loss in fine-tuning.

As shown in Fig. 11, in experiments of multimodal image translation, we pre-
train our model on the cat domain and it achieves satisfactory performance
when applied to the task of dog2cat translation. Next, our model still
performs well if we change the target domain to the bigcats. Therefore, we
believe with pre-training, our model is capable of translating the images from
one unseen domain to another unseen domain. If we continue to fine-tune on the
dataset of bigcats, we can see that there is much room for improvement of
generated image quality. As a result, we believe our model has a great
potential for transfer learning based on large datasets like ImageNet[8].

###  4.6 Failure Cases

Our method fails on some cases where the context is complicated such as Fig
12. For instance, on the dataset of horse2zebra, our model sometimes
erroneously puts the zebra’s stripes on the background.

Figure 12: Failure Cases. The context is so complicated that our model cannot
integrate style and content information well.

##  5 Discussion

The method proposed in this paper has achieved satisfactory results in the
task of image-to-image translation, but we have not conducted experiments on
other types of translation tasks. For example, we believe our method can work
on language processing as well. Furthermore, due to limited resources, we only
tested the potential of our method on transfer learning with small datasets.
If we can pre-train on a larger data set, our model may be able to achieve
better results in image translation. In this article, we just use a very
simple model structure, but our approach is also applicable to other models,
such as ResNet [16], Vision Transformer[9] and Diffusion Model [33]. We
believe that better results can be achieved if our methods are combined with
these further efforts.

##  6 Conclusions

In order to conquer the challenge coming from learning the mapping
relationship in image-to-image translation, we propose a new objective, which
is to translate images by learning the similarities and differences without
learning any mappings or joint distributions. Additionally, we propose a
simple model structure called Auto-Contrastive-Encoder to solve this problem,
and it has achieved satisfactory results. We have also shown the potential of
our model in transfer learning. It is promising that our method can make the
task of image-to-image translation move forward.

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E-mail<EMAIL_ADDRESS>
# Integrals of the difference of subharmonic functions
over discs and planar small sets

B. N. Khabibullin  Bashkir State University, Zaki Validi Str. 32,
Bashkortostan, 420076 Russian Federation

(November 25, 2020)

###### Abstract

The maximum of the modulus of a meromorphic function cannot be restricted from
above by the Nevanlinna characteristic of this meromorphic function. But
integrals from the logarithm of the module of a meromorphic function allow
similar restrictions from above. This is illustrated by one of the important
theorems of Rolf Nevanlinna in the classical monograph by A. A. Goldberg and
I. V. Ostrovskii on meromorphic functions, as well as by the Edrei – Fuchs
Lemma on small arcs and its versions for small intervals in articles by A. F.
Grishin, M. L. Sodin, T. I. Malyutina. Similar results for integrals of
differences of subharmonic functions even with weights were recently obtained
by B.N. Khabiblullin, L.A. Gabdrakhmanova. All these results are on integrals
over subsets on a ray. In this article, we establish such results for
integrals of the logarithm of the modulus of a meromorphic function and the
difference of subharmonic functions over discs and planar small sets. Our
estimates are uniform in the sense that the constants in these estimates are
explicitly written out and do not depend on meromorphic functions and the
difference of subharmonic functions provided that these functions has an
integral normalization near zero.

meromorphic function, Nevanlinna characteristic, subharmonic function,
$\delta$-subharmonic function, Riesz measure

##  1 Introduction. Definitions and notation

Upper estimates of integrals for meromorphic functions in the complex plane
${\mathbb{C}}$ over segments or small sets on a ray via the Nevanlinna
characteristic are considered in works (RNevanlinna, , pp. 24–27), (GOe, ,
Notes, Ch. 1), (GOe, , Ch. 1, Theorem 7.2), (GrS, , Lemma 3.1), (GrM, ,
Theorem 8), (GabKha20, , Theorem 1 (on small intervals), Remark 1.1,
Conclusion of the Grishin – Malyutina Theorem), (KhaMst, , Main Theorem). Our
article KhaMst  contains all these listed results as special cases (see
Theorem 1 below). We give in (GabKha20, , Inrtoduction) and (KhaMst, ,
1.1–1.2) a detailed history of the issue with full formulations of all
previous results. In this paper, we obtain similar upper estimates for
integrals already over discs or small planar sets. As in [1], we establish the
main theorem of the paper in a more general subharmonic version. Let’s move on
to precise definitions.

As usual, $\mathbb{R}$ is the real line, or the real axis of the complex plane
$\mathbb{C}$, and ${\mathbb{R}}^{+}:=\\{r\in{\mathbb{R}}\colon 0\leq r\\}$ is
the positive closed semiaxis. We denote singleton sets by a symbol without
curly brackets. So, ${\mathbb{R}}^{+}\\!\setminus\\!0$ is the positive open
semiaxis, $\overline{\mathbb{R}}$ is the extended real axis. Besides,
$D(z,r):=\\{z^{\prime}\in{\mathbb{C}}\colon|z^{\prime}-z|<r\\}$ is an open
disc, $\overline{D}(z,r):=\\{z^{\prime}\in{\mathbb{C}}\colon|z^{\prime}-z|\leq
r\\}$ is a closed disc,
$\partial\overline{D}(z,r):=\\{z^{\prime}\in{\mathbb{C}}\colon|z^{\prime}-z|=r\\}$
is a circle with center $z\in{\mathbb{C}}$ of radius
$r\in\overline{\mathbb{R}}^{+}$; $D(z,0)=\varnothing$,
$\overline{D}(z,0)=\partial\overline{D}(z,0)=z$, $D(z,+\infty)={\mathbb{C}}$,
$D(r):=D(0,r),\overline{D}(r):=\overline{D}(0,r)$,
$\partial\overline{D}(r):=\partial\overline{D}(0,r)$.

Given a function $f\colon X\to{\overline{\mathbb{R}}}$, $f^{+}:=\sup\\{0,f\\}$
and $f^{-}:=(-f)^{+}$ are positive and negative parts of function $f$,
respectively; $|f|:=f^{+}+f^{-}$.

Given $S\subset{\mathbb{C}}$, ${\sf sbh}(S)$ is the class of all subharmonic
on an open neighbourhood of $S$. We set ${\sf sbh}_{*}(S):=\bigl{\\{}v\in{\sf
sbh}(S)\colon v\not\equiv-\infty\bigr{\\}}$.

By $\uplambda$ we denote the planar Lebesgue measure on ${\mathbb{C}}$. We
also use the notation ${\rm mes}$ for the linear Lebesgue measure on
${\mathbb{R}}$.

For $r\in{\mathbb{R}}^{+}$ and a function
$v\colon\partial\overline{D}(r)\to{\overline{\mathbb{R}}}$, we define

$\displaystyle{\sf M}_{v}(r)$ $\displaystyle:=\sup_{|z|=r}v(z),\quad
r\in{\mathbb{R}}^{+},$ (1M) $\displaystyle{\sf C}_{v}(r)$
$\displaystyle:=\frac{1}{2\pi}\int_{0}^{2\pi}v(re^{i\varphi}){\,{\rm
d}}\varphi,\quad r\in{\mathbb{R}}^{+}\\!\setminus\\!0,$ (1C)

where ${\sf C}_{v}(r)$ is the average over the circle
$\partial\overline{D}(0,r)$, if this integral exists. For a
$\uplambda$-measurable function
$v\colon\overline{D}(r)\to\overline{\mathbb{R}}$, we use also the average over
the disc $\overline{D}(r)$ defined as

${\sf B}_{v}(r):=\frac{1}{\pi r^{2}}\int_{\overline{D}(r)}v{\,{\rm
d}}\uplambda,\quad{\sf B}_{v}(0):={\sf
M}_{v}(0)\overset{\eqref{{Cu}M}}{=}v(0)=:{\sf C}_{v}(0).$ (2)

See (Rans, , 2.6), (HK, , 2.7), (BaiKhaKha17, , 3) on properties of ${\sf
M}_{v}$, ${\sf C}_{v}$, ${\sf B}_{v}$ in the case of a subharmonic function
$v$.

For a Borel subset $S\subset{\mathbb{C}}$, the set of all Borel, or Radon,
positive measures $\mu\geq 0$ on $S$ is denoted by ${\sf Meas\,}^{+}(S)$, and
${\sf Meas\,}(S):={\sf Meas\,}^{+}(S)-{\sf Meas\,}^{+}(S)$ is the set of all
charges, or signed measures, on $S$. For a charge $\nu\in{\sf Meas\,}(S)$, we
denote by

$\nu^{+}\in{\sf Meas\,}^{+}(S),\quad\nu^{-}:=(-\nu)^{+}\in{\sf
Meas\,}^{+}(S),\quad|\nu|:=\nu^{+}+\nu^{-}\in{\sf Meas\,}^{+}(S)$

the upper, lower, total variations of this charge $\nu$, respectively. We set

$\displaystyle\nu(z,r)$
$\displaystyle:=\nu\bigl{(}\overline{D}(z,r)\bigr{)}\in\overline{\mathbb{R}}\quad\text{if
}\overline{D}(z,r)\subset S,\quad 0\leq r\leq R,$ (3z)
$\displaystyle\nu^{{\text{\tiny\rm rad}}}(r)$
$\displaystyle:=\nu(0,r)=\nu\bigl{(}\overline{D}(r)\bigr{)}\in\overline{\mathbb{R}}\quad\text{if
}\overline{D}(r)\subset S,\quad 0\leq r\leq R,$ (3r) $\displaystyle{\sf
N}_{\nu}(r,R)$ $\displaystyle:=\int_{r}^{R}\frac{\nu^{{\text{\tiny\rm
rad}}}(t)}{t}{\,{\rm d}}t\in\overline{\mathbb{R}}^{+}\quad\text{if
}\overline{D}(R)\subset S,\quad 0\leq r\leq R,$ (3N)

provided that the last integral is well defined. For $0\leq r\leq
R\in{\mathbb{R}}^{+}$ and functions

$v\colon\partial\overline{D}(r)\cup\partial\overline{D}(R)\to{\overline{\mathbb{R}}},$

we define

${\sf C}_{v}(r,R)\overset{\eqref{{Cu}C}}{:=}{\sf C}_{v}(R)-{\sf
C}_{v}(r)=\frac{1}{2\pi}\int_{0}^{2\pi}\bigl{(}v(Re^{i\varphi})-v(re^{i\varphi})\bigr{)}{\,{\rm
d}}\varphi$ (4)

provided that ${\sf C}_{v}(R)$ and ${\sf C}_{v}(r)$ are well defined.

If $D\subset{\mathbb{C}}$ is a domain and $u\in{\sf sbh}_{*}(D)$, then there
is its Riesz measure

$\varDelta_{u}:=\frac{1}{2\pi}{\bigtriangleup}u\in{\sf Meas\,}^{+}(D),$ (5)

where ${\bigtriangleup}$ is the Laplace operator acting in the sense of the
theory of distribution or generalized functions. This definition of the Riesz
measures carries over naturally to $u\in{\sf sbh}_{*}(S)$ for connected
subsets $S\subset{\mathbb{C}}$. By the Poisson – Jensen – Privalov formula
Rans , HK , we have

${\sf C}_{v}(r,R)={\sf N}_{\varDelta_{v}}(r,R)\quad\text{for all
$0<r<R<+\infty$ if $v\in{\sf sbh}_{*}\bigl{(}\overline{D}(R)\bigr{)}$}.$ (6)

Let $U=u-v$ be a difference of subharmonic functions $u,v\in{\sf
sbh}_{*}\bigl{(}\overline{D}(0,R)\bigr{)}$, i. e., a $\delta$-subharmonic non-
trivial ($\not\equiv\pm\infty$) function Arsove53 , Arsove53p , Gr ,
(KhaRoz18, , 3.1) on $\overline{D}(R)$ with the Riesz charge

$\varDelta_{U}\overset{\eqref{df:cm}}{:=}\varDelta_{u}-\varDelta_{v}\overset{\eqref{df:cm}}{:=}\frac{1}{2\pi}{\bigtriangleup}u-\frac{1}{2\pi}{\bigtriangleup}v\in{\sf
Meas\,}\bigl{(}\overline{D}(0,R)\bigr{)},\text{ and
}\varDelta^{-}_{U}:=(\varDelta_{U})^{-}$

is the lower variation of the Riesz charge $\varDelta_{U}$ of $U$. Now we can
determine the difference Nevanlinna characteristic ${\sf T}$ of
$\delta$-subharmonic non-trivial ($\not\equiv\pm\infty$) function $U$ as a
function of two variables

${\sf T}_{U}(r,R):={\sf C}_{U^{+}}(r,R)+{\sf N}_{\varDelta_{U}^{-}}(r,R),\quad
0<r\leq R\in{\mathbb{R}}^{+}.$ (7)

A representation $U=u_{U}-v_{U}$ with $u_{U},v_{U}\in{\sf
sbh}_{*}\bigl{(}\overline{D}(0,R)\bigr{)}$ is canonical if the Riesz measure
$\varDelta_{u_{U}}$ of $u_{U}$ is the upper variation $\varDelta_{U}^{+}$ of
$\varDelta_{U}$ and the Riesz measure $\varDelta_{v_{U}}$ of $v_{U}$ is the
lower variation $\varDelta_{U}^{-}:=(\varDelta_{U})^{-}$ of $\varDelta_{U}$.
The canonical representation for $U$ is defined up to the harmonic function
added simultaneously to each of the representing subharmonic functions $u_{U}$
and $v_{U}$, and

${\sf T}_{U}(r,R)\overset{\eqref{CN},\eqref{T0}}{=}{\sf C}_{U^{+}}(r,R)+{\sf
C}_{v_{U}}(r,R)={\sf C}_{\sup\\{u_{U},v_{U}\\}}(r,R),\quad 0<r\leq
R\in{\mathbb{R}}^{+}.$ (8)

By (8), the difference Nevanlinna characteristic $T_{U}$ is already uniquely
defined for all values $0<r\leq R<+\infty$ by positive values in
${\mathbb{R}}^{+}$, and is also increasing and convex with respect to the
logarithmic function $\ln$ in the second variable $R$, but is decreasing in
the first variable $r\leq R$. Recall that the following notation is used for
the meromorphic function $F\not\equiv 0$ on ${\mathbb{C}}$ in the classic
monograph by A. A. Goldberg and I. V. Ostrovskii GOe  for the maximum of
module

$\displaystyle M(r,F)$
$\displaystyle\underset{\text{\tiny$r\in{\mathbb{R}}^{+}$}}{:=}\sup\bigl{\\{}\bigl{|}F(z)\bigr{|}\colon|z|=r\bigr{\\}},$
(9M) and for the Nevanlinna characteristic $\displaystyle T(r,F)$
$\displaystyle\underset{\text{\tiny$r\in{\mathbb{R}}^{+}$}}{:=}m(r,F)+N(r,F),$
(9T) $\displaystyle m(r,F)$
$\displaystyle\underset{\text{\tiny$r\in{\mathbb{R}}^{+}$}}{:=}\frac{1}{2\pi}\int_{0}^{2\pi}\ln^{+}\bigl{|}F(re^{i\varphi})\bigr{|}{\,{\rm
d}}\varphi,$ (9m) $\displaystyle N(r,F)$
$\displaystyle\underset{\text{\tiny$r\in{\mathbb{R}}^{+}$}}{:=}\int_{0}^{r}\frac{n(t,F)-n(0,F)}{t}{\,{\rm
d}}t+n(0,F)\ln r,$ (9N)

where $n(r,F)$ is the number of poles of $F$ in the closed disc
$\overline{D}(r):=\\{z\in{\mathbb{C}}\colon|z|\leq r\\}$, taking into account
the multiplicity. The function $\ln|F|$ is non-trivial $\delta$-subharmonic on
${\mathbb{C}}$, and

$\displaystyle\ln M(r,F)$
$\displaystyle\overset{\eqref{{TN}M},\eqref{{Cu}M}}{=}{\sf
M}_{\ln|F|}(r),\quad r\in{\mathbb{R}}^{+},$ (10M) $\displaystyle m(r,f)$
$\displaystyle\overset{\eqref{{TN}m},\eqref{{Cu}C}}{=}{\sf
C}_{\ln^{+}|F|}(r),\quad r\in{\mathbb{R}}^{+},$ (10m) $\displaystyle
N(R,F)-N(r,F)$ $\displaystyle\overset{\eqref{{TN}N},\eqref{{murad}N}}{=}{\sf
N}_{\varDelta_{\ln|F|}^{-}}(r,R),\quad 0<r<R\in{\mathbb{R}}^{+},$ (10N)
$\displaystyle T(R,F)-T(r,F)$
$\displaystyle\overset{\eqref{{TN}T},\eqref{T0}}{=}{\sf T}_{\ln|F|}(r,R).\quad
0<r<R\in{\mathbb{R}}^{+},$ (10T)

##  2 Recent and new results

Let us formulate the main result from KhaMst  for the ‘‘one-dimensional’’
subset $E\subset[0,r]\subset{\mathbb{R}}^{+}$ in the special case without a
weight function-multiplier $g\in L^{p}(E)$, $1<p\leq+\infty$, in the
integrand.

######  Theorem 1 ((KhaMst, , Main Theorem)).

Let $0<r_{0}<r<+\infty$, $1<k\in{\mathbb{R}}^{+}$, $E\subset[0,r]$ be ${\rm
mes}$-measurable, $g\in L^{p}(E)$, where $1<p\leq\infty$ and $q\in[1,+\infty)$
is defined by $\dfrac{1}{p}+\dfrac{1}{q}=1$. If $U\not\equiv\pm\infty$ is a
non-trivial $\delta$-subharmonic functions on ${\mathbb{C}}$, and
$u\not\equiv-\infty$ is a subharmonic function on ${\mathbb{C}}$, then

$\displaystyle\int_{E}{\sf M}_{U^{+}}(t)g(t){\,{\rm d}}t$
$\displaystyle\leq\frac{4qk}{k-1}\bigl{(}{\sf T}_{U}(r_{0},kr)+{\sf
C}_{U^{+}}(r_{0})\bigr{)}\|g\|_{L^{p}(E)}\sqrt[q]{{\rm
mes}\,E}\ln\frac{4kr}{{\rm mes}\,E},$ (11T) $\displaystyle\int_{E}{\sf
M}_{|u|}(t)g(t){\,{\rm d}}t$ $\displaystyle\leq\frac{5qk}{k-1}\bigl{(}{\sf
M}_{u^{+}}(kr)+{\sf C}_{u^{-}}(r_{0})\bigr{)}\sqrt[q]{{\rm
mes}\,E}\ln\frac{4kr}{{\rm mes}\,E}.$ (11M)

In particular, by (10M)–(10T) we have for meromorphic functions the following

######  Corollary 1.

Let, under the conditions of Theorem 1, $F\not\equiv 0$ be a meromorphic
function, and $f\not\equiv 0$ be an entire function. Then, in the traditional
notation (9M)–(9N), we have

$\displaystyle\int_{E}\bigl{(}\ln^{+}M(t,F)\bigr{)}g(t){\,{\rm d}}t$
$\displaystyle\leq\frac{4qk}{k-1}\bigl{(}T(kr,F)-N(r_{0},F)\bigr{)}\|g\|_{L^{p}(E)}\sqrt[q]{{\rm
mes}\,E}\ln\frac{4kr}{{\rm mes}\,E},$ (11T) $\displaystyle\int_{E}\bigl{|}\ln
M(t,f)\bigr{|}g(t){\,{\rm d}}t$
$\displaystyle\leq\frac{5qk}{k-1}\bigl{(}\ln^{+}M(kr,f)+m(r_{0},1/f)\bigr{)}\sqrt[q]{{\rm
mes}\,E}\ln\frac{4kr}{{\rm mes}\,E}.$ (11M)

###### Proof.

By (10M)–(10T) we have for $U:=\ln|F|$

${\sf T}_{U}(r_{0},kr)+{\sf
C}_{U^{+}}(r_{0})\overset{\eqref{{cs}T},\eqref{{cs}m}}{=}T(kr,F)-T(r_{0},F)+m(r_{0},f)\overset{\eqref{{TN}T}}{=}T(kr,F)-N(r_{0},F).$
(13)

We can replace the bracket on the right-hand side of (11T) with the right side
of these equalities. Then we obtain (11T) by (10M) for the integrand in the
left-hand side of (11T).

For $u:=\ln|f|$, we have

${\sf M}_{u^{+}}(kr)+{\sf
C}_{u^{-}}(r_{0})\overset{\eqref{{cs}M}}{=}\ln^{+}M(kr,f)+{\sf
C}_{\ln^{-}|f|}(r_{0})\overset{\eqref{{cs}m}}{=}\ln^{+}M(kr,f)+m(r_{0},1/f).$
(14)

We can replace the bracket on the right-hand side of (11M) with the right side
of these equalities. Then we obtain (11M) by (10M) for the integrand in the
left-hand side of (11M). ∎

######  Remark 1.

Our elementary example (KhaMst, , 1.1, (3)) shows that it is impossible to
discard the terms $-N(r_{0},F)$ and $m(r_{0},1/f)$ in parentheses on the
right-hand sides of inequality (11T) and (11M), respectively. In particular,
the classical Rolf Nevanlinna Theorem (GOe, , Ch. 1, Theorem 7.2) can be
formulated in the following correct form: for each meromorphic function
$F\not\equiv 0$ and $1<k\in{\mathbb{R}}^{+}$

$\frac{1}{r}\int_{0}^{r}\ln^{+}M(t,F){\,{\rm d}}t\leq\frac{4k\ln
4k}{k-1}\bigl{(}T(kr,F)-N(r_{0},F)\bigr{)}\quad\text{\it for all\/
$0<r_{0}\leq r\in{\mathbb{R}}^{+}$.}$ (15)

Indeed, it is sufficient to choose $E:=[0,r]$, $g\equiv 1$ with ${\rm
mes}\,E=r$, and $p:=\infty$ with $q=1$ in (11T).

Our main result is established for the case of a planar ‘‘two-dimensional’’
subset $E\subset\overline{D}(r)\subset{\mathbb{C}}$:

######  Theorem 2.

Let $0<r_{0}<r<+\infty$, $1<k\in{\mathbb{R}}^{+}$, $E\subset\overline{D}(r)$
be a $\uplambda$-measurable subset. If $U\not\equiv\pm\infty$ is a
$\delta$-subharmonic functions on ${\mathbb{C}}$, and $u\not\equiv-\infty$ is
a subharmonic function on ${\mathbb{C}}$, then

$\displaystyle\int_{E}U^{+}{\,{\rm d}}\uplambda\leq\frac{2k}{k-1}\bigl{(}{\sf
T}_{U}(r_{0},kr)+{\sf
C}_{U^{+}}(r_{0})\bigr{)}\uplambda(E)\ln\frac{100kr^{2}}{\uplambda(E)},$ (16T)
$\displaystyle\int_{E}|u|{\,{\rm d}}\uplambda\leq\frac{3k}{k-1}\bigl{(}{\sf
M}_{u^{+}}(kr)+{\sf
C}_{u^{-}}(r_{0})\bigr{)}\uplambda(E)\ln\frac{100kr^{2}}{\uplambda(E)}.$ (16M)

Theorem 2 is proved at the end of Sec. 3 after some preparation.

We have not seen before these estimates (16T) and (16M) even for the case
$E=\overline{D}(r)$:

######  Corollary 2.

If $U\not\equiv\pm\infty$ be a $\delta$-subharmonic functions on
${\mathbb{C}}$, and $u\not\equiv-\infty$ be a subharmonic function on
${\mathbb{C}}$, then

$\displaystyle{\sf B}_{U^{+}}(r)$
$\displaystyle\leq\frac{7k\ln(ek)}{k-1}\bigl{(}{\sf T}_{U}(r_{0},kr)+{\sf
C}_{U^{+}}(r_{0})\bigr{)}\quad\text{for all\/ $0<r_{0}<r\in\mathbb{R}^{+}$ and
$1<k\in{\mathbb{R}}^{+}$},$ (17T) $\displaystyle{\sf B}_{|u|}(r)$
$\displaystyle\leq\frac{11k\ln(ek)}{k-1}\bigl{(}{\sf M}_{u^{+}}(kr)+{\sf
C}_{u^{-}}(r_{0})\bigr{)}\quad\text{for all\/ $0<r_{0}<r\in\mathbb{R}^{+}$ and
$1<k\in{\mathbb{R}}^{+}$}.$ (17M)

###### Proof.

Let $E:=\overline{D}(r)$ in (16). Then $\uplambda(E)=\pi r^{2}$,

$\ln\frac{100kr^{2}}{\uplambda(E)}=\ln\frac{100k}{\pi}\leq\frac{7}{2}\ln
ek,\quad{\sf
B}_{U^{+}}(r)\overset{\eqref{Bu}}{=}\frac{1}{\uplambda(E)}\int_{E}U^{+}{\,{\rm
d}}\uplambda,\quad{\sf
B}_{|u|}(r)\overset{\eqref{Bu}}{=}\frac{1}{\uplambda(E)}\int_{E}|u|{\,{\rm
d}}\uplambda,$

and by (16) we obtain (17). ∎

For a meromorphic function $F\not\equiv 0$ on ${\mathbb{C}}$, in the frame of
traditional notation (9), we denote the average of $\ln^{+}|F|$ over a disc
$\overline{D}(r)$ as

${m^{[2]}}(r,F)\underset{r\in{\mathbb{R}}^{+}}{:=}\frac{2}{r^{2}}\int_{0}^{r}\left(\frac{1}{2\pi}\int_{0}^{2\pi}\ln^{+}\bigl{|}F(te^{i\varphi})\bigr{|}{\,{\rm
d}}\varphi\right)\,t{\,{\rm
d}}t\overset{\eqref{{Cu}C}}{=}\frac{2}{r^{2}}\int_{0}^{r}m(t,F)\,t\\!{\,{\rm
d}}t\overset{\eqref{Bu}}{=}{\sf B}_{\ln^{+}|F|}(r).$ (18)

######  Corollary 3.

Let, under the conditions of Theorem 2, $F\not\equiv 0$ be a meromorphic
function, and $f\not\equiv 0$ be an entire function. Then, in the traditional
notation (9M)–(9N) and (18), we have

$\displaystyle\int_{E}\ln^{+}\bigl{|}F(z)\bigr{|}{\,{\rm d}}\uplambda(z)$
$\displaystyle\leq\frac{2k}{k-1}\bigl{(}T(kr,F)-N(r_{0},F)\bigr{)}\uplambda(E)\ln\frac{100kr^{2}}{\uplambda(E)},$
(19T) $\displaystyle\int_{E}\Bigl{|}\ln\bigl{|}f(z)\bigr{|}\Bigr{|}{\,{\rm
d}}\uplambda(z)$
$\displaystyle\leq\frac{3k}{k-1}\bigl{(}\ln^{+}M(kr,f)+m(r_{0},1/f)\bigr{)}\ln\frac{100kr^{2}}{\uplambda(E)}.$
(19M) $\displaystyle{m^{[2]}}(r,F)$
$\displaystyle\leq\frac{7k\ln(ek)}{k-1}\bigl{(}T(kr,F)-N(r_{0},F)\bigr{)},$
(19F) $\displaystyle{m^{[2]}}(r,f)+{m^{[2]}}(r,1/f)$
$\displaystyle\leq\frac{11k\ln(ek)}{k-1}\bigl{(}{\sf M}_{u^{+}}(kr)+{\sf
C}_{u^{-}}(r_{0})\bigr{)}$ (19f)

for all $0<r_{0}<r\in\mathbb{R}^{+}$ and $1<k\in{\mathbb{R}}^{+}$.

###### Proof.

For $U:=\ln|F|$, we obtain (19T) by (16T), (13), (10), and also (19F) by
(17T), (13), (18).

For $u:=\ln|f|$ we obtain (19M) by (16M), (14), (10), and also (19f) by (17M),
(14), (18) since ${m^{[2]}}(r,f)+{m^{[2]}}(r,1/f)\overset{\eqref{BF}}{=}{\sf
B}_{|\ln|f||}(r)$ for all $r\in{\mathbb{R}}^{+}$. ∎

##  3 Lemmata and Proof of Theorem 2

######  Lemma 1.

Let $0\leq r<R<+\infty$, $E\subset\overline{D}(r)$ be $\uplambda$-measurable,
$U=u-v$ be a difference of subharmonic functions $u,v\in{\sf
sbh}_{*}\bigl{(}\overline{D}(R)\bigr{)}$, $\varDelta_{v}$ be the Riesz measure
of $v$. Then

$\int_{E}U^{+}{\,{\rm d}}\uplambda\leq\frac{1}{2}\Bigl{(}\frac{R+r}{R-r}{\sf
C}_{U^{+}}(R)+\varDelta_{v}^{{\text{\tiny\rm
rad}}}(R)\Bigr{)}\uplambda(E)\ln\frac{(10R)^{2}}{\uplambda(E)}.$ (20)

###### Proof.

For $w\in E\subset\overline{D}(r)$, by the Poisson – Jensen formula (Rans, ,
4.5), we have

$U(w)=\frac{1}{2\pi}\int_{0}^{2\pi}U(Re^{i\varphi}){\rm
Re}\frac{Re^{i\varphi}+w}{Re^{i\varphi}-w}{\,{\rm
d}}\varphi-\int_{D(R)}\ln\Bigl{|}\frac{R^{2}-z\bar{w}}{R(w-z)}\Bigr{|}{\,{\rm
d}}\varDelta_{u}(z)\\\
+\int_{D(R)}\ln\Bigl{|}\frac{R^{2}-z\bar{w}}{R(w-z)}\Bigr{|}{\,{\rm
d}}\varDelta_{v}(z)\leq\frac{R+r}{R-r}{\sf
C}_{U^{+}}(R)+\int_{D(R)}\ln\frac{2R}{|w-z|}{\,{\rm d}}\varDelta_{v}(z)$

where the right-hand side of the inequality is positive. Hence, by
integrating, we get

$\int_{E}U^{+}{\,{\rm d}}\uplambda\leq\frac{R+r}{R-r}{\sf
C}_{U^{+}}(R)\uplambda(E)+\int_{D(R)}\int_{E}\ln\frac{2R}{|w-z|}{\,{\rm
d}}\uplambda(w){\,{\rm d}}\varDelta_{v}(z)\\\ \leq\frac{R+r}{R-r}{\sf
C}_{U^{+}}(R)\uplambda(E)+\varDelta_{v}\bigl{(}\overline{D}(R)\bigr{)}\int_{E}\ln\frac{2R}{|w-z|}{\,{\rm
d}}\uplambda(w).$ (21)

Denote by $\uplambda_{E}$ the restriction of the Lebesgue measure $\uplambda$
to $\uplambda$-measurable set $E\subset\overline{D}(r)$. Obviously,

$\displaystyle{\rm supp\,}\uplambda_{E}$
$\displaystyle\subset\overline{D}(r)\subset D(R),$ (22s)
$\displaystyle\uplambda_{E}(w,t)$
$\displaystyle\overset{\eqref{{murad}z}}{\leq}\pi t^{2}\quad\text{for each
$w\in{\mathbb{C}}$ and $t\in{\mathbb{R}}^{+}$},$ (22t)
$\displaystyle\uplambda_{E}({\mathbb{C}})$ $\displaystyle=\uplambda(E)\leq\pi
r^{2}\leq\pi R^{2}.$ (22E)

Consider the last integral in (21):

$\int_{E}\ln\frac{2R}{|w-z|}{\,{\rm
d}}\uplambda(w)=\int_{{\mathbb{C}}}\ln\frac{2R}{|w-z|}{\,{\rm
d}}\uplambda_{E}(w)\overset{\eqref{{lE}s}}{=}\int_{0}^{2R}\ln\frac{2R}{t}{\,{\rm
d}}\uplambda_{E}(z;t)\overset{\eqref{{lE}t}}{=}\int_{0}^{2R}\frac{\uplambda_{E}(z;t)}{t}{\,{\rm
d}}t.$ (23)

By (22E) we have
$\sqrt{\uplambda(E)}\overset{\eqref{{lE}E}}{\leq}\sqrt{\pi}r<2r\leq 2R$. From
here we can split the last integral in (23) into the sum of two positive
integrals:

$\int_{0}^{2R}\frac{\uplambda_{E}(z;t)}{t}{\,{\rm
d}}t=\int_{0}^{\sqrt{\uplambda(E)}}\frac{\uplambda_{E}(z;t)}{t}{\,{\rm
d}}t+\int_{\sqrt{\uplambda(E)}}^{2R}\frac{\uplambda_{E}(z;t)}{t}{\,{\rm d}}t$
(24)

Using (22t), we have for first integral on the right-hand side of this
equality (24) the estimate

$\int_{0}^{\sqrt{\uplambda(E)}}\frac{\uplambda_{E}(z;t)}{t}{\,{\rm
d}}t\overset{\eqref{{lE}t}}{\leq}\int_{0}^{\sqrt{\uplambda(E)}}\frac{\pi
t^{2}}{t}{\,{\rm d}}t=\frac{\pi}{2}\uplambda(E).$ (25)

Using (22E), we have for second integral on the right-hand side of (24) the
estimate

$\int_{\sqrt{\uplambda(E)}}^{2R}\frac{\uplambda_{E}(z;t)}{t}{\,{\rm
d}}t\leq\uplambda({\mathbb{C}})\int_{\sqrt{\uplambda(E)}}^{2R}\frac{\uplambda_{E}(z;t)}{t}{\,{\rm
d}}t\overset{\eqref{{lE}E}}{=}\frac{1}{2}\uplambda(E)\ln\frac{4R^{2}}{\uplambda(E)}.$
(26)

Thus, it follows from (25)–(26) and (24) that

$\int_{0}^{2R}\frac{\uplambda_{E}(z;t)}{t}{\,{\rm
d}}t\leq\frac{\pi}{2}\uplambda(E)+\frac{1}{2}\uplambda(E)\ln\frac{4R^{2}}{\uplambda(E)}=\frac{1}{2}\uplambda(E)\ln\frac{4e^{\pi}R^{2}}{\uplambda(E)}\leq\frac{1}{2}\uplambda(E)\ln\frac{(10R)^{2}}{\uplambda(E)},$
(27)

where

$\ln\frac{(10R)^{2}}{\uplambda(E)}\geq 2,\text{ since
}\uplambda(E)\overset{\eqref{{lE}E}}{\leq}\pi R^{2},$

and estimate (27) together with (23) and (21) gives (20). ∎

######  Lemma 2 ((KhaMst, , Lemma 1)).

Let $\mu\in{\sf Meas\,}^{+}\bigl{(}\overline{D}(R)\bigr{)}$. Then

$\mu^{{\text{\tiny\rm rad}}}(r)\leq\frac{R}{R-r}{\sf
N}_{\mu}(r,R)\quad\text{for each $0\leq r\leq R$.}$ (28)

######  Lemma 3.

Let $0<r<+\infty$, $0<b\in{\mathbb{R}}^{+}$, $E\subset\overline{D}(r)$ be
$\uplambda$-measurable, $U=u-v$ be a difference of subharmonic functions
$u,v\in{\sf sbh}_{*}\Bigl{(}\overline{D}\bigl{(}(1+b)^{2}r\bigr{)}\Bigr{)}$.
Then

$\int_{E}U^{+}{\,{\rm d}}\uplambda\leq\frac{2+b}{2b}\Bigl{(}{\sf
C}_{U^{+}}\bigl{(}(1+b)r\bigr{)}+{\sf
N}_{\varDelta_{v}}\bigl{(}(1+b)r,(1+b)^{2}r\bigr{)}\Bigr{)}\uplambda(E)\ln\frac{\bigl{(}10(1+b)r\bigr{)}^{2}}{\uplambda(E)}.$
(29)

###### Proof.

By Lemma 1 with $R:=(1+b)r$ we have

$\int_{E}U^{+}{\,{\rm d}}\uplambda\leq\frac{1}{2}\Bigl{(}\frac{2+b}{b}{\sf
C}_{U^{+}}(R)+\varDelta_{v}^{{\text{\tiny\rm
rad}}}\bigl{(}(1+b)r\bigr{)}\Bigr{)}\uplambda(E)\ln\frac{\bigl{(}10(1+b)r\bigr{)}^{2}}{\uplambda(E)}.$

Hence, by Lemma 2 with $(1+b)^{2}r$ instead of $R$ and $(1+b)r$ instead of
$r$, we obtain

$\int_{E}U^{+}{\,{\rm d}}\uplambda\leq\frac{1}{2}\frac{2+b}{b}\Bigl{(}{\sf
C}_{U^{+}}\bigl{(}(1+b)r\bigr{)}+{\sf
N}_{\varDelta_{v}}\bigl{(}(1+b)r,(1+b)^{2}r\bigr{)}\Bigr{)}\uplambda(E)\ln\frac{\bigl{(}10(1+b)r\bigr{)}^{2}}{\uplambda(E)},$

which gives (29). ∎

###### Proof of Theorem 2.

We can assume that $U=u-v$ is the canonical representation of $U$. Consider a
number $b>0$ such that $(1+b)^{2}=k$. By Lemma 3, we have

$\int_{E}U^{+}{\,{\rm
d}}\uplambda\leq\frac{1}{2}\frac{\sqrt{k}+1}{\sqrt{k}-1}\Bigl{(}{\sf
C}_{U^{+}}(\sqrt{k}r)+{\sf
N}_{\varDelta_{v}}(\sqrt{k}r,kr)\Bigr{)}\uplambda(E)\ln\frac{\bigl{(}10\sqrt{k}r\bigr{)}^{2}}{\uplambda(E)}\\\
\leq\frac{k}{k-1}\Bigl{(}\underset{{\sf
T}_{U}(r_{0},\sqrt{k}r)}{\underbrace{{\sf C}_{U^{+}}(r_{0},\sqrt{k}r)+{\sf
N}_{\varDelta_{v}}(r_{0},\sqrt{k}r)}}+{\sf C}_{U^{+}}(r_{0})+\underset{{\sf
T}_{U}(r_{0},kr)}{\underbrace{{\sf C}_{U^{+}}(r_{0},kr)+{\sf
N}_{\varDelta_{v}}(r_{0},kr)}}\Bigr{)}\uplambda(E)\ln\frac{\bigl{(}10r\bigr{)}^{2}k}{\uplambda(E)}\\\
\leq\frac{k}{k-1}\Bigl{(}2{\sf T}_{U}(r_{0},kr)+{\sf
C}_{U^{+}}(r_{0})\Bigr{)}\uplambda(E)\ln\frac{\bigl{(}10r\bigr{)}^{2}k}{\uplambda(E)}\leq\frac{2k}{k-1}\Bigl{(}{\sf
T}_{U}(r_{0},kr)+{\sf
C}_{U^{+}}(r_{0})\Bigr{)}\uplambda(E)\ln\frac{\bigl{(}100kr^{2}}{\uplambda(E)}$

and obtain (16T).

If $u\in{\sf sbh}_{*}({\mathbb{C}})$, then the function ${\sf M}_{u}^{+}$ is
increasing, and

$\int_{E}u^{+}{\,{\rm d}}\uplambda\overset{\eqref{{Cu}M}}{\leq}{\sf
M}_{u^{+}}(r)\uplambda(E)\quad\text{for $E\subset\overline{D}(r)$.}$ (30)

For $U_{u}:=0-u$, the difference $0-u$ is the canonical representation of
$\delta$-subharmonic non-trivial function $U_{u}$ and we have

${\sf T}_{U_{u}}(r,R)\overset{\eqref{T}}{=}{\sf C}_{\sup\\{0,u\\}}(r,R)={\sf
C}_{u^{+}}(r,R)\leq{\sf C}_{u^{+}}(R)\leq{\sf M}_{u^{+}}(R).$ (31)

Hence, by Theorem 2 in part (16T) for $U_{u}$ in the role of $U$, we obtain

$\int_{E}(-u)^{+}{\,{\rm
d}}\uplambda\overset{\eqref{inDl+}}{\leq}\frac{2k}{k-1}\Bigl{(}{\sf
T}_{U_{u}}(r_{0},kr)+{\sf
C}_{U_{u}^{+}}(r_{0})\Bigr{)}\uplambda(E)\ln\frac{100kr^{2}}{\uplambda(E)}\\\
\overset{\eqref{TCM}}{\leq}\frac{2k}{k-1}\Bigl{(}{\sf M}_{u^{+}}(kr)+{\sf
C}_{(-u)^{+}}(r_{0})\Bigr{)}\uplambda(E)\ln\frac{100kr^{2}}{\uplambda(E)}.$

The latter together with (30) gives (16M). ∎

Acknowledgments. The work was supported by a Grant of the Russian Science
Foundation (Project No. 18-11-00002).

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