# Multimodal Resonance in Strongly Coupled Inductor Arrays

Robert R. Hughes, James Treisman, Alexis Hernandez Arroyo, Anthony J.
Mulholland

(DRAFT )

###### Abstract

Magnetic resonance coupling (MRC) is widely used for wireless power transfer
(WPT) applications, but little work has explored how MRC phenomena could be
exploited for sensing applications. This paper introduces, validates and
evaluates the unique multi-resonant phenomena predicted by circuit theory for
over-coupled inductive arrays, and presents eigen-formulae for calculating
resonant frequencies and voltage modes within passively excited arrays.
Finite-element simulations and experimental results demonstrate the validity
of the multi-modal resonant principles for strongly-coupled inductor arrays.
The results confirm the distinctive multi-modal resonant frequencies these
arrays exhibit, corresponding to the specific magnetic excitation "modes"
(comparable to vibrational modes in multi-degree-of-freedom systems). The
theoretical and finite element models presented offer a framework for
designing and optimizing novel inductive sensing arrays, capitalizing on the
unique resonant effects of over-coupling and exploiting their potential
magnetic field shaping.

##  1 Introduction

Electrical resonance is a fundamental effect used in electrical engineering,
physics and electronics that has been widely researched and utilized for many
applications. The phenomenon refers to the ability of an electrical system,
typically containing components that store and discharge energy between
electric (capacitors) and magnetic (inductors) fields, to respond to an
alternating voltage or current, and exhibit resonant behavior at a specific
frequency [7]. Over the past century, electrical resonance has been used in a
wide variety of applications, ranging from simple devices such as radio and
television tuners [4], to near-field sensing applications [16, 9], as well as
more advanced applications such as wireless power transfer [13].

It is in wireless power transfer (WPT) systems in particular that the concepts
of multi-coil magnetic resonance coupling (MRC) systems have been explored
predominatly for the transmission of power over greater distances [20] and
between multiple inductive elements [14, 8, 15]. It is in such systems that
resonance splitting phenomena have been most commonly observed and exploited.
In this paper, these same frequency splitting effects are considered for
parallel LC measurements of coplanar inductor arrays, specifically evaluating
the phenomenona in mm-scale inductive array sensors, comparable to sensor
arrays used in non-destructive testing (NDT) applications.

When it comes to inductive sensor arrays, such as those used in non-
destructive testing (NDT) applications, resonance is strictly avoided. This is
due to the electronically unstable nature of resonance, while expensive
electronic multiplexing is employed in order to avoid the unpredictable mutual
interactions (cross-talk) between neighbouring coils. While the stability of
avoiding resonance is beneficial, the potential advantages and novel
technology enabled by understanding the behaviour of resonance in arrays
sensors are ignored.

Over the past decade, the resonance principles of WPT in multi-coil sensors
has gained increasing interest as a potential area for novel technology
development in near-field sensing applications [10, 2, 5, 17]. Researchers
have exploited the resonance splitting phenomenon to evaluate displacement
between sensing coils [3, 1], and even explored the effects of series resonant
array sensors for non-destructive applications [6]. However, while these
effects have been recognised and exploited in series resonant arrays, a
detailed evaluation of the phenomena experienced by arrayed resonant inductors
is yet to be available from the literature for measurements for parallel LC
resonators. This limits the potential for resonant multi-coil sensor design
and optimisation to advance the field of resonant inductor array sensing.

In this paper, we present the theory, experimental validation and key
observations behind the unique phenomena found in magnetically over-coupled
inductive arrays exhibiting multiple resonant frequencies (multi-modal) for
parallel resonant driver coils in passive LC resonant arrays. The paper covers
both the theoretical principles of multi-modal electrical resonance arrays,
comparison against finite element simulations and validation of experimental
results. While the study limits itself to the evaluation of co-planar array
elements, the principles presented would naturally extend into two and three-
dimensional array configurations.

##  2 Theory

Electrical resonance occurs when an electrical system oscillates at its
resonant frequency, $f_{0}$, as determined by its electrical properties of
capacitance, $C$, inductance, $L$, and resistance, $R$. The classical
expression for the natural resonant frequency of a electrical resonator with
either series or parallel $L$ and $C$ components is defined as,

$\omega_{0}=2\pi f_{0}\approx\sqrt{\frac{1}{LC}},$ (1)

where $\omega_{0}$ is the natural resonant angular frequency of the system
[7]. The principles of exploiting the resonant frequency shifting behaviour of
single inductive sensors has been explored by many authors [16, 12], however
little has been done to explore the resonance phenomena of multi-sensor
arrays.

The influence of neighbouring inductor resonators on the electrical impedance
spectra of a driven coil has been highlighted as a point of interest for
applications in array sensing [10, 6]. The following sections detail the
theory behind the resonance phenomena of magnetically-coupled multi-coil
sensors.

###  2.1 Two Coil Model

When close to another comparable coil, a primary coil will inductively couple
to a secondary coil. The effect of the secondary coil can be modelled as an
inductor, $L_{2}$, in series with a resistor, $R_{2}$, and a capacitor,
$C_{2}$. This coupling, parameterised by the coupling coefficient, $k$, will
alter the effective inductance and resistance ($L_{1}^{\prime}$ and
$R_{1}^{\prime}$ respectively) of the primary measurement circuit and will
distort the impedance. The full formulae for the impedance of the bimodal coil
system can be found in [1, 11]. The general expression for the resonant
frequencies of a two-coil system can be defined as [1],

$\omega_{\pm}^{\prime}{}^{2}=\frac{1}{2}\frac{\omega_{1}^{2}}{\left[1-k^{2}\right]}\left[1+\frac{\omega_{2}^{2}}{\omega_{1}^{2}}\left(1\pm\frac{\omega_{1}}{\omega_{2}}\sqrt{\frac{\omega_{2}^{2}}{\omega_{1}^{2}}+\frac{\omega_{1}^{2}}{\omega_{2}^{2}}+4k^{2}-2}\right)\right],$
(2)

where $\omega_{n}=\sqrt{1/L_{n}C_{n}}$. Equation 2 therefore predicts two
unique solutions to the resonant frequency of the coupled system. Therefore,
by knowing the natural resonant frequency of any two coils in their uncoupled
($k=0$) state, values for these two unique resonant frequencies can be
determined for a given value of $k$. Equation 2 can also be written in its
dimensionless form for the frequency ratio relative to the resonant frequency
of the driver coil $r_{1}=\omega/\omega_{1}$,

$r_{1}^{2}=\frac{\left[1+\frac{\omega_{2}^{2}}{\omega_{1}^{2}}\right]\pm\sqrt{\left[1+\frac{\omega_{2}^{2}}{\omega_{1}^{2}}\right]^{2}-4(1-k^{2})}}{2\left[1-k^{2}\right]}.$
(3)

Figure 1: Equivalent circuit simulated impedance spectra for coupled resonant
inductor circuits showing a) the dual resonant peaks of the frequency spectra
and b) coupling coefficient verses frequency dispersion curves, representing
the resonant vibrational modes of the system.

####  2.1.1 Identical Coils

In the high-frequency case where circuit components of coils 1 and 2 are
identical, $R_{1}=R_{2}=R$, $L_{1}=L_{2}=L$, and $C_{1}=C_{2}=C$, such that
$M=kL$, and the resonant frequencies $\omega_{1}=\omega_{2}=\omega_{0}$, the
resonant frequencies and coupling coefficient respectively can be simplified
to [1],

$\displaystyle\omega_{\pm}^{\prime}$
$\displaystyle\approx\sqrt{\frac{1}{LC\left[1\pm k\right]}},$ (4)
$\displaystyle k$
$\displaystyle\approx\frac{\omega_{-}^{\prime}{}^{2}-\omega_{+}^{\prime}{}^{2}}{\omega_{-}^{\prime}{}^{2}+\omega_{+}^{\prime}{}^{2}}.$
(5)

Figure 1.a shows the frequency spectra of strongly-coupled identical coils and
shows how the frequency splitting effect occurs as a function of $k$. The
specific impedance equation for a two-coil system cna be found in [1].
Equation 4 can be used to predict the dispersion of the two resonant
frequencies, shown as white dotted lines in Figure 1.b, as a function of the
coupling coefficient. The system can be thought of as exhibiting independent
vibrational modes. Figure 1.b also shows a red dotted line representing the
dispersion separation threshold; the coupling coefficient of this system above
which independent resonant peaks can be resolved. This threshold is dependent
on the Q-factor of the systems and as such lower resistance systems exhibit a
sharper, more easily resolvable dispersion at lower coupling coefficients.
There is also a practical upper threshold to the coupling coefficient of a
realistic inductively coupled system which is dependent on the geometry of the
system and the permeability of the cores used within the coils.

Equation 5 matches the generalised Cohn-Matthaei formulae for coupled
resonators derived by Tyurnev 2008 [18], and means that the coupling
coefficient can be calculated by measurement of the resonant frequencies.
Comparable formula are regularly applied in the modelling of microwave
bandpass networks and meta-material design, but the features and properties of
coupled resonators are yet to be explored in sensing applications.

Figure 2: Equivalent circuit diagram of an N-dimensional array

###  2.2 Multi-coil Circuit Model

Figure 2 shows an example circuit diagram for a N-dimensional (ND) linear
array of resonant coils. To consider a mutli-coil system (i.e. when the number
of coils N > 2) we will evaluate a 1D linear array of three coils before
extending to ND. For the 3-coil array, Kirchhoff’s laws provide the equations
[19],

$\displaystyle V_{1}$ $\displaystyle=I_{1}[R_{1}+i\omega
M_{11}]+i\omega[M_{21}I_{2}+M_{31}I_{3}],$ (6) $\displaystyle 0$
$\displaystyle=I_{2}Z_{2}+i\omega[M_{12}I_{1}+M_{32}I_{3}],$ (7)
$\displaystyle 0$ $\displaystyle=I_{3}Z_{3}+i\omega[M_{13}I_{1}+M_{23}I_{2}],$
(8)

which can be written in matrix form,

$\displaystyle\textbf{V}=\mathbb{Z}\textbf{I}\xrightarrow{}\begin{vmatrix}V_{1}\\\
0\\\ 0\end{vmatrix}=\begin{vmatrix}Z_{1}&i\omega M_{12}&i\omega M_{13}\\\
i\omega M_{21}&Z_{2}&i\omega M_{23}\\\ i\omega M_{31}&i\omega
M_{32}&Z_{3}\end{vmatrix}\begin{vmatrix}I_{1}\\\ I_{2}\\\ I_{3}\end{vmatrix},$
(9)

where the impedance of the passive circuits ($m>1$) is defined as,

$\displaystyle\begin{split}Z_{m}=R_{m}+\frac{1}{i\omega C_{m}}+i\omega
M_{mm},\end{split}$ (10)

and the mutual inductance between coils is,

$\displaystyle\mathbb{M}=\begin{vmatrix}M_{11}&M_{12}&M_{13}\\\
M_{21}&M_{22}&M_{23}\\\ M_{31}&M_{32}&M_{33}\end{vmatrix},$ (11)

where $M_{nm}=k_{nm}\sqrt{L_{n}L_{m}}$ is the mutual inductance, between coils
$n$ and $m$, with the coupling coefficient
$k_{n,m}=\\{k_{n,m}\in\mathbb{R},0<k<1\\}$, where $n$ is the index of the
excitation circuit, and $m$ is the index of the passive circuit. We can
therefore define a coupling coefficient matrix,

$\displaystyle\mathbb{K}=\begin{vmatrix}k_{11}&k_{12}&k_{13}\\\
k_{21}&k_{22}&k_{23}\\\
k_{31}&k_{32}&k_{33}\end{vmatrix}=\begin{vmatrix}1&k_{12}&k_{13}\\\
k_{21}&1&k_{23}\\\ k_{31}&k_{32}&1\end{vmatrix},$ (12)

where $k_{nm}=k_{mn}$. When $n=m$, $k_{nn}=1$ and therefore from Equation 11
$M_{nn}=L_{n}$.

Equations 7 and 8 can be redefined into a general expression for the current
within each circuit,

$\displaystyle\begin{split}I_{m}&=\frac{i\omega\left[\sum_{n\neq
m}^{N}M_{nm}I_{n}\right]}{Z_{m}},\end{split}$ (13)

where $M_{1m}$ and $M_{2m}$ are the nearest and second nearest neighbour
mutual inductances experienced by the coil $m$. The voltage experienced by
coil $n$ can therefore be defined as,

$\displaystyle\begin{split}V_{n}&=I_{n}\left[R_{n}+i\omega
M_{nn}\right]+i\omega\left[\sum_{m\neq n}^{M}M_{mn}I_{m}\right].\end{split}$
(14)

These expressions will remain valid for a coil in a 1D chain for all N and M,
representing the number of coils in the chain.

###  2.3 Resonant Modes in ND Arrays

To formulate an eigenvalue matrix for the coupled inductor system, we must
recognise that the system is being measured as a parallel $LC$ resonant
circuit. As such, we must consider the measured admittance,
$\mathbb{Y}_{0}=\mathbb{Z}_{0}^{-1}$, of the coupled system,

$\displaystyle\mathbb{Y}_{0}\textbf{V}=\textbf{I}.$ (15)

We can define $\mathbb{Y}_{0}$ via the reciprocal rule as,

$\displaystyle\mathbb{Y}_{0}=\frac{1}{\mathbb{Z}_{0}}$
$\displaystyle=\frac{1}{\mathbb{Z}_{L}}+\frac{1}{\mathbb{Z}_{c}},$ (16)
$\displaystyle=\mathbb{Z}_{L}^{-1}+{i\omega\mathbb{C}},$ (17)

where $\mathbb{C}$ is the $N\times N$ parallel capacitance matrix,

$\displaystyle\mathbb{C}=\begin{vmatrix}C_{1}&0&\cdots&0\\\
0&C_{2}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&C_{N}\end{vmatrix},$ (18)

with $C_{n}$ representing the parallel capacitances in inductor circuit, $n$.
$\mathbb{Z}_{L}$ is the $N\times N$ coupled inductive impedance matrix, which
can be approximated in the high-frequency case ($R\ll\omega L$) to,

$\displaystyle\mathbb{Z}_{L}\approx
i\omega\mathbb{M}=i\omega\begin{vmatrix}M_{11}&M_{12}&\cdots&M_{1N}\\\
M_{21}&M_{22}&\cdots&M_{2N}\\\ \vdots&\vdots&\ddots&\vdots\\\
M_{N1}&M_{N2}&\cdots&M_{NN}\end{vmatrix},$ (19)

where $M_{nm}=k_{nm}\sqrt{L_{n}L_{m}}$, and where $k$ and $L$ have their usual
meaning as the coupling coefficient and inductance respectively. We can write
Equation 17 as,

$\displaystyle\mathbb{Y}_{0}$
$\displaystyle=\frac{1}{i\omega\mathbb{M}}+{i\omega\mathbb{C}},$ (20)
$\displaystyle=\frac{1}{i\omega}\left(\mathbb{M}^{-1}-\omega^{2}\mathbb{C}\right).$
(21)

Inputting Equation 21 into Equation 15 and rearranging gives,

$\displaystyle\left[\mathbb{C}\left(\mathbb{\Omega}-\omega^{2}\mathbb{1}\right)\right]\textbf{V}={i\omega}\textbf{I},$
(22)

where $\mathbb{\Omega}$ is the characteristic eigen-matrix with the form,

$\mathbb{\Omega}=\left[\mathbf{C}\mathbb{M}\mathbf{C}^{T}\right]^{-1}=\Omega\mathbb{K}^{-1}\Omega^{T},$
(23)

$\mathbf{C}$ and $\Omega$ are $N\times N$ diagonal matrices with elements
$\sqrt{C_{n}}$ and $\omega_{n}$ respectively, and

$\displaystyle\Omega=\begin{vmatrix}\omega_{1}&0&\cdots&0\\\
0&\omega_{2}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&\omega_{N}\end{vmatrix},$ (24)

$\omega_{n}=\sqrt{L_{n}C_{n}}^{-1}$ is the natural resonant frequency of an
uncoupled coil $n$. The inverse coupling coefficient matrix $\mathbb{K}^{-1}$
can be calculated using the standard inverse matrix identity
($\mathbb{K}^{-1}=\nicefrac{{Adj.\mathbb{K}}}{{|\mathbb{K}|}}$).

Given that the impedance $Z_{0}$ is a maximum (infinite in the idealised
cases) at resonance, we can conclude that the admittance $Y_{0}$ will be a
minimum at resonance (zero in idealised case), such that the current
$\mathbf{I}$ becomes zero at resonance. Equation 22 can therefore be
rearranged into the form,

$\displaystyle\left(\mathbb{\Omega}-\omega_{i}^{2}\mathbb{1}\right)\vec{v}_{i}=0,$
(25)

which takes the form of a traditional eigen-equation, where $\vec{v}_{i}$ is
the $i^{th}$ voltage eigenvector, and $\omega_{i}^{2}$ is the eigenvalue of
the characteristic matrix, $\mathbb{\Omega}$, and satisfies the expression,

$\displaystyle\mathbb{\Omega}\vec{v}_{i}=\omega_{i}^{2}\vec{v}_{i}.$ (26)

We can therefore calculate the eigenvalues and vectors of the coupled system
by solving,

$\displaystyle\det{\left(\mathbb{\Omega}-\omega_{i}^{2}\mathbb{1}\right)}$
$\displaystyle=0.$ (27)

With all electronic components in the system known, and if the coupling
coefficients, $k_{nm}$, between array elements can be estimated, we can
predict the eigenvalues (resonant frequencies) of any ND-coupled system. From
this matrix analysis, it is clear that an overcoupled system will exhibit as
many resonant modes as there are array elements.

Figure 3 shows the the equivalent circuit theory impedance magnitude spectra
calculated using equations 13-14 for a linear 3-coil array with increasing
resistance, demonstrating the difference between the eigenmode predicted
resonant frequencies and the actual peak impedance frequencies as the high
Q-factor assumption breaks down.

Figure 3: Circuit model predicted impedance magnitude profiles of a 3-coil
linear array systems of identical coil inductance ($10~{}\mu H$) and
capacitance ($150~{}pF$) with varying series resistance, $R$, showing the
resonant damping effect and disparity between the eigen-matrix calculated
resonant frequencies (black dotted lines) and the circuit predicted peak
frequencies (red solid lines).

##  3 Materials & Methods

The significance of the above impedance formulation and eigenvalue prediction
is evaluated in the following section for some example array configurations;
qualitatively comparing circuit model predicted impedance profiles with 2D
finite element model (2DFEM) simulated results and experimental measurements.

###  3.1 2D Finite Element Simulation

Two-dimensional (2D) Finite element models (FEM) were created in COMSOL 6.1
using the AC/DC module. Models were developed using Magnetic Fields and
Electrical Circuit studies to simulate the equivalent circuit of resonating
coils and were evaluated in the frequency domain. The models simulated
infinitely long (extending into the page) coil arrays of height $10~{}mm$,
core radius $2~{}mm$, with homogenised multi-turn coil windings of thickness
$0.6~{}mm$. The core material was ferrite, with a relative permeability
$\mu_{r}=25$. Each coil was capacitively loaded with a capacitor of $50~{}pF$
and a series resistance of $200~{}\Omega$. Due to the 2D nature of the
simulation, the inductances of the coil array elements were unrealistically
high, but the prediction of the relative resonances in the arrays remains
valid.

From these simulations, the electrical impedance spectra, $Z(f)$, the array
element voltages, $V_{n}$, and the spatial distribution of the magnetic flux
density, $B(f,x,y)$, can be evaluated as a function of the excitation
frequency.

###  3.2 Experimental Validation

Quantitative experimental validation is performed by evaluating the impedance
spectra of a simple linear array of $2~{}mm$ outer radius inductors, of
inductance $10~{}\mu H$ and circuit capacitance of $1.7~{}nF$, giving each
element a natural resonant frequency of $1.2~{}\mbox{MHz}$. Each coil has a
ferrite core (Grade 61, FairRite) of diameter $1~{}mm$, and height $10~{}mm$.
The impedance spectra of the linear array was measured using an Impedance
Analyser (Trewmac TE3001) between $0.5-1.5~{}\mbox{MHz}$ across each of the
unique coil array elements. The voltages across each array element were
measured at each resonant frequency using a Picoscope 4824A USB oscilloscope,
while a SonEMAT Howland Current Source was used to convert a voltage input
waveform from a RS Pro RSDG 830 Arbitrary Function Generator into a controlled
alternating current supplied to the central array element.

##  4 Results & Discussion

A number of specific array configurations are considered in this study,
including identical resonator arrays in linear and close-packed arrangements,
as well as disparate resonant arrays where each element exhibits a unique
resonant frequency achieved via capacitive tuning. In each case, circuit
models, 2D FE simulated and experimental results are compared and contrasted
with specific attention to the qualitative phenomenon and features of the
array’s impedance and magnetic flux density. While the theoretical models
presented in section 2 can be applied to any physical configuration of coils,
even those with disparate circuit parameters and resonances, in this paper we
focus on arrays of nominally identical coplanar inductors as an example
validation case, i.e. with identical coil properties and capacitive loads such
that each coil element before installation in the array has identical resonant
frequencies $f_{n}=f_{m}=f_{0}$.

###  4.1 Three-Coil Arrays

Examples of predicted impedance profiles, $Z_{0}(f)$, generated using the
general circuit model in Section 2.2 are shown in Figure 4. These are compared
to the predicted eigen-frequencies, $f_{n}$ (dashed vertical lines), from
equation 27 for three different 3-coil configurations; a linear array measured
at the end and central coil respectively and a close-packed array. Each coil
is an identical resonator with inductance $L=10~{}\mu H$, capacitance
$C=150~{}pF$, and resistance $R=10~{}\Omega$.

Figure 4: Predicted impedance spectra for three 3-coil over-coupled arrays
configurations of identical coil inductance ($10~{}\mu H$), capacitance
($150~{}pF$), series resistance ($10~{}\Omega$) and coupling coefficient
($k=0.14$) showing i) impedance magnitude spectra $|Z(f)|$ for three coil
measurement configurations; Linear (A and B), and iii) Close-packed (C)
arrays. Plots in blue and red represent measurements of a linear array from
the end and centre coils respectively. The yellow curve shows the measurement
for a close-packed coil configuration (C). Dotted lines correspond to the
theoretically predicted resonant eigen-frequencies with colours corresponding
to the predicted eigen-modes shown in the ii) and iii) for linear and close-
packed arrays respectively.

Figure 4 also shows the eigenvectors (representing the array excitation modes)
found from Equation 26, providing information on which elements are excited,
and to what extent, at each resonant frequency, as well as which elements are
unexcited (i.e. array nodes).

The simulated results show that although the eigen-matrix predicts identical
resonant frequencies and modes regardless of which element is measured, only
the impedance profile of measurements made on the end coil exhibit three
distinct resonant peaks. This is due to the measurement asymmetry enabling
three unique modes, while the symmetry of the linear array measured at the
central coil leads to only 2 unique resonant peaks. This is shown by the
second mode ($f_{2}$) where the central coil has an eigenvalue of 0.

The mode-shapes shown in Figure 4.ii and iii represent the excitation states
of each element at each resonant frequency. Both linear arrays (Figure 4.a &
b) exhibit the same eigen-modes and resonant frequencies, however when
measuring from the central coil, only two resonant peaks are observed with the
2nd resonant frequency corresponding to a mode where the central coil exhibits
a node state in the eigen-modes. In comparison, Figure 4.iii shows that a
close-packed 3-coil array experiences different resonant frequencies to the
linear array configuration, and there exists two simultaneous modes at the
same resonant frequency. This is believed to correspond to the equal chance of
coil 1 being in phase with either coil 2 or 3.

This eigen-mode analysis has the potential to be extremely powerful for
exciting arrays in specific spatial coil configurations by selecting
appropriate frequencies. For example, in a linear array, the lowest resonant
frequency corresponds to an alternating eigenmode between coil elements
corresponding to a spatially alternating magnetic field direction. Conversely,
exciting the array at the highest frequency will excite a mode whereby all
elements are in phase such that the array behaves like a single larger coil.

####  4.1.1 2D FEM Analysis

Figure 5 shows the impedance spectra predicted using a 2D FEM of a linear
3-coil array as measured from an active (a) end and (b) central coil. The
results show the same resonant splitting phenomena predicted by the circuit
model (see Figure 4), with 3 unique modes predicted for the end-excited coil,
and only 2 unique resonant peaks for the centrally-excited coil. The FEM
predicts the same patterns of each resonant peak as the circuit model, with
key features being; 1) the first and second peaks occurring in close proximity
in the frequency domain, 2) the second peak having the highest amplitude, and
3) the third resonant peak having the lowest amplitude resonance mode. The
centrally-excited array also exhibits the same resonant spectra as predicted
by the circuit theory, with the lowest frequency having the greatest
prominence.

The spatial distributions of the magnetic flux density on the bottom tiles in
Figure 5 incorporate arrows that show the magnitude and direction of the
magnetic flux density in the middle of each coil element at each resonant
frequency of the array. This analysis shows the spatial patterns of the
magnetic flux in these co-planar arrays is shared between coil elements in
different mode-shapes mimicking the eigenvectors predicted by the circuit
model.

Figure 5: 2D FEM impedance spectra for two linear 3-coil over-coupled arrays
configurations of identical resonance showing impedance magnitude $|Z|$
spectra and the spatial distribution of the magnetic flux density at each
resonant frequency (i-iii). Showing a) linear array measured from active end
coil, and b) measured from an active central coil.

###  4.2 Five-Coil Linear Array

Figure 6 shows how the addition of further array elements leads to additional
resonant modes. The impedance spectra show how the number of peaks changes
depending on which coil is used to excite the array, in a similar way to the
3-coil array above.

a)

b)

c)

Figure 6: Circuit model prediction - Five identically resonating coils in a
linear array, showing; a) linear resonant array configuration (blue = coil 1
active, red = coil 2 active, yello = coil 3 active), b) magnitude $|Z|$ and
phase $\Phi$ of impedance spectra as measured across coils 1-3, c) predicted
eigenvector modeshapes of the array coil elements at the eigen-matrix
predicted resonant frequencies $f_{n}$.

a)

b)

c)

Figure 7: 2D FE five-coil linear resonant array spectra - Measured across LC
resonator element 1, 2 & 3 (a), simulated array element voltages (b) and
magnetic flux density at the resonant frequencies of the simulated array (c)
when excited by coil element 1.

Figure 7.a shows the impedance spectra predicted using a 2D FEM of a linear
five-coil array as measured from coils 1 (blue), 2 (red) and 3 (yellow). The
results show the same resonant splitting phenomena predicted by the circuit
model (see Figure 6), with five unique modes predicted for the end-excited
array, and only three unique resonant peaks for the centrally-excited array.
The FEM predicts the same patterns of each resonant peak as the circuit model,
with key features being; 1) the first (i) and second (ii) peaks occurring in
close proximity in the frequency domain, 2) the third (iii) peak being the
most prominent, and 3) the fifth (v) resonant peak being the least prominent
resonance mode. The centrally-excited array also exhibits the same resonant
spectra as predicted by the circuit theory, with the lowest frequency having
the greatest prominence (yellow lines in Figures 6 and 7). These resonant
rules are consistent with the three-coil case.

Figure 7.b shows the predicted voltage predictions for for each coil element
at each resonant frequency. The voltage distributions between coils
corresponds directly to the equivalent eigen-mode shapes predicted in Figure
7.

Figure 7.c show the magnitude and direction of the magnetic flux density in
the middle of each coil element at each resonant frequency of the simulated
array. This analysis shows the spatial patterns of the magnetic flux in these
co-planar arrays as it is shared between coil elements in different mode-
shapes mimicking the eigenvectors predicted by the circuit model.

###  4.3 Experimental Validation

In order to experimentally validate the predictions of both the circuit theory
and the FEM, an experimental five-coil linear array was manufactured. The
array consisted of five identical cylindrical coils of free-space inductance
$L=10\pm 0.1~{}\mu H$ and dimensions shown in Figure 8.a (all dimensions are
given with a measurement error of $\pm 0.1mm$). Each coil was positioned flush
with one end of a ferrite (Grade 61, FairRite) rod of diameter $1~{}mm$ and
length $10~{}mm$, and each was housed in a linear configuration in a SLA
3D-printed plastic array housing, containing five $2~{}mm$ diameter holes
separated by a $200~{}\mu m$ wall.

Each array element was connected in parallel to a $1.4\pm 0.1~{}nF$ surface
mount capacitor, a surface mount SMA connector, and a $0.1m$ SMA to BNC
coaxial cable. The total capacitance and resistance of each array element were
measured using a digital multi-meter and displayed in Table 4.3.

a)

b)
i) Experimental ii) Circuit Theory

c)

Figure 8: Experimental and Circuit Model Comparison for a five-coil array,
showing a) the experimental five-coil array configuration and, b) the
impedance spectra $|Z(f)|$ showing the experimental (i) and circuit model
predicted spectra (ii) when the active coil is changed between each coil
element in the configuration. c) Shows i) the experimentally measured voltage
outputs of each array element at each resonant frequency (excitation coil
independent), compared to the ii) predicted eigenmodes from circuit theory.

A TREWMAC TE3001 Impedance Analyser was used to measure the resulting
impedance spectra of each element within the array and the results are plotted
in Figure 8.b (left). In comparison to the idealised symmetric theoretical
impedance spectra shown in Figure 6.b for coil 1, the experimental results do
not show the five distinct resonant peaks predicted by the circuit theory of
an identical resonator array. They instead show a maximum of four identifiable
peaks with the central two peaks being so close in proximity that they are
rendered indistinguishable when measured from coil 5 (which should have an
identical spectra to coil 1). This comparison between coil 1 and 5 spectra
demonstrates the asymmetry of the physical array caused by natural variability
between the circuit components, and inconsistencies in the positioning of the
elements within the array. However, this asymmetry also helps to explain why
the spectra are not identical to the idealised theoretical case shown in
Figure 6.b.

The right-hand spectra shown in Figure 8.b are the circuit theory predicted
impedance spectra for an asymmetric array, specifically produced via manually
applying asymmetry to the inductance of each array element with values used
shown in Table 4.3.

Table 1:  Circuit parameters in equivalent circuit model of experimental 5-coil linear array Circuit | $1$ | $2$ | $3$ | $4$ | $5$
---|---|---|---|---|---
$L\ (\mu H)$ | $16.7$ | $17.4$ | $12.7$ | $17.3$ | $17.9$
$C\ (nF)$ | $1.72$ | $1.72$ | $1.73$ | $1.72$ | $1.72$
$R\ (\Omega)$ | $3.73$ | $4.0$ | $3.79$ | $3.88$ | $3.96$

Completely identical inductors in an array theoretically result in the central
coil having the highest effective inductance (as indicated by the steeper
gradients in impedance at low frequencies for the middle coils), seen in the
theoretical three and five coil arrays (Figures 4 and 6 respectively).
Meanwhile, dissimilar array elements seem to result in the central coil
exhibiting the lowest effective inductance. The mechanism for this is not
clear.

There is an additional effect which has not been accounted for in the
theoretical circuit models and which is likely to be contributing to the
disparity in impedance spectra between the experiment and the circuit theory
when the coils are identical. That is, that placing ferrite cored coils next
to each other will impact on the inductance of each array element. If the
manually modified values of the circuit modelled asymmetric array are similar
to the actual values, then the pattern of inductances within the linear array
suggests that the coils at the end have a smaller increase in inductance when
placed in the array, compared to coils 2 and 4. This makes physical sense
given that coils 2 and 4 have ferrite cores to both sides of them, while coils
1 and 5 have ferrite coils only on one side. Future studies will explore how
this can be mitigated through the use of dummy cores at the ends of the array
and through close-packing these arrays, and parameter optimisation used to
match theory to experimental results.

Figure 8.c shows the experimentally measured voltages across the array
elements at each resonant frequency (i) compared to the eigenmodes predicted
by the asymmetric array circuit model (ii). In spite of the asymmetric array
configuration, the modes remain recognisable to those shown for the idealised
identical element system shown in Figure 6.c, but exhibit some distorted
features. The experimental and approximated circuit model results exhibit
highly comparable modes at their respective resonant frequencies, with a few
minor exceptions occurring for coils 1 and 2 in mode $f_{2}$, and coils 4 and
5 in mode $f_{4}$. This is likely in part related to the close proximity of
these resonant frequencies to other resonant modes.

##  5 Conclusion

This paper presents and validates a circuit theory for predicting and
evaluating the resonant phenomena observed in over-coupled resonant inductive
arrays. The mathematical formalisation is presented for electrical
measurements across a single parallel resonant coil element in an array of
passive series resonant inductors, and critically evaluated against finite
element simulations and experimental results of linear resonant inductor
arrays.

The circuit model, FE simulations and experimental results all confirm the
unique multi-modal resonant phenomena observed in measurements of individual
array elements, with these resonant frequencies corresponding to specific
magnetic excitation "modes"; equivalent to vibrational modes in multi-degree-
of-freedom systems.

With the models presented, novel inductive arrays can be designed and
optimised to exploit the unique resonant effects of over-coupling for
applications in wireless power transfer and array sensing. The ability to
identify and predict resonant modes in arrays opens the door for array sensing
without electronic multiplexing of the whole array, but rather through the
excitation of individual coils within the array by excitation at specific
frequencies. This interpretation of magnetic modes has the potential to enable
magnetic field shaping through simple frequency selection.

Further work is required to explore the many and varied possible spatial array
configurations and to apply these resonant array principles to novel
applications. It is anticipated that the theory and analysis presented herein
has the potential to initiate a unique field of research and a profound change
in how arrayed inductor systems are designed and used.

## References

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[label2]organization=Saint Petersburg State University,addressline=University
Avenue, 28D, city=Saint Petersburg, postcode=198504, country=Russian
Federation

# The Bohr Radius of the Weighted Bloch Spaces

Khasyanov R.Sh.11footnotemark: 1<EMAIL_ADDRESS>
###### Abstract

The concept of the Bohr radius of a pair of Banach spaces is introduced. The
lower estimate for the value of the Bohr radius from the Bloch space to the
space of bounded functions obtained by I. Kayumov, S. Ponnusamy and N.
Shakirov is slightly improved. It is shown that for any weighted Bloch space
the Bohr radius is not less than $1/\sqrt{2}$. A criterion for the sharpness
of the Bohr inequality in the weighted Bloch space with $R=1/\sqrt{2}$ is
obtained. Using this criterion, the examples of the weights for which the
inequality is sharp is given.

###### keywords:

Bohr radius, weighted Bloch space, finite Blaschke product

##  1 Introduction and main results

In 1914, H. Bohr, studying Dirichlet series, noticed [7] the following
interesting fact in complex analysis, which is now called the Bohr phenomenon:

######  Theorem A.

Let $f(z)=\sum_{n\geq 0}a_{n}z^{n}$ and
$\|f\|_{\infty}:=\sup_{z\in\mathbb{D}}|f(z)|\leq 1$ in the unit disc
$\mathbb{D}=\\{|z|<1\\}$. Then

$\sum_{n\geq 0}|a_{n}|r^{n}\leq 1,\quad 0\leq r\leq 1/3.$

The constant $1/3$ is sharp.

In fact, Bohr proved this for $r\leq 1/6$. The best constant $1/3$ was
obtained independently in the same year by M. Riesz, I. Schur and F. Wiener
(for different proofs see e.g. appendix in [11]). This theorem is equivalent
to the inequality known as the Bohr inequality:

$\sum_{n\geq 0}|a_{n}|r^{n}\leq\|f\|_{\infty},\quad 0\leq r\leq 1/3.$

An active study of various modifications and generalizations of the Bohr
inequality began in the middle of 1990s since P. Dixon, using the Bohr
inequality, solved a long-standing problem about the characterization of
Banach algebras [11]. Part of the subsequent research in this area is directed
towards extending the Bohr phenomenon in multidimensional framework and in
more abstract settings (see e.g. [2], [5], [6], [16]). In 2018 B. Bhowmik and
N. Das applied the Bohr inequality to the question of comparing majorant
series of subordinating functions [4]. For studies related to the Bohr
inequality, see, for example, [1], [4]-[10], [12]-[16].

We consider the following questions:

######  Problem 1.

Fix a holomorphic function $f(z)=\sum_{n\geq 0}a_{n}z^{n}$ and consider the
class $\mathcal{F}$ of all analytic functions of the form $\sum_{n\geq
0}b_{n}z^{n}$ such that $|b_{n}|=|a_{n}|$ for all $n\geq 0.$ Which properties
of $f$ are inherited by all functions in the class $\mathcal{F}$? In other
words, what properties of a holomorphic function can be detected from the
moduli of its Maclaurin series coefficients?

For instance, the radius of convergence of the Maclaurin series and the Hardy
space $H^{2}$ norm depend only on the moduli of the coefficients. On the other
hand, there are examples of properties that may hold for some members of
$\mathcal{F}$ but not for others: the series $\sum_{n\geq 1}\dfrac{\pm
z^{n}}{n}$ are bounded in the $\mathbb{D}$ for almost every choice of plus and
minus signs ([17], Chapter V, Theorem 8.34), but if all plus signs are taken,
the series is unbounded in the unit disk. Bohr’s theorem implies that if
$|f(z)|\leq 1$ in the unit disc $\mathbb{D}$, then $|g(z)|\leq 1$ in the disk
of radius $1/3$ for all functions $g\in\mathcal{F}$.

######  Problem 2.

Let $X,Y$ be Banach spaces of analytic functions in the disc
$\mathbb{D}=\left\\{|z|<1\right\\}$. Let $f(z)=\sum_{n\geq 0}a_{n}z^{n}$. We
need to find the maximal $R$ for which

$\|f\|_{X}\leq 1\Longrightarrow\|\sum_{n\geq 0}|a_{n}|(Rz)^{n}\|_{Y}\leq 1.$

We call such $R$ the Bohr radius from $X$ to $Y$ and denote $R_{X\rightarrow
Y}$. If $X$ and $Y$ coincide, we write $R_{X}.$

The classical Bohr theorem states that $R_{H^{\infty}}=1/3.$ We study Bohr
radius in Bloch spaces.

######  Definition 1.

We say that the analytical function in $\mathbb{D}$ $f(z)=\sum_{n\geq
0}a_{n}z^{n}$ belongs to the weighted Bloch space $\mathcal{B}(\omega)$ with
the weight $\omega(r)\geq 0$ if

$\|f\|_{\mathcal{B}(\omega)}:=|a_{0}|+\sup_{z\in\mathbb{D}}\omega(|z|)|f^{\prime}(z)|<\infty.$

For the standart weight $\omega(r)=1-r^{2}$ we just write $\mathcal{B}$.

We start with the question of finding the value of $R_{\mathcal{B}\rightarrow
H^{\infty}}$. This problem has already been studied in [13], where the
following lower and upper estimates are obtained: $0.624162\ldots\geq
R_{\mathcal{B}\rightarrow H^{\infty}}\geq R$, where $R\approx 0.55356\ldots$
is the solution of the equation $1-r+r\log(1-r)=0$. We noticed that the proof
of the theorem can be slightly improved and we get a lower estimate about
$0.01$ better. Note that in [1] V. Allu and H. Halder considered this problem
for functions defined on simply connected domains. As a consequence, they
obtained a slightly less sharp lower estimate for the Bohr radius from the
Bloch space to $H^{\infty}:R_{\mathcal{B}\rightarrow H^{\infty}}\geq
0.546679$.

######  Theorem 1.

$R_{\mathcal{B}\rightarrow H^{\infty}}\geq 0.563777.$

Next, we consider the problem of finding the Bohr radius of the weighted Bloch
spaces. The following theorem immediately follows from the Cauchy inequality.

######  Theorem 2.

Let $\mathcal{B}(\omega)$ be Bloch space with the weight $\omega(r)\geq 0$.
Then

$R_{\mathcal{B}(\omega)}\geq 1/\sqrt{2}.$

We ask a natural question: for a given weight, is the Bohr inequality with
$R=1/\sqrt{2}$ sharp? Using the method of E. Bombieri and J. Bourgain [8], we
give a criterion for the sharpness of such an inequality.

######  Theorem 3.

The inequality

$\Big{|}\Big{|}\sum_{n\geq
0}|a_{n}|\Big{(}\dfrac{z}{\sqrt{2}}\Big{)}^{n}\Big{|}\Big{|}_{\mathcal{B}(\omega)}\leq||f||_{\mathcal{B}(\omega)}$
$None$

is sharp, i.e. turns to equality for some nonzero $f\in\mathcal{B}(\omega)$,
if and only if there exists $r_{0}\in\Big{[}\dfrac{1}{\sqrt{2}},1\Big{]}$,
such that for all $r\in[0,1)$ the inequality

$\dfrac{\omega(r)}{\omega(r_{0})}\leq\min{\Big{(}2-\dfrac{r}{r_{0}};\dfrac{\sqrt{2}r_{0}+r}{\sqrt{2}r+r_{0}}\Big{)}}.$
$None$

holds. Extreme functions up to multiplication by a constant have the following
form:

$\int_{c}^{z}\dfrac{z/r_{0}-e^{i\phi}/\sqrt{2}}{1-e^{-i\phi}z/(\sqrt{2}r_{0})}dz+C,\quad\phi\in[0,2\pi),\>c\in\mathbb{D},\>C\in\mathbb{C}.$

Using Theorem 3, we can find the examples of the weights for which the
inequality (1) is sharp. We fix $r_{0}\in[1/\sqrt{2},1]$ and denote
$\omega_{1}(r):=2-\dfrac{r}{r_{0}},\>\omega_{2}(r):=\dfrac{\sqrt{2}r_{0}+r}{\sqrt{2}r+r_{0}}.$
Note that $\omega_{2}(r)\leq\omega_{1}(r)$ for $r\leq r_{0}$ and
$\omega_{1}(r)\leq\omega_{2}(r)$ for $r\geq r_{0}$. Let
$\tilde{\omega}(r):=\dfrac{\omega(r)}{\omega(r_{0})},$ then
$\tilde{\omega}(r)\leq h(r):=\min{(\omega_{1}(r),\omega_{2}(r))}$ and
$\tilde{\omega}(r_{0})=1.$ It is easy to see, that $h(r)$ is a decreasing and
convex function for $r\in[0,r_{0}).$ So we can drawn a plot of the function
$h(r):$

We see that there is no smooth function $\tilde{\omega}(r)$ with the necessary
properties and $r_{0}<1$. That is why any smooth weight does not satisfy the
condition (2). However, we can give examples of weights that satisfy condition
(2):

Examples.

1\. $\omega(r)\equiv 1,\>r_{0}=1.$

Fix $r_{0}\in[1/\sqrt{2},1)$ and $\alpha\geq 1.$

2\. The weight

$\omega(r)=\begin{cases}1,&r\leq r_{0}\\\
\Big{(}\dfrac{1-r}{1-r_{0}}\Big{)}^{\alpha},&r\geq r_{0}\end{cases}$

satisfies the condition (2).

3\. The weight

$\omega(r)=\Big{(}1-\Big{|}\dfrac{r-r_{0}}{1-r_{0}r}\Big{|}\Big{)}^{\alpha}$

also satisfies the condition (2).

Next we get the upper bound for the Bohr radius in classical Bloch space,
using a function from the work of F. Avkhadiev and I. Kayumov [3].

######  Theorem 4.

Let $\mathcal{B}$ be the classical Bloch space with the weight
$\omega(r)=1-r^{2}$. Then

$0.707...\leq R_{\mathcal{B}}\leq 0.769...$

######  Problem 3.

Let $X$ be Banach space of analytic functions in the unit disc,
$f(z)=\sum_{n\geq 0}a_{n}z^{n}$. We need to find

$m_{X}(r):=\sup_{f\in X}\dfrac{\Big{\|}\sum_{n\geq
0}|a_{n}|(rz)^{n}\Big{\|}_{X}}{\|f\|_{X}}.$

In 1962 E. Bombieri showed ([9] or [12, p. 170–178]) that for $X=H^{\infty}$

$m_{\infty}(r):=m_{H^{\infty}}(r)=\dfrac{3-\sqrt{8(1-r^{2})}}{r},\quad 1/3\leq
r\leq 1/\sqrt{2}.$

This theorem was proved by I. Kayumov and S. Ponnusamy in a simpler way in the
work [14].

It immediately follows from the Cauchy inequality that
$m_{{\infty}}(r)\leq\dfrac{1}{\sqrt{1-r^{2}}}:$

$\sum_{n\geq 0}|a_{n}|r^{n}\leq\Big{(}{{\sum_{n\geq
0}|a_{n}|^{2}}}\Big{)}^{1/2}\Big{(}{\sum_{n\geq
0}r^{2n}}\Big{)}^{1/2}=\dfrac{\|f\|_{2}}{\sqrt{1-r^{2}}}\leq\dfrac{\|f\|_{\infty}}{\sqrt{1-r^{2}}}.$

In the 2004, E. Bombieri and J. Bourgain proved [8] the following two profound
theorems related to the inequality above.

######  Theorem B.

For all $r\in(0;1)$ except $r=1/\sqrt{2}$

$m_{{\infty}}(r)<\dfrac{1}{\sqrt{1-r^{2}}},\>\>\>\>\>m_{{\infty}}(1/\sqrt{2})=\sqrt{2}.$

######  Theorem C.

For all $\varepsilon>0$ there exists $C(\varepsilon)>0$, such that as
$r\rightarrow 1$ the lower bound

$m_{{\infty}}(r)\geq\dfrac{1}{\sqrt{1-r^{2}}}-C(\varepsilon)\Big{(}\log\dfrac{1}{1-r}\Big{)}^{\varepsilon+3/2}$

holds

Theorem C implies that $m_{{\infty}}(r)\rightarrow\infty$ when $r\rightarrow
1$. It is clear from the proof of Theorem 1 that
$m_{\mathcal{B}}(r)\leq\dfrac{r}{\sqrt{1-r^{2}}}$. Using the construction
already discussed in the proof of Theorem 2, we prove an analogue of Theorem B
for the Bloch space:

######  Theorem 5.

$m_{\mathcal{B}}(R)<\dfrac{R}{\sqrt{1-R^{2}}},\quad 0<R<1.$

##  2 Proofs

###### Proof of Theorem 1.

If $\|f\|_{\mathcal{B}}\leq 1$, then
$|f^{\prime}(z)|^{2}\leq\dfrac{1}{(1-|z|^{2})^{2}},\>z\in\mathbb{D}$.

We integrate this inequality over the circle $|z|=r$ and obtain from Parceval
formula

$\sum_{n\geq 1}n^{2}|a_{n}|^{2}r^{2n}\leq\dfrac{r^{2}}{(1-r^{2})^{2}},\quad
0<r<1.$

Hence

$\sum_{n\geq 1}n^{2}|a_{n}|^{2}r^{n}\leq\dfrac{r}{(1-r)^{2}}.$

We may now divide this inequality by r and integrate with respect to $r$ with
limit from $0$ to $r$:

$\sum_{n\geq 1}n|a_{n}|^{2}r^{n}\leq\dfrac{r}{1-r}.$

Replace $r$ with $r^{s}$:

$\sum_{n\geq 1}n|a_{n}|^{2}r^{ns}\leq\dfrac{r^{s}}{1-r^{s}},\quad 0<r,s<1.$

Thus,

$\sum_{n\geq 1}|a_{n}|r^{n}\leq\Big{(}{\sum_{n\geq
1}\dfrac{r^{2ns}}{n}}\Big{)}^{1/2}\Big{(}{\sum_{n\geq
1}n|a_{n}|^{2}r^{2n(1-s)}}\Big{)}^{1/2}\leq\Big{(}{\dfrac{r^{2(1-s)}\log(1-r^{2s})}{r^{2(1-s)}-1}}\Big{)}^{1/2}.$

Consequently, $\sum_{n\geq 1}|a_{n}|r^{n}\leq 1$ with $r\leq R$, where $R$ is
solution of the equation

$\log(1-r^{2s})=\dfrac{r^{2(1-s)}-1}{r^{2(1-s)}}.$

Using Wolfram Mathematica, we deduce that the best possible $r$, which can be
achieved in this way is obtained at $s=0.333771$ and is equal to $0.563777.$ ∎

Unfortunately, there is no extreme function for $R=0.563777$, since if such a
function $f(z)=\sum_{n\geq 1}a_{n}z^{n}$ existed, then

$|a_{n}|=C\dfrac{R^{n(2s-1)}}{n},\quad n\geq 1$

Then the function $f(z)$ is analytic in $\mathbb{D}$ only for $s\geq 1/2$.

###### Proof of Theorem 2.

In what follows we will solve an equivalent problem for weighted norms of
functions in place of their derivatives. Note that
$(f(Rz))^{\prime}=Rf^{\prime}(Rz)$. Thus, to find the Bohr radius in the Bloch
space, it is sufficient to solve the following problem: find the maximal $R$
at which

$\sup_{{\begin{subarray}{c}r\in(0;1)\end{subarray}}}\omega(r)\|f(re^{i\theta})\|_{L^{\infty}[0,2\pi)}\leq
1\Longrightarrow\\\ \sup_{r\in(0;1)}\omega(r)R\sum_{n\geq
0}|a_{n}|(Rr)^{n}\leq 1,$

where $f(z)=\sum_{n\geq 0}a_{n}z^{n}$.

Let us show that $R_{\mathcal{B}(\omega)}\geq\dfrac{1}{\sqrt{2}}:$

$\omega(r)R\sum_{n\geq 0}|a_{n}|(Rr)^{n}\leq\omega(r)R\Big{(}{\sum_{n\geq
0}|a_{n}|^{2}r^{2n}}\Big{)}^{1/2}\Big{(}{\sum_{n\geq
0}R^{2n}}\Big{)}^{1/2}=\hskip 147.95433pt(4)\\\
=\omega(r)\|f(re^{i\theta})\|_{L^{2}[0,2\pi]}\dfrac{R}{\sqrt{{1-R^{2}}}}\leq\omega(r)\|f(re^{i\theta})\|_{L^{\infty}[0,2\pi]}\dfrac{R}{\sqrt{{1-R^{2}}}}.$

The inequality $\dfrac{R}{\sqrt{{1-R^{2}}}}\leq 1$ is equivalent to
$R\leq\dfrac{1}{\sqrt{2}}$, hence
$R_{\mathcal{B}(\omega)}\geq\dfrac{1}{\sqrt{2}}.$ ∎

###### Proof of Theorem 3.

As before, instead of derivatives of functions, we consider the functions
themselves. Thus, we need to show that a nonzero function $f(z)=\sum_{n\geq
0}a_{n}z^{n}$ such that

$\sup_{r\in(0;1)}\dfrac{\omega(r)}{\sqrt{2}}\sum_{n\geq
0}|a_{n}|\Big{(}\dfrac{r}{\sqrt{2}}\Big{)}^{n}=\sup_{{\begin{subarray}{c}r\in(0;1)\end{subarray}}}\omega(r)\|f(re^{i\theta})\|_{L^{\infty}[0,2\pi)}<\infty.\hskip
8.53581pt$ $None$

exist if and only if the weight $\omega$ satisfies (2).

Necessity. Assume that equality (5) holds for some $f$. Let us show that there
exists $r_{0}\in[0,1]$ with the following properties:

1\. $r_{0}$ is the point where the supremum of the left-side in (5) is
attained:

$\omega(r)\sum_{n\geq
0}|a_{n}|\Big{(}\dfrac{r}{\sqrt{2}}\Big{)}^{n}\leq\omega(r_{0})\sum_{n\geq
0}|a_{n}|\Big{(}\dfrac{r_{0}}{\sqrt{2}}\Big{)}^{n},\quad 0\leq r<1.$

2\. The norm of $f$ is also attained at $r_{0}:$

$\omega(r)\|f(re^{i\theta})\|_{L^{\infty}[0,2\pi)}\leq\omega(r_{0})\|f(r_{0}e^{i\theta})\|_{L^{\infty}[0,2\pi)},\quad
0\leq r<1.$

3\. The Cauchy inequality in the chain (4) turns into equality which is
equivalent to

$|a_{n}|r_{0}^{n}=C(r_{0})\Big{(}\dfrac{1}{\sqrt{2}}\Big{)}^{n},\quad n\geq
0,$

where $C(r_{0})$ is a constant depending only on $r_{0}$.

4\. The last inequality in the chain (4) is equality, that is

$\|f(r_{0}e^{i\theta})\|_{L^{2}[0,2\pi)}=\|f(r_{0}e^{i\theta})\|_{L^{\infty}[0,2\pi)}.$

Define $r_{0}$ by the condition 1. We substitute $r=r_{0}$ and $R=1/\sqrt{2}$
in (4) and use the condition 1 and equality (5):

$\omega(r_{0})\|f(r_{0}e^{i\theta})\|_{L^{\infty}[0,2\pi)}\geq\dfrac{\omega(r_{0})}{\sqrt{2}}\sum_{n\geq
0}|a_{n}|\Big{(}\dfrac{r_{0}}{\sqrt{2}}\Big{)}^{n}\geq\\\
\geq\sup_{r\in(0;1)}\dfrac{\omega(r)}{\sqrt{2}}\sum_{n\geq
0}|a_{n}|\Big{(}\dfrac{r}{\sqrt{2}}\Big{)}^{n}=\sup_{r\in(0;1)}\omega(r)\|f(re^{i\theta})\|_{L^{\infty}[0,2\pi)}.$

Thus, condition 2 is also satisfied. Conditions 3 and 4 following from the
fact that all inequalities in (4) are equalities for $r=r_{0}$ and
$R=1/\sqrt{2}.$

It follows from the condition 3 that the radius of convergence for $f$ is
$\sqrt{2}r_{0}$, where $r_{0}\geq 1/\sqrt{2}.$ It is known that the equality
$\|f(r_{0}e^{i\theta})\|_{L^{2}[0,2\pi)}=\|f(r_{0}e^{i\theta})\|_{L^{\infty}[0,2\pi)}$
holds only for functions

$f(z)=K(r_{0})B\Big{(}\dfrac{z}{r_{0}}\Big{)},$

where the function $B(z)$ is a finite Blaschke product (see, for example,
[12]). Without loss of generality, we can assume that $|K(r_{0})|=1$.

We find $C(r_{0})$. Let

$B\Big{(}\dfrac{z}{r_{0}}\Big{)}=\prod_{i=1}^{K}\Big{(}\dfrac{z/r_{0}-\alpha_{i}}{1-\bar{\alpha_{i}}z/r_{0}}\Big{)}^{d_{i}},\hskip
19.91692pt\sum_{i=1}^{K}d_{i}=d,\hskip
14.22636pt\alpha_{i}\in\mathbb{D}\>(1\leq i\leq K).$

Denote by $dA(z)$ the standard Lebesgue measure on the plane. From the formula
for the area of the image of the finite Blaschke product, we can calculate its
degree:

$\displaystyle
d=\dfrac{1}{\pi}\int_{\mathbb{D}}\big{|}B^{\prime}(z)\big{|}^{2}dA(z)=\dfrac{1}{\pi}\int_{\mathbb{D}}\big{|}(f(r_{0}z))^{\prime}\big{|}^{2}dA(z)=$
$\displaystyle=\sum_{n\geq 1}n|a_{n}|^{2}r_{0}^{2n}=\sum_{n\geq
1}\dfrac{n}{2^{n}}C^{2}(r_{0})=2C^{2}(r_{0}).$

Therefore $C(r_{0})=\sqrt{d/2}.$ Note that

$\sqrt{d/2}=C(r_{0})=|a_{0}|=|f(0)|=\Big{|}\prod_{i=1}^{K}\alpha_{i}^{d_{i}}\Big{|}<1,$
$None$

hence $d=1$. If $d=1$, then from (6), $|\alpha_{1}|=1/\sqrt{2}$ and

$f(z)=\dfrac{z/r_{0}-e^{i\phi}/\sqrt{2}}{1-e^{-i\phi}z/(\sqrt{2}r_{0})}.$

Without loss of generality, we consider $\phi=0$. In this case, the supremum
$|f(re^{i\theta})|$ is reached when $\theta=\pi$,

$\|f(rz)\|_{L^{\infty}[0,2\pi)}=|f(-r)|=\dfrac{r/r_{0}+1/\sqrt{2}}{1+r/(\sqrt{2}r_{0})}$

and

$\sum_{n\geq
0}|a_{n}|\Big{(}\dfrac{r}{\sqrt{2}}\Big{)}^{n}=\dfrac{1}{\sqrt{2}}\dfrac{1}{1-r/(2r_{0})}.$

Now it follows from the condition 1 that

$\dfrac{\omega(r)}{\omega(r_{0})}\leq 2-\dfrac{r}{r_{0}},$

while condition 2 implies

$\dfrac{\omega(r)}{\omega(r_{0})}\leq\dfrac{\sqrt{2}r_{0}+r}{\sqrt{2}r+r_{0}}.$

We showed that the weight $\omega$ satisfies (2).

Sufficiency. Define $f$ by

$f(z)=\dfrac{z/r_{0}-e^{i\phi}/\sqrt{2}}{1-e^{-i\phi}z/(\sqrt{2}r_{0})}.$

Then $f$ automatically satisfies condition 3 and 4, while condition (2) on the
weight implies that $f$ also satisfies condition 1 and 2. Hence for $f$
condition 1 holds with both suprema attained at $r=r_{0}.$∎

###### Proof of Theorem 4.

We consider the function $f(z)=\sum_{n\geq 0}a_{n}z^{n}$ from [3]:

$f(z)=\dfrac{3\sqrt{3}}{2}\dfrac{1-a^{2}}{(1-az)^{2}}\Big{(}\dfrac{z-a}{1-az}\Big{)},\quad
0<a<1/\sqrt{3}.$

It is not difficult to show that $\sup_{z\in\mathbb{D}}(1-|z|^{2})|f(z)|=1$.
Obviously, $a_{0}=-\frac{3\sqrt{3}a(1-a^{2})}{2}<0$. Let us show that if
$0<a<1/\sqrt{3}$, then $a_{n}>0,\>n\geq 1$. Denote
$\tilde{a}=\dfrac{a^{2}}{1-a^{2}}$ and let

$g(w)=\sum_{n\geq
0}b_{n}w^{n}=\dfrac{1}{(1-w)^{2}}\Big{(}\dfrac{w}{1-w}-\tilde{a}\Big{)}=\sum_{n\geq
0}(n+1)(n/2-\tilde{a})w^{n}.$

Since $f(z)=C(a)g(az),\>C(a)>0,$ the condition $a_{n}>0$ is equivalent to
$b_{n}>0.$ But $b_{n}=(n+1)(n/2-\tilde{a})>0$ for all $n\geq 1$ if and only if
$0<a<1/\sqrt{3}.$

Thus,

$\sum_{n\geq
0}|a_{n}|z^{n}=\dfrac{3\sqrt{3}(1-a^{2})}{2}\Big{(}\dfrac{z-a}{(1-az)^{3}}+2a\Big{)}$

and

$R(1-r^{2})\sum_{n\geq
0}|a_{n}|(Rr)^{n}=R(1-r^{2})\dfrac{3\sqrt{3}(1-a^{2})}{2}\Big{(}\dfrac{Rr-a}{(1-aRr)^{3}}+2a\Big{)}.$

Using Wolfram Mathematica again, we find that the last expression is greater
than one, for example, at $a=0.35$ and $R=0.769$, so $R_{\mathcal{B}}\leq
0.769$. ∎

###### Proof of Theorem 5.

Suppose that the expression $(1-r^{2})\sum_{n\geq
0}|a_{n}|\Big{(}\dfrac{r}{\sqrt{2}}\Big{)}^{n}$ reaches the supremum at the
point $r_{0}$. We replace all the inequalities in the chain (4) with
equalities for $r=r_{0}$. Then, as in the proof of Theorem 3, it will follows
from the sharpness condition of the Cauchy’s inequality that

$|a_{n}|r_{0}^{n}=C(r_{0})R^{n},\quad n\geq 0,$

so $r_{0}\geq R$. It follows from the equality of norms that
$f(z)=B\Big{(}\dfrac{z}{r_{0}}\Big{)}$, where $B(z)$ is finite Blaschke
product that is

$f(z)=B\Big{(}\dfrac{z}{r_{0}}\Big{)}=\prod_{i=1}^{K}\Big{(}\dfrac{z/r_{0}-\alpha_{i}}{1-\bar{\alpha_{i}}z/r_{0}}\Big{)}^{d_{i}},\>\>\>\sum_{i=1}^{K}d_{i}=d.$

It follows from the formula for the area of the image of the finite Blaschke
product that

$C(r_{0})=\sqrt{d}\Big{(}\dfrac{1-R^{2}}{R}\Big{)}.$

Since we have replaced all the inequalities with equalities, then

$(1-r^{2})|f(r)|\leq(1-r_{0}^{2})|f(r_{0})|=1-r_{0}^{2},\quad 0\leq r<1.$

We substitute $r=0$ into the left part of the inequality above:

$|f(0)|=C(r_{0})=\sqrt{d}\Big{(}\dfrac{1-R^{2}}{R}\Big{)}\leq 1-r_{0}^{2}\leq
1-R^{2},$

which is impossible for all $R\in(0;1)$. ∎

Acknowledgments. I would like to thank Professor A.D. Baranov for formulation
of the problem and for valuable discussions, Professor I.R. Kayumov for useful
comments and M.S. Ivanova for help with examples of weights.

The work is supported by Ministry of Science and Higher Education of the
Russian Federation (agreement No 075-15-2021-602) and by Theoretical Physics
and Mathematics Advancement Foundation ”BASIS”.

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