# All-fibre wavefront sensor

T. A. Wright,1 S. Yerolatsitis,1,2 K. Harrington,1 R. J. Harris,3 and T. A.
Birks1
1Centre for Photonics and Photonic Materials, Department of Physics,
University of Bath, Bath, BA2 7AY, UK
2Now at CREOL, The College of Optics and Photonics, University of Central
Florida, Orlando, Florida 32816, USA
3Max-Planck-Institute for Astronomy, Königstuhl 17, 69117, Heidelberg, Germany
E-mail<EMAIL_ADDRESS>E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We report on a tapered three-core optical fibre that can be used as a tip-tilt
wavefront sensor. In this device, a coupled region of a few millimetres at the
sensing tip of the fibre converts fragile phase information from an incoming
wavefront into robust intensity information within each of the cores. The
intensity information can be easily converted to linear wavefront error over
small ranges, making it ideal for closed loop systems.The sensor uses minimal
information to infer tip-tilt and is compatible with remote detector arrays.
We explore its application within adaptive optics and present a validation
case to show its applicability to astronomy.

###### keywords:

instrumentation: adaptive optics

††pubyear: 2022††pagerange: All-fibre wavefront sensor–A

##  1 Introduction

To overcome the limits imposed by the atmosphere, it is common for ground-
based telescopes use adaptive optics (AO) to correct the effect of atmospheric
turbulence on incoming light from astronomical objects. This implementation of
AO systems has led to numerous previously unachievable scientific discoveries,
including the determination of stellar orbits around in the centre of the
Milky Way (Ghez et al., 1998) and direct imaging detection of planetary
systems around other stars (Chauvin, G. et al., 2004). AO systems also offer
significant advantages to applications outside of astronomy. AO can correct
for atmospheric aberrations in other applications, such as in free-space
optical communications (Weyrauch & Vorontsov, 2005; Gregory et al., 2012), or
even produce ultrafast laser filamentation for remote location sensing (Daigle
et al., 2009). AO has also been used to mitigate aberrations caused by the
refractive index structure of samples in a range of biomedical applications
such as ophthalmology (Akyol et al., 2021), in-vivo imaging (Zawadzki et al.,
2005) and microscopy (Booth, 2014).

For the AO system to construct the optimal image in the focal plane, two key
components are required: a method for sensing the incoming wavefront, and then
a method for correcting it. Corrections to the incident wavefronts are
commonly enacted by AO systems using one or more deformable mirrors (DMs). For
large telescope installations with strict wavefront correction requirements, a
state-of-the-art DM may be driven by thousands of actuators, each able to
apply a small local perturbation to the mirror on millisecond time scales
(Guyon, 2018, table 4).

The specific instructions for these actuators must be rapidly generated in
response to the incoming wavefront’s existing state as measured by the
wavefront sensor (WFS). Accordingly, the removal of wavefront distortions is
limited by how well the WFS describes the current state of the wavefront and
how quickly the DM receives and acts on this information. It is common
practice to position the WFS in the pupil plane by reimaging the incident
wavefront via a dichroic beam splitter. One problem with this approach is that
it can introduce path-difference aberrations caused by non-common optical
components in the wavefront-sensing and science beams (Sauvage et al., 2007).
However, for convenience most WFSs in use do sense the wavefront in the pupil
plane, including the Shack–Hartmann (SH) WFS (Platt & Shack, 2001), the
pyramid WFS (Ragazzoni, 1996), and the curvature WFS (Roddier, 1988). Phase-
diversity WFSs, while uncommon in astronomy, are an exception to this trend
and operate in the focal plane, thus avoiding path-difference errors
(Gonsalves & Chidlaw, 1979; Paxman et al., 2007). We refer the interested
reader to the reviews of AO systems given by Davies & Kasper (2012) and Rigaut
(2015).

Recent studies have detailed attempts to exploit advances in fibre-optic and
waveguide technologies to introduce alternative WFS implementations (Corrigan
et al., 2016; Corrigan et al., 2018; Norris et al., 2020). These proposed
techniques make use of photonic lanterns (Birks et al., 2015) to acquire both
phase and amplitude information directly from the focal plane, although the
recovery of the wavefront is non-trivial. Norris et al., for example, use a
deep neural network to reconstruct information on the input phase from the
output intensities, such that it is difficult to recover the performance
metrics for WFS. More straightforward approaches to recovering tip-tilt
information using a photonic lantern are possible (Cruz-Delgado et al., 2021).
However, in such cases, the complex transfer matrix for each lantern still
needs to be solved numerically. Furthermore, photonic lanterns are not
wavelength agnostic and will likely prove challenging to scale up to measure
complex wavefront distortions and to have larger detection windows. This
requirement will necessitate using higher mode numbers, making each device
harder to calibrate, and using multiple lanterns that would each require
individual calibration. Nevertheless, fibre- and waveguide-based techniques
provide further advantages, such as allowing users to remap the output of the
WFS into any shape, including 1:1 mode-to-pixel configurations. In contrast to
the bulky two-dimensional pixel arrays currently used for Shack–Hartmann WFSs
(SH-WFSs), these devices are compatible with linear detector arrays located
far from the pupil or focal plane.

We report an easily-scalable all-fibre tip-tilt sensor design based on a
locally tapered three-core fibre and demonstrate operation in the pupil plane.
At the tapered tip, our sensor uses minimal information to infer tip-tilt and
converts phase information to amplitude information. This information is then
guided through the uncoupled three-core fibre, making the sensor compatible
with remote detector arrays. We present experimental verification of the
photonics by detecting a diverging beam at a large distance, approximating a
flat wavefront. We make these measurements without focusing optics ahead of
the fibre probe to make the experimental results more general and not specific
to a particular telescope arrangement. Later, we discuss the WFS’s response to
light in the context of real-world telescope integration and, particularly,
how the angular sensitivity is relevant to astronomy. Furthermore, we indicate
how to upscale our WFS design to an array of tip-tilt sensors that can deduce
higher-order wavefront aberrations.

##  2 Principle of Operation

In an optical system, phase measurements at any three non-collinear points are
sufficient to determine the wavefront tip and tilt. In principle, we can use a
three-core fibre to transport these phases to a remote location, where they
can be analysed, e.g., interferometrically. In practice, the phase information
will be lost due to time-dependent and effectively-random inhomogeneities,
bends and environmental disturbances to the optical fibre. However, we can
exploit core-to-core coupling in a short input section to convert the phase
information into amplitude information, which is robustly preserved along the
arbitrarily-disturbed fibre length if the fibre is designed so that the cores
are uncoupled everywhere except at the input.

Depicted in Fig. 1 (a), our design consists of a fibre with three identical
cores at the corners of an equilateral triangle. We taper (narrow down a short
length of) the fibre to locally induce coupling, then cleave it to provide an
input end-face: the sensing tip. The spacing of the three cores is otherwise
sufficiently large that no coupling occurs between them along the fibre beyond
this short tapered region. At the output, we can then monitor the powers in
the cores using an image sensor or, in principle, three photodiodes. Figure 1
(a) also shows parameters describing the tilt of the wavefront incident on the
fibre end-face: the angle of tilt is $\theta$, and the azimuth of the plane of
incidence is $\alpha$. The device’s response, manifested as different powers
at each core’s output, can be represented by an intensity-weighted ‘centroid’
position

$\mathbf{r}=\frac{P_{1}\mathbf{r}_{1}+P_{2}\mathbf{r}_{2}+P_{3}\mathbf{r}_{3}}{P_{1}+P_{2}+P_{3}}.$
(1)

This abstract space is spanned by core position vectors
$\mathbf{r}_{\mathrm{n}}$ on a unit circle, as shown in Fig. 1 (b) where
$\mathbf{r}$ is expressed in polar coordinates $(\rho,\phi)$. The full
analytical relation between the parameters sought $(\theta,\alpha)$ and
measured $(\rho,\phi)$ is described in the Appendix. For sufficiently small
tilt angles $\theta$ it can be shown that

$\displaystyle\rho=\delta,$ $\displaystyle\phi=\alpha,$ (2)

and thus, the full expression reduces to

$\delta=\left[\frac{2\pi}{\sqrt{3}}\frac{\Lambda}{\lambda}\sin\left(\frac{2\pi
z}{L_{\mathrm{C}}}\right)\right]\theta.$ (3)

Here $\delta$ is the tilt angle $\theta$ scaled by a function of the tapered
fibre’s core separation $\Lambda$, length $z$ and coupling length
$L_{\mathrm{C}}$, along with the free-space wavelength $\lambda$ of the light.
That is, to a first-order approximation, the displacement, $\rho$, of the
centroid is proportional to the input tilt, $\theta$, and the azimuths of the
centroid and the plane of incidence are the same. The sensitivity of $\rho$ to
$\theta$ is maximised when the length is 1/4 of the coupling length:
$z/L_{\mathrm{C}}=1/4$. From here on, we will assume this to be the case.

Figure 1: (a) The all-fibre wavefront sensor. The coupled and transition
sections are about a centimetre long, while the uniform fibre with uncoupled
cores can be several metres long. (b) The centroid of the output powers (red
cross) with polar coordinates $(\rho,\phi)$ in a space spanned by normalised
core positions, which matches the weighted centroid of a near-field image at
the fibre output. (c) Simulated polar plots of output centroid $(\rho,\phi)$,
in red, for conically precessing inputs $(\delta,\alpha)$, in blue. (d)
Simulated polar plots of output centroid $(\rho,\phi)$, in red, for a square
array of input tilts $(\delta,\alpha)$, in blue. We assume
$z/L_{\mathrm{C}}=1/4$.

To illustrate, Fig. 1 (c) depicts the loci of a conically precessing input
(i.e. fixed $\delta$ and varying $\alpha$) in blue and the resulting output
centroid in red. Five input values of $\delta$, labelled, are plotted. Figure
1 (d) shows how a square grid of input states maps onto a grid of output
centroids. As seen in Fig. 1 (c) and Fig. 1 (d), the first-order approximation
starts to fail around $\delta=0.2$, but the relationship remains analytic.
Beyond $\delta=0.8$, we find that the measured centroid folds in on itself,
and by $\delta=1.6$, the sign of the centroid switches. Therefore, we have
three correction regimes, the first of which is linear $(\delta<0.2)$. The
second regime $(0.2<\delta<0.8)$ is no longer linear, but the correction is
analytically derivable and the magnitude is correct. The third regime
$(0.8<\delta<1.6)$ is also analytical, and the sign of correction is correct,
but the magnitude is not. Beyond these three regimes, when $\delta>1.6$, the
correction sign changes making the system unstable. Such limits are not unique
to our device; this effect is similar to spots wandering between subapertures
in a Shack-Hartmann, or non-illumination of one or more quadrants on a pyramid
WFS.

The operation of our WFS is notable for using the least information possible
(three powers) to infer tip-tilt. Furthermore, the mapping between the input
wavefront perturbation and the output parameters is computationally
straightforward. The simplicity of this mapping is in contrast to other fibre
wavefront sensors (Corrigan et al., 2016; Corrigan et al., 2018; Norris et
al., 2020), and would be advantageous when attempting to mitigate latency
within an integrated astronomical AO system.

##  3 Fabrication of the Fibre and WFS

We fabricated the three-core fibre using the stack-and-draw technique
(Russell, 2003). Each core has a diameter of 9.5 $\upmu$m and an NA of 0.11
(single-mode for $\lambda>1.37\,\upmu$m) in the final fibre. For all of our
optical measurements, we used a light source with a wavelength of 1.55
$\upmu$m. The separation between each core is 38 $\upmu$m, which is sufficient
for us to observe no coupling between cores along a 100-m length of the fibre.
This length exceeds any requirement for remote detection in astronomy as it is
also vital to minimise latency in the data transmission from a WFS to the
deformable mirror. An optical micrograph of the fibre cross-section is inset
within Fig. 3(b).

Figure 2: (a) Schematic of the WFS fabrication principle. (b) Simulated
variation with length of the output powers in the cores, for input in core 1
only. We taper the three-core fibre whilst monitoring the output using a
camera. We couple light into a single core at the input and then taper the
fibre until the power in this core at the output is 1/9 of the total, and the
power in each of the two remaining cores is 4/9 of the total. At this point,
indicated "Optimal taper length" in (b), the symmetric taper has a length of
$L_{\mathrm{C}}/2$. We then cleave halfway along the length to form a sensing
tip with a length of $L_{\mathrm{C}}/4$.

To form the tapered section, we heat and stretch the fibre, cleaving it such
that the length of the coupled region, $z$, is $L_{\mathrm{C}}/4$. To do this,
we couple light into a single core and monitor the output of the fibre whilst
it is being tapered. As shown in Fig. 2 (b), the power distribution between
cores at the output changes with taper elongation. By stopping the tapering
process when the power in core 1 is minimised (1/9 of the maximum value), we
ensure that the tapered section’s total length is $L_{\mathrm{C}}/2$. Cleaving
the tapered fibre at its midpoint, Fig. 2 (a), produces two equal sections of
length $L_{\mathrm{C}}/4$. If we were to cleave at a point offset from the
centre, the WFS would still work, although would operate but with a reduced
sensitivity to wavefront tilt as described by Eq. (3) when $z\neq
L_{\mathrm{C}}/4$.

For the device presented, the combined length of the coupled region and
transition was 12 mm, and the remaining length of the fibre, where the cores
were uncoupled, was 1.2 m. The core-to-core separation in the tip, $\Lambda$,
was 24 $\upmu$m.

##  4 Experimental Verification

To experimentally validate our design, we launched 1.55-$\upmu$m laser light
into free space from a single-mode fibre (SMF) mounted on an x-y translation
stage positioned 0.6 m from the sensing tip. At this distance, compared with
the 24-$\upmu$m separation of the fibre cores, the light emitted from the
launch fibre approximated a plane wave at the input end face. Therefore, by
adjusting the stage’s transverse position, we could control the tip and tilt
of this incoming plane wave relative to the device (see in Fig. 3 (a)).

Figure 3: (a) Schematic showing the experimental setup. (b) A typical near-
field image used to calculate the centroid $(\rho,\phi)$ of the powers in each
core. Inset: Optical micrograph of the untapered three-core fibre. (c) Polar
$(\rho,\phi)$ plot of experimental and simulated response of centroid position
to input tilts on a square array with pitch 2.9 arcminutes; the plotted range
$(\rho<0.25)$ is roughly equivalent to the central 9 points in Fig. 1 (d).

To evaluate the centroid positions $(\rho,\phi)$, we collected near-field
images of the fibre output using a ×10 lens and a short-wave infrared camera
(Raptor Photonics, OWL 640 M). This was for convenience: to use a three-pixel
linear detector array, we would first need to reformat the fibre output. An
example image is shown in Fig. 3 (b). We adjusted the x-y translation stage to
49 positions within a $7\\!\times\\!7$ grid, producing step sizes in $\theta$
equal to 2.9 arcminutes, and captured output images at each point. The grid
alignment to the fibre was at an arbitrary offset in $\theta$ and $\alpha$. We
evaluated the centroid positions from the acquired images and plotted these in
Fig. 3 (c), along with analytical values of $\rho$ and $\phi$. We calculated
the centroid positions $(\rho,\phi)$ of the analytical grid using equations 2
and 3, acting on a perfectly square input array $(\delta,\alpha)$. We
determined the centre offset and orientation of this square input grid
relative to the fibre end-face by taking the mean position of the experimental
values of $\rho$ and $\phi$, as well as the gradients of lines drawn through
each row and column. In calculating the $\delta$ component of the analytical
centroid coordinates, we used the measured values of $\Lambda$ (24 $\upmu$m)
and $\lambda$ (1.55 $\upmu$m), and assumed the sinusoidal term was equal to 1
(i.e. $z=L_{\mathrm{C}}/4$).

Figure 4: Experimental and simulated response of centroid position to input
tilts on a square array with pitch (a) 1.7 arcminutes, and (b) 5.7 arcminutes.

Further to this, we repeated the measurement to record our devices’ response
to input tilts on grids with tilt spacings of both 1.7 and 5.7 arcminutes,
shown in Fig. 4 (a) and Fig. 4 (b), respectively. In all cases, we see good
agreement between our experimental and analytical results so that the
wavefront tip-tilt can be inferred from the measured centroid positions. The
plotted uncertainties relate to the mechanical precision of the x-y stage that
we used to control the relative tip-tilt of the plane wave. We measured the
change in centroid positions after repeated adjustments of this transverse
stage from, and back to, an arbitrarily selected position and found that the
recorded centroids were distributed around this point with a standard
deviation in $\rho$ of $\sigma$ = 0.0055, which corresponds to $\sim 20$
arcseconds. In Fig. 3 (c), Fig. 4 (a) and Fig. 4 (b), the error bars
correspond to $2\sigma=0.011$.

We have shown that, in our verification setup, our WFS can resolve input tilt
separations of $\leq$ 1.7 arcminutes. However, in principle, the tilt
resolution of our WFS is limited only by the detector’s capability to
determine the power difference between each core. This characteristic
demonstrates an optical improvement over a SH-WFS, where the angular
resolution at the focal plane is limited by the pixel period and microlens’
focal length (Valente et al., 2015). For example, the Thorlabs WFS40-5C is a
commercial WFS with an effective focal length of 4.1 mm and a pixel pitch of
5.5 $\upmu$m, thus giving an angular resolution of $\delta\theta=2.56$
arcminutes. Similarly, the SH-WFS on the CANARY AO testbed (Gendron et al.,
2011) uses a camera with 24-$\upmu$m pixels and microlenses with a focal
length of 29 mm; this amounts to an angular resolution of $\delta\theta\sim
2.84$ arcminutes, although higher resolutions can be obtained at a sub-pixel
level (Tian & Huhns, 1986).

##  5 Use with Astronomical Systems

###  5.1 Telescope integration

In our experiments we used a diverging beam at a large distance, approximating
a flat wavefront with the translation of the beam giving the incident angle on
the sensing tip of the fibre. This means our wavefront filled the lab, in a
beam configuration that was neither relevant nor interesting for astronomy. To
validate the applicability of the wavefront sensor to astronomy, we need to
estimate the response to incoming light in these systems. To do this we take a
model telescope, arranged to retain telecentricty (see Fig. 5), and subdivide
the pupil into subapertures.

Figure 5: Our model wavefront sensing system arranged in a 4f configuration.
On the left a pupil is split into subapertures $D_{\mathrm{A}}$ (here
hexagonal segments, though this is frequently square). This is imaged by the
telescope of focal length $f_{\mathrm{T}}$, and then the pupil is re-imaged
using a microlens of focal length $f_{\mathrm{L}}$. A tilted beam is shown as
green dashed lines, with angles at the pupil, $\theta_{\mathrm{pupil}}$, and
the fibre, $\theta_{\mathrm{fibre}}$, shown.

A subaperture $(D_{\mathrm{A}})$ is focussed by the telescope (here
represented by a lens of focal length $f_{\mathrm{T}}$). After a further
distance $f_{\mathrm{L}}$, a microlens of focal length $f_{\mathrm{L}}$
reimages a pupil subaperture to match the size of the sensing tip of the
sensing fibre. This is in many ways similar to how a single element of a
Shack-Hartmann WFS is used. The equilateral positioning of the cores in our
WFS mean that it will not be sensitive to aberrations that are cylindrically
symmetric. Furthermore, the input cores are close enough that higher-order
aberrations such as coma and astigmatism will simply act as modifications to
the measured tip-tilt of a plane wave. The magnification of the system,
$f_{\mathrm{T}}/f_{\mathrm{L}}$, yields the relation between the angle
$\theta_{\mathrm{pupil}}$ on-sky and the angle $\theta_{\mathrm{fibre}}$
detected by the sensor

$\theta_{\mathrm{pupil}}=\frac{n\Lambda}{D_{\mathrm{A}}}\theta_{\mathrm{fibre}}$
(4)

Here, $n$ is the number of sub-samples of the pupil, i.e. the diameter of the
subaperture’s image on the sensing tip divided by the core spacing, $\Lambda$.
Combining this with our previous knowledge of the sensor’s response in
different regimes, we can translate this into limits of how and where the
wavefront sensor can be used. Using this we can then work out the maximum
incoming tilt angle from the pupil for each of these regimes.

For astronomical wavefront sensing, the throughput of the device is also
important. For the case where the subaperture’s image covers the tapered
fibre’s cores and extends one mode radius beyond their centres $(n=1.75)$, we
calculate that 54 % of the light through the subaperture is captured by the
cores and detected. This coupling efficiency reduces to 40 % for a subaperture
image that extends two mode-radii beyond the centres of the cores $(n=2.35)$.
These efficiency values were obtained numerically by solving the overlap
integral between a plane wave passing through the subaperture and the
electric-field distribution of the light confined by the three-core structure
at the sensing tip. Figure 6 (a) shows the electric-field distribution at the
sensing tip as simulated in the commercial finite-element analysis software,
COMSOL Multiphysics.

Figure 6: (a) Density plot of the simulated field distribution of the
fundamental supermode (the equal superposition of the fields in the three
cores), with (black circles in order of increasing size) the core boundaries,
the mode radii $r_{\mathrm{I}}$ and a subaperture image of diameter $n\lambda$
that lies one mode radius beyond the centres of the cores. (b) Schematic
diagram of a ‘leopard-print’ multicore fibre containing many independent
three-core elements, overlaid with the pupil subapertures the cores are
sensing. (c) Schematic diagram showing how our proposed fibre facilitates
measurements of higher-order wavefront aberrations. An array of three-core
sensors measure tip-tilt at points across the wavefront, which can be
interpolated to construct a single measure of the higher-order wavefront
aberrations. For simplicity, we have depicted this in one dimension, but using
all of the nineteen sets of cores would enable a two-dimensional
representation of the wavefront to be recovered.

The figure shows the modes extending beyond the core diameters when the fibre
is tapered, leading to higher coupling efficiencies. For our experimental
tapered fibre, the calculated mode field diameter of 14.3 $\upmu$m is
substantial compared to the core separation of 24 $\upmu$m. It would be
possible to increase the coupling efficiency by tapering the sensing tip of
the fibre to a smaller diameter. However, the length of the tapered region
would need to decrease for the device to maintain the same number of coupling
lengths, which at some point would be difficult to achieve.

###  5.2 On-sky validation example

With our laboratory validation of the sensor and theoretical framework, the
next step will be to validate the concept on-sky. It would be straightforward
to scale the geometry of our WFS design to optimise performance at any
wavelength, for example to take advantage of lower noise detectors at visible
wavelengths. However, we optimised our prototype for 1550 nm (the astronomical
H-band). Therefore, we consider an AO system capable of providing an IR light
path and simultaneously monitoring our performance using its existing
infrastructure. We imagine the CANARY AO system on the William Herschel
Telescope (WHT) (Gendron et al., 2011) as our host AO system. CANARY is an AO
demonstrator for the Extremely Large Telescope (ELT), designed with the
flexibility to try out different AO flavours and new wavefront sensor types.
It has a history of hosting experimental instruments (Harris et al., 2015;
Haffert et al., 2020; Sinquin et al., 2020) and crucially is designed to
record test data from them in conjunction with AO measurements.

We emulate the CANARY SH-WFS setup, which divides the pupil using a
$7\\!\times\\!7$ microlens array (Gendron et al., 2011), but we perform our
wavefront sensing in the IR instead of the visible. Accordingly, our measure
of beam coherence, the Fried parameter, $r_{0}$, will be larger. We estimate
that $r_{0}=26$ cm at 1550 nm by assuming $r_{0}=15$ cm at 500 nm (similar to
Gendron et al.). This means our sensitivity to shorter wavelength aberrations
will be lower, limiting our achievable correction when correcting for visible
wavelengths.

The WHT has a 4.2-m aperture and a focal ratio of 10.8, giving a focal length
of $f_{\mathrm{T}}=45.36$ m and CANARY has a pupil subaperture of DA = 60 cm.
According to Eq. 4, this necessitates a microlens with a focal length of
$f_{\mathrm{L}}\sim 1.8$ mm. Since our fibre diameter is not unusually large,
this is well within current manufacturing capabilities. In future designs,
this microlens could be replaced by 3D printed technologies (Dietrich et al.,
2017), allowing faster integration and custom design.

Using the above numbers, and by choosing $n=1.75$, we can show how our sensor
would respond if integrated into CANARY. As with our laboratory experiment, we
use CANARY’s H-band. On-sky, the wavefront sensor fabricated for our
experiment would be linear to 51.4 mas ($\delta=0.2$), then would be
analytically derived until 205.6 mas ($\delta=0.8$) and function with the
correct sign until 411.3 mas ($\delta=1.6$). Accordingly, the sensor is highly
sensitive to tilts in the incoming wavefront but with an operating range too
small for open-loop operation (Glindemann et al., 2000). As such, we would
need the AO loop to be closed with a conventional wavefront sensor before we
could begin sensing, though a sufficiently bright star could provide accurate
wavefront measurements for small tilts. The sensor can, of course, be made
less sensitive by reducing the separation between cores at the sensing tip,
$\Lambda$. Making this change would not necessarily diminish the devices’
suitability for measuring small angles if the detector setup was still able to
distinguish small differences in power between each core.

The sensor does have a slow wavelength dependence, as shown by the explicit
dependence on $\lambda$ in Eq. 3. This can be compensated somewhat by
departing from the $z=L_{\mathrm{C}}/4$ length condition within the sinusoid
in Eq. 3, since $L_{\mathrm{C}}$ has its own wavelength dependence via the
coupling coefficient $C$. In any case the condition of an untilted wavefront
$(\theta=0)$, and hence the sign of the feedback correction needed to attain
it, does not depend on wavelength. For the fibre of our experiments,
calculations show that the sensitivity of centroid displacement, $\mathbf{r}$,
to tilt angle, $\theta$, in Eqs. 2 and 3 can be kept greater than 50 % of its
maximum value over a 350 nm wavelength range centred on 1550 nm. In contrast,
we expect more strongly-coupled systems like the photonic lantern of Cruz-
Delgado et al. (2021) to necessarily have many more coupling cycles,
$z/L_{\mathrm{C}}$, along the tapered section than the $\sim 1/4$ of our
wavefront sensor, resulting in a strong wavelength dependence and a narrow
bandwidth.

We have considered a single wavefront sensor element measuring tip-tilt across
a subaperture of the pupil. To access the full pupil we need an array of
several three-core elements. This can be provided monolithically by a
multicore fibre containing a ‘leopard-print’ pattern of sets of three cores,
shown schematically in Figure 6 (b). The sets of three cores should be far
enough apart that light does not couple from one set to its neighbours. The
whole fibre can then be tapered to form the array of tip-tilt sensors from
which higher-order aberrations can be deduced. We illustrate this concept in
Figure 6 (c), where tip-tilt, as measured by many three-core elements in a
single fibre, can be used to reconstruct more complex wavefront aberrations.

##  6 Future Astronomical Use

The sensor we have detailed here is compact and shows high accuracy over a
small operational range and requires only three detector pixels to reconstruct
each sensed subaperture. Due to the limited operational range, it would need
to be implemented as a closed loop wavefront sensor, sitting behind a
correcting DM and performing optimisations on a partially corrected point
spread function. The sampling is also set by the Fried parameter, as higher-
order aberrations cannot be sensed by the three cores. This makes the sensor
dependent on the seeing at the site, with the number of subapertures
increasing for larger telescopes.

Due to the small size of the tapered-fibre sensing tip, our all-fibre WFS
could be exploited for ELT instrumentation, allowing crowded focal planes to
be used simultaneously for both science fibres, integral-field-unit fibres and
wavefront sensing fibres without the need for a dichroic to split light
between the two. This would allow a reduction in non-common path aberrations
caused by vibrations and differences in optics, which is necessary for
efficient use of single mode instrumentation.

##  7 Conclusions

We have experimentally demonstrated a tip-tilt wavefront sensor based on a
tapered three-core fibre. In this device, a coupled region at the fibre tip
region acts as a lens, converting phase information from an incoming wavefront
into amplitude information within each of the cores, which can then be routed
away from the telescope focal plane. The sensor uses minimal information to
infer tip-tilt and is compatible with remote detector arrays. We have also
theoretically derived the equations to allow its use in astronomy.

We have shown that our wavefront sensor is particularly sensitive to small tip
and tilt perturbations of the subaperture. Only requiring three cores means
that they system can have lower noise than conventional wavefront sensors. In
principle, the tilt resolution of our WFS is limited only by the capability of
the detector setup to resolve the power difference between each core.

Future work will be conducted on two fronts. First, to theoretically optimize
the sensor, with detailed calculations of throughput, signal to noise of the
system and analysis of the optimal operating conditions. (Our 3-core wavefront
sensor is a proof of concept and has by no means been optimised.) Second, to
test a prototype in a telescope simulator and on sky. This will allow us to
confirm the laboratory performance in realistic conditions.

## Acknowledgements

This work was supported by the European Union’s Horizon 2020 research and
innovation program under grant no. 730890 (OPTICON – Optical Infrared
Coordination Network for Astronomy).

## Data Availability

Data underlying the results presented are available at (Wright et al., 2022).

## References

  * Akyol et al. (2021) Akyol E., Hagag A. M., Sivaprasad S., Lotery A. J., 2021, Eye, 35, 244
  * Birks et al. (2015) Birks T. A., Gris-Sánchez I., Yerolatsitis S., Leon-Saval S., Thomson R. R., 2015, Adv. Opt. Photonics, 7, 107
  * Booth (2014) Booth M. J., 2014, Light Sci. Appl., 3, e165
  * Chauvin, G. et al. (2004) Chauvin, G. Lagrange, A.-M. Dumas, C. Zuckerman, B. Mouillet, D. Song, I. Beuzit, J.-L. Lowrance, P. 2004, A&A, 425, L29
  * Corrigan et al. (2016) Corrigan M., Harris R. J., Thomson R. R., MacLachlan D. G., Allington-Smith J., Myers R., Morris T., 2016, in Marchetti E., Close L. M., Véran J.-P., eds, Proc. SPIE Conf. Ser. Vol. 9909, Adaptive Optics Systems V. SPIE, pp 1848 – 1855, doi:10.1117/12.2230568
  * Corrigan et al. (2018) Corrigan M. K., Morris T. J., Harris R. J., Anagnos T., 2018, in Close L. M., Schreiber L., Schmidt D., eds, Proc. SPIE Conf. Ser. Vol. 10703, Adaptive Optics Systems VI. SPIE, pp 1313 – 1320, doi:10.1117/12.2311336
  * Cruz-Delgado et al. (2021) Cruz-Delgado D., et al., 2021, Opt, Lett., 46, 3292
  * Daigle et al. (2009) Daigle J.-F., Kamali Y., Châteauneuf M., Tremblay G., Théberge F., Dubois J., Roy G., Chin S., 2009, Appl. Phys. B, 97, 701
  * Davies & Kasper (2012) Davies R., Kasper M., 2012, Annu. Rev. Astron. Astr., 50, 305
  * Dietrich et al. (2017) Dietrich P.-I., Harris R. J., Blaicher M., Corrigan M. K., Morris T. J., Freude W., Quirrenbach A., Koos C., 2017, Opt. Express, 25, 18288
  * Gendron et al. (2011) Gendron E., et al., 2011, A&A, 529, L2
  * Ghez et al. (1998) Ghez A. M., Klein B. L., Morris M., Becklin E. E., 1998, ApJ, 509, 678
  * Glindemann et al. (2000) Glindemann A., Hippler S., Berkefeld T., Hackenberg W., 2000, Exp. Astron., 10, 5
  * Gonsalves & Chidlaw (1979) Gonsalves R. A., Chidlaw R., 1979, in Tescher A. G., ed., Proc. SPIE Conf. Ser. Vol. 0207, Applications of Digital Image Processing III. SPIE, pp 32 – 39, doi:10.1117/12.958223
  * Gregory et al. (2012) Gregory M., Heine F. F., Kämpfner H., Lange R., Lutzer M., Meyer R., 2012, Opt. Eng., 51, 031202
  * Guyon (2018) Guyon O., 2018, Annu. Rev. Astron. Astr., 56, 315
  * Haffert et al. (2020) Haffert S. Y., et al., 2020, J. Astron. Telesc. Instrum. Syst., 6, 1
  * Harris et al. (2015) Harris R. J., et al., 2015, MNRAS, 450, 428
  * Norris et al. (2020) Norris B. R., Wei J., Betters C. H., Wong A., Leon-Saval S. G., 2020, Nat. Commun., 11, 1
  * Paxman et al. (2007) Paxman R. G., Thelen B. J., Murphy R. J., Gleichman K. W., III J. A. G., 2007, in Carreras R. A., Gonglewski J. D., Rhoadarmer T. A., eds, Proc. SPIE Conf. Ser. Vol. 6711, Advanced Wavefront Control: Methods, Devices, and Applications V. SPIE, pp 8 – 22, doi:10.1117/12.734665
  * Platt & Shack (2001) Platt B. C., Shack R., 2001, J. Refract. Surg., 17, S573
  * Ragazzoni (1996) Ragazzoni R., 1996, J. Mod. Optic., 43, 289
  * Rigaut (2015) Rigaut F., 2015, Publ. Astron. Soc. Pac., 127, 1197
  * Roddier (1988) Roddier F., 1988, Appl. Opt., 27, 1223
  * Russell (2003) Russell P., 2003, Science, 299, 358
  * Sauvage et al. (2007) Sauvage J.-F., Fusco T., Rousset G., Petit C., 2007, JOSA A, 24, 2334
  * Sinquin et al. (2020) Sinquin B., et al., 2020, MNRAS, 498, 3228
  * Tian & Huhns (1986) Tian Q., Huhns M. N., 1986, Comput. graph. image process., 35, 220
  * Valente et al. (2015) Valente D., Rativa D., Vohnsen B., 2015, Opt. Express, 23, 13005
  * Weyrauch & Vorontsov (2005) Weyrauch T., Vorontsov M. A., 2005, Appl. Opt., 44, 6388
  * Wright et al. (2022) Wright T. A., Yerolatsitis S., Kerrianne H., Harris R. J., Birks T. A., 2022, Dataset for "All-fibre wavefront sensor", doi:10.15125/BATH-01135
  * Zawadzki et al. (2005) Zawadzki R. J., et al., 2005, Opt. Express, 13, 8532

##  Appendix A Analytical relationship between wavefront tip-tilt and centroid
position

The behaviour of the coupling section at the input of the fibre wavefront
sensor can be analysed using its supermodes. These are the combinations of
amplitudes and phases in (the fundamental modes of) each core which propagate
along the fibre without change except for an overall phase $\sim e^{i\beta
z}$, where $\beta$ is the supermode’s propagation constant. For three
identical weakly-coupled equally-spaced cores, the state vectors describing
the amplitudes and phases in the three cores for each supermode A, B and C can
be written

$\mathbf{a}_{\mathrm{A}}=\frac{1}{\sqrt{3}}\begin{pmatrix}1\\\ 1\\\
1\end{pmatrix}\quad\mathbf{a}_{\mathrm{B}}=\frac{1}{\sqrt{6}}\begin{pmatrix}2\\\
-1\\\
-1\end{pmatrix}\quad\mathbf{a}_{\mathrm{C}}=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
1\\\ -1\end{pmatrix},$ (5)

with the corresponding propagation constants

$\displaystyle\beta_{\mathrm{A}}=\beta_{0}+2C,$ (6)

and

$\beta_{\mathrm{B}}=\beta_{\mathrm{C}}=\beta_{0}-C,$ (7)

where $\beta_{0}$ is the propagation constant of one core in isolation, and
$C$ is the coupling coefficient between a pair of cores.

The input field is written as a superposition of the three supermodes, which
are allowed to propagate through distance $z$ to the end of the coupling
region and thereby accumulate different phases according to their $\beta$
values. The square modulus of the output amplitude in each core gives the
normalised power $P$ that emerges from that core.

To deduce the coupling characteristic in Fig. 2 that were used to ensure that
the coupling region was the right length, we take the input state vector to be
$\mathbf{a}=(1,0,0)$, i.e. light in core 1 only. The output powers are then

$\displaystyle P_{1}$ $\displaystyle=\left[5+4\cos(3Cz)\right]/9,$ (8)
$\displaystyle P_{2}$ $\displaystyle=P_{3}=\left[2-2\cos(3Cz)\right]/9.$ (9)

The coupling length $L_{\mathrm{C}}$ is the distance, $z$, where the powers
return to the input condition of $P_{1}$ = 1. It is given by
$3CL_{\mathrm{C}}$ = $2\pi$, and can be determined readily from the measured
output powers using Eq. (8–9). $P_{1}$ is minimised when $3Cz=\pi$,
corresponding to $z=L_{\mathrm{C}}/2$.

In a non-uniform structure, such as a real fibre taper with a transition along
which coupling gradually falls to zero, $C$ is position-dependent and
$L_{\mathrm{C}}$ becomes ill-defined. However, the expressions in this paper
remain correct if $Cz$ is generalised to the integral of $C\mathrm{d}z$, with
the ratio $z/L_{\mathrm{C}}=3Cz/2\pi$ being well-defined as the overall number
of complete coupling cycles along the whole tapered section.

To deduce the response of the structure to a tilted wavefront, we use the
transverse wavevector
$\mathbf{k}_{\mathrm{T}}=k_{0}\theta(\cos\alpha,\sin\alpha)$ in the plane of
the fibre endface with the geometry of Fig. 1 (a). Here, $k_{0}$ is the free-
space wavenumber, $k_{0}=2\pi/\lambda$. We assume small $\theta$, and evaluate
the phases $\mathbf{k}_{\mathrm{T}}\cdot\mathbf{r}$ at the centres of the
cores to determine the input state vector

$\mathbf{a}=\begin{pmatrix}e^{i\delta_{0}\cos\alpha}\\\
e^{i\delta_{0}\cos(\alpha-2\pi/3)}\\\
e^{i\delta_{0}\cos(\alpha-2\pi/3)}\end{pmatrix},$ (10)

where

$\delta_{0}=\frac{k_{0}\theta\Lambda}{\sqrt{3}}.$ (11)

After some effort, the supermode analysis yields an analytic expression for
the output power in core 1:

$\displaystyle P_{1}=1+\frac{2}{9}\left[1-\cos\left(\frac{2\pi
z}{L_{C}}\right)\right]\big{\\{}2\cos\left[\delta_{0}\left(\chi_{-}-\chi_{+}\right)\right]-$
(12)
$\displaystyle\cos\left[\delta_{0}\left(\chi_{0}-\chi_{-}\right)\right]-\cos\left[\delta_{0}\left(\chi_{+}-\chi_{0}\right)\right]\big{\\}}$
$\displaystyle+\frac{2}{3}\sin\left(\frac{2\pi
z}{L_{C}}\right)\big{\\{}\sin\left[\delta_{0}\left(\chi_{0}-\chi_{-}\right)\right]-\sin\left[\delta_{0}\left(\chi_{+}-\chi_{0}\right)\right]\big{\\}},$

where

$\displaystyle\chi_{0}$ $\displaystyle=\cos\alpha,$ (13)
$\displaystyle\chi_{\pm}$ $\displaystyle=\cos(\alpha\pm 2\pi/3).$ (14)

The powers in cores 2 and 3 are given in turn by the cyclic rotation
$\chi_{0}\\!\rightarrow\\!\chi_{-}\\!\rightarrow\\!\chi_{+}\\!\rightarrow\\!\chi_{0}$,
which is equivalent to subtracting $2\pi/3$ from each occurrence of $\alpha$.
These powers can be expressed using the centroid of Eq. (1) in polar
coordinates as

$\displaystyle\rho$
$\displaystyle=\frac{\sqrt{\left(P_{1}-P_{2}\right)^{2}+\left(P_{2}-P_{3}\right)^{2}+\left(P_{3}-P_{1}\right)^{2}}}{\sqrt{2}\,\left(P_{1}+P_{2}+P_{3}\right)},$
(15) $\displaystyle\cos\phi$
$\displaystyle=\frac{\left(P_{1}-P_{2}\right)+\left(P_{1}-P_{3}\right)}{\sqrt{2}\,\sqrt{\left(P_{1}-P_{2}\right)^{2}+\left(P_{2}-P_{3}\right)^{2}+\left(P_{3}-P_{1}\right)^{2}}},$
(16) $\displaystyle\sin\phi$
$\displaystyle=\frac{\sqrt{3}\,\left(P_{1}-P_{3}\right)}{\sqrt{2}\,\sqrt{\left(P_{1}-P_{2}\right)^{2}+\left(P_{2}-P_{3}\right)^{2}+\left(P_{3}-P_{1}\right)^{2}}}.$
(17)

Calculation of $\phi$ is simpler via $\tan\phi$, although the sign of
$\cos\phi$ or $\sin\phi$ is needed to get $\phi$ in the right quadrant.

To first order in $\delta_{0}$, Eq. (12) becomes

$\displaystyle P_{1}$ $\displaystyle=1+2\delta_{0}\cos\alpha\sin\left(2\pi
z/L_{C}\right),$ (18) $\displaystyle P_{2}$
$\displaystyle=1+2\delta_{0}\cos(\alpha-2\pi/3)\sin\left(2\pi z/L_{C}\right),$
(19) $\displaystyle P_{3}$
$\displaystyle=1+2\delta_{0}\cos(\alpha+2\pi/3)\sin\left(2\pi z/L_{C}\right),$
(20)

and application of Eq. (15–17) yields Eq. (2) and Eq. (3).

Expressions of similar form can be derived for cases where the coupling
between the cores is not weak. For example, in a typical photonic lantern, the
fibre is tapered small enough that the fibre cores lose their identities and
the tapered section acts like a single multi-mode fibre core (Birks et al.,
2015). In such cases, $C$ is not defined but is replaced in the analysis by
$(\beta_{01}-\beta_{11})/3$, where $\beta_{lm}$ is the propagation constant of
the $\mathrm{LP}_{lm}$ mode. This is typically much greater than $C$ for a
pair of well-separated cores, meaning that the multimode section of any
realistic photonic lantern will be considerably longer than its
$L_{\mathrm{C}}$. This makes it difficult to meet the $z=L_{\mathrm{C}}/4$
condition, and produces a strongly ‘beating’ wavelength dependence. For this
reason, we do not consider this case further.


# Certain properties of the class of univalent functions with real
coefficients

Milutin Obradović  Department of Mathematics, Faculty of Civil Engineering,
University of Belgrade, Bulevar Kralja Aleksandra 73, 11000, Belgrade, Serbia
<EMAIL_ADDRESS>and  Nikola Tuneski  Department of Mathematics and
Informatics, Faculty of Mechanical Engineering, Ss. Cyril and Methodius
University in Skopje, Karpoš II b.b., 1000 Skopje, Republic of Macedonia.
<EMAIL_ADDRESS>
###### Abstract.

Let ${\mathcal{U}}^{+}$ be the class of analytic functions $f$ such that
$\frac{z}{f(z)}$ has real and positive coefficients and $f^{-1}$ be its
inverse. In this paper we give sharp estimates of the initial coefficients and
initial logarithmic coefficients for $f$, as well as, sharp estimates of the
second and the third Hankel determinant for $f$ and $f^{-1}$. We also show
that the Zalcman conjecture holds for functions $f$ from ${\mathcal{U}}^{+}$.

###### Key words and phrases:

univalent, real coefficients, logarithmic coefficients, coefficient estimates,
Hankel determinant, Zalcman conjecture

###### 1991 Mathematics Subject Classification:

30C45, 30C50, 30C55

##  1\. Introduction

Let $\mathcal{A}$ be the class of functions $f$ that are analytic in the open
unit disc ${\mathbb{D}}=\\{z:|z|<1\\}$ of the form

(1) $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots,$

let $\mathcal{S}$ be its subclass consisting of univalent functions from
${\mathcal{A}}$, and $\mathcal{S}^{+}$ consists of functions $f$ from
${\mathcal{S}}$ with representation

(2) $\frac{z}{f(z)}=1+b_{1}z+b_{2}z^{2}+\cdots,\qquad b_{n}\geq 0,\quad
n=1,2,3,\ldots.$

Note that the Silverman class of univalent functions with negative
coefficients, i.e., with expansion

$f(z)=z-a_{2}z^{2}-a_{3}z^{3}+\cdots,\qquad a_{n}\geq 0,\quad n=1,2,3,\ldots,$

is subclass of ${\mathcal{S}}^{+}$ since $z/f(z)$ satisfies (2). Also, the
Koebe function $k(z)=\frac{z}{(1+z)^{2}}$ is in ${\mathcal{S}}^{+}$.

Further, with ${\mathcal{U}}(\lambda)$, $0\leq\lambda<1$, we will denote the
class of functions $f$ from ${\mathcal{A}}$ satisfying the condition

(3) $\left|\left(\frac{z}{f(z)}\right)^{2}f^{\prime}(z)-1\right|<\lambda,\quad
z\in{\mathbb{D}},$

while ${\mathcal{U}}\equiv{\mathcal{U}}(1)$. This class of functions attracts
significant interest in the past decades. The class is intriguing because it
doesn’t follow the usual pattern to embed or be embedded in the class of
starlike functions (functions that map the unit disk onto a starlike region).
A collection of the more significant once can be found in [23, Chapter 12].

If we denote with ${\mathcal{U}}^{+}(\lambda)$ the class of functions that
satisfy (2) and (3), and additionally
${\mathcal{U}}^{+}\equiv{\mathcal{U}}^{+}(1)$, then [15] we have the following
equivalence

$f\in{\mathcal{S}}^{+}\quad\Leftrightarrow\quad
f\in{\mathcal{U}}^{+}\quad\Leftrightarrow\quad\sum_{n=2}^{\infty}(n-1)b_{n}\leq
1.$

Next lemma makes an extension of the second equivalence.

######  Lemma 1.

$f\in{\mathcal{U}}^{+}(\lambda)\Leftrightarrow\sum_{n=2}^{\infty}(n-1)b_{n}\leq\lambda.$

###### Proof.

If $f\in{\mathcal{U}}^{+}(\lambda)$, then by definition

$\begin{split}\quad\left|\left(\frac{z}{f(z)}\right)^{2}f^{\prime}(z)-1\right|=\left|\frac{z}{f(z)}-z\left(\frac{z}{f(z)}\right)^{\prime}-1\right|=\left|-\sum_{n=2}^{\infty}(n-1)b_{n}z^{n}\right|<\lambda,\end{split}$

$z\in{\mathbb{D}}.$ For real $z$ and $z\rightarrow 1$ from left, the last
inequality gives $\sum_{n=2}^{\infty}(n-1)b_{n}\leq\lambda.$

If $\sum_{n=2}^{\infty}(n-1)b_{n}\leq\lambda$, then

$\left|\left(\frac{z}{f(z)}\right)^{2}f^{\prime}(z)-1\right|=\left|-\sum_{n=2}^{\infty}(n-1)b_{n}z^{n}\right|\leq\sum_{n=2}^{\infty}(n-1)b_{n}|z|^{n}<\sum_{n=2}^{\infty}(n-1)b_{n}\leq\lambda,$

$z\in{\mathbb{D}}$, which mens that $f\in{\mathcal{U}}^{+}(\lambda)$. ∎

In [20] the authors gave sharp bounds of the first five coefficients in the
expansion of $f$. The result for the case $\lambda=1$ reduces to the
following.

Theorem A. Let
$f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots\in{\mathcal{S}}^{+}={\mathcal{U}}^{+}$.
Then the following estimates are sharp

$-2\leq a_{2}\leq 0,\quad-1\leq a_{3}\leq 3,\quad-4\leq
a_{4}\leq\frac{4}{3}\sqrt{\frac{2}{3}},\quad-\frac{9}{4}\leq a_{5}\leq 5.$

Let note that for the general class ${\mathcal{U}}$ we have $|a_{n}|\leq n$,
$n=2,3,\ldots$, independently from the de Branges theorem. Namely, for the
functions $f$ from ${\mathcal{U}}$, we have

$\frac{f(z)}{z}\prec\frac{1}{(1-z)^{2}}=1+\sum_{n=1}^{\infty}nz^{n-1},$

and the rest follows from the Rogosinski theorem ([23, Theorem 3.2.9, p.35]).

Further, in [16], the authors proved that for $f$ in the general class
${\mathcal{U}}(\lambda)$, $0<\lambda\leq 1$, $|a_{2}|\leq 1+\lambda$, and if
$|a_{2}|=1+\lambda$, then $f$ must be of the form

$f_{\theta}(z)=\frac{z}{1-(1+\lambda)e^{i\theta}z+\lambda e^{2i\theta}z^{2}}.$

For the case $f\in{\mathcal{U}}^{+}(\lambda)$ we have the following
corresponding result.

Theorem B. If $f$ is given by (1) with (2), and
$f\in{\mathcal{U}}^{+}(\lambda)$, $0<\lambda\leq 1$, then $-(1+\lambda)\leq
a_{2}\leq 0$ (or $0\leq b_{1}\leq 1+\lambda$). Moreover, if
$a_{2}=-(1+\lambda)$, then $f$ must be of the form

(4) $f_{\lambda}(z)=\frac{z}{1+(1+\lambda)z+\lambda z^{2}},$

i.e., $b_{2}=\lambda$, and $b_{3}=b_{4}=\cdots=0$,

In this paper we study coefficient problems for inverse functions of functions
in ${\mathcal{U}}^{+}(\lambda)$, more precisely we will give sharp upper
bounds of the leading coefficients and leading logarithmic coefficients of the
inverse functions, as well as estimates of the modulus of the second and the
third Hankel determinant.

##  2\. Coefficient estimates

The famous Koebe 1/4 theorem guaranties that each function $f$ from
${\mathcal{S}}$ has an inverse at least on a disk with radius 1/4. Let the
inverse function has an expansion

(5) $f^{-1}(w)=w+A_{2}w^{2}+A_{3}w^{3}+\cdots$

on the disk $|w|<\frac{1}{4}$. By using the identity $f(f^{-1}(w))=w$, and
representations (1) and (5), we obtain the following relations:

(6) $\begin{array}[]{l}A_{2}=-a_{2},\\\\[5.69054pt]
A_{3}=-a_{3}+2a_{2}^{2},\\\\[5.69054pt]
A_{4}=-a_{4}+5a_{2}a_{3}-5a_{2}^{3}.\end{array}$

We now give sharp bounds of $A_{2}$, $A_{3}$ and $A_{4}$.

######  Theorem 1.

Let $f\in{\mathcal{U}}^{+}(\lambda)$, $0<\lambda\leq 1$, and $f^{-1}$ is given
by (5). Then, the following estimates are sharp:

(7) $\begin{array}[]{l}0\leq A_{2}\leq 1+\lambda,\\\\[5.69054pt] 0\leq
A_{3}\leq 1+3\lambda+\lambda^{2},\\\\[5.69054pt] 0\leq
A_{4}\leq(1+\lambda)(1+5\lambda+\lambda^{2}).\end{array}$

###### Proof.

Since $f\in{\mathcal{U}}^{+}(\lambda)$, $0<\lambda\leq 1$, then by definition

$f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots=\frac{z}{1+b_{1}z+b_{2}z^{2}+\cdots},$

where $b_{n}\geq 0,\quad n=1,2,3,\ldots.$ After comparing the coefficients we
receive,

(8) $\begin{array}[]{l}a_{2}=-b_{1},\\\\[5.69054pt]
a_{3}=-b_{2}+b_{1}^{2},\\\\[5.69054pt]
a_{4}=-b_{3}+2b_{1}b_{2}-b_{1}^{3},\\\\[5.69054pt]
a_{5}=-b_{4}+b_{2}^{2}+2b_{1}b_{3}-3b_{1}^{2}b_{2}+b_{1}^{4},\end{array}$

and by combining (8) with (6),

(9) $\begin{array}[]{l}A_{2}=b_{1},\\\\[5.69054pt]
A_{3}=b_{2}+b_{1}^{2},\\\\[5.69054pt]
A_{4}=b_{3}+3b_{1}b_{2}+b_{1}^{3}.\end{array}$

Since $a_{2}$ is real (so are $a_{3}$, $a_{4}$,…), and $|a_{2}|\leq 1+\lambda$
for all functions from ${\mathcal{U}}(\lambda)$ (see [24] or [23, Theorem
12.3.1, p.188]), we receive $-(1+\lambda)\leq a_{2}\leq 1+\lambda$.

From $A_{2}=-a_{2}=b_{1}$, $b_{1}\geq 0$ and $a_{2}\leq 1+\lambda$, easily
follows the estimate

(10) $0\leq A_{2}=b_{1}\leq 1+\lambda.$

Further, from Lemma 1 we have

$b_{2}+2b_{3}+3b_{4}+\cdots\leq\lambda,$

which implies

(11) $\begin{split}0\leq b_{2}&\leq\lambda\\\
b_{2}+2b_{3}&\leq\lambda\quad\left(\Leftrightarrow
b_{3}\leq\frac{1}{2}(\lambda-b_{2})\right)\\\
b_{2}+2b_{3}+3b_{4}&\leq\lambda\quad\left(\Leftrightarrow
b_{4}\leq\frac{1}{3}(\lambda-2b_{3}-b_{2})\right).\end{split}$

Combining (10), (11) and (9), we receive

$0\leq A_{3}\leq\lambda+(1+\lambda)^{2}=1+3\lambda+\lambda^{2},$

and for $A_{4}$,

$\begin{split}0\leq A_{4}&\leq\frac{1}{2}(\lambda-
b_{2})+3(1+\lambda)b_{2}+(1+\lambda)^{3}\\\
&=\frac{\lambda}{2}+\left(\frac{5}{2}+3\lambda\right)b_{2}+(1+\lambda)^{3}\\\
&\leq\frac{\lambda}{2}+\left(\frac{5}{2}+3\lambda\right)\lambda+(1+\lambda)^{3}\\\
&=(1+\lambda)(1+5\lambda+\lambda^{2}).\end{split}$

For the sharpness of the upper bounds, for the function $f_{\lambda}(z)$
defined by (4), we have

$\begin{split}f_{\lambda}(z)&=\frac{z}{1+(1+\lambda)z+\lambda z^{2}}\\\
&=z-(1+\lambda)z^{2}+(1+\lambda+\lambda^{2})z^{3}-(1+\lambda+\lambda^{2}+\lambda^{3})z^{4}+\cdots,\end{split}$

i.e., $f_{\lambda}\in{\mathcal{U}}^{+}(\lambda)$ and

$z=f_{\lambda}^{-1}(w)=w+(1+\lambda)w^{2}+(1+3\lambda+\lambda^{2})w^{3}+(1+\lambda)(1+5\lambda+\lambda^{2})w^{4}+\cdots.$

For the sharpness of the lower bounds it is enough to consider the inverse
function of the function

$f(z)=\frac{z}{1+\lambda z^{4}/3},$

where $b_{1}=b_{2}=b_{3}=0$. ∎

######  Remark 1.

In [21], the authors obtained the same result, but with an additional
condition

(12) $\frac{f(z)}{z}\prec\frac{1}{(1+z)(1+\lambda z)}.$

In our proof we didn’t use that condition.

######  Remark 2.

From (8) we have $a_{3}=b_{1}^{2}-b_{2}\equiv\psi_{1}(b_{1})$, $0\leq
b_{1}\leq 1+\lambda$. It is evident that $\psi_{1}(b_{1})\geq-
b_{2}\geq-\lambda$, while $\max\psi_{1}=\psi_{1}(1+\lambda)$. But,
$b_{1}=1+\lambda$ implies $b_{2}=\lambda$, so

$\psi_{1}(1+\lambda)=(1+\lambda)^{2}-\lambda=1+\lambda+\lambda^{2}.$

Similarly, by (8) we have

$-a_{4}=b_{1}^{3}-2b_{1}b_{2}+b_{3}\equiv\psi_{2}(b_{1}),$

$0\leq b_{1}\leq 1+\lambda$, and by using the previous comments,

$\max\psi_{2}=\psi_{2}(1+\lambda)=(1+\lambda)^{3}-2(1+\lambda)\lambda=1+\lambda+\lambda^{2}+\lambda^{3}.$

It means that $a_{4}\geq-(1+\lambda+\lambda^{2}+\lambda^{3})$. If we use the
upper bound for $a_{4}$ given in [20], then

$|a_{4}|\leq 1+\lambda+\lambda^{2}+\lambda^{3}$

for $f\in{\mathcal{U}}^{+}(\lambda)$. It means that the results given in
Theorem 3 from [20] are valid independently of the condition (12).

This is the reason we pose the next conjecture.

######  Conjecture 1.

For the functions in the class ${\mathcal{U}}^{+}(\lambda)$,

$|a_{n}|\leq\frac{1-\lambda^{n}}{1-\lambda}=1+\lambda+\lambda^{2}+\lambda^{3}+\cdots+\lambda^{n},\quad
n=2,3,4,\ldots.$

This was proven to hold for the general class ${\mathcal{U}}(\lambda)$ under
the condition (12) in the cases $n=2,3,4$ ([17] or [23, Theorem 12.3.2,
p.191].

In [20] the authors studied the logarithmic coefficients for the class
${\mathcal{U}}^{+}$ (${\mathcal{S}}^{+}$), and here we will continue and give
estimates for the logarithmic coefficients of the inverse functions of
functions in the classes ${\mathcal{U}}^{+}(\lambda)$. For the logarithmic
coefficients ($\Gamma_{1}$, $\Gamma_{2}$, $\Gamma_{3}$,…) we have

$\log\frac{f^{-1}(w)}{w}=\log(1+A_{2}w+A_{3}w^{2}+\cdots),$

or

$\sum_{n=1}^{\infty}2\Gamma_{n}w^{n}=A_{2}w+\left(A_{3}-\frac{1}{2}A_{2}^{2}\right)w^{2}+\left(A_{4}-A_{2}A_{3}+\frac{1}{3}A_{2}^{3}\right)w^{3}+\cdots.$

From the last relation we get

(13)
$\begin{array}[]{l}\Gamma_{1}=\frac{1}{2}A_{2}=\frac{1}{2}b_{1},\\\\[5.69054pt]
\Gamma_{2}=\frac{1}{2}\left(A_{3}-\frac{1}{2}A_{2}^{2}\right)=\frac{1}{2}\left(b_{2}+\frac{1}{2}b_{1}^{2}\right),\\\\[5.69054pt]
\Gamma_{3}=\frac{1}{2}\left(A_{4}-A_{2}A_{3}+\frac{1}{3}A_{2}^{3}\right)=\frac{1}{2}\left(b_{3}+2b_{1}b_{2}+\frac{1}{3}b_{1}^{3}\right),\end{array}$

where we used relations (9).

######  Theorem 2.

Let $f\in{\mathcal{U}}^{+}(\lambda)$, $0<\lambda\leq 1$. Then:

  * (a)

$0\leq\Gamma_{1}\leq\frac{1+\lambda}{2}$;

  * (b)

$0\leq\Gamma_{2}\leq\frac{1}{4}(1+4\lambda+\lambda^{2})$;

  * (c)

$0\leq\Gamma_{3}\leq\frac{1}{6}(1+\lambda)(1+8\lambda+\lambda^{2})$,

and all results are sharp.

###### Proof.

The proof is similar to the one of Theorem 1. ∎

######  Corollary 1.

For $\lambda=1$ we receive sharp estimates for the inverse function of
functions from ${\mathcal{U}}^{+}(1)={\mathcal{U}}^{+}={\mathcal{S}}^{+}$:

$0\leq\Gamma_{1}\leq 1,\quad 0\leq\Gamma_{2}\leq\frac{3}{2},\quad
0\leq\Gamma_{3}\leq\frac{10}{3}.$

##  3\. The second and the third Hankel determinant

Another concept, rediscovered few years ago, attracting the attention of
mathematicians working in the field of univalent functions is so called Hankel
determinant $H_{q}(n)(f)$ of a given function
$f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots,$, for $q\geq 1$ and $n\geq 1$, defined
by

$H_{q}(n)(f)=\left|\begin{array}[]{cccc}a_{n}&a_{n+1}&\ldots&a_{n+q-1}\\\
a_{n+1}&a_{n+2}&\ldots&a_{n+q}\\\ \vdots&\vdots&{}\hfil&\vdots\\\
a_{n+q-1}&a_{n+q}&\ldots&a_{n+2q-2}\\\ \end{array}\right|.$

Main interest is to find upper bound (preferably sharp) of the modulus of
$H_{q}(n)(f)$. The general Hankel determinant is hard to deal with, so the
second and the third ones,

$H_{2}(2)(f)=\left|\begin{array}[]{cc}a_{2}&a_{3}\\\
a_{3}&a_{4}\end{array}\right|=a_{2}a_{4}-a_{3}^{2}$

and

$H_{3}(1)(f)=\left|\begin{array}[]{ccc}1&a_{2}&a_{3}\\\ a_{2}&a_{3}&a_{4}\\\
a_{3}&a_{4}&a_{5}\\\
\end{array}\right|=a_{3}(a_{2}a_{4}-a_{3}^{2})-a_{4}(a_{4}-a_{2}a_{3})+a_{5}(a_{3}-a_{2}^{2}),$

respectively, are studied instead. The research is focused on the subclasses
of univalent functions (starlike, convex, $\alpha$-convex, close-to-convex,
spirallike,…) since the general class of normalised univalent functions is
also hard to deal with. Some of the more significant results can be found in
[1, 2, 4, 5, 6, 7, 10, 9, 11, 13, 25].

In this section we will give sharp bounds of the second and the third Hankel
determinant for the functions in ${\mathcal{U}}^{+}(\lambda)$ and for their
inverse.

Using (8), (10) and (11), after some calculations, we receive

(14) $\begin{split}H_{2}(2)(f)&=b_{1}b_{3}-b_{2}^{2},\\\
H_{3}(1)(f)&=b_{2}b_{4}-b_{3}^{2}.\end{split}$

######  Theorem 3.

Let $f\in{\mathcal{U}}^{+}(\lambda)$, $0<\lambda\leq 1$. Then the following
estimates are sharp:

  * ($i$)

$-\lambda^{2}\leq
H_{2}(2)(f)\leq\left(1-\frac{\lambda}{2}\right)\frac{\lambda}{2}$;

  * ($ii$)

$-\frac{\lambda^{2}}{4}\leq H_{3}(1)(f)\leq\frac{\lambda^{2}}{12}$.

###### Proof.

  * ($i$)

First, let note that from (10) and (11), we have $b_{1},b_{3}\geq 0$ and
$b_{2}\leq\lambda$, which easily implies

$H_{2}(2)(f)=b_{1}b_{3}-b_{2}^{2}\geq-b_{2}^{2}\geq-\lambda^{2}.$

The estimate is sharp due to the function $f(z)=\frac{z}{1+\lambda z^{2}}$.

For the upper bound, from (14) we have $H_{2}(2)(f)=b_{1}b_{3}-b_{2}^{2}\leq
b_{1}b_{3}$, and further from $0\leq b_{1}\leq 1+\lambda$ and $0\leq
b_{3}\leq\frac{\lambda}{2}$ (see (10) and (11)) we need to find
$\max\\{b_{1}b_{3}\\}$. For $b_{3}=\frac{\lambda}{2}$ ($\Rightarrow$
$b_{2}=b_{4}=\cdots=0$) we have

$H_{2}(2)(f)\leq\frac{\lambda}{2}b_{1}.$

If we consider the function $f$ defined by

$\frac{z}{f(z)}=1+b_{1}z+\frac{\lambda}{2}z^{3},$

then $f$ is in ${\mathcal{U}}^{+}(\lambda)$ if
$1+b_{1}z+\frac{\lambda}{2}z^{3}\neq 0$. Since

$\left.\frac{z}{f(z)}\right|_{z=1}=1+b_{1}+\frac{\lambda}{2}>0\quad\text{and}\quad\left.\frac{z}{f(z)}\right|_{z=-1}=1-b_{1}-\frac{\lambda}{2},$

then it is necessary $1-b_{1}-\frac{\lambda}{2}\geq 0$. In contrary, if
$1-b_{1}-\frac{\lambda}{2}<0$, then for $z=r$ real in $-1<r<1$, we have

$\left.\frac{z}{f(z)}\right|_{z=r}=1+b_{1}r+\frac{\lambda}{2}r^{3},$

has zero in $(-1,1)$, i.e., $f$ is not analytic on the unit disk.

The previous condition is also a sufficient one. Namely, for $0\leq b_{1}\leq
1-\frac{\lambda}{2}$ and $0\leq b_{3}\leq\frac{\lambda}{2}$:

$\begin{split}\left|\frac{z}{f(z)}\right|&=\left|1+b_{1}z+\frac{\lambda}{2}z^{3}\right|\geq
1-b_{1}|z|-\frac{\lambda}{2}|z|^{3}\\\
&>1-\left(1-\frac{\lambda}{2}\right)-\frac{\lambda}{2}=0.\end{split}$

So, $0\leq b_{1}\leq 1-\frac{\lambda}{2}$ and
$H_{2}(2)(f)\leq\left(1-\frac{\lambda}{2}\right)\frac{\lambda}{2}$. The upper
bound is sharp due to the function
$f(z)=\frac{z}{1+(1-\lambda/2)z+\lambda/2z^{3}}$.

  * ($ii$)

Similarly as in the proof of part (a), from (14) and (11),

$\begin{split}H_{3}(1)(f)&\leq b_{2}\cdot\frac{1}{3}(\lambda-
b_{2}-2b_{3})-b_{3}^{2}=\frac{1}{3}\lambda
b_{2}-\frac{1}{3}b_{2}^{2}-\frac{2}{3}b_{2}b_{3}-b_{3}^{2}\\\
&\leq\frac{1}{3}\lambda
b_{2}-\frac{1}{3}b_{2}^{2}\leq\frac{\lambda^{2}}{12}.\end{split}$

Also, $H_{3}(1)(f)\geq-
b_{3}^{2}\geq-\left(\frac{\lambda}{2}\right)^{2}=-\frac{\lambda^{2}}{4}$,
since by (11), $b_{3}\leq\frac{\lambda-b_{2}}{2}\leq\frac{\lambda}{2}.$ The
functions $f(z)=\frac{z}{1+\lambda/2z^{3}}$ and
$f(z)=\frac{z}{1+\lambda/2z^{2}+\lambda/6z^{4}}$ show that the estimates are
sharp.

∎

For the sharp estimates of the Hankel determinant for the inverse of the
functions from ${\mathcal{U}}^{+}(\lambda)$ we have the following theorem.

######  Theorem 4.

Let $f\in{\mathcal{U}}^{+}(\lambda)$, $0<\lambda\leq 1$. Then the following
estimates are sharp:

  * ($i$)

$-\lambda^{2}\leq H_{2}(2)(f^{-1})\leq\lambda(1+\lambda+\lambda^{2})$;

  * ($ii$)

$-\frac{\lambda^{2}}{4}\leq H_{3}(1)(f^{-1})\leq\lambda^{3}$.

###### Proof.

  * ($i$)

From (9) we have

$H_{2}(2)(f^{-1})=A_{2}A_{4}-A_{3}^{2}=b_{1}b_{3}+b_{1}^{2}b_{2}-b_{2}^{2}\equiv\psi_{1}(b_{1}),$

where $\psi_{1}$ is an increasing function of $b_{1}$, $0\leq b_{1}\leq
1+\lambda$, and $\psi_{1}(b_{1})\leq\psi(1+\lambda)$. But, Lemma 1, implies
$b_{2}=\lambda$ and $b_{3}=0$ when for $b_{1}=1+\lambda$. So,
$\psi_{1}(b_{1})\leq\psi(1+\lambda)=(1+\lambda)\cdot
0+(1+\lambda)^{2}\lambda-\lambda^{2}=\lambda(1+\lambda+\lambda^{2})$, and the
result follows. This upper bound is sharp due to the function $f_{\lambda}$
defined in (4).

Also, $H_{2}(2)(f^{-1})\geq-b_{2}^{2}\geq-\lambda^{2}$. This bound is also
sharp as the function $f(z)=\frac{z}{1+\lambda z^{2}}$ shows.

  * ($ii$)

For the third Hankel determinant of $f^{-1}$ we have

$H_{3}(1)(f^{-1})=H_{3}(1)(f)-(a_{3}-a_{2}^{2})^{3}=b_{2}b_{4}-b_{3}^{2}+b_{2}^{3},$

which is an increasing function of $b_{2}$ on the interval $(0,\lambda)$,
reaching the maximum value on the interval for $b_{2}=\lambda$. At the same
moment, due to Lemma 1 we receive that $b_{3}=b_{4}=0$, leading to

$H_{3}(1)(f^{-1})\leq\lambda\cdot 0-0^{2}+\lambda^{3}=\lambda^{3}.$

The function $\frac{z}{1+\lambda z^{3}}$ shows that this estimate is the best
possible.

Also, $H_{3}(1)(f^{-1})\geq-b_{3}^{2}\geq-\frac{\lambda^{2}}{4}$ since from
Lemma 1, under the condition $b_{n}=0$ for all positive integers $n\neq 3$, we
receive $b_{3}\leq\frac{\lambda}{2}$. This estimate is also the best possible
with an extremal function $\frac{z}{1+\lambda/2z^{3}}$.

∎

##  4\. Zalcman conjecture

In the early 1970’s Zalcman posed the following conjecture for the class of
univalent functions:

$|a_{n}^{2}-a_{2n-1}|\leq(n-1)^{2}\quad\quad(n\in{\mathbb{N}},n\geq 2).$

There is a manuscript by Krushkal ([8]) that uses complex geometry of the
universal Teichmüller space claiming to have proven the conjecture, but this
work is not widely reckognized to be correct. Later, in 1999, Ma ([12])
proposed a generalized Zalcman conjecture,

$|a_{m}a_{n}-a_{m+n-1}|\leq(m-1)(n-1)\quad\quad(m,n\in{\mathbb{N}},m\geq
2,n\geq 2),$

and closed it for the class of starlike functions and for the class of
univalent functions with real coefficients. The general case is still an open
problem.

The Zalcman conjecture for the class ${\mathcal{U}}$ when $n=2$ and $n=3$ was
proven in [14].

Theorem C. Let $f\in{\mathcal{U}}$ be of the form (1). Then

  * $(i)$

$|a_{2}a_{3}-a_{4}|\leq 2$;

  * $(ii)$

$|a_{2}a_{4}-a_{5}|\leq 3$.

These inequalities are sharp with equality for the Koebe function
$k(z)=\frac{z}{(1-z)^{2}}=z+\sum_{n=2}nz^{n}$ and its rotations.

We now give direct proof of the Zalcman conjecture for the class
${\mathcal{U}^{+}(\lambda)}$ for the cases when $n=2$ and $n=3$.

######  Theorem 5.

Let $f\in\mathcal{U}^{+}(\lambda)$, $0<\lambda\leq 1$, be of the form (1).
Then

  * $(i)$

$-(1+\lambda)\lambda\leq a_{2}a_{3}-a_{4}\leq\frac{1}{2}\lambda$;

  * $(ii)$

$|a_{2}a_{4}-a_{5}|\leq\lambda+\lambda^{2}+\lambda^{3}$.

These inequalities are sharp.

###### Proof.

  * ($i$)

Using (8) we have

$a_{2}a_{3}-a_{4}=-b_{1}b_{2}+b_{3}\leq b_{3}\leq\frac{1}{2}\lambda,$

(where we used Lemma 1 or (11)). Also,

$-(a_{2}a_{3}-a_{4})=b_{1}b_{2}-b_{3}\leq b_{3}\leq(1+\lambda)\lambda,$

(see Remark 2), i.e., $a_{2}a_{3}-a_{4}\leq(1+\lambda)\lambda$. The functions
$f_{\lambda}$ and $f(z)=\frac{z}{1+\frac{\lambda}{2}z^{3}}$ show that the
result is sharp.

  * ($ii$)

Using (8) we obtain that

$a_{2}a_{4}-a_{5}=b_{1}^{2}b_{2}-b_{1}b_{3}-b_{2}^{2}+b_{4}\equiv\psi_{2}(b_{1}),$

$0\leq b_{1}\leq 1+\lambda$. Since,
$\max\psi_{2}=\psi_{2}(1+\lambda)=(1+\lambda)^{2}\lambda-\lambda^{2}=\lambda+\lambda^{2}+\lambda^{3}$,
then

$a_{2}a_{4}-a_{5}\leq\lambda+\lambda^{2}+\lambda^{3}.$

On the other hand,

$a_{2}a_{4}-a_{5}\geq-
b_{1}b_{3}-b_{2}^{2}\geq-(1+\lambda)\frac{\lambda}{2}-\lambda^{2}>-(\lambda+\lambda^{2}+\lambda^{3}).$

The result is sharp due to the function $f_{\lambda}$.

∎

## References

  * [1] Cho, N.E., Kowalczyk, B., Kwon, O.S., Lecko, A., Sim, Y.J.: Some coefficient inequalities related to the Hankel determinant for strongly starlike functions of order alpha. J. Math. Inequal. 11 (2), 429–439 (2017)
  * [2] Cho, N.E., Kowalczyk, B., Kwon, O.S., Lecko, A., Sim, Y.J.: The bounds of some determinants for starlike functions of order alpha. Bull. Malays. Math. Sci. Soc. 41, 523–535 (2018)
  * [3] Duren P.L., Univalent function, Springer-Verlag, New York, 1983.
  * [4] Hayman W.K., On the second Hankel determinant of mean univalent functions, Proc. London Math. Soc. (3), 18:77–94, 1968.
  * [5] A. Janteng, S.A. Halim, M. Darus, Coefficient inequality for a function whose derivative has a positive real part. J Inequal Pure Appl Math. 7 vol.2 (2006), 1–5.
  * [6] A. Janteng, S.A. Halim, M. Darus, Hankel determinant for starlike and convex functions. Int J Math Anal. 1 vol.13 (2007), 619–625.
  * [7] Kowalczyk B., Lecko A., Sim Y.J., The sharp bound of the Hankel determinant of the third kind for convex functions. Bull. Aust. Math. Soc., 97 (2018), no. 3, 435–445.
  * [8] Krushkal, S. A short geometric proof of the zalcman and bieberbach conjectures, 1–19. https://arxiv.org/abs/1408.1948v1
  * [9] Kwon, O.S.; Lecko, A.; Sim, Y.J. The bound of the Hankel determinant of the third kind for starlike functions. Bull. Malays. Math. Sci. Soc. 2019, 42, 767–780.
  * [10] Lecko A., Sim Y.J., Śmiarowska B., The sharp bound of the Hankel determinant of the third kind for starlike functions of order 1/2, Complex Analysis and Operator Theory https://doi.org/10.1007/s11785-018-0819-0.
  * [11] S.K. Lee, V. Ravichandran, S. Supramaniam, Bounds for the second Hankel determinant of certain univalent functions. J Inequal Appl. (2013) art.281.
  * [12] Ma, W. Generalized Zalcman conjecture for starlike and typically real functions, J. Math. Anal. Appl. 234, 1 (1999), 328–339.
  * [13] Obradović M., Tuneski N., Some properties of the class $\mathcal{U}$, Annales. Universitatis Mariae Curie-Skłodowska. Sectio A - Mathematica, 73 1 (2019), 49–56.
  * [14] M. Obradovic, N. Tuneski, Zalcman and generalized Zalcman conjecture for the class U, Novi Sad Journal of Mathematics, published online. https://doi.org/10.30755/NSJOM.12436
  * [15] Obradovic M., Ponnusamy S., Coefficient characterization for certain classes of univalent functions, Bull. Belg. Math. Soc. Simon Stevin 16 (2009), no. 2, 251–263.
  * [16] Obradovic M., Ponnusamy S., Wirths K.J., Geometric studies on the class $\mathcal{U}(\lambda)$, Bull. Malays. Math. Sci. Soc. 39 (2016), no. 3, 1259–1284.
  * [17] Obradovic M., Ponnusamy S., Wirths K.-J., Logarithmic coefficients and a coefficient conjecture for univalent functions, Monatsh. Math., 1–13, 2017.
  * [18] Obradovic M., Ponnusamy S., Wirths K.J., Logarithmic coefficients and a coefficient conjecture for univalent functions, Monatsh. Math. 185 (2018), no. 3, 489–501.
  * [19] Obradovic M., Ponnusamy S., On the class $\mathcal{U}$, Proc. 21st Annual Conference of the Jammu Math. Soc. and a National Seminar on Analysis and its Application, 11-–26, 2011.
  * [20] Obradovic M., Tuneski N., A class of univalent functions with real coefficients, Bull. Malays. Math. Sci. Soc., 43 (2020), 2875–2886.
  * [21] Obradovic M., Tuneski N., Coefficients of the inverse of functions for the subclass of the class ${\mathcal{U}}(\lambda)$, The Journal of Analysis, (2021). https://doi.org/10.1007/s41478-021-00326-5
  * [22] Reade M.O., Silverman H., Todorov P.G., Classes of rational functions, Topics in complex analysis (Fairfield, Conn., 1983), 99–103, Contemp. Math., 38, Amer. Math. Soc., Providence, RI, 1985.
  * [23] Thomas D.K., Tuneski N., Vasudevarao A., Univalent functions. A primer, De Gruyter Studies in Mathematics, 69. De Gruyter, Berlin, 2018.
  * [24] Vasudevarao A., Yanagihara H., On the growth of analytic functions in the class ${\mathcal{U}}(\lambda)$, Comput. Methods Funct. Theory, 13(4) (2013), 613–634.
  * [25] Zaprawa P., Third Hankel determinants for subclasses of univalent functions, Mediterr. J. Math. 14 (2017), no. 1, Art. 19, 10 pp.