11institutetext: Department of Space, Earth and Environment, Chalmers
University of Technology, Onsala Space Observatory, SE-439 92 Onsala, Sweden
22institutetext: CAS Key Laboratory for Research in Galaxies and Cosmology,
Department of Astronomy, University of Science and Technology of China, Hefei
230026 33institutetext: School of Astronomy and Space Sciences, University of
Science and Technology of China, Hefei, Anhui 230026, People’s Republic of
China 44institutetext: Joint ALMA Observatory (JAO) Alonso de Córdova 3107,
Santiago, Chile 55institutetext: European Southern Observatory, Alonso de
Córdova 3107, Casilla 19001, Vitacura, Santiago, Chile

# Detection of H2O and OH+ in $z>3$ Hot Dust-Obscured Galaxies

F. Stanley  K. K. Knudsen<EMAIL_ADDRESS>S. Aalto 11 L. Fan 11 N.
Falstad 2233 E. Humphreys 114455

###### Abstract

Aims. In this paper we present the detection of H2O and OH+ emission in $z>3$
hot dust-obscured galaxies (Hot DOGs).

Methods. Using ALMA Band-6 observations of two Hot DOGs, we have detected
H2O($2_{02}-1_{11}$) in W0149$+$2350, and H2O($3_{12}-3_{03}$) and the
multiplet OH+($1_{1}-0_{1}$) in W0410$-$0913\. These detections were
serendipitous, falling within the side-bands of Band-6 observations aimed to
study CO(9$-$8) in these Hot DOGs.

Results. We find that both sources have luminous H2O emission with line
luminosities of $L_{\rm H_{2}O(2_{02}-1_{11})}>2.2\times 10^{8}L_{\odot}$ and
$L_{\rm H_{2}O(3_{12}-3_{03})}=8.7\times 10^{8}L_{\odot}$ for W0149$+$2350 and
W0410$-$0913, respectively. The H2O line profiles are similar to those seen
for the neighbouring CO(9$-$8) line, with linewidths of FWHM $\sim 800-1000$
km s-1. However, the H2O emission seems to be more compact than the CO(9$-$8).
OH+($1_{1}-0_{1}$) is detected in emission for W0410$-$0913, with a
FWHM$=1000$ km s-1 and a line luminosity of $L_{\rm
OH^{+}(1_{1}-0_{1})}=6.92\times 10^{8}L_{\odot}$. The ratio of the observed
H2O line luminosity over the IR luminosity, for both Hot DOGs, is consistent
with previously observed star forming galaxies and AGN. The H2O/CO line ratio
of both Hot DOGs and the OH+/H2O line ratio of W0410$-$0913 are comparable to
those of luminous AGN found in the literature.

Conclusions. The bright H2O($2_{02}-1_{11}$), and H2O($3_{12}-3_{03}$)
emission lines are likely due to the combined high star-formation levels and
luminous AGN in these sources. The presence of OH+ in emission, and the
agreement of the observed line ratios of the Hot DOGs with luminous AGN in the
literature, would suggest that the AGN emission is dominating the radiative
output of these galaxies. However, follow-up multi-transition observations are
needed to better constrain the properties of these systems.

###### Key Words.:

galaxy evolution – galaxies: high-redshift – galaxies: ISM

##  1 Introduction

Energetic feedback from active galactic nuclei (AGN) has the potential to heat
and remove the cold molecular gas from the interstellar medium (ISM),
consequently suppressing the star-formation of the host galaxy. The most
successful models of galaxy formation and evolution require AGN feedback to
explain many of the puzzling properties of local massive galaxies and the
intergalactic medium (e.g., red colours, steep luminosity function, black-hole
– spheroid relationship, metal enrichment of the intergalactic medium; see
Alexander & Hickox 2012, and Fabian 2012 for reviews). Understanding the
properties of the molecular gas in the ISM of active galaxies is key to
constraining and disentangling the effects of AGN and star formation.

Within the past few years the Wide-Infrared Survey Explorer (WISE; Wright et
al., 2010) has led to the discovery of a population of luminous, dust-obscured
AGN (e.g. Eisenhardt et al., 2012; Bridge et al., 2013; Lonsdale et al.,
2015). Selected to be the reddest sources in the WISE colour-colour plot
([3.4-4.6] $\mu$m vs [4.6-12] $\mu$m), they are by definition highly obscured
and dusty galaxies. They are commonly referred to as Hot Dust Obscured
Galaxies (Hot DOGs). This new class of galaxies is a rare sample with only
1000 over the entire sky (e.g., Eisenhardt et al., 2012). They are mostly
high-redshift objects, at $1<z<4$ (e.g. Wu et al., 2012; Tsai et al., 2015),
and extremely luminous with $L_{bol}>10^{13}L_{\odot}$ (e.g. Jones et al.,
2014; Wu et al., 2014). Based on X-ray observations and spectral energy
distribution (SED) studies, there is clear evidence that they host highly
dust-obscured AGN (e.g. Stern et al., 2014; Piconcelli et al., 2015; Assef et
al., 2015, 2016; Fan et al., 2016b).

Hot DOGs are found in predominantly over-dense regions (e.g. Jones et al.,
2017), and the most luminous among them show evidence of a high merger
fraction (62% Fan et al., 2016a). This supports a scenario where Hot DOGs
could represent the post-merger transitional phase from a dusty starburst
dominated phase to an optically bright quasar phase (e.g., Eisenhardt et al.,
2012; Wu et al., 2012; Fan et al., 2016a). Their rarity could be a result of
rare events such as major mergers, the relative briefness of the transitional
phase they represent, and/or the population is tracing the tail end of the
mass function.

In this paper we present the detection of H2O, and OH+ emission lines for the
first time in Hot DOGs. In the past few years there have been a number of
detections of H2O transitions in the far-infrared (FIR) and sub-mm wavelengths
with the advent of observatories such as Herschel, NOEMA, and ALMA. Studies of
nearby galaxies have shown that H2O is the third most abundant species in
dense warm star-forming regions, and shock heated regions (e.g. Cernicharo et
al., 2006; Bergin et al., 2003; González-Alfonso et al., 2013). Detections of
H2O transitions in high-$z$ and lensed sub-mm galaxies confirm that they are
among the strongest molecular lines in starburst galaxies and also show
similar line profiles and spatial distribution as the high-$J$ CO emission
(e.g. Lis et al., 2011; van der Werf et al., 2011; Bradford et al., 2011;
Combes et al., 2012; Lupu et al., 2012; Bothwell et al., 2013; Omont et al.,
2013; Riechers et al., 2013; Weiß et al., 2013; Yang et al., 2016; Oteo et
al., 2017; Yang et al., 2019; Apostolovski et al., 2019; Casey et al., 2019;
Jarugula et al., 2019; Yang et al., 2020). Furthermore, there is a correlation
between the line luminosity and the IR luminosity, extending over three orders
of magnitude (e.g. Omont et al., 2013; Yang et al., 2013, 2016). Modelling of
H2O emission in extragalactic sources has shown that the combination of
different transitions can help disentangle the different components of the
molecular ISM (e.g. González-Alfonso et al., 2014; Liu et al., 2017).

OH+ is an important part of the chemical network for the formation of H2O, and
can trace the ionisation rate of atomic/molecular gas (e.g. Neufeld et al.,
2010; Hollenbach et al., 2012), as well as the turbulent gas component (e.g.
González-Alfonso et al., 2018). OH+ is observed both in emission and
absorption in both low and high-$z$ galaxies (e.g. van der Werf et al., 2010;
Spinoglio et al., 2012; Kamenetzky et al., 2012; Pereira-Santaella et al.,
2013; Riechers et al., 2013; Gallerani et al., 2014; van der Tak et al., 2016;
Li et al., 2020). Interestingly, OH+ is observed in emission for sources with
a strong AGN component, (e.g. van der Werf et al., 2010; Spinoglio et al.,
2012; Pereira-Santaella et al., 2013; Li et al., 2020), while starburst
galaxies show OH+ in absorption (e.g. Kamenetzky et al., 2012; Riechers et
al., 2013; van der Tak et al., 2016).

The two Hot DOGs discussed in this paper, W0149+2350 and W0410-0913, were
selected to be among the most starbursting Hot DOGs, with massive molecular
gas reservoirs discovered in CO(4$-$3) (Fan et al. 2018). Both sources have
exceptionally wide CO lines in multiple transitions, with FWHM$\approx
750-950$ km s-1 (Fan et al., 2018, Knudsen et al. in prep; see Table 1) and
large molecular gas masses of $10^{10}-10^{11}M_{\odot}$.

In section 2 we present the ALMA observations and the extraction of the
spectra. In section 3 we present our results for the two sources, and in
section 4 we discuss the cause of excitation of the observed emission and
compare it to the literature. Finally, in section 5 we give a summary and
conclusions. Throughout this paper we assume a WMAP7 cosmology.

##  2 ALMA observations and analysis

We obtained multi-band observations of W0149$+$2350 and W0410$-$0913 in
project 2017.1.00123.S to study CO rotational transitions, and these data
yielded three serendipitous detections of other molecular lines. Observations
were carried out using the band-6 receivers, where one side-band was tuned to
the redshifted CO(9-8) with the correlator used in frequency domain mode with
each spectral window (spw) having a bandwidth of 1.875 GHz, the other sideband
used a continuum setup. Observations of W0149$+$2350 were carried out on 24th
of September 2018, with a total of 49 min of on-source integration time. The
observed frequencies cover the ranges of 229.82–233.60 GHz, and 243.68–245.55
GHz, which include the redshifted H2O($2_{02}-1_{11}$) transition in addition
to the targeted CO(9-8). Observations of W0410$-$0913 were carried out on the
07th of September 2018, with a total of 6 min of on-source integration time.
The observed frequencies cover the ranges of 221.08–224.81 GHz, and
236.28–240.57 GHz, which include the H2O($3_{12}-3_{03}$) transition, and the
OH+($1_{1}-0_{1}$) multiplet in addition to the targeted CO(9-8). Flux and
bandpass calibration was done using J0006$-$0623 and J0423$-$0120, and gain
calibration was done with J0152+2207 and J0407$-$1211, for W0149$+$2350 and
W0410$-$0913, respectively. Beam sizes correspond to
$0.53\hbox{${}^{\prime\prime}$}\times 0.32\hbox{${}^{\prime\prime}$}$, and
$0.67\hbox{${}^{\prime\prime}$}\times 0.52\hbox{${}^{\prime\prime}$}$, for
W0149$+$2350 and W0410$-$0913 respectively.

Reduction, calibration, and imaging was done using CASA (Common Astronomy
Software Application111https://casa.nrao.edu; McMullin et al. 2007). The
pipeline reduced data delivered from the observatory were of sufficient
quality, no or little additional flagging and further calibration was
necessary. The steps required for standard reduction and calibration included
in the pipeline include flagging, bandpass calibration, as well as flux and
gain calibration. For our imaging process we applied natural weighting. A
conservative estimate of the error on the absolute flux calibration is $10\%$.
Continuum emission was subtracted using the line-free channels identified in
both sidebands for each source. For both sources, the channel width was set to
60 km s-1.

To extract the spectrum we used the following procedure. We created the
moment-0 map of each emission line, and used the imfit tool of casa to get an
estimate of the extent of the emission. We then extracted the spectrum from a
region that is equivalent to two times the size estimate of the source.

In this paper we also show the CO(9$-$8) emission for comparison. This data
has been extracted and analysed in the same way as the H2O and OH+ emission
that are the focus of this paper. However, the CO(9$-$8) data will be
presented in more detail in Knudsen et al. (in prep) along with other CO
transitions.

Table 1: Source properties as presented in Fan et al. (2018).  Source | RA | DEC | $z_{\rm CO(4-3)}$ | $L^{\prime}_{\rm CO(4-3)}$ | $L_{\rm IR,AGN}$ | $L_{\rm IR,SF}$
---|---|---|---|---|---|---
| [J2000] | [J2000] |  | [K km s-1 pc2] | [$L_{\odot}$] | [$L_{\odot}$]
W0149$+$2350 | 01:49:46.16 |  $+$23:50:14.6 | 3.237 | $2.4\times 10^{10}$ | $7.6\times 10^{13}$ | $1.14\times 10^{13}$
W0410$-$0913 | 04:10:10.60 |  $-$09:13:05.2 | 3.631 | $1.8\times 10^{11}$ | $1.5\times 10^{14}$ | $4.7\times 10^{13}$

##  3 Results

Table 2: Identified lines based on the CDMS333https://cdms.astro.uni-koeln.decatalogue entries, and corresponding RMS of their spectrum (RMSspec). Line | $\nu_{\rm rest}$ | RMSspec
---|---|---
| [GHz] | [mJy]
H2O($2_{02}-1_{11}$) | 987.937 | 0.28
H2O($3_{12}-3_{03}$) | 1097.365 | 0.30
OH+($1_{1}-0_{1}$) multiplet: |  | 0.57
OH+($1_{11}-0_{11}$) | 1032.998 |
OH+($1_{12}-0_{11}$) | 1033.004 |
OH+($1_{11}-0_{12}$) | 1033.113 |
OH+($1_{12}-0_{12}$) | 1033.118 |

###  3.1 H2O(202–111) in W0149$+$2350

For W0149$+$2350 we have detected H2O emission in the transition of
H2O(202–111), see Fig. 1 (in yellow). Although the line is truncated, it
covers $\rm\sim 1000\,km\,s^{-1}$ with a line profile that cannot be described
by a single Gaussian. This type of line profile is also seen in their CO(9-8)
line emission, see Fig. 1 (in grey; Knudsen et al. in prep).

In Fig. 1 we show the line, with the fit of a single Gaussian, and a double
Gaussian. A double Gaussian fit to the line improves the $\chi^{2}$ by 22%
compared to the single Gaussian fit. In Table 3 we give the fit results for
the two component fit to the line. However, as the line is truncated, we
integrate the measured flux of the line, to calculate a lower limit of the
total integrated line flux (see Table 3).

In Fig. 2 we show the HST map from Fan et al. (2016a), with the moment-0 map
contours of the H2O emission in blue, and the CO contours in yellow for
comparison. To estimate the extent of the emission observed, we use the imfit
tool of casa, to fit to the moment-0 maps of each line. However, as the H2O
line is truncated, to have a meaningful comparison of the extent of the
emission we create moment-0 maps limited to the velocity range of
$-300<\upsilon<750$ km s-1, for both the H2O and CO emission. We find that the
H2O emission of W0149$+$2350 extends across $2.3\pm 0.6$ kpc, while the
CO(9$-$8) emission extends across $3.3\pm 0.5$ kpc (see Table 4).

Figure 1: The detected H2O line (yellow) of W0149$+$2350, with the CO($9-8$)
emission (grey) shown in the background for comparison. The fitted single and
double Gaussians are shown with the red and blue curves respectively.
Figure 2: HST WFC3 cutout of W0149+2350. Overplotted are the contours corresponding to 3, 4, 5, 7, and 9$\sigma$, for the H2O line (blue), and CO($9-8$) line (yellow). The RMS values of the moment maps correspond to 0.046 Jy km s-1 and 0.055 Jy km s-1 for the H2O and CO($9-8$) lines, respectively. Negative contours corresponding to -3$\sigma$ are also plotted with dashed lines. Table 3: Detected line properties. For each emission line we give the central velocity ($\upsilon_{\rm cen}$), peak flux ($S_{\rm peak}$), FWHM, and integrated flux ($S_{\rm int}$), based on the Gaussian fits to the line. We also give the corresponding line luminosity ($L_{\rm line}$). The H2O emission for both sources is best fit by a double Gaussian, therefore we give the parameters of the two components corresponding to the fit. In the case of W0149$+$2350, where the line is truncated, we give a lower limit on the total $S_{\rm int}$ and $L_{\rm line}$ based on the integration of the observed spectral line.  Source | line | $\upsilon_{\rm cen}$ | $S_{\rm peak}$ | FWHM | $S_{\rm int}$ | $L_{\rm line}$
---|---|---|---|---|---|---
|  | [km s-1] | [mJy] | [km s-1] | [Jy km s-1] | [$\times 10^{8}L_{\odot}$]
W0149$+$2350 | H2O($2_{02}-1_{11}$) |  |  |  |  |
| comp1 | $-219\pm 103$ | $1.3\pm 0.2$ | $564\pm 413$ | $0.80\pm 0.60$ | $1.58\pm 1.17$
| comp2 | $396\pm 58$ | $1.4\pm 0.2$ | $421\pm 99$ | $0.63\pm 0.18$ | $1.24\pm 0.36$
| total |  |  |  |  $>$1.12 | $>2.2$
W0410$-$0913 | H2O($3_{12}-3_{03}$) |  |  |  |  |
| comp1 | $-307\pm 44$ | $1.2\pm 0.7$ | $256\pm 167$ | $0.31\pm 0.29$ | $0.83\pm 0.76$
| comp2 | $99\pm 54$ | $3.8\pm 0.2$ | $737\pm 103$ | $2.99\pm 0.46$ | $7.90\pm 1.23$
| total |  |  |  | $3.30\pm 0.54$ | $8.73\pm 1.44$
| OH+($1_{1}-0_{1}$) | $-100\pm 83$ | $2.6\pm 0.3$ | $1000\pm 196$ | $2.78\pm 0.62$ | $6.92\pm 1.55$

###  3.2 H2O(312–303) and OH+($1_{1}-0_{1}$) in W0410$-$0913

For W0410$-$0913 we have detected the H2O($3_{12}-3_{03}$), and the
OH+($1_{1}-0_{1}$) multiplet with transitions at 1032.998–1033.118 GHz (see
Table 3, although unresolved). Similar to what is seen for W0149$+$2350, the
H2O($3_{12}-3_{03}$) line also shows a very wide profile, that is similar to
that of the CO(9$-$8).

In Fig. 3 we show the H2O($3_{12}-3_{03}$) line, with the fit of a single
Gaussian, and a double Gaussian. A double Gaussian fit to the line improves
the $\chi^{2}$ by 12% compared to the single Gaussian fit. In Table 3 we give
the results for the two component fit to the line.

Due to their large linewidths the CO(9$-$8) and OH+ lines are partially
blended. We perform a simultaneous fit of the CO(9$-$8) and OH+ emission. We
do this by using a double Gaussian fit for the CO(9$-$8), and a single
Gaussian for the OH+ emission. We allow all three parameters of the Gaussians
to vary during the fit, restricting the central velocity ($\upsilon_{\rm
cen}$) of each component of the CO line within a range of 400 km s-1, and for
the OH+ line within 200 km s-1 of the expected $\upsilon_{\rm cen}$. In Fig. 5
we show the double Gaussian fit to the CO(9$-$8) line (blue dotted curve), and
the resulting fit to the OH+ emission (blue dashed curve). In Table 3 we give
line properties based on the fit. In this case the integrated line flux
($S_{\rm int}$) is based on the fit. We note that in our analysis we have
assumed that the OH+ is in emission based on the observed spectrum; however,
it is possible that it has a different line profile, such as a P-Cygni
profile, that is not visible due to the blending with the CO(9$-$8) line. In
some starburst galaxies OH+ has been found to have a P-Cygni line profile. An
example is NGC253, for which it is concluded that the OH+ is primarily arising
in cold diffuse foreground gas (van der Tak et al., 2016). If indeed the OH+
multiplet observed here had a P-Cygni profile, it would require a strongly
asymmetric line profile for the CO(9-8) in order to compensate for the
superimposed absorption feature of the OH+. This would require luminous
CO(9$-$8) emission at $\sim$1000 km s-1 from the line centre. Such a peculiar
line profile is unlikely, and is not seen for the lower$-J$ CO (Fan et al.,
2018, Knudsen et al. in prep) and H2O transitions. It could also be possible
for the emission identified as OH+ to be alternatively interpreted as a
massive and highly excited molecular outflow. However, such a scenario would
require a combination of orientation and foreground absorption that would
allow us to see the red component of the outflow while not seeing emission
from the blue component. Furthermore, there is no evidence for such a massive
outflow in the lower$-J$ CO transitions (Fan et al., 2018, Knudsen et al. in
prep).

In Fig. 4 we show the HST map from Fan et al. (2016a), with the moment-0 map
contours of the H2O emission in blue, the OH+ emission in white, and the CO
contours in yellow for comparison. The OH+ emission follows closely the extent
seen in the CO(9$-$8), that shows an extended non-uniform structure, but the
H2O emission is more compact. We estimate the extent of the emission observed
using the imfit tool of casa. Following the analysis of W0149$+$2350, we
create moment-0 maps limited to the velocity range of $-500<\upsilon<500$ km
s-1, for the H2O, CO, and OH+ emission, to match to the velocity range covered
by the H2O line. The OH+ is extended across $9.3\pm 2.1$ kpc, similar to the
CO($9-8$) emission extending over $5.7\pm 0.6$ kpc, while the H2O is more
compact covering $3.0\pm 0.6$ kpc. The extent of the CO(9$-$8) and OH+
emission is significantly larger than the H2O; however, deeper and higher
resolution observations are needed to better constrain these properties and to
further interpret these results.

Table 4: The estimated size, based on a 2D fit to the moment-0 maps of each emission line. The size given corresponds to the FWHM of the emission, deconvolved from the beam.For W0149$+$2350 sizes were calculated for the restricted velocity range of $-300<\upsilon<750$ km s-1, to match the observed velocities of the truncated H2O($2_{02}-1_{11}$) line. For W0410$-$0913, sizes were calculated for the velocity range of $-500<\upsilon<500$ km s-1, to match the observed velocities of the H2O($3_{12}-3_{03}$) line. We note that the sizes do not change significantly if we do not apply these restrictions. Source | line | size | PA
---|---|---|---
|  | [arcsec2] | [deg]
W0149$+$2350: | H2O($2_{02}-1_{11}$) | $(0.30\pm 0.08)\times(0.22\pm 0.08)$ | $43\pm 89$
| CO(9$-$8) | $(0.43\pm 0.07)\times(0.31\pm 0.06)$ | $4\pm 26$
W0410$-$0913: | H2O($3_{12}-3_{03}$) | $(0.41\pm 0.08)\times(0.33\pm 0.09)$ | $126\pm 86$
| OH+($1_{1}-0_{1}$) | $(1.3\pm 0.3)\times(0.8\pm 0.2)$ | $107\pm 30$
| CO(9$-$8) | $(0.77\pm 0.08)\times(0.43\pm 0.05)$ | $108\pm 8$
Figure 3: The detected H2O line (yellow) of W0410$-$0913, with the CO($9-8$)
emission (grey) shown in the background for comparison. The fitted single and
double Gaussians are shown with the red and blue curves respectively.
Figure 4: HST WFC3 cutout of W0410-0913. Overplotted are the contours
corresponding to 3, 4, 5, 7, and 9$\sigma$, for the H2O line (blue), OH+ line
(white), and CO($9-8$) line (yellow). The RMS values of the moment maps
correspond to 0.12 Jy km s-1, 0.146 Jy km s-1, and 0.116 Jy km s-1 for the
H2O, OH+ and CO($9-8$) lines, respectively. Negative contours corresponding to
-3$\sigma$ are also plotted with dashed lines. Figure 5: The detected OH+ line
(yellow) of W0410$-$0913, next to the CO(9$-$8) line (grey). The grey colour
also shows the channels used to fit the CO(9$-$8) line. The blue dotted curve
shows the double Gaussian fit to the CO($9-8$), the blue dashed curve shows
the single Gaussian fit to the OH+ emission, and the solid curve is the
combination of the above.

For both sources we find that the high-density gas traced by H2O extends over
a few kpc scales. Based on the total gas masses estimated by Fan et al.
(2018), and if we assume a spherical geometry444We note that if we instead
assume a cylindrical geometry with a radius based on the estimated sizes from
Table 4 and a height of 0.5 kpc, our results remain the same. with radii of
1-2 kpc and a volume filling factor of 0.01-0.1%, then the estimated gas
density is of the range of $n(H_{2})\sim 10^{5}-10^{6}$ cm-3. This is in
agreement with what is required for the excitation of the observed H2O
emission. Therefore, the extent of the observed emission is consistent with
the gas mass estimated for these galaxies.

##  4 Discussion

###  4.1 Excitation of the H2O and OH+ emission

In a detailed study of H2O submillimetre lines in the nuclei of nearby star-
forming galaxies, Liu et al. (2017) used modelling to determine the physical
conditions and excitation mechanisms needed for the H2O transitions of low,
medium, and high energy transition. Liu et al. (2017) examined three
components of the molecular ISM: a cold component with densities of
$n(H_{2})\sim 10^{4}-10^{6}$ cm-3, dust temperatures of $T_{d}\sim 20-30$ K,
and column densities of $N(H_{2})\sim 10^{23}$ cm-2; a warm component with
$n(H_{2})\sim 10^{5}-10^{6}$ cm-3, $T_{d}\sim 40-70$ K, and $N(H_{2})\sim
1-4\times 10^{24}$ cm-2; and a hot component with $n(H_{2})\geq 10^{6}$ cm-3,
$T_{d}\sim 100-200$ K, $N(H_{2})\geq 5\times 10^{24}$ cm-2. Interestingly, the
H2O($2_{02}-1_{11}$) transition can trace both the warm and hot components,
but the H2O($3_{12}-3_{03}$) transition is stronger in the warm component of
the ISM, and is not strongly produced in hot ISM conditions. Based on the
radiative-transfer analysis of Liu et al. (2017) the H2O($2_{02}-1_{11}$) and
H2O($3_{12}-3_{03}$) transitions can be produced by collisional excitation
alone, while higher energy transitions characteristic of the hot ISM component
require IR pumping to reach the observed intensities. However, González-
Alfonso et al. (2014), that also modelled the H2O submillimetre lines for IR
galaxies, argue that IR pumping has a significant role in producing the
H2O($2_{02}-1_{11}$) and H2O($3_{12}-3_{03}$) line emission. In both cases,
higher energy transitions are more likely to exclusively trace the hot
component of the ISM surrounding AGN. Indeed, there was recently a detection
of H2O($4_{14}-3_{21}$) from the circumnuclear disk of a lensed quasar (Stacey
et al., 2020).

As we only have a single H2O line detection for each source, it is not
possible to make an extensive analysis. But we take a simplistic modelling
approach using the non-LTE molecular radiative transfer code radex (van der
Tak et al., 2007), to provide some constraints on the possible excitation of
the observed H2O emission.

We use the peak fluxes for each of the observed H2O lines and the
corresponding CO($9-8$) line, for each source. We assume a spherical symmetry,
and CO column densities in the range of $10^{16}-10^{17}$ cm-2 to avoid
optically thin emission at the low end and too high opacities at the high end.
Based on the SED fit and decomposition of these sources (Tsai et al., 2015;
Fan et al., 2016b), we expect an AGN torus dust temperature of 450 K, and a
dust temperature due to star-formation of 51 K and 63 K.

For W0149$+$2350 we find that the H2O($2_{02}-1_{11}$) transition, at 987.94
GHz (E${}_{\rm L}=$53 K), may be collisionally excited. For intermediate
densities ($\leq 10^{6}$ cm-3) relatively high H2O($2_{02}-1_{11}$) abundances
($>10^{-6}$) are required. At higher densities ($>10^{6}$ cm-3), lower
abundances of $10^{-7}-10^{-8}$ are possible. For W0410$-$0913 we find that
the H2O($3_{12}-3_{03}$) transition, at 1097.36 GHz (E${}_{\rm L}=$196 K), may
be collisionally excited, but that would require densities $>10^{6}$ cm-3.
Even for these high densities, the H2O($3_{12}-3_{03}$) abundance is rather
high ($10^{-5}-10^{-6}$). However, for both sources, having such high
densities filling the region corresponding to the beam at these distances
(i.e. $\sim$4–5 kpc) is unlikely. This can either be solved with a small
filling factor of dense clumps and/or the emission emerging from a dense
nuclear structure (e.g. the AGN torus). Alternatively, all or parts of the H2O
emission are being influenced by IR pumping, which would allow it to arise
also in lower density gas. This is in agreement with what is argued by
González-Alfonso et al. (2014).

Based on the results of our simplistic RADEX modelling and what has been found
by González-Alfonso et al. (2014) and Liu et al. (2017), we expect that the
transitions that we have detected are due to the high levels of star-formation
in our sources, with a possibly significant contribution from the AGN (through
IR pumping).

So far in the literature, sources where the OH+ line is in emission have been
galaxies with significant AGN contributions such as Mrk231 (van der Werf et
al., 2010), NGC1068 (Spinoglio et al., 2012), and NGC7130 (Pereira-Santaella
et al., 2013). Therefore, the detection of OH+ in emission for W0410$-$0913,
could be indicative of excitation from cosmic rays and X-rays from the AGN
(e.g. van der Werf et al., 2010; Spinoglio et al., 2012; Pereira-Santaella et
al., 2013; Li et al., 2020). The relative line intensities between the
CO($9-8$), the OH+($1_{1}-0_{1}$) multiplet, and the H2O($3_{12}-3_{03}$)
observed for W0410$-$0913 are consistent with those of Mrk231 (van der Werf et
al., 2010), for which it was shown that the combination of strong high$-J$ CO,
and OH+ emission indicate X-ray driven excitation from the AGN of Mrk231.
However, due to the limitations of our data, and the lack of other HnO+
transitions, we cannot confirm that the OH+ emission is due to the AGN through
radiative transfer modelling.

###  4.2 The H2O–IR luminosity ratio of Hot DOGs

The relationship between the H2O line luminosity with IR luminosity (from 8 to
1000 $\mu m$; $L_{\rm IR}$), and the ratio of the two, has been well studied
for low and high-$z$ dusty star-forming galaxies (see e.g. Yang et al., 2013,
2016; Jarugula et al., 2019). Here we compare the two detected Hot-DOGs of
this paper to low and high-$z$ ultra-luminous IR galaxies (ULIRGs), sub-
millimetre galaxies (SMGs) and AGN in the literature. We only compare to
sources with the same H2O transition available, as each of the sources.

As our sources have both a strong AGN and star formation contribution to the
total $L_{\rm IR}$, it is important to consider how we compare the H2O line
luminosity and the IR luminosity. For example, if the observed H2O emission is
primarily tracing the star formation in our source, then we should take the
ratio with the IR luminosity due to star formation ($L_{\rm IR,SF}$); however,
if the emission is tracing both AGN and star formation regions then the total
IR luminosity should be considered ($L_{\rm IR,tot}$). It is not possible in
our analysis to confidently distinguish if the H2O emission is only due to the
star-formation of the source, as excitation from the AGN could also be
contributing significantly. For this reason we examine both of the possible
scenarios when comparing to the literature.

In Fig. 6(top) we show the ratio of the line luminosity for the $\rm
H_{2}O(2_{02}-1_{11})$ transition over $L_{\rm IR}$ ($L_{\rm
H_{2}O(2_{02}-1_{11})}$/$L_{\rm IR}$) as a function of $L_{\rm IR}$. Here we
plot two values for W0149$+$2350, $L_{\rm H_{2}O(2_{02}-1_{11})}$/$L_{\rm
IR,tot}>0.25\times 10^{-5}$ (solid star) and $L_{\rm
H_{2}O(2_{02}-1_{11})}$/$L_{\rm IR,SF}>2\times 10^{-5}$ (hollow star), in
comparison to both strong AGN and ULIRGs/SMGs that have been detected in the
same transition. As we only have a lower limit for W0149$+$2350 the $L_{\rm
H_{2}O(2_{02}-1_{11})}$/$L_{\rm IR,tot}$ ratio, although low, is consistent
with both AGN and ULIRGs/SMGs at low and high-$z$. However, the lower limit
for $L_{\rm H_{2}O(2_{02}-1_{11})}$/$L_{\rm IR,SF}$, is only consistent with
luminous high-$z$ ULIRGs/SMGs, as can be expected if the source of the H2O
emission are the star formation regions.

In Fig. 6(bottom) we show the ratio for the $\rm H_{2}O(3_{12}-3_{03})$
transition ($L_{\rm H_{2}O(3_{12}-3_{03})}$/$L_{\rm IR}$) as a function of
$L_{\rm IR}$. Again, we plot two values corresponding to W0410$-$0913, $L_{\rm
H_{2}O(3_{12}-3_{03})}$/$L_{\rm IR,tot}=0.44\times 10^{-5}$ (solid star) and
$L_{\rm H_{2}O(3_{12}-3_{03})}$/$L_{\rm IR,SF}=1.86\times 10^{-5}$ (hollow
star). In this case the $L_{\rm H_{2}O(3_{12}-3_{03})}$/$L_{\rm IR,tot}$ ratio
of W0410$-$0913 is consistent with low-$z$ AGN and ULIRGs, but is
significantly below what is seen for high-$z$ ULIRGs/SMGs, while the $L_{\rm
H_{2}O(3_{12}-3_{03})}$/$L_{\rm IR,SF}$ ratio is consistent with the high-$z$
ULIRGs/SMGs.

Figure 6: The ratio of the H2O line luminosity over the IR luminosity ($L_{\rm
H_{2}O}$/$L_{\rm IR}$), as a function of $L_{\rm IR}$, for the two Hot DOGs.
(top) W0149$+$2350 in comparison to literature sources with detected $\rm
H_{2}O(2_{02}-1_{11})$ emission, labelled based on their classification (van
der Werf et al., 2011; Combes et al., 2012; Bothwell et al., 2013; Omont et
al., 2013; Riechers et al., 2013; Falstad et al., 2015, 2017; Yang et al.,
2016; Liu et al., 2017; Oteo et al., 2017; Apostolovski et al., 2019; Jarugula
et al., 2019). Also shown are the average ratio found by Yang et al. (2013)
for nearby IR galaxies. (bottom) W0410$-$0913 in comparison to literature
sources with detected $\rm H_{2}O(3_{12}-3_{03})$ emission, labelled based on
their classification (Riechers et al., 2013; Falstad et al., 2015, 2017; Liu
et al., 2017; Oteo et al., 2017), with the average ratio found by Yang et al.
(2013) for nearby IR galaxies also shown.

###  4.3 The H2O and OH+ line ratios of Hot DOGs

In this section we compare the observed H2O and OH+ line ratios of our sources
to the ratios in other galaxies found in the literature.

We find that for W0149$+$2350 the $H_{2}O(2_{02}-1_{11})$/CO(9$-$8) ratio is
$>0.5$. This is in agreement with what is seen in the literature for high–$z$
ULIRGs/SMGs and AGN (0.5–0.8 for ULIRGs/SMGs, and 0.4–0.7 for AGN; e.g.
Apostolovski et al., 2019; Bradford et al., 2009; Jarugula et al., 2019; Li et
al., 2020; Oteo et al., 2017; Riechers et al., 2013; van der Werf et al.,
2011; Weiß et al., 2007). However, the ratio of
$H_{2}O(2_{02}-1_{11})$/CO(4$-$3) is $>1.4$, which is significantly higher
than what has been observed for high–$z$ ULIRGs/SMGs (0.4–0.85; e.g. Bradford
et al., 2009; Jarugula et al., 2019; Omont et al., 2013; Yang et al., 2016,
2017), although consistent with the luminous AGN APM08279$+$5255 that has a
$H_{2}O(2_{02}-1_{11})$/CO(4$-$3) of 2.4 (van der Werf et al., 2011; Weiß et
al., 2007).

For W0410$-$0913 we calculate line ratios of $\rm
H_{2}O(3_{12}-3_{03})/CO(9-8)=0.8\pm 0.2$, $\rm
H_{2}O(3_{12}-3_{03})/CO(4-3)=0.6\pm 0.1$, and $\rm
OH^{+}(1_{1}-0_{1})/H_{2}O(3_{12}-3_{03})=0.8\pm 0.2$. In this case the
available similar measurements in the literature are limited. However, the
$\rm H_{2}O(3_{12}-3_{03})/CO(9-8)$ ratio of W0410$-$0913 is consistent if
somewhat higher than high–$z$ ULIRGs/SMGs (0.4–0.75; e.g. Oteo et al., 2017;
Riechers et al., 2013). The $\rm OH^{+}(1_{1}-0_{1})/H_{2}O(3_{12}-3_{03})$
ratio is consistent with low–$z$ luminous AGN (e.g. Mrk231 and NGC1068; van
der Werf et al., 2010; Liu et al., 2017; Spinoglio et al., 2012).

The nearby ULIRG Mrk231 is particularly interesting to compare to as it has a
luminous AGN that contributes 70% of the IR luminosity, similar to what is
found for our Hot DOGs with 87% and 77% in W0149$+$2350 and W0410$-$0913,
respectively (Fan et al., 2016b). Mrk231 has $\rm
H_{2}O(2_{02}-1_{11})$/CO(4$-$3) and $\rm H_{2}O(3_{12}-3_{03})/CO(4-3)$ line
ratios of 0.4 (e.g. van der Werf et al., 2010; Papadopoulos et al., 2007), and
an $\rm OH^{+}(1_{1}-0_{1})/H_{2}O(3_{12}-3_{03})$ ratio of 0.65 (van der Werf
et al., 2010; Liu et al., 2017). Overall, W0410$-$0913 seems most consistent
with what is seen for Mrk231, while for W0149$+$2350 the $\rm
H_{2}O(2_{02}-1_{11})$/CO(4$-$3) ratio is significantly larger than for
Mrk231. Interestingly, W0149$+$2350 also has a significantly larger AGN
contribution to the IR luminosity than W0410$-$0913 and Mrk231.

##  5 Summary and Conclusions

In this paper we have presented the recent detection of H2O and OH+ in $z>3$
Hot DOGs. This is the first detection of these emission lines for this class
of galaxies, and one of the few in high-$z$ non-lensed galaxies (e.g. Casey et
al., 2019). Specifically, we have detected the $\rm H_{2}O(2_{02}-1_{11})$
transition from W0149$+$2350, and the $\rm H_{2}O(3_{12}-3_{03})$, and the
OH+($1_{1}-0_{1}$) multiplet from W0410$-$0913\. These were serendipitous
detections in an observation program targeting the CO(9$-$8) emission of these
sources (Knudsen et al. in prep).

Similar to what has been found for previously observed high-$z$ ULIRGs and
SMGs, the line profile of the H2O emission lines follows that of the high-$J$
CO(9$-$8) emission. But the extent of the H2O emission seems to be more
compact than that of CO(9$-$8), and in the case of W0410$-$0913 it is more
compact than the OH+($1_{1}-0_{1}$) emission too. However, deeper and higher
resolution observations are needed to better constrain the emission properties
and to further interpret these results.

A single H2O line detection does not allow for the disentangling of the ISM
components and determining the molecular gas properties. However, the luminous
H2O emission in these two Hot DOGs indicate the existence of warm dense
molecular gas conditions ($n(H_{2})\sim 10^{5}-10^{6}$ cm-3, $T_{d}\sim 40-70$
K), possibly dominated by collisional excitation, with likely contribution
from IR pumping from the AGN. We cannot confidently distinguish between
excitation due to the AGN or star-formation or both for the H2O emission.
However, the detection of OH+($1_{1}-0_{1}$) in emission for W0410$-$0913, and
the agreement of the observed line ratios with luminous AGN in the literature
indicate that the energy output from the AGN is dominating the radiative
output of this galaxy, even though there is significant ongoing star formation
(1000–5000 M⊙ yr-1). This is consistent with the fact that these galaxies host
AGN that dominate the IR luminosity, contributing $\gtrsim 70\%$ (Fan et al.,
2016b). This would also be consistent with the scenario that Hot DOGs are
going through a transitional phase from a starburst–dominated to an
AGN–dominated phase (e.g., Eisenhardt et al., 2012; Wu et al., 2012; Fan et
al., 2016a).

In order to break the degeneracies and disentangle the relative contributions
from the AGN and the star formation, a multi-transitional approach is
required. Based on the results of modelling studies (e.g. González-Alfonso et
al., 2014; Liu et al., 2017), it is possible to design observational programs
targeting a combination of H2O transitions that trace different components of
the molecular gas. Transitions such as $\rm H_{2}O(2_{02}-1_{11})$ and $\rm
H_{2}O(3_{12}-3_{03})$, with energies of $E_{up}<250-350$ K, can be produced
by collisional excitation alone or a combination of collisional excitation and
IR pumping. However, higher energy transitions, such as H2O($4_{14}-3_{21}$)
or $\rm H_{2}O(4_{22}-4_{13})$, require IR pumping and are therefore more
likely to directly trace excitation from the AGN. A combination of H2O
transitions from low, medium, and high energy levels, could be a useful tool
for disentangling the AGN and star formation contributions to the excitation
of the dense molecular gas. Furthermore, targeting additional HnO+
transitions, such as H2O+ and H3O+ in combination with OH+, would allow for
constraints on the chemistry and excitation of the diffuse gas of these
galaxies, possibly directly connected to the AGN (see e.g., González-Alfonso
et al., 2013, 2018, for detailed analysis).

## Acknowledgements

We thank the anonymous referee for constructive comments. We thank the staff
of the Nordic ALMA Regional Center node for their support and helpful
discussions. We thank Dr Hannah Calcutt and Dr Pierre Cox for helpful
discussions. KK acknowledges support from the Knut and Alice Wallenberg
Foundation. SA acknowledges support from the European Research Council (ERC)
and the Swedish Research Council. LF acknowledges the support from the
National Natural Science Foundation of China (NSFC Nos. 11822303, 11773020 and
11421303) and Shandong Provincial Natural Science Foundation (JQ201801). This
paper makes use of the following ALMA data: ADS/JAO.ALMA#2017.1.00123.S. ALMA
is a partnership of ESO (representing its member states), NSF (USA) and NINS
(Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan) and KASI
(Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA
Observatory is operated by ESO, AUI/NRAO and NAOJ.

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††footnotetext:  File: Final_version.tex, printed: 2024-8-29, 18.09

# Region of variability for certain subclass of univalent functions

Jnana Preeti Parlapalli  Jnana Preeti Parlapalli, Department of Computer
Science, SRM University, Amaravati, Andhra Pradesh 522502, India.
<EMAIL_ADDRESS>and  Vasudevarao Allu  Vasudevarao Allu, School of
Basic Science, Indian Institute of Technology Bhubaneswar, Bhubaneswar-752050,
Odisha, India<EMAIL_ADDRESS>
###### Abstract.

Let $\mathbb{D}:=\\{z\in\mathbb{C}:|z|<1\\}$ be the unit disk. For
$0<\alpha<1$, let $f_{\alpha}(z)=z/(1-z^{\alpha})$ for $z\in\mathbb{D}$. We
consider the class $\mathcal{F}$ of analytic functions $f_{\alpha}$ which
satisfy
$\left(1+zf^{\prime\prime}_{\alpha}(z)/f^{\prime}_{\alpha}(z)\right)>\beta$
for $0<\beta<1$. In this paper, we determine the region of variability of
$\log f^{\prime}_{\alpha}(z_{0})$ for fixed $z_{0}\in\mathbb{D}$ when $f$
varies over the class
${\mathcal{F}}(\lambda):=\\{f_{\alpha}\in\mathcal{F}:f_{\alpha}(0)=0,f^{\prime}_{\alpha}(0)=1\,\mbox{and}\,f^{\prime\prime}_{\alpha}(0)=2\lambda(1-\beta)\,\,\,\mbox{for}\,\,0\leq\lambda\leq
1\\}$.

###### Key words and phrases:

Analytic, univalent, starlike, convex, close-to-convex functions; Region of
variability

###### AMS Subject Classification: Mathematics Subject Classification:

Primary 30C45, 30C50, 30C80

##  1\. Introduction

Let $\mathbb{D}:=\\{z:\,|z|<1\\}$ be the unit disk in the complex plane
$\mathbb{C}$ and $\mathcal{H}$ denote the space of all analytic functions on
${\mathbb{D}}$. Here we think of $\mathcal{H}$ as a topological vector space
endowed with the topology of uniform convergence over compact subsets of
$\mathbb{D}$. A function $f$ is said to be univalent if it is one-to-one.
Further, let $\mathcal{A}:=\\{f\in\mathcal{H}:\,f(0)=f^{\prime}(0)-1=0\\}$ and
$\mathcal{S}$ denote the class of univalent functions in $\mathcal{A}$. A
domain $\Omega\subseteq\mathbb{C}$ is called a starlike domain with respect to
a point $z_{0}\in\Omega$ if the line segment $tw+(1-t)z_{0}$ joining $z_{0}$
to any other point $w$ in $\Omega$ lies entirely in $\Omega$. If $z_{0}=0$
then the domain $\Omega$ is called a starlike domain. A function
$f\in\mathcal{S}$ is said to be starlike with respect to $w_{0}\in
f(\mathbb{D})$ if the image domain $f(\mathbb{D})$ is starlike with respect to
$w_{0}$. If $w_{0}=0$, $f$ is simply called a starlike function. The class of
univalent starlike functions is denoted by ${\mathcal{S}}^{*}$. A domain
$\Omega\subseteq\mathbb{C}$ is said to be convex if the line segment joining
any two points of the domain $\Omega$ is contained in $\Omega$. A function $f$
is said to be convex if the image of the domain $\Omega$ under $f$ is a convex
domain. The class of univalent convex functions is denoted by ${\mathcal{C}}$.

A function $f\in\mathcal{A}$ is called close-to-convex if there exists a
convex (univalent) function $g$ and a number $\phi\in{\mathbb{R}}$ such that
${\rm Re\,}(e^{i\phi}f^{\prime}(z)/g^{\prime}(z))>0$ for $z\in{\mathbb{D}}$.
Each univalent starlike function $f$ is characterized by the analytic
condition ${\rm Re\,}(zf^{\prime}(z)/f(z))>0$ in ${\mathbb{D}}$. Due to
Alexender’s theorem it is known that $zf^{\prime}$ is starlike if, and only
if, $f$ is convex (see [5]). In $1952$, Kaplan [6] proved that every close-to-
convex function is univalent in ${\mathbb{D}}$. A function $f\in\mathcal{S}$
is said to be starlike function of order $\alpha,-1/2\leq\alpha<1$ if it
satisfies the condition

${\rm
Re\,}\left(\frac{zf^{\prime}(z)}{f(z)}\right)>\alpha\,\,\,\,\,\mbox{for}\quad
z\in\mathbb{D}.$

Let $\mathcal{S}^{*}(\alpha)$ be the class of all starlike functions of order
$\alpha$ in $\mathbb{D}$. A function $f\in\mathcal{S}$ is said to be convex
function of order $\alpha,-1/2\leq\alpha<1$ if it satisfies the condition

${\rm
Re\,}\left(1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right)>\alpha\,\,\,\,\mbox{for}\quad
z\in\mathbb{D}.$

Let $\mathcal{C}(\alpha)$ be the class of all convex functions of order
$\alpha$ in $\mathbb{D}$.

Let $\mathcal{F}\subset\mathcal{A}$ and $z_{0}\in\mathbb{D}$. Then upper and
lower estimates of the form

$K_{1}\leq f(z_{0})\leq K_{2},\,\,\,M_{1}\leq|f^{\prime}(z_{0})|\leq
M_{2},\,\,\,m_{1}\leq\mbox{Arg}f^{\prime}(z_{0})\leq m_{2}\,\,\mbox{for
all}\,\,f\in\mathcal{F},$

are respectively called a growth theorem, a distortion theorem and a rotation
theorem at $z_{0}$ for $\mathcal{F}$, where $K_{i},M_{i}$ and $m_{i}$
$(i=1,2)$ are some non-negative constants. These estimates deal only with the
absolute values of $f(z_{0})$ and $f^{\prime}(z_{0})$ or with the argument of
$f^{\prime}(z_{0})$. If one wants to study the exact value of $f(z_{0})$ and
$f^{\prime}(z_{0})$, then it is necessary to consider the region of
variability of $f(z_{0})$ and $f^{\prime}(z_{0})$ when $f$ ranges over the
class $\mathcal{F}$.

Region of variability for several classes of univalent functions has become an
interesting area of research in geometric function theory. Several authors
have studied the region of variability for several classes of functions. In
$2005$, Yanagihara [15] studied region of variability for functions of bounded
derivatives. In $2006$, Yanagihara [16] proved the region of variability for
convex functions. In $2007$, Ponnusamy and Vasudevarao [7] studied the region
of variability of two subclases of univalent functions. In $2009$, Ponnusamy
et al. [10] have studied the region of variability for the class
$\mathcal{B}(\alpha,\beta,M)$ of analytic and univalent funtions in the unit
disk $\mathbb{D}$ with $f(0)=0$, $f^{\prime}(0)=\alpha$, and
$f^{\prime\prime}(0)=M\beta$ satisfying $|zf^{\prime\prime}(z)|\leq M$ for
$\alpha,\beta\in\mathbb{C}$ and $M\in\mathbb{R}$ with $0<M\leq|\alpha|$ and
$|\beta|\leq 1$. Furthermore, Ponnusamy et al. [10] have also studied the
region of variability for the class $\mathcal{P}(\alpha,M)$ of analytic and
univalent funtions in $\mathbb{D}$ with $f(0)=0$, $f^{\prime}(0)=\alpha$, and
$f^{\prime\prime}(0)=M\beta$ satisfying $zf^{\prime\prime}(z)>-M$ in
$\mathbb{D}$, where $\alpha\in\mathbb{C}\setminus\\{0\\}$ and $0<M\leq
1/\log\,4$. In $2009$, Ponnusamy et al. [11] studied the region of variability
for spiral-like functions with respect to a boundary point. In $2010$,
Ponnusamy and Vasudevarao [8] extensively studied the region of variability
for functions with positive real part. In $2011$, Ponnusamy et al. [12]
investigated the region of variability for exponentially convex univalent
functions. For other interesting results on region of variability, we refer to
[7, 9, 13].

If $f$ is locally univalent function, then pre-Schwarzian norm is defined by

(1.1)
$\|f\|=\sup_{z\in\mathbb{D}}\,(1-|z|^{2})\bigg{|}\dfrac{f^{\prime\prime}(z)}{f^{\prime}(z)}\bigg{|}.$

The class of univalent functions is preserved under a number of elementary
transformations. The preservation of $\mathcal{S}$ under the disk automorphism
leads to the study of the behaviour of the pre-Schwarzian norm of $f$. The
pre-Schwarzian norm has the close connection with functions in $\mathcal{S}$.
Hence the study of region of variability problems together with pre-Schwarzian
norm is very classic and worthy. It is well known that $||f||\leq 6$ if f is
univalent in $\mathbb{D}$, and conversely if $||f||\leq 1$ then $f$ is
univalent in $\mathbb{D}$, and these bounds are sharp (see [3]).

The following lemma is useful to prove our main result.

######  Lemma 1.2.

[7] For $\theta\in\mathbb{R}$ and $0<\lambda<1$, the function

$\displaystyle
G(z)=\int_{0}^{z}\frac{e^{i\theta}z}{\left(1+\left(e^{i\theta}-1\right)\lambda
z-e^{i\theta}z\right)^{2}}\,dz,\;z\in\mathbb{D}$

has a double zero at the origin and no zeros elsewhere in $\mathbb{D}$.
Furthermore, these exists a starlike univalent function $G_{0}$ in
$\mathbb{D}$ such that $G=2^{-1}e^{i\theta}G^{2}_{0}$ and
$G_{0}(0)=G^{\prime}_{0}(0)-1$.

For a positive integer $p$, let

$\left(\mathcal{S}^{*}\right)^{p}=\\{f=f_{0}^{p}:f_{0}\in\mathcal{S}^{*}\\}.$

######  Lemma 1.3.

[15] Let $f$ be an analytic function in $\mathbb{D}$ with $f(z)=z^{p}+\cdots.$
If

$\left(1+z\frac{f^{\prime\prime}(z)}{f^{\prime}(z)}\right)>0,\,\,\,z\in\mathbb{D},$

then $f\in\left(\mathcal{S}^{*}\right)^{p}$.

##  2\. Preliminaries

For $0<\alpha<1$ and $0<\beta<1$, let

(2.1) $f_{\alpha}(z)=\displaystyle\frac{z}{1-z^{\alpha}},\,\,z\in\mathbb{D}$

and

$\displaystyle\mathcal{F}=\bigg{\\{}f_{\alpha}\in\mathcal{A}:{\rm
Re}\left(1+\frac{zf^{\prime\prime}_{\alpha}(z)}{f^{\prime}_{\alpha}(z)}\right)>\beta\bigg{\\}}.$

Let
${\mathcal{F}}(\lambda)=\\{f_{\alpha}\in\mathcal{F}:\,f_{\alpha}^{\prime\prime}(0)=2\lambda(1-\beta)\,\,\,\mbox{for}\,\,\,0\leq\lambda\leq
1\\}$. For $f_{\alpha}\in\mathcal{F}$, let

(2.2)
$P_{f}(z)=1+\displaystyle\frac{zf^{\prime\prime}_{\alpha}(z)}{f^{\prime}_{\alpha}(z)}-\beta.$

Clearly, $P_{f}(0)=1-\beta$. Let $\mathcal{B}_{0}$ be the class of analytic
functions $\omega$ in the unit disk $\mathbb{D}$ such that $|\omega(z)|\leq 1$
in $\mathbb{D}$ and $\omega(0)=0$. Then for each $f\in\mathcal{F}(\lambda)$,
there exists an $\omega_{f}\in\mathcal{B}_{0}$ of the form

(2.3)
$\displaystyle\omega_{f}(z):=\frac{P_{f}(z)-(1-\beta)}{P_{f}(z)+(1-\beta)}.$

A simple computation shows that

$\displaystyle\omega^{\prime}_{f}(0)=\lim_{z\rightarrow
0}\frac{\omega_{f}(z)-\omega_{f}(0)}{z-0}=\lim_{z\rightarrow
0}\frac{\omega_{f}(z)}{z}.$

Let $\omega_{f}^{\prime}(0)=\lambda$, where $|\lambda|\leq 1$. From (2.3), we
have

$\omega_{f}^{\prime}(z)=\dfrac{2(1-\beta)P^{\prime}_{f}(z)}{\left(P_{f}(z)+(1-\beta)\right)^{2}}.$

Since $P_{f}(0)=1-\beta$, we have

(2.4) $\omega^{\prime}_{f}(0)=\frac{P^{\prime}_{f}(0)}{2(1-\beta)}.$

Therefore

(2.5) $\lambda=\frac{P^{\prime}_{f}(0)}{2(1-\beta)}.$

From (2.2), we have

(2.6)
$P_{f}(z)f^{\prime}_{\alpha}(z)=(1-\beta)f^{\prime}_{\alpha}(z)+zf^{\prime\prime}_{\alpha}(z).$

Differentiating (2.6) gives

(2.7)
$P^{\prime}_{f}(z)f^{\prime}_{\alpha}(z)+\left(P_{f}(z)-(1-\beta)-1\right)f^{\prime\prime}_{\alpha}(z)+zf^{\prime\prime}_{\alpha}(z)=0.$

A simple computation of (2.7) at $z=0$ shows that

(2.8) $P^{\prime}_{f}(0)=f^{\prime\prime}_{\alpha}(0).$

From (2.5) and (2.8), we obtain

(2.9) $f^{\prime\prime}_{\alpha}(0)=2\lambda(1-\beta).$

Define $g:\mathbb{D}\rightarrow\mathbb{D}$ by

$g(z)=\displaystyle\frac{\frac{\omega_{f}(z)}{z}-\lambda}{1-\frac{\overline{\lambda}\omega_{f}(z)}{z}},$

equivalently

$g(z)=\dfrac{\omega_{f}(z)-\lambda z}{z-\overline{\lambda}\omega_{f}(z)}.$

Then, clearly $g(0)=0$ and $g$ is an analytic function in $\mathbb{D}$.
Therefore $g\in\mathcal{B}_{0}$.

First we prove that the class $\mathcal{F}(\lambda)$ is rotationally
invariant. The class $\mathcal{F}(\lambda)$ is rotationally invariant if, and
only if, $f_{\theta}\in\mathcal{F}(\lambda)$, where
$f_{\theta}:=e^{-i\theta}f_{\alpha}(e^{i\theta}z)$, where $f_{\alpha}$ is
defined by (2.1). A simple computation shows that

$\displaystyle
f^{\prime}_{\theta}(z)=f_{\alpha}^{\prime}\left(e^{i\theta}z\right)\;\;\mbox{and}\,\,f^{\prime\prime}_{\theta}(z)=e^{i\theta}f_{\alpha}^{\prime\prime}\left(e^{i\theta}z\right).$

Then, it is easy to see that

$\displaystyle{\rm
Re}\left(1+\frac{zf^{\prime\prime}_{\theta}(z)}{f^{\prime}_{\theta}(z)}\right)={\rm
Re}\left(1+\frac{ze^{i\theta}f^{\prime\prime}\left(e^{i\theta}z\right)}{f^{\prime}\left(e^{i\theta}z\right)}\right)={\rm
Re}\left(1+\frac{\xi f^{\prime\prime}(\xi)}{f^{\prime}(\xi)}\right)>\beta,$

where $\xi=ze^{i\theta}$. Since $f_{\alpha}\in\mathcal{F}(\lambda)$, it is
easy to see that $f_{\theta}(0)=0$, $f^{\prime}_{\theta}(0)=1$, and
$f^{\prime\prime}_{\theta}(0)=2\lambda(1-\beta)e^{i\theta}$. Therefore,
$f_{\theta}\in\mathcal{F}(\lambda)$. Hence the class $\mathcal{F}(\lambda)$ is
rotational invariant. Since the class $\mathcal{F}(\lambda)$ is rotationally
invariant, without loss of generality we may assume that $0\leq\lambda\leq 1$
instead of $\lambda\in\mathbb{D}$.

We wish to construct an extremal function $F_{a,\lambda}(z)$ where
$a\in\overline{\mathbb{D}}=\\{z\in\mathbb{C}:|z|\leq 1\\}$ and it belongs to
the class $\mathcal{F}(\lambda)$. From the second condition of the Schwarz
lemma, we have $g(z)=az$, where $|a|=1$. Therefore, we see that

$\displaystyle g(z)=\dfrac{\omega_{f}(z)-\lambda
z}{z-\overline{\lambda}\omega_{f}(z)}=\frac{P_{f}(z)-(1-\beta)\left(\dfrac{1+\lambda
z}{1-\lambda
z}\right)}{P_{f}(z)+(1-\beta)\left(\dfrac{z+\lambda}{z-\lambda}\right)}=az$

which implies that

(2.10)
$\displaystyle\dfrac{\displaystyle\frac{f_{\alpha}^{\prime\prime}(z)}{f_{\alpha}^{\prime}(z)}-\frac{2(1-\beta)\lambda}{1-\lambda
z}}{\displaystyle\frac{f_{\alpha}^{\prime\prime}(z)}{f_{\alpha}^{\prime}(z)}+\frac{2(1-\beta)}{z-\lambda}}=az\left(\frac{z-\lambda}{1-\lambda
z}\right):=az\,\delta(z,\lambda),\,\,\mbox{where}\,\,\delta(z,\lambda)=\frac{z-\lambda}{1-\lambda
z}.$

Further simplification of (2.10) gives

(2.11)
$\displaystyle\dfrac{f_{\alpha}^{\prime\prime}(z)}{f_{\alpha}^{\prime}(z)}=2(1-\beta)\dfrac{\dfrac{\lambda}{1-\lambda}+\dfrac{az\delta(z,\lambda)}{z-\lambda}}{1-az\delta(z,\lambda)}=\dfrac{2(1-\beta)(az+\lambda)}{1-\lambda
z-az^{2}+\lambda az}.$

From (2.11), we obtain

(2.12)
$\displaystyle\frac{f_{\alpha}^{\prime\prime}(z)}{f_{\alpha}^{\prime}(z)}=2(1-\beta)\frac{\delta(az,\lambda)}{1-\delta(az,\lambda)z}.$

Integrating both sides of (2.12), we obtain

(2.13) $\displaystyle
F_{a,\lambda}(z):=f_{\alpha}(z)=\int_{0}^{z}\exp\left(\int_{0}^{\xi}\frac{2(1-\beta)\delta(a\xi,\lambda)}{1-\delta(a\xi,\lambda)}\,\,d\xi\right)d\xi.$

Clearly, $F_{a,\lambda}(0)=0,F_{a,\lambda}^{\prime}(0)=1$ and
$F_{a,\lambda}^{\prime\prime}(0)=2\lambda(1-\beta)$. Hence
$F_{a,\lambda}\in\mathcal{F}(\lambda)$.

If $f\in\mathcal{F}(\lambda)$, we define

(2.14) $h(z)=\frac{P_{f}(z)}{1-\beta}.$

where $P_{f}(z)=1+zf_{\alpha}^{\prime\prime}/{f_{\alpha}^{\prime}}-\beta.$ So
that ${\rm Re\,}h(z)>0$ and $h(0)=1$. Therefore by Herglotz representation for
functions with positive real part there exists unit probability measure $\mu$
on $(-\pi,\pi]$, i.e.,

$\int_{-\pi}^{\pi}\,d\mu(t)=1,$

such that

(2.15) $h(z)=\int_{-\pi}^{\pi}\frac{1+ze^{-it}}{1-ze^{-it}}\,\,d\mu(t).$

In view of (2.14), we have

$P_{f}(z)=(1-\beta)\int_{-\pi}^{\pi}\frac{1+ze^{-it}}{1-ze^{-it}}\,\,d\mu(t).$

From (LABEL:e-0.1), we obtain

(2.16)
$1+\frac{zf_{\alpha}^{\prime\prime}}{f_{\alpha}^{\prime}}-\beta=(1-\beta)\int_{-\pi}^{\pi}\frac{1+ze^{-it}}{1-ze^{-it}}\,\,d\mu(t),$

which implies that

$\frac{zf_{\alpha}^{\prime\prime}}{f_{\alpha}^{\prime}}=(1-\beta)\int_{-\pi}^{\pi}\bigg{(}\frac{1+ze^{-it}}{1-ze^{-it}}-1\bigg{)}\,\,d\mu(t).$

Therefore,

$\frac{f_{\alpha}^{\prime\prime}}{f_{\alpha}^{\prime}}=2(1-\beta)\int_{-\pi}^{\pi}\frac{e^{-it}}{1-ze^{-it}}\,\,d\mu(t).$

Integrating on both sides with respect to $z$, we obtain

$\log
f_{\alpha}^{\prime}(z)=2(1-\beta)\int_{-\pi}^{\pi}-\log(1-ze^{-it})\,\,d\mu(t),$

and hence

(2.17)
$f_{\alpha}(z)=\int_{0}^{t}\exp\bigg{(}2(1-\beta)\int_{-\pi}^{\pi}-\log(1-ze^{-it})\,d\mu(t)\bigg{)}.$

The main aim of this paper is to obtain the region of variability of $\log
f^{\prime}(z_{0})$ where $f$ runs over the class $\mathcal{F}(\lambda)$.
Further, we demonstrate the region of variability of $\log f^{\prime}(z_{0})$
for the class $\mathcal{F}(\lambda)$ by using Mathematica.

##  3\. Main Results

Consider the following set

$V(z_{0},\lambda,\beta)=\\{\log
f^{\prime}(z_{0}):f\in\mathcal{F}(\lambda)\\}.$

Now we mention some basic properties of the set $V(z_{0},\lambda,\beta)$.

######  Proposition 3.1.

We have

  * (1)

$V(z_{0},\lambda,\beta)$ is compact.

  * (2)

$V(z_{0},\lambda,\beta)$ is convex.

  * (3)

For $|\lambda|=1$ or $z_{0}=0$, we have

(3.2) $V(z_{0},\lambda,\beta)=\\{-2(1-\beta)\log(1-\lambda z)\\}$

  * (4)

For $|\lambda|<1$ and $z_{0}\in\mathbb{D}\setminus\\{0\\}$,
$V(z_{0},\lambda,\beta)$ has $-2(1-\beta)\log(1-\lambda z)$ as an interior
point.

######  Proposition 3.3.

For $f\in\mathcal{F}(\lambda)$ where $0\leq\lambda<1$, we have

$\displaystyle\left|\frac{f^{\prime\prime}(z)}{f^{\prime}(z)}-c(z,\lambda,\beta)\right|\leq
r(z,\lambda,\beta),\;z\in\mathbb{D}.$

where

$c(z,\lambda,\beta)=\frac{2(1-\beta)\left(\lambda(1-|z|^{2})+(|z|^{2}-\lambda^{2})\right)\bar{z}}{(1-|z|^{2})(1-2\lambda({\rm
Re}z)+|z|^{2})}$

and

$r(z,\lambda,\beta)=\frac{2(1-\beta)(1-\lambda^{2})|z|}{(1-|z|^{2})(1-2\lambda({\rm
Re}z)+|z|^{2})}$

For each $z\in\mathbb{D}\setminus\\{0\\}$,equality hold if, and only if,
$f=F_{e^{i\theta},\lambda}$ for some $\theta\in\mathbb{R}$.

######  Corollary 3.4.

Let $\gamma:=z(t)$, where $0\leq t\leq 1$ be a $C^{1}$ curve in $\mathbb{D}$
with $z(0)=0$ and $z(1)=z_{0}$. Then, we have

$\displaystyle
V(z_{0},\lambda,\beta)\subset\bigg{\\{}w\in\mathbb{C}:|\omega-C(\lambda,\gamma,\beta)|\bigg{\\}}\leq
R(\lambda,\gamma,\beta),$

where
$C(\lambda,\gamma,\beta)=\displaystyle\int_{0}^{1}c(z(t),\lambda,\beta)z^{\prime}(t)dt$
and
$R(\lambda,\gamma,\beta)=\displaystyle\int_{0}^{1}r(z(t),\lambda,\beta)|z^{\prime}(t)|dt$.

Now we state the following result which is vital to prove our main result.

######  Proposition 3.5.

(Uniqueness of the curve) Let $z_{0}\in\mathbb{D}\setminus\\{0\\}$. Then, for
$\theta\in(-\pi,\pi]$, we have

$\displaystyle\log F^{\prime}_{e^{i\theta},\lambda}(z_{0})\in\partial
V(z_{0},\lambda,\beta).$

Furthermore, if $\log f^{\prime}(z_{0})=\log
F^{\prime}_{e^{i\theta},\lambda}(z_{0})$ for some $f\in\mathcal{F}(\lambda)$
and $\theta\in(-\pi,\pi]$, then $f=F_{e^{i\theta},\lambda}$.

Finally, we state our main result to obtain the region of variability of $\log
f^{\prime}(z_{0})$ where $f$ runs over the class $\mathcal{F}(\lambda)$.

######  Theorem 3.6.

For $0\leq\lambda<1$ and $z_{0}\in\mathbb{D}\setminus\\{0\\}$, the boundary
$\partial{V(z_{0},\lambda,\beta)}$ is the Jordan curve given by

(3.7) $(-\pi,\pi]\ni\theta\mapsto\log
F^{\prime}_{e^{i\theta},\lambda}(z_{0})=\int_{0}^{z_{0}}\frac{\delta(e^{i\theta}z,\lambda)}{z\delta(e^{i\theta}z,\lambda)-1}\,dz,\quad
z\in\mathbb{D}.$

If $\log f^{\prime}(z_{0})=\log F^{\prime}_{e^{i\theta},\lambda}(z_{0})$ for
some $f\in\mathcal{F}(\lambda)$ and $\theta\in(-\pi,\pi]$, then
$f(z)=F_{e^{i\theta},\lambda}(z)$. Here $F_{e^{i\theta},\lambda}(z)$ is given
by

(3.8)
$F_{e^{i\theta},\lambda}(z)=f_{\alpha}(z)=\int_{0}^{z}\exp\left(\int_{0}^{\xi}\frac{2(1-\beta)\delta(e^{i\theta}\xi,\lambda)}{1-\delta(e^{i\theta}\xi,\lambda)}d\xi\right)d\xi.$

##  4\. Proof of the main results

Proof of Proposition 3.1. (1) To show that $V(z_{0},\lambda,\beta)$ we need to
show that the set $\mathcal{F}(\lambda)$ is compact. For this, we need to show
that $\mathcal{F}(\lambda)$ is closed and locally bounded. Let $f_{n}$ be a
sequence of functions in $F(\lambda)$ such that $f_{n}\rightarrow f$ uniformly
on every compact subsets of the unit disk. Then to show that $F(\lambda)$ is
compact, we show that $f\in F(\lambda)$. Since $f_{n}\rightarrow f$ uniformly
on every compact subset of $\mathbb{D}$, we have $f^{\prime}_{n}\rightarrow
f^{\prime}$ uniformly on every compact subsets of $\mathbb{D}$. Therefore, if
$f_{n}\rightarrow f$ uniformly, then $f_{n}(0)\rightarrow f(0)$ pointwise
which gives that $f(0)=0$. Since, $f^{\prime}_{n}\rightarrow f^{\prime}$
uniformly on every compact subsets of $\mathbb{D}$, we get

(4.1) $f^{\prime}_{n}(0)\rightarrow
f^{\prime}(0),\,\,f^{\prime\prime}_{n}(0)\rightarrow f^{\prime\prime}(0)$

pointwise. Since $f_{n}\in\mathcal{F}(\lambda)$, $f^{\prime}_{n}(0)=1$ and
$f^{\prime\prime}_{n}(0)=2\lambda(1-\beta)$. In view of (4.1), we have
$f^{\prime}(0)=1$ and $f^{\prime\prime}(0)=2\lambda(1-\beta)$. In view of the
above discussion, it is easy to see that

$1+\dfrac{zf^{\prime\prime}_{n}(z)}{f^{\prime}_{n}(z)}\longrightarrow
1+\dfrac{zf^{\prime\prime}(z)}{f^{\prime}(z)}$

uniformly on every compact subsets of $\mathbb{D}$. Since
$f_{n}\in\mathcal{F}(\lambda)$, we have

$0<\left(1+\dfrac{zf^{\prime\prime}_{n}(z)}{f^{\prime}_{n}(z)}\right)\longrightarrow\left(1+\dfrac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right)\geq
0$

in $\mathbb{D}$. Therefore, by the maximum modulus principle, it follows that

$\left(1+\dfrac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right)>0$

in $\mathbb{D}$. Locally boundedness follows from the definition itself, thus
$\mathcal{F}(\lambda)$ is compact and in view of the continuous function
$\psi:\mathcal{F}(\lambda)\to V(z_{0},\lambda,\beta)$ defined by

$\psi(f)=log(f^{\prime}(z_{0}))$

we conclude that the class $V(z_{0},\lambda,\beta)$ is also compact.
(2) Let $f_{1},f_{2}\in\mathcal{F}(\lambda)$ and $0\leq t\leq 1$. Then it is
easy to see that the function

(4.2) $f_{t}(z)=\int_{0}^{z}\exp\left\\{(1-t)\log f^{\prime}_{1}(\xi)+t\log
f^{\prime}_{2}(\xi)\right\\}\,d\xi$

belongs to $\mathcal{F}(\lambda)$ and hence $V(z_{0},\lambda,\beta)$ is
convex.
(3) If $z_{0}=0$, then (3.2) is trivially true. If
$|\lambda|=|\omega^{\prime}_{f}(0)|=1$, then by an application of the Schwarz
lemma, we have $\omega_{f}(z)=\lambda z$. Therefore using (2.3), we obtain

(4.3) $P_{f}(z)=(1-\beta)\left(\frac{1+\lambda z}{1-\lambda z}\right).$

Further simplification of (4.3) by using (2.2) shows that

$\log f^{\prime}(z)=2(1-\beta)\frac{\lambda}{1-\lambda z}$

which implies that

$f(z)=\int_{0}^{z}\exp\left(2(1-\beta)\frac{\lambda}{1-\lambda\xi}\right)\,d\xi.$

Conversely,

$V(z_{0},\lambda,\beta)=\\{-2(1-\beta)\log(1-\lambda z)\\}.$

(4) For $|\lambda|<1$ and $a\in\overline{\mathbb{D}}$, we define

$\delta(z,\lambda)=\frac{z+\lambda}{1+\overline{\lambda}z}$

and

(4.4)
$F_{a,\lambda}(z)=f_{\alpha}(z)=\int_{0}^{z}\exp\left(\int_{0}^{\xi}\frac{2(1-\beta)\delta(a\xi,\lambda)}{1-\delta(a\xi,\lambda)}d\xi\right)d\xi.$

An observation shows that

(4.5) $\omega_{F_{a,\lambda}}(z)=z\delta(az,\lambda).$

We want to prove that the mapping $\mathbb{D}\ni a\mapsto\log
F^{\prime}_{a,\lambda}(z_{0})$ is a non-constant analytic function of "$a$"
for each fixed $z_{0}\in\mathbb{D}\setminus\\{0\\}$ and
$\lambda\in\mathbb{D}$. To show this, we let

(4.6) $\displaystyle h(z)$
$\displaystyle=\left.\frac{1}{(1-\lambda^{2})}\frac{\partial}{\partial
a}\left\\{\frac{}{}\log F^{\prime}_{a,\lambda}(z)\right\\}\right|_{a=0}$
$\displaystyle=2(1-\beta)\int_{0}^{z}\frac{\xi}{(1-\lambda\xi)^{2}}\,\,d\xi$
$\displaystyle=z^{2}+\cdots.$

A simple computation shows that

$\left\\{\frac{zh^{\prime\prime}(z)}{h^{\prime}(z)}\right\\}=2(1-\beta)\left\\{\frac{1+\lambda
z}{1-\lambda z}\right\\}>0\,\,\mbox{for}\,\,z\in\mathbb{D}.$

By Lemma 1.3, there exists a function $h_{0}\in\mathcal{S}^{*}$ with
$h=h_{0}^{2}$. The univalence of $h_{0}$ together with the condition
$h_{0}(0)=0$ implies that $h(z_{0})\neq 0$ for
$z_{0}\in\mathbb{D}\setminus\\{0\\}$. Consequently, the mapping $\mathbb{D}\ni
a\mapsto\log F^{\prime}_{a,\lambda}(z_{0})$ is a non-constant analytic
function of "$a$", and hence it is an open mapping. Therefore
$V(z_{0},\lambda,\beta)$ contains the open set $\\{\log
F^{\prime}_{a,\lambda}(z_{0}):|a|<1\\}$. In particular,

$\log F^{\prime}_{0,\lambda}(z_{0})=-2(1-\beta)\log(1-\lambda z)$

is an interior point of $\\{\log F^{\prime}_{a,\lambda}(z_{0}):|a|<1\\}\subset
V(z_{0},\lambda,\beta)$. This completes the proof. ∎

Proof of Proposition 3.3. For $f\in\mathcal{F}(\lambda)$, define $g(z)$ by

$g(z)=\frac{\omega_{f}(z)-\lambda z}{z-\overline{\lambda}\omega_{f}(z)}.$

where $w_{f}$ is defined by (2.3).

From (2.3), it is easy to see that

(4.7) $\displaystyle\omega_{f}(z)-\lambda z=\frac{P_{f}(z)\left(1-\lambda
z\right)-(1-\beta)(1+2\lambda)}{P_{f}(z)+(1-\beta)}$

and

(4.8) $\displaystyle
z-\overline{\lambda}\omega_{f}(z)=\frac{(z-\lambda)P_{f}(z)+(1-\beta)(z+\lambda)}{P_{f}(z)+(1-\beta)}.$

Therefore, in view (4.7) and (4.8), we obtain

$\displaystyle g(z)=\frac{\omega_{f}(z)-\lambda
z}{z-\overline{\lambda}\omega_{f}(z)}=\frac{P_{f}(z)(1-\lambda
z)-(1-\beta)(1+\lambda z)}{(z-\lambda)P_{f}(z)+(1-\beta)(z+\lambda)}.$

Again, $|g(z)|\leq|z|$ shows that

$\displaystyle\displaystyle\left|\frac{P_{f}(z)(1-\lambda
z)-(1-\beta)(1+\lambda
z)}{(z-\lambda)P_{f}(z)+(1-\beta)(z+\lambda)}\right|\leq|z|$

which can be written as

(4.9) $\displaystyle\left|\frac{P_{f}(z)-(1-\beta)\left(\dfrac{1+\lambda
z}{1-\lambda
z}\right)}{P_{f}(z)+(1-\beta)\left(\dfrac{z+\lambda}{z-\lambda}\right)}\right|\leq|z|\left|\dfrac{z-\lambda}{1-\lambda
z}\right|.$

Further simplification of (4.9) by using (2.2), we obtain

$\displaystyle\left|\frac{(1-\beta)+\dfrac{zf^{\prime\prime}_{\alpha}(z)}{f_{\alpha}^{\prime}(z)}-(1-\beta)\left(\dfrac{1+\lambda
z}{1-\lambda
z}\right)}{(1-\beta)+\dfrac{zf^{\prime\prime}_{\alpha}(z)}{f_{\alpha}^{\prime}(z)}+(1-\beta)\left(\dfrac{z+\lambda}{z-\lambda}\right)}\right|\leq|z|\left|\frac{z-\lambda}{1-\lambda
z}\right|$

which implies that

(4.10)
$\displaystyle\left|\dfrac{\dfrac{zf^{\prime\prime}_{\alpha}(z)}{f_{\alpha}^{\prime}(z)}-(1-\beta)\left(\dfrac{2\lambda
z}{1-\lambda
z}\right)}{\dfrac{zf^{\prime\prime}_{\alpha}(z)}{f_{\alpha}^{\prime}(z)}+(1-\beta)\left(\frac{2z}{z-\lambda}\right)}\right|\leq|z|\left|\dfrac{z-\lambda}{1-\lambda
z}\right|.$

A simple computation shows that (4.10) is equivalent to

(4.11)
$\displaystyle\left|\displaystyle\dfrac{\dfrac{zf^{\prime\prime}_{\alpha}(z)}{f_{\alpha}^{\prime}(z)}-A(z,\lambda,\beta)}{\displaystyle\dfrac{zf^{\prime\prime}_{\alpha}(z)}{f_{\alpha}^{\prime}(z)}+B(z,\lambda,\beta)}\right|\leq|z|\left|\tau\left(z,\lambda\right)\right|,$

where

$A(z,\lambda,\beta)=\frac{2(1-\beta)\lambda}{1-\lambda
z},B(z,\lambda,\beta)=\frac{2(1-\beta)z}{z-\lambda},\,\,\mbox{and}\,\,\tau(z,\lambda)=\frac{z-\lambda}{1-\lambda
z}.$

Further simplification of (4.11) implies that

$\displaystyle\left|\frac{f^{\prime\prime}_{\alpha}(z)}{f_{\alpha}^{\prime}(z)}-\frac{A(z,\lambda,\beta)+|z|^{2}|\tau(z,\lambda)|^{2}B(z,\lambda,\beta)}{1-|z|^{2}|\tau(z,\lambda)|^{2}}\right|$
$\displaystyle\leq\dfrac{|z||\tau(z,\lambda)||A(z,\lambda,\beta)+B(z,\lambda)|}{1-|z|^{2}|\tau(z,\lambda)|^{2}},$

where

$\displaystyle
A(z,\lambda,\beta)+|z|^{2}|\tau(z,\lambda)|^{2}B(z,\lambda,\beta)$
$\displaystyle=2(1-\beta)\left(\frac{\lambda}{1-\lambda
z}+|z|^{2}\frac{(z-\lambda)(\bar{z}-\lambda)}{|1-\lambda
z|^{2}}\frac{1}{z-\lambda}\right)$
$\displaystyle=2(1-\beta)\left(\frac{\lambda(1-|z|^{2})+\left(|z|^{2}-\lambda^{2}\right)\bar{z}}{|1-\lambda
z|^{2}}\right).$

A simple computation shows that

$\displaystyle
A(z,\lambda,\beta)+B(z,\lambda,\beta)=2(1-\beta)\frac{1-\lambda^{2}}{(1-\lambda
z)(z-\lambda)}.$

Therefore, it is easy to see that

$\displaystyle|z||\tau(z,\lambda)||A(z,\lambda,\beta)+B(z,\lambda,\beta)|=\frac{2(1-\beta)(1-\lambda^{2})|z|}{|1-\lambda
z|^{2}}$

which shows that

$\displaystyle 1-|z|^{2}|\tau(z,\lambda)|^{2}=\frac{(1-|z|^{2})(1-2\lambda
z)+|z|^{2}}{|1-\lambda z|^{2}}.$

Thus, we have

$\displaystyle\frac{A(z,\lambda,\beta)+|z|^{2}|\tau(z,\lambda)B(z,\lambda,\beta)|}{1-|z|^{2}|\tau(z,\lambda)|}$
$\displaystyle=\frac{2(1-\beta)\left(\lambda(1-|z|^{2})+(|z|^{2}-\lambda^{2})\right)\bar{z}}{(1-|z|^{2})(1-2\lambda({\rm
Re}z)+|z|^{2})}$ $\displaystyle:=c(z,\lambda,\beta).$

Further,

$\displaystyle\cfrac{|z||\tau(z,\lambda)||A(z,\lambda,\beta)+B(z,\lambda,\beta)|}{1-|z|^{2}|\tau(z,\lambda)|}$
$\displaystyle=\frac{2(1-\beta)(1-\lambda^{2})|z|}{(1-|z|^{2})(1-2\lambda({\rm
Re}z)+|z|^{2})}$ $\displaystyle:=r(z,\lambda,\beta).$

This completes the proof. ∎

We note that if $f\in\mathcal{F}(0)$, then pre-Schwarzian norm $\|f\|\leq
2(1-\beta).$ Indeed,

(4.12)
$\bigg{|}\frac{f^{\prime\prime}(z)}{f^{\prime}(z)}-c(z,0,\beta)\bigg{|}\leq
r(z,0,\beta),$

where

$c(z,0,\beta)=\dfrac{2(1-\beta)|z|^{2}\bar{z}}{(1-|z|^{2})(1+|z|^{2})}\,\,\,\,\mbox{and}\,\,\,\,r(z,0,\beta)=\dfrac{2(1-\beta)|z|}{(1-|z|^{2})(1+|z|^{2})}.$

From (4.12), we have

$\bigg{|}\dfrac{f^{\prime\prime}(z)}{f^{\prime}(z)}-\dfrac{2(1-\beta)|z|^{2}\bar{z}}{(1-|z|^{2})(1+|z|^{2})}\bigg{|}\leq\dfrac{2(1-\beta)|z|}{(1-|z|^{2})(1+|z|^{2})}$

which implies

$\bigg{|}\dfrac{f^{\prime\prime}(z)}{f^{\prime}(z)}\bigg{|}-\bigg{|}\dfrac{2(1-\beta)|z|^{2}\bar{z}}{1-|z|^{4}}\bigg{|}\leq\dfrac{2(1-\beta)|z|}{1-|z|^{4}}.$

Therefore,

$\bigg{|}\dfrac{f^{\prime\prime}(z)}{f^{\prime}(z)}\bigg{|}\leq\dfrac{2(1-\beta)|z|}{1-|z|^{4}}+\dfrac{2(1-\beta)|z|^{3}}{1-|z|^{4}}=\dfrac{2(1-\beta)|z|}{1-|z|^{2}}$

which implies that

$(1-|z|^{2})\bigg{|}\dfrac{f^{\prime\prime}(z)}{f^{\prime}(z)}\bigg{|}\leq
2(1-\beta)|z|,$

and hence

$\sup_{z\in\mathbb{D}}\,(1-|z|^{2})\bigg{|}\dfrac{f^{\prime\prime}(z)}{f^{\prime}(z)}\bigg{|}\leq
2(1-\beta)\,\,\text{for}\,\,z\in\mathbb{D}.$

Therefore, if $f\in\mathcal{F}(0)$, then

$\|f\|\leq 2(1-\beta).$

###### Proof of Corollary 3.4.

A simple computation using the proposition 3.3 shows that

$\displaystyle\left|\log
f^{\prime}_{\alpha}(z)-C(\lambda,\gamma,\beta)\right|$
$\displaystyle=\left|\log
f^{\prime}_{\alpha}(z)-\int_{0}^{1}c(z(t),\lambda,\beta)z^{\prime}(t)\,dt\right|$
$\displaystyle=\left|\bigintsss_{0}^{1}\frac{f^{\prime\prime}_{\alpha}(z(t))}{f^{\prime}_{\alpha}(z(t))}z^{\prime}(t)dt-\bigintsss_{0}^{1}c(z(t),\lambda,\beta)z^{\prime}(t)\,dt\right|$
$\displaystyle\leq\bigintsss_{0}^{1}\left|\left(\frac{f^{\prime\prime}_{\alpha}(z(t))}{f^{\prime}_{\alpha}(z(t))}-c(z(t),\lambda,\beta)\right)\right||z^{\prime}(t)|\,dt$
$\displaystyle\leq\int_{0}^{1}r(z(t),\lambda,\beta)|z^{\prime}(t)|\,dt:=R(\lambda,\gamma,\beta).$

This completes the proof. ∎

Proof of Proposition 3.5. From (2.13), we have

$F_{a,\lambda}(z)=\displaystyle\int_{0}^{z}\exp\left(\int_{0}^{\xi_{2}}\frac{2(1-\beta)\delta(a\xi_{1},\lambda)}{1-\delta(a\xi_{1},\lambda)}\,\,d\xi_{1}\right)d\xi_{2}.$

Then, it is easy to see that

$\displaystyle
F^{\prime}_{e^{i\theta},\lambda}=\exp\left(\int_{0}^{z}\frac{2(1-\beta)\delta(a\xi_{1},\lambda)}{1-\delta(a\xi_{1},\lambda)}\,\,d\xi_{1}\right)$

and hence,

$\displaystyle\frac{F^{\prime\prime}_{e^{i\theta},\lambda}}{F^{\prime}_{e^{i\theta},\lambda}}=\frac{2(1-\beta)\delta(az,\lambda)}{1-\delta(az,\lambda)z}.$

A simple computation shows that

(4.13)
$\displaystyle\frac{F^{\prime\prime}_{e^{i\theta},\lambda}}{F^{\prime}_{e^{i\theta},\lambda}}-A(z,\lambda,\beta)$
$\displaystyle=2(1-\beta)\left(\frac{az(1-\lambda^{2})}{(1-\lambda
z)(1+\lambda(a-1))z-az^{2}}\right),$ (4.14)
$\displaystyle\frac{F^{\prime\prime}_{e^{i\theta},\lambda}}{F^{\prime}_{e^{i\theta},\lambda}}+B(z,\lambda,\beta)=2(1-\beta)\left(\frac{1-\lambda^{2}}{(1+\lambda(a-1)z-az^{2})(z-\lambda)}\right)$

and

(4.15)
$\displaystyle\frac{F^{\prime\prime}_{e^{i\theta},\lambda}}{F^{\prime}_{e^{i\theta},\lambda}}-c(z,\lambda,\beta)=\frac{2(1-\beta)(1-\lambda^{2})(az-|z|^{2})(\lambda
a+\bar{z}+\lambda)}{(1-|z|^{2})((1+|z|^{2})-2\lambda(z))(1+\lambda(a-1)z-az^{2})}.$

By replacing $a$ by $e^{i\theta}$ (i.e., substituting $a=e^{i\theta}$), we
obtain

(4.16)
$\displaystyle\frac{F^{\prime\prime}_{e^{i\theta},\lambda}}{F^{\prime}_{e^{i\theta},\lambda}}-c(z,\lambda,\beta)=r(z,\lambda,\beta)\frac{e^{i\theta}z\left(1+\lambda\left(e^{-i\theta}-1\right)\bar{z}\right)-a^{-i\theta}z^{-2}}{\left(1+\lambda\left(e^{i\theta}-1\right)z-e^{i\theta}z^{2}\right)|z|}.$

From Lemma 1.2, we have

$\displaystyle
G^{\prime}(z)=\frac{e^{i\theta}z}{\left(1+\lambda\left(e^{i\theta}-1\right)z-e^{i\theta}z^{2}\right)^{2}}.$

Thus it is easy to see that

(4.17)
$\displaystyle\frac{G^{\prime}(z)}{|G^{\prime}(z)|}=\frac{\left|1+\lambda\left(e^{i\theta}-1\right)-e^{i\theta}z^{2}\right|^{2}}{\left(1+\lambda\left(e^{i\theta}-1\right)z-e^{i\theta}z^{2}\right)^{2}}\frac{e^{i\theta}z}{|z|}.$

Applying (4.17) in (4.16), we obtain

(4.18)
$\displaystyle\frac{F^{\prime\prime}_{e^{i\theta},\lambda}}{F^{\prime}_{e^{i\theta},\lambda}}-c(z,\lambda,\beta)=r(z,\lambda,\beta)\frac{\left|1+\lambda\left(e^{i\theta}-1\right)-e^{i\theta}z^{2}\right|^{2}}{\left(1+\lambda\left(e^{i\theta}-1\right)z-e^{i\theta}z^{2}\right)^{2}}\frac{e^{i\theta}z}{|z|}.$

Therefore from (4.17) and (4.18) we have

(4.19)
$\displaystyle\frac{F^{\prime\prime}_{e^{i\theta},\lambda}}{F^{\prime}_{e^{i\theta},\lambda}}-c(z,\lambda,\beta)=r(z,\lambda,\beta)\frac{G^{\prime}(z)}{|G^{\prime}(z)|}.$

Here $G$ is a starlike function as defined in Lemma 1.2. Therefore, for any
$z_{0}\in\mathbb{D}\setminus\\{0\\}$, the line segment joining $0$ and
$G_{0}(z_{0})$ lies entirely in $G_{0}(\mathbb{D})$. Define $\gamma_{0}$ by

(4.20)
$\displaystyle\gamma_{0}=z(t)=G_{0}^{-1}\left(tG_{0}(z_{0})\right)\;\;\mbox{for}\,\,0\leq
t\leq 1.$

It is easy to see that

$\displaystyle G(z(t))$
$\displaystyle=2^{-1}e^{i\theta}\left(G_{0}(z(t))\right)^{2}$
$\displaystyle=2^{-1}e^{i\theta}\left(tG_{0}(z_{0})\right)^{2}$
$\displaystyle=t^{2}\,2^{-1}e^{i\theta}G_{0}(z_{0})^{2}=t^{2}\,G(z_{0}).$

Therefore,

(4.21) $\displaystyle
G^{\prime}(z(t))z^{\prime}(t)=2tG(z_{0})\,\,\mbox{for}\,\,0\leq t\leq 1.$

We note that $z(0)=0$ and $z(1)=z_{0}$. A simple computation using (4.19)
shows that

(4.22) $\displaystyle\log
F^{\prime}_{e^{i\theta},\lambda}(z_{0})-C(\lambda,\gamma_{0},\beta)$
$\displaystyle=\bigintsss_{0}^{1}\frac{F^{\prime\prime}_{e^{i\theta},\lambda}(z(t))}{F^{\prime}_{e^{i\theta},\lambda}(z(t))}z^{\prime}(t)\,dt-\bigintsss_{0}^{1}c(z(t),\lambda,\beta)z^{\prime}(t)\,dt$
$\displaystyle=\bigintsss_{0}^{1}\left(\frac{F^{\prime\prime}_{e^{i\theta},\lambda}(z(t))}{F^{\prime}_{e^{i\theta},\lambda}(z(t))}-c(z(t),\lambda,\beta)\right)z^{\prime}(t)\,dt$
$\displaystyle=\bigintsss_{0}^{1}r(z(t),\lambda,\beta)\frac{G^{\prime}(z(t))|z^{\prime}(t)|}{|G^{\prime}(z(t))z^{\prime}(t)|}|z^{\prime}(t)|\,dt$
$\displaystyle=\frac{G(z_{0})}{|G(z_{0})|}\int_{0}^{1}r(z(t),\lambda,\beta)|z^{\prime}(t)|\,dt$
$\displaystyle=\frac{G(z_{0})}{|G(z_{0})|}\,r(\lambda,\gamma_{0},\beta).$

Hence from (4.22), we have

$\displaystyle\left|\log
F^{\prime}_{e^{i\theta},\lambda}(z_{0})-C(\lambda,\gamma_{0},\beta)\right|=R(\lambda,\gamma_{0},\beta).$

This shows that $\log
F^{\prime}_{e^{i\theta},\lambda}\in\partial\overline{\mathbb{D}}\left(C(\lambda,\gamma_{0},\beta),R(\lambda,\gamma_{0},\beta)\right)$.

Let $V(z_{0},\lambda,\beta)=\\{\log
f^{\prime}(z_{0}):f\in\mathcal{F}(\lambda)\\}.$ From Corollary 3.4, it is easy
to see that

$\displaystyle\log
F^{\prime}_{e^{i\theta},\lambda}\in\overline{\mathbb{D}}(C(\lambda,\gamma,\beta),R(\lambda,\gamma,\beta))\subset
V(z_{0},\lambda,\beta).$

Then, we have $\log F^{\prime}_{e^{i\theta},\lambda}\in\partial
V(z_{0},\lambda,\beta)$. Suppose there exists $f$ such that

$\displaystyle\log F^{\prime}_{e^{i\theta},\lambda}(z_{0})=\log
f^{\prime}(z_{0})$

for $z_{0}\in\mathbb{D}$. Then we claim that $F_{e^{i\theta},\lambda}=f$. We
set

$\displaystyle
h(t)=\frac{\overline{G(z_{0})}}{{G(z_{0})}}\left(\frac{f^{\prime\prime}(z(t))}{f^{\prime}(z(t))}-c(z(t),\lambda,\beta)\right)z^{\prime}(t),$

where $\gamma_{0}:z(t)$, $0\leq t\leq 1$ as in (4.20). Then it is easy to see
that $h(t)$ is a continuous function on $[0,1]$ and satisfies

$\displaystyle|h(t)|$
$\displaystyle=\frac{|\overline{G(z_{0})}|}{|{G(z_{0})}|}\left|\frac{f^{\prime\prime}(z(t))}{f^{\prime}(z(t))}-c(z(t),\lambda,\beta)\right||z^{\prime}(t)|$
$\displaystyle=\left|\frac{f^{\prime\prime}(z(t))}{f^{\prime}(z(t))}-c(z(t),\lambda,\beta)\right||z^{\prime}(t)|$
$\displaystyle\leq
r(z(t),\lambda,\beta)|z^{\prime}(t)|\,\,\,\,\mbox{for}\,\,0\leq t\leq 1.$

Further, we see that

$\displaystyle\int_{0}^{1}{\rm Re}(h(t))dt$ $\displaystyle=\int_{0}^{1}{\rm
Re}\left(\frac{\overline{G(z_{0})}}{{G(z_{0})}}\left(\frac{f^{\prime\prime}(z(t))}{f^{\prime}(z(t))}-c(z(t),\lambda,\beta)\right)z^{\prime}(t)\right)dt$
$\displaystyle={\rm
Re}\left(\frac{\overline{G(z_{0})}}{{G(z_{0})}}\int_{0}^{1}\frac{f^{\prime\prime}(z(t))}{f^{\prime}(z(t))}z^{\prime}(t)dt-\int_{0}^{1}c(z(t),\lambda,\beta)z^{\prime}(t)dt\right)$
$\displaystyle={\rm Re}\left(\frac{\overline{G(z_{0})}}{{G(z_{0})}}\left(\log
f^{\prime}(z_{0})-C(\lambda,\gamma_{0},\beta)\right)\right)$
$\displaystyle={\rm Re}\left(\frac{\overline{G(z_{0})}}{{G(z_{0})}}\left(\log
F^{\prime}_{e^{i\theta,\lambda}}(z_{0})-C(\lambda,\gamma_{0},\beta)\right)\right)$
$\displaystyle=\int_{0}^{1}r(z(t),\lambda,\beta)|z^{\prime}(t)|dt.$

Therefore, $h(t)=r(z(t),\lambda,\beta)|z^{\prime}(t)|$. Thus we have

$\displaystyle\frac{f^{\prime\prime}}{f^{\prime}}=\frac{F^{\prime\prime}_{e^{i\theta,\lambda}}}{F^{\prime}_{e^{i\theta,\lambda}}}$

on the curve $\gamma_{0}$. Therefore, we must have $\log f^{\prime}=\log
F^{\prime}_{e^{i\theta,\lambda}}$ which shows that $f\equiv
F_{e^{i\theta,\lambda}}$. This completes the proof. ∎

###### Proof of Proposition 3.6.

We wish to show that the closed curve

$(-\pi,\pi]\ni\theta\mapsto\log F^{\prime}_{e^{i\theta},\lambda}(z_{0})$

is a simple curve. Suppose that $\log
F^{\prime}_{e^{i\theta_{1},\lambda}}(z_{0})=\log
F^{\prime}_{e^{i\theta_{2}\lambda}}(z_{0})$ for some
$\theta_{1},\theta_{2}\in(-\pi,\pi]$ with $\theta_{1}\neq\theta_{2}$. Then in
view of (2.8), we have

$F_{e^{i\theta_{1},\lambda}}=F_{e^{i\theta_{2}\lambda}}.$

Using (4.5), it is easy to see that

$e^{i\theta_{1}}z=\tau\left(\dfrac{\omega_{F_{e^{i\theta_{1}}\lambda}}}{z},\lambda\right)=\tau\left(\dfrac{\omega_{F_{e^{i\theta_{2}}\lambda}}}{z},\lambda\right)=e^{i\theta_{2}}z,$

which is a contradiction. Thus the curve $(-\pi,\pi]\ni\theta\mapsto\log
F^{\prime}_{e^{i\theta},\lambda}(z_{0})$ must be a simple curve. Since
$V(z_{0},\lambda,\beta)$ is a compact, convex subset of $\mathbb{C}$ and has
non-empty interior, the boundary $\partial V(z_{0},\lambda,\beta)$ is a simple
closed curve. From Proposition 3.3, the curve $\partial
V(z_{0},\lambda,\beta)$ contains the curve $(-\pi,\pi]\ni\theta\mapsto\log
F^{\prime}_{e^{i\theta},\lambda}(z_{0})$. We note that a simple closed curve
cannot contain any simple closed curve other than itself. Thus, $\partial
V(z_{0},\lambda,\beta)$ is given by $(-\pi,\pi]\ni\theta\mapsto\log
F^{\prime}_{e^{i\theta},\lambda}(z_{0})$. ∎

The following figures 1-5 show that the boundary of $V(z_{0},\lambda,\beta)$
for the values of $z_{0}\in\mathbb{D}\setminus\\{0\\},0\leq\lambda\leq 1$ and
$0<\beta<1$. We notice that according to Proposition 3.1, the region bounded
by the Jordan curve $\partial V(z_{0},\lambda,\beta)$ is compact and convex.

$\partial V(z_{0},\lambda,\beta)$ $\partial V(z_{0},\lambda,\beta)$

Figure 1. Region of variability of $\log f^{\prime}(z_{0})$ when
$f\in{\mathcal{F}}(\lambda)$

$\begin{array}[]{ll}z_{0}=0.0778577+0.803506i&z_{0}=0.734813+0.272699i\\\
\lambda=0.886156&\lambda=0.751387\\\
\beta=0.951601&\beta=0.0264723\end{array}$

$\partial V(z_{0},\lambda,\beta)$ $\partial V(z_{0},\lambda,\beta)$

Figure 2. Region of variability of $\log f^{\prime}(z_{0})$ when
$f\in{\mathcal{F}}(\lambda)$

$\begin{array}[]{ll}z_{0}=0.49963+0.23489i&z_{0}=-0.36336+0.539691i\\\
\lambda=0.716587&\lambda=0.71442\\\ \beta=0.59288&\beta=0.105526\end{array}$

$\partial V(z_{0},\lambda,\beta)$ $\partial V(z_{0},\lambda,\beta)$

Figure 3. Region of variability of $\log f^{\prime}(z_{0})$ when
$f\in{\mathcal{F}}(\lambda)$

$\begin{array}[]{ll}z_{0}=0.588369+0.0292795i&z_{0}=0.269304+0.355388i\\\
\lambda=0.884117&\lambda=0.529987\\\ \beta=0.200402&\beta=0.629449\end{array}$

$\partial V(z_{0},\lambda,\beta))$ $\partial V(z_{0},\lambda,\beta)$

Figure 4. Region of variability of $\log f^{\prime}(z_{0})$ when
$f\in{\mathcal{F}}(\lambda)$

$\begin{array}[]{ll}z_{0}=0.441545+0.437309i&z_{0}=0.138243-0.208931i\\\
\lambda=0.277434&\lambda=0.395356\\\ \beta=0.830942&\beta=0.381739\end{array}$

$\partial V(z_{0},\lambda,\beta)$ $\partial V(z_{0},\lambda,\beta)$

Figure 5. Region of variability of $\log f^{\prime}(z_{0})$ when
$f\in{\mathcal{F}}(\lambda)$

$\begin{array}[]{ll}z_{0}=-0.533249-0.0367948i&z_{0}=0.69143-0.596334i\\\
\lambda=0.626941&\lambda=0.349507\\\ \beta=0.593493&\beta=0.762552\end{array}$

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