11institutetext:

# Compiling Turing Machines into
Storage Modification Machines

J.-M. Chauvet

###### Abstract

It is well known that Schönhage’s Storage Modification Machines (SMM) can
simulate Turing Machines (TM) since Schönhage’s original proof of the Turing
completeness of the eponymous machines. We propose a simple transformation of
TM into SMM, setting the base for a straightforward TM-to-SMM compiler.

##  1 Introduction

It is well known that Schönhage’s Storage Modification Machines [4] can
simulate Turing Machines (TM) [5]. Simulation of small universal TM, or other
simple universal models such as Post’s tag systems [1] and the cellular
automaton Rule 110 [6], is by now a standard way to prove that a large number
of alternate models of computation, including a variety of physically-inspired
systems, are computationally universal. In the following, we consider a
slightly revised version of Schönhage’s Storage Modification Machine (SMM) and
offer a simple construction to simulate the behavior of a TM. This
construction sets the base for a generic TM-to-SMM compiler.

##  2 Images of Computation: Turing Machines and Storage Modification Machines

_Turing Machines_ , masquerading as read-write-head-and-tape mechanic
contraptions, are in fact quite mathematical, theoretical models of
computation. They are best viewed as abstract machines (metaphorically)
manipulating symbols on a strip of _tape_ according to a table of rules. In a
simplified design, the infinite tape is divided into contiguous cells, each
cell having a right- and a left-neighbor, which may or not exhibit a _symbol_
(from a given, usually finite, alphabet). The _head_ when positioned over a
cell can read or write this cell’s symbol. Finally the machine is
characterized by a _state_ , also from a given, usually finite, state-set.

An elementary execution step of such a TM is summarized as follows:

  1. 1.

_Read_ the current cell’s symbol $\sigma$ (or blank, if no symbol is present).

According then to the combination of current state and symbol in the
transition table, $(S,\sigma)$:

  1. 2.

_Write_ back a symbol $\sigma^{\prime}$ on the current cell.

  2. 3.

_Move_ head to one of the two neighbors of the current cell.

  3. 4.

_Change_ internal state to $S^{\prime}$.

The behavior of this simple TM is entirely specified by its transition table
which states, for each relevant $(S,\sigma)$ combination, the data required to
perform steps 2, 3 and 4 above.

In our construction, this data is simply a string which concatenates the
symbol $\sigma^{\prime}$, the constant $L$ (left) or $R$ (right) to indicate
which contiguous neighbor to target in step 3, and finally the new state
$S^{\prime}$ of the TM.

Running a TM involves (i) positioning the head on a tape with some initial
symbols in some cells, (ii) setting some state as the initial one, and (iii)
start consecutively executing the above steps. By convention when no
combination of state and symbol is found in the transition table, the machine
halts111The whole point of Turing in his original 1936 paper was to provide a
mathematical description of a very simple device capable of arbitrary
computations, and prove properties of computation in general – and in
particular, the uncomputability of the _Entscheidungsproblem_ (decision
problem) of predicting whether a so specified TM would halt or move forever,
reading and writing symbols..

_Storage Modification Machines_ presented here are from a variant in [2] where
they are used to implement population protocol models. Like a TM, a SMM
represents a single computing agent. It is equipped with memory and a
processing unit. Its memory stores a finite directed graph of equal out-degree
nodes [4], with a distinguished node called the _center_. (Edges of this graph
are also called _pointers_.) Edges leading out of each node are uniquely
labelled by distinct _directions_ , drawn from a finite set $D$.

Any string $x\in D^{*}$ refers to the node $p(x)$ reached from the center by
following the sequence of directions labelled by $x$. (Note that this is
somehow similar to moving the head in a TM.) In the variant used here, nodes
may have different out-degrees, and we set $p(x)=\emptyset$ when $x$ is not a
valid path in the graph.

SMM are furthermore characterized by a _program_ , or _control list_ , which
is a finite list of consecutively numbered instructions (reminiscent of the
transition tables in TM). The restricted instruction set is as follows:

  * •

_new label_ creates a new labelled node and makes it the center, setting all
its outgoing edges to the previous center.

  * •

_set xd to y_ where $x,y$ are paths in $D^{*}$ and $d\in D$ is a direction,
redirects the $d$ edge of $p(x)$ to point to $p(y)$.

  * •

_center x_ where $x$ is a path, moves the center to $p(x)$.

  * •

_if x y then ln_ where $x,y$ are paths and _ln_ a line number, jumps to line
_ln_ if $p(x)=p(y)$ and skips to the next line if not. Line numbers can be
absolute, _ln_ , or relative to the current line number, _+ln_ or _-ln_.

  * •

_stop message_ halts the SMM, printing out _message_.

The similarities highlighted above between TM and SMM are our guidelines to
the specification of a generic transformation of the TM transition table to
appropriate sequences of SMM instructions.

##  3 A TM-to-SMM Compiler

More specifically, elaborating on these similarities: each TM tape cell and
each position of the head over it are represented by a node in the compiled
SMM, with one dedicated direction, say f, pointing from head to cell and cell
to head.

The symbol on the tape cell is simply the binary representation of its index
in the alphabet, using $n$ specialized directions in the tape node, where $n$
is the integer immediately superior to the base-2 log of the alphabet size.
The state of the TM is encoded in a head node in the same way, the binary
representation of its index in the state-set using $m$ specialized directions
(which may be in common with the former $n$ ones, without confusion as a node
is either a tape or a head).

A special direction in both type of nodes, say o, always points to a fixed
initial node, the _Origin_. The center of the SMM is set to the current
position of the TM head, i.e. the SMM head node linked to the current tape
cell tape node.

By convention, the binary representation bits in nodes are $0$ when their
direction points to self, and $1$ when their direction points to the fixed
Origin.

Finally tape nodes are doubly-linked using two specialized directions, e and w
(for east and west); their corresponding head nodes (in the f direction) are
doubly-linked in the same way. The compiled SMM has then $4+max(n,m)$
directions.

The TM compiler is built as a sequence of code generation/decoration passes:

  * •

_State Selection_. Build a decision tree based on the state binary
representation over the $m$ bit directions, using SMM controls if <B> o to
test each <B> bit direction.

  * •

_Symbol Detection_. At each leaf of the previous tree, build a decision tree
based on the symbol binary representation over the $n$ bit directions. At each
leaf of each of these new trees, add the control list implementing the
$(\sigma,S)$ transition found in the TM table: binary encode the new symbol on
the current tape node; move to the e or w head node as indicated in the
transition (possibly creating new SMM nodes in the process, if they do not
exist, hence simulating an infinite tape); finally encode the new state in
this head cell, and make it the center. This reproduces the steps 2-4 of the
TM.

  * •

_Prologue_. Prefix the control list resulting from the previous passes with a
specific control list setting up the initial tape and symbols, the initial
head position and the initial state.

The compiled SMM is ran stepwise: the initial prologue to set up the TM as a
first mandatory step, followed by a sequence of calls to the above transition
control list. Each call to the SMM execution step implements a full transition
in the TM execution.

###  3.1 SMM simulation of the TM simulation of the Collatz $3x+1$ function

We exercize the TM-to-SMM compiler on the compact 3-4 TM, simulating the
Collatz $3x+1$ function, given in [3] and defined as:

$(S,\sigma)$ | b(lank) | 0 | 1 | 2
---|---|---|---|---
$A$ | bLC | 0RA | 0RB | 1RA
$B$ | 2LC | 1RB | 2RA | 2RB
$C$ | bRA | 0LC | 1LC | 2LC

where $R$, $L$ represent right and left, captured as e, w directions in the
compiled SMM. This TM operates on integers represented in base 3: the symbol
alphabet is $\\{b,0,1,2\\}$ and the state set is $\\{A,B,C\\}$. The initial
tape is the base-3 representation of $x_{0}$, and the initial state is set to
$A$. Note that this TM never halts as it ends up – if the Collatz conjecture
is true – looping over the $\\{1,4,2\\}$ cycle.

The following page displays twelve steps of the compiled SMM execution,
starting from the initial value $x_{0}=19$. On the right-hand side, top to
bottom, the SMM growth is graphed with the center highlighted in gray. (The o
direction and all binary representations of symbols and states in the SMM
nodes are omitted for clarity.) On the left-hand side a more conventional TM
graphical representation of the SMM computation is presented; symbols and
states are shown and color-coded, with the position of the head. The initial
tape cell is circled twice.

The repeated execution of the SMM control list results in values which are
read (in base 3) when the TM state is $C$ and the head is at its leftmost
position, over a blank, b, tape cell. Note that the compact Collatz 3-4 TM
computes the only odd values in the Collatz series starting at $x_{0}$,
skipping over all the intermediate divisions by $2$. Here in the trace
displayed next page:

T | Base 3 | Decimal
---|---|---
0 | 201 | 19
7 | 1002 | 29
… | … | …

Of course the delay between successive printings of the iterated values on the
tape lengthens as the tape is progressively expanded right, forcing a longer
trip of the head back to its leftmost position in state $C$.

This compiled SMM has 6 directions, 2 of which are used for the binary
representation of states and symbols. At any given time it counts $2n+1$
nodes, where $n$ is the length of the tape. The prologue control list is 43
lines long; the transition control list is 398 lines long (including compiler-
generated comments).

(Listings for the compiled SMM program, and other Turing Machines simulations
is on-line at https://github.com/CRTandKDU/SMM.)

Figure 1: Twelve steps of the SMM compiled from the compact Collatz 3-4 TM.

##  4 Conclusion

Schönhage’s Storage Modification Machines can generally simulate Turing
Machines. This paper provides an alternate construction of such SMM
simulations which finds its use in a Turing Machine to SMM compiler . Related
questions on the minimal (space) complexity SMM required for simulation of
complex Turing Machines may then be addressed by looking into optimizing this
simple TM-to-SMM compiler.

## References

  * [1] John Cocke and Marvin Minsky.  Universality of Tag Systems with P = 2.  J. ACM, 11(1):15–20, January 1964.
  * [2] Rachid Guerraoui and Eric Ruppert.  Names Trump Malice: Tiny Mobile Agents Can Tolerate Byzantine Failures.  In Proceedings of the 36th Internatilonal Collogquium on Automata, Languages and Programming: Part II, ICALP ’09, page 484–495, Berlin, Heidelberg, 2009. Springer-Verlag.
  * [3] Pascal Michel.  Simulation of the Collatz 3x+1 function by Turing machines, 2014.  https://arxiv.org/abs/1409.7322, last accessed on 9/25/2021.
  * [4] A. Schönhage.  Storage Modification Machines.  SIAM Journal on Computing, 9(3):490–508, 1980.
  * [5] A. M. Turing.  On computable numbers, with an application to the Entscheidungsproblem.  Proceedings of the London Mathematical Society. Second Series, 42:230–265, 1936.  This is the paper that introduced what is now called the Universal Turing Machine.
  * [6] Stephen Wolfram.  A New Kind of Science.  Wolfram Media, 2002.


# Non-fungible token transactions: data and challenges

Jason B. Cho<EMAIL_ADDRESS>Department of Statistics and Data Science,
Cornell University, Ithaca, NY, USA  Sven Serneels<EMAIL_ADDRESS>Gallop
Data, Inc., Denver, CO, USA  Department of Mathematics, University of Antwerp,
Antwerp, Belgium  David S. Matteson<EMAIL_ADDRESS>Department of Statistics
and Data Science, Cornell University, Ithaca, NY, USA

###### Abstract

Non-fungible tokens (NFT) have recently emerged as a novel blockchain hosted
financial asset class that has attracted major transaction volumes. Investment
decisions rely on data and adequate preprocessing and application of analytics
to them. Both owing to the non-fungible nature of the tokens and to a
blockchain being the primary data source, NFT transaction data pose several
challenges not commonly encountered in traditional financial data. Using data
that consist of the transaction history of eight highly valued NFT
collections, a selection of such challenges is illustrated. These are: price
differentiation by token traits, the possible existence of lateral swaps and
wash trades in the transaction history and finally, severe volatility. While
this paper merely scratches the surface of how data analytics can be applied
in this context, the data and challenges laid out here may present
opportunities for future research on the topic.

Keywords: Non-fungible tokens, NFT, Financial transactions, wash trading,
volatility, rarity, data analytics

##  1 Introduction

The blockchain technology has attracted a lot of attention over the past
decade as a decentralized ledger. The blockchain technology was initially
launched in 2009 as the Bitcoin Network based on a whitepaper with sole focus
on the transfer and storage of digital currency (Nakamoto, 2008). Bitcoin’s
success as a cryptocurrency sparked a wave of development in blockchain
technologies in the wake of which many new blockchains were launched. Some of
these concerned refining the Bitcoin approach to digital currencies. However,
it was soon recognized that the blockchain technology could have utility much
broader than digital currency. The Ethereum network was established to fill
that void: it was designed to be a blockchain that enabled to settle digital
contracts and run decentralized applications (Buterin, 2014). Several other
networks have followed Ethereum’s footsteps since.

Digital, or smart, contracts are applied to various fields. One such area that
stands out is a contract that transfers a digital right of ownership to a
unique asset, which can be both digital or physical. From the blockchain
perspective, such an asset is a token, yet it is unique and therefore non-
fungible. Again, the concept of non-fungible tokens (NFTs) can be applied to
many real world applications, ranging from supply chain to digital sports
cards and tokens with utility in games, just to name a few. However, NFTs have
probably become most hyped in the realm of digital art, which has received
significant mainstream media attention over the past four years. The hype
around NFTs has attracted investors individual and institutional, which has
caused certain tokens and even entire collections to be very highly valued.

Owing to the sheer amount of money that circulates in the NFT markets, high
valued NFTs are widely seen as a novel financial asset class. A broad set of
financial services have emerged around NFTs that span lending protocols, hedge
and index funds, NFT fractionalization, buy now pay later services and many
more. This phenomenon is referred to as the financialization of NFT. In
addition, there has been a significant increase in studies exploring the
financial aspects of NFT markets in the recent years (Bao and Roubaud (2022)).
Studies such as Kong and Lin (2022), Mazur (2021), Borri et al. (2022) and Ko
et al. (2022) analyzed risk-return of NFT in relation to other asset-class
such as bond, equity and commodity. All four studies conclude NFT market to be
extremely volatile and behave differently from the traditional asset class.

A few studies also have explored various potential predictors for NFT price.
Dowling (2022a) and Ante (2022) explored the correlation between the price of
NFT collections and cryptocurrency such as Bitcoin and Ether. Both studies
concluded cryptocurrency pricing to be weakly correlated with NFT pricing. The
effect of media on NFT price was also explored by Umar et al. (2022) and
Kapoor et al. (2022). Particularly, Umar et al. (2022) analyzed how media
coverage of COVID-19 affects the price and liquidity in the NFT market. Kapoor
et al. (2022) also found features from social media with regards to a NFT
collection, such as number of likes, are significant factors in predicting the
price of a NFT token.

Success of financial services and researches on the NFT markets cannot take
place without accurate data. The study by Oh et al. (2022) claims that the
experienced investors outperform inexperienced investors by 10 percentage
points per trade, suggesting high degree of informational inefficiency in the
NFT market. Data on NFT transactions, however, come with a set of challenges
not shared with traditional financial data. These are both the result of the
process that generates the data and of the nature of the asset. While studies
on NFT often uses such data, challenges and difficulties in analyzing NFT
collections data are rarely explored. This paper will highlight these unique
challenges by illustrating them based on the transaction history for a set of
eight commonly traded, high valued collections of NFTs and will suggest how to
pre-process the raw data so that they become amenable to statistical and
financial analysis.

The paper is organized as follows: in Section 2, the data themselves are
introduced, along with a discussion on specific characteristics of NFT
transaction data and some summary statistics. Section 3 introduces a selection
of challenges associated with NFT transactional data and some suggestions to
pre-process such data appropriately. Eventually, the final Section provides an
outlook into the emerging field of analytics for said data.

##  2 The Data

NFTs have diverse fields of application in the real world. However, the most
financialized segments of the broader NFT market are digital art and gaming.
Specific aspects of the tokens and their application need to be considered to
analyze data for each segment. To narrow down the scope of the paper, the data
described here all resort to the digital art segment of profile picture
collections (PFP). These collections are designed in a similar way: a artist
creates a base image, to which attributes are added. By varying these
attributes in a way that each realized combination occurs only once, a
collection of unique images is created, each of which follow the base pattern,
yet are clearly distinguishable. Ownership for each of these images is then
minted on the blockchain of choice as an NFT. An illustration of how varying
traits lead to unique images is presented in Figure 1 for the collection Bored
Ape Yacht Club.

Figure 1: Sample of six Bored Ape Yacht Club tokens with Blue Beam, Cyborg, 3D
(top) and Bored, Holographic, X-Eyes (bottom) Eye traits, among other trait
differences.

###  2.1 Sources and Structure

In theory, all blockchain transactions are publicly available on the
blockchain itself. However, querying raw blockchain transactions can be a
daunting task and this not only because of the information technical
challenges. Transactions that involve one or more smart contracts will be
stored on the blockchain as bytecode, which then needs to be decoded and such
decoding will be specific to the contract deployed. Because of these technical
challenges, several companies have recently spun up data services that provide
structred, decoded transactional data for NFTs. One such company is
Gallop111https://www.higallop.com, whose set of APIs is the source for the
data in this paper. As of this writing, Gallop provides data of 156 Ethereum
and 295 Solana NFT collections. The data include transaction histories such as
the transacted price of each token, timestamp of the transaction, and the
associated wallet addresses, as well as visual traits such as background
color, body type, or accessories of the token within the collection. The data
selected for this paper correspond to six PFP collections on the Ethereum
blockchain (Bored Ape Yacht Club, Bored Ape Kennel Club, Mutant Ape Yacht
Club, Cryptopunks, Meebits, and Autoglyphs) and two on the Solana blockchain
(Degenerate Ape Academy and Aurory), all of which are highly valued at the
time of writing. An overview of the collections selected, is presented in
Table 1. The data contain transaction histories from collection launch up to
March 31, 2022. Data were compiled from the Gallop API responses into a
structured table and were analyzed using R. Accompanying this manuscript, the
tabular data for these eight collections have been made publicly available
(Serneels et al., 2022).

|  | Aurory | Autoglyphs | Bored Ape Kennel Club | Bored Ape Yacht Club
---|---|---|---|---|---
Currency | Solana | Ethereum | Ethereum | Ethereum
Project Launch | August 2021 | April 2019 | June 2021 | April 2021
Token Count | 10,000 | 512 | 10,000 | 10,000
Transactions |  |  |  |
| Daily Average | 68.78 | 0.45 | 70.54 | 79.85
| Total | 14,582 | 490 | 20,176 | 26,751
Floor Price |  |  |  |
| Daily Minimum | 0.005478 | 0.005300 | 0.000010 | 0.000050
| Daily Average | 21.29 | 51.81 | 1.80 | 5.72
| Daily Maximum | 35.01 | 449.00 | 8.40 | 102.69
Total Trading Volume |  |  |  |
| Daily Average | 3,032.63 | 17.95 | 303.64 | 1,654.62
| Total | 642,917.79 | 19,496.99 | 86,841.95 | 554,296.23
|  |  |  |  |
|  | Cryptopunks | Degenerate Ape Academy | Meebits | Mutant Ape Yacht Club
Currency | Ethereum | Solana | Ethereum | Ethereum
Project Launch | June 2017 | May 2021 | August 2021 | August 2021
Token Count | 9,999 | 10,000 | 20,000 | 19,414
Transactions |  |  |  |
| Daily Average | 10.86 | 85.60 | 77.01 | 139.87
| Total | 18,923 | 19,516 | 25,566 | 30,073
Floor Price |  |  |  |
| Daily Minimum | 0.000050 | 0.052039 | 0.000900 | 0.000025
| Daily Average | 15.59 | 42.60 | 2.26 | 0.76
| Daily Maximum | 217.00 | 88.01 | 6.15 | 15.01
Total Trading Volume |  |  |  |
| Daily Average | 428.46 | 5,222.89 | 9,131.63 | 1,511.11
| Total | 746,379.75 | 1,190,818.21 | 3,031,702.79 | 324,889.44

Table 1: Associated blockchain, project launch date, token counts, transaction
summary, floor price, and trading volume of the eight profile picture token
collections

###  2.2 Exploratory Data Analysis

(a) Aurory

(b) Autoglyphs

(c) Bored Ape Kennel Club

(d) Bored Ape Yacht Club

(e) Cryptopunks

(f) Degenerate Ape Academy

(g) Meebits

(h) Mutant Ape Yacht Club

Figure 2: Log(price) of Eight Profile Picture Collections (PFP) in USD from
the Project Launch and March 31, 2022.

The average price of a collection, as well as its trend within a given time
span, can vary greatly among collections. Price histories are plotted for the
eight collections on a logarithmic scale in Figure 2. The average prices of
Aurory, Autoglphys, Bored Ape Kennel Club, Bored Ape Yacht Club, Cryptopunks,
Degenerate Ape Academy, Meebits, and Mutant Ape Yacht club were 3582.05,
97096.10, 12832.39, 66391.96, 113939.13, 7465.30, 341651.02, and 35896.81 USD,
respectively. Only Autoglyphs and Cryptopunks show a steady increase in price.
Bored Ape Yacht Club had a steady increase during the first few months after
collection launch, but became approximately stationary around August 2021.
However, the Aurory, Bored Ape Kennel Club and Mutant Ape Yacht Club
collections were stationary during the given period, with the exception of a
few outliers. Moreover, it is interesting to note that seemingly bimodal price
distributions exist for several collections, most notoriously so for Meebits,
an effect that will be discussed in more detail in Sections 3.2 and 3.3.

(a) Aurory

(b) Autoglyphs

(c) Bored Ape Kennel Club

(d) Bored Ape Yacht Club

(e) Cryptopunks

(f) Degenerate Ape Academy

(g) Meebits

(h) Mutant Ape Yacht Club

Figure 3: Number of Transactions of Eight Profile Picture Collections (PFP) in
USD from the Project Launch and March 31, 2022.

In addition to the price, trading volume of a financial asset is another
technical indicator that traders look to determine liquidity when making
trading decisions. Similar to high trading volume of a stock after the Initial
Public Offering (IPO), collections such as Bored Ape Kennel Club, Bored Ape
Yacht Club, Meebits, and Mutant Ape Yacht Club hit their highest trading
volumes right after project launch as shown in Figure 3. When compared against
traditional financial instruments, the trading volumes for all eight
collections were low. While some collections reached a thousand to few
thousand daily transactions, the median daily transactions of Aurory,
Autoglphys, Bored Ape Kennel Club, Bored Ape Yacht Club, Cryptopunks,
Degenerate Ape Academy, Meebits, and Mutant Ape Yacht club were 16, 1, 36, 33,
5, 13, 33 and 80 respectively. Trading volume also had high day-to-day
variation with the standard deviation of 15.99, 3.80, 104.63, 167.80, 28.60,
25.44, 123.02, and 392.53 respectively.

##  3 Challenges and Limitations

###  3.1 Traits

(a) Price Change by Eye Type (Bored Ape Yacht Club)

(b) Price Change by Body Type (Cryptopunks)

(c) ”None” vs ”Striped Tee” (Bored Ape Yacht Club)

Figure 4: Log(price) of Bored Ape Yacht Club and Cryptopunks by Various
Traits.

One of the main challenges to analyze the price of NFTs at the collection
level is the wide price range among tokens, as shown for two collections in
Figure 2. Such differences can partially be explained by the design of each
token. Tokens are generated according to a base pattern that is then
complemented by giving it certain traits. Most collections are designed in
such a way that a certain (combination of) traits is much rarer than other
combinations. For instance, the Cryptopunks collection consists of 10,000
tokens, distributed across six body types. However, there are only 6 Aliens,
14 Apes and 24 Zombies, while the remainder of the tokens have either Male or
Female body type. The price of a token with Alien, Ape and Zombie body types
was generally considerably higher than the ones with Male and Female body
types (Figure 4(b)). Similarly, for the Bored Ape Yacht Club (BAYC) by
YugaLabs, only 118 of 10,000 tokens have laser eyes or blue beams radiating
from their eyes, and these were traded at prices noticeably higher than tokens
with others Eyes traits (Figure 4(a)).

However, not all design traits seem to contribute to the price of a token. For
example, in the BAYC collection, the most common Cloth trait is ”none”: 1,886
of the 10,000 tokens have this trait. The next common trait is ”Striped Tee”,
which 412 tokens have. While the tokens with ”None” trait are about 4.5 times
more common than the ones with ”Striped Tee” trait, no noticeable differences
were found in their prices, as is illustrated in Figure 4(c).

Figure 5: Price vs Rarity of a Token (Bored Ape Yacht Club).

The relationship between the price of a token and its rarity was explored by
calculating the rarity score and analyzing the linear relationship between the
score and the average price of the token. The rarity score of a token was
calculated by averaging the rarity of a token over multiple traits (hair,
background, accessories, etc.) a token is designed to have, where rarity is
simply the fraction of the number of tokens with the trait to the total number
of tokens in the collection. A negative correlation between rarity and price
of a token was detected (Figure 5), suggesting that rarer tokens are indeed
traded at higher prices than more common tokens. However, the R-squared of
only around 0.001 suggests that the correlation is rather weak, and that there
are other meaningful factors that contribute to the price of a token. Here we
also note that average rarity is only one way to compute token rarity222The
company https://rarity.tools specializes in token rarity and offers two
version of highly customizable rarity models, the details of which would range
beyond the scope of this paper. and other measures may lead to more
significant correlations. For instance, Kong and Lin (2022) arrive at similar
findings using a more sophisticated hedonic regression model for Cryptopunks
transactions though May 2021.

###  3.2 Lateral swaps

(a) Price Change by Exchange Type
---
(Bored Ape Kennel Club)

(b) Price Change by Exchange Type
---
(Bored Ape Yacht Club)

(c) Price Change by Exchange Type
---
(Mutant Ape Yacht Club)

(d) Price Change by Exchange Type
---
(Meebits)

Figure 6: Price Change by Exchange Type

One challenge to decode NFT transactions on Ethereum resides in the fact that
while transactions have a value field, the value contained in that field
reflects the value paid into the top level contract, which will only
correspond to the price paid for the token in very simple transactions, such
as peer-to-peer Cryptopunk transactions. In transactions that involve more
than one smart contract, the transaction price can only be retrieved by
decoding further layers of smart contract execution. In fact, many
transactions will have a value equal to zero at the top level, but contain
non-zero transaction prices in the token contract execution.

All of the above is accounted for in Gallop’s data aggregation, yet some
atypical transactions may still be found in decoded transaction data. For
instance, when users use a smart contract to swap tokens laterally between
wallet addresses, they may still have to pay the execution price of the swap
contract, which will appear on the blockchain as the value. This results in a
set of transactions recorded at a price of 0.005 ETH, which is considerably
lower than the fair value token transaction price, as strikes the eye for
collections such as Bored Ape Kennel Club, Bored Ape Yacht Club, and Mutant
Ape Yacht Club, plotted in Figure 6. As these transaction records are not
indicative of the fair value of the token, they should be excluded when
performing token or collection valuation analysis333The public Gallop API will
filter these anomalous transactions out, yet they were retained in the data
described here as they represent a challenge common in NFT transaction data..

###  3.3 Wash Trading

Wash trading is a process in which a market participant sells and buys the
same token back multiple times within a short period to deceive other market
participants about its price and liquidity. If such malicious market
participants are successful, they can completely distort transaction price
statistics. A good illustration of that effect is shown in the transaction
price history for Meebits (Figure 6(d)). Meebits’ transaction history follows
an approximately smooth and continuous trend through January 2022, but then a
vast amount of transactions occur at a wide range of prices deviating from
that trend. Unusual transactions of the Meebits tokens garnered media
attention and a few articles have been published on this phenomenon (Howcroft
(2022)). Many suspect that most, if not all, of the transactions at
significantly higher prices compared to the trend to be wash trades. As can be
seen from Figure 6, which groups transactions by exchange protocol, almost all
transactions at elevated prices have been executed using the LooksRare
protocol.

LooksRare444https://looksrare.org/ launched in January 2022 as a new
decentralized marketplace, aiming to take on competition with the (then and
now) dominant marketplace OpenSea555https://opensea.io/. In order to grow
market share, the LooksRare protocol rewards users by paying them back a
fraction of the transacted amount in their native LOOKS token. However, the
LooksRare protocol thereby inherently encourages wash trading. By using
LooksRare, wash traders reap a double benefit: they not only succeed at
artificially raising average sales prices and volumes for their tokens, but
they also get rewarded in LOOKS token for doing so. Moreover, they can then
decide to stake the LOOKS token, for which they will reap a second round of
rewards in the form of Wrapped Ether (WETH) tokens. An example of how trading
a single token in this way can lead to profits of approximately 3.5 million
USD, is described in Coffman (2022).

(a) Transactions between
---
2021-07-11 and 2022-01-03
(Bored Ape Kennel Club #282)

(b) Transactions between
---
2021-07-11 and 2022-02-12
(Meebit #17021)

Figure 7: Transactions History of Bored Ape Kennel Club #282 and Meebits
#17021

While wash trading can clearly distort market prices, it is an entire
different question how it can be detected. To propose a detailed algorithm for
that purpose would lead us beyond the scope of this Data Note, yet we will
provide a hint as to how either wallets or collections can be flagged for
suspicious wash trading activity.

At first, a wash trade is by definition a a trade executed between different
wallets owned by the same person or entity. On the blockchain, wallets are
anonymous and in theory, users can create new wallet addresses each time they
trade. However, at least for non-automated traders or less sophisticated wash
trading algorithms, to create a new wallet address each time is a cumbersome
step. Therefore, one can expect that in such cases, the token traded would
return to the originating wallet address at a given point in time. In several
cases, the latter occurs a striking amount of times, as illustrated in Figure
7(b). Herein, the trade graph is plotted for two example tokens. On the one
hand, Bored Ape Kennel Club #282 has unique wallets involved in all of its
five transactions (Figure 7(a)), whereas Meebit #17021 was transacted 32 times
on January 17, 2022, between the same pair of wallets , notably via LooksRare,
before eventually being transacted on Opensea into a different wallet on
February 12, 2022.

(a) Bored Ape Kennel Club

(b) Bored Ape Yacht Club

(c) Cryptopunks

(d) Meebits

(e) Mutant Ape Yacht Club

Figure 8: Distribution of the Wallet to Transactions Ratio.

By the anonymous nature of NFT wallets, one can not detect wash trades with
absolute certainty. Even though transactions involving BAKC #282 involve
unique wallet addresses, it is possible that all these wallets are owned by
the same party. Nevertheless, the 32 transactions between a pair of wallets
involving Meebit #17021 are most likely the result of wash trading activity.
Many other examples of repeated transfers between wallet pairs can be found in
the Meebits transaction history. The latter would lead us to believe that a
lot of the wash trading activity has been executed with a low degree of
automation. Two holistic approaches investigating wash trading for the entire
blockchain arrive at similar conclusions. At first, Tariq and Sifat (2022)
corroborate the existence of wash trades on the Ethereum and Wax blockchains
by performing hypothesis tests on the entirety of NFT transactions. They test
for adherence to Benford’s law, existence of clusters in pricing and test if
transaction prices fit Pareto—Levy’s law, all of which significantly point to
wash trades being present. Secondly, von Wachter et al. (2022) store the
entirety of Ethereum transactions that involve NFTs as a directed graph and
then extract all closed loops from it, which leads them to conclude that
approximately two percent of all NFT transactions are suspect to being wash
trades and that the vast majority of these only involve up to a handful of
wallet addresses. To identify such closed loop trades, they employ Johnson
(1975)’s exhaustive algorithm to find elementary circuits in a directed graph.

While von Wachter et al. (2022) state to observe suspected wash trading in
most collections, some collections have been prone to a higher degree of
suspect activity. Figure 8 shows the ratio of the number of unique wallets to
the total number of transactions within the data history. A low unique wallet-
to-transaction ratio would indicate a large number of transactions occurring
between only a few wallet addresses, which is indicative of suspected wash
trading. Meebits stands out as a collection with a comparably low unique
wallets to transaction ratio (Figure 8(d)). Despite LooksRare’s effort to
discourage wash trading by raising transaction fees and limiting the amount of
LOOKS token to be rewarded in a day, the data suggest a that many continue to
exploit the system to this day.

###  3.4 Market Volatility

(a) Aurory

(b) Autoglyphs

(c) Bored Ape Kennel Club

(d) Bored Ape Yacht Club

(e) Cryptopunks

(f) Degenerate Ape Academy

(g) Meebits

(h) Mutant Ape Yacht Club

Figure 9: Percentage of the Number of Transactions per Token.

Another challenge associated with the analysis of a NFT collections is its
extreme volatility, as has been pointed out by Dowling (2022b), Borri et al.
(2022), Kong and Lin (2022) and Mazur (2021). Unlike cryptocurrencies or
equity markets where a large number of historical data points exist at a
regular interval, the NFT market is highly illiquid with sales occurring
sporadically over time as seen in Figure 3. This may be one driver that causes
the transacted prices to be extremely volatile, as suggested by Kapoor et al.
(2022). Moreover, a large proportion of tokens are transacted only once after
the collection launch. Figure 9 shows the distribution of the number of
transactions of each token during the studied period and it also suggests that
only a small fraction of tokens are traded more than three times.

(a) Aurory

(b) Autoglyphs

(c) Bored Ape Kennel Club

(d) Bored Ape Yacht Club

(e) Cryptopunks

(f) Degenerate Ape Academy

(g) Meebits

(h) Mutant Ape Yacht Club

Figure 10: Average Daily Log Return by Collections.

Indeed, the volatility of all eight studied collections during the studied
period was extremely high (Figure 10). The average daily log returns at the
collection level were rather stationary except for Mutant Ape Yacht Club.
Average returns for Mutant Ape Yacht club were especially volatile for the
first few days but became stationary with relatively constant variance
throughout the studied period. 12-month realized volatility, the standard
deviation of the daily log return times the square root of the number of
trading days of 252, was computed for the eight collections based on the daily
average transacted price of tokens (2). Cryptopunks had the highest realized
volatility of 1,140% and Aurory had the lowest with 291%. For comparison, the
annualized volatility of S&P 500 was 14.95% as of March 31st of 2022. The
52-week high of volatility for crude oil, a commodity that is known to be
volatile, in the past year was around 102%, according to CBOE Crude Oil
Volatility Index (OVX). So far, studies have addressed this issue by either
averaging the data or aggregating the data to a weekly or monthly level. For
example, Kapoor et al. (2022) studied the relationship between the social
media mentions of a given NFT token and its price. In the analysis, the token
value was defined to be the average selling price of the asset over all its
historic sales. Dowling (2022b) studied the pricing of NFT of land in a
virtual world. Due to extreme volatility and the low number of daily
transactions at the daily level, data were aggregated at the weekly level and
were used for the analysis.

| Aurory | Autoglphs | Bored Ape Kennel Club | Bored Ape Yacht Club |
---|---|---|---|---|---
12-month Realized Volatility | 291% | 1,084% | 420% | 340% |
|  |  |  |  |
| Cryptopunks | Degenerate Ape Academy | Meebits | Mutant Ape Yacht Club |
12-month Realized Volatility | 1,140% | 464% | 1,013% | 896% |

Table 2: 12 Month Realized Volatility of eight profile picture collections

##  4 Conclusions and Outlook

The emergence of NFTs initially spurred interest from blockchain enthusiasts
and individual collectors. In recent years, though, as a growing number of NFT
collections have become highly valued, the NFT ecosystem has financialized and
NFTs can now be regarded as a novel financial asset class. Financial decisions
rely on data and analysis thereof. This paper has described characteristics
common to transactional data for many NFT collections and has illustrated that
such data entail unique analytical challenges. Owing to these and other
challenges, several authors have identified NFTs as an emerging field of
research. Beyond doubt, the existing body of academic literature is still
limited and many research questions regarding NFTs remain unanswered. Baals et
al. (2022) recognize this fact and propose a research agenda forward with
focus on the financial economics for NFTs. While research on financial
economics for NFTs will encompass advanced data analytics, future research NFT
transactions may offer a set of opportunities beyond that field as well,
particularly in data science.

In fact, each of the challenges described in Section 3 may lead to novel data
science methods and/or applications being proposed. For instance, to account
for the high price differentiation according to certain, but not all, traits,
one can imagine to construct a machine learning model that uses information
from both transactions and token metadata, possibly even from the images
themselves, to obtain accurate token specific price predictions. Section 3.3
hinted to ways to detect wash trading at the transaction and collection level.
These ideas could be used to construct a robust detection algorithm. Moreover,
NFT data that involve wash trades can be a fertile ground for classification
and anomaly detection models, such as robust statistics. At the height of the
Meebits wash trade wave caused by LooksRare’s reward system, more than half of
transactions were suspect wash trades. A fraction of over 50% of anomalies
generally poses challenges to be detected by robust statistics as the latter
will break down, which provides an interesting avenue for research. Finally,
the volatility challenges highlighted in Section 3.4 require deeper
investigation, should one want to apply downstream financial analytics, such
as risk/reward quantification or NFT portfolio optimization.

As presented, NFT data provide a fertile ground for future data science
research with its own unique challenges. We hope that the material presented
here can be a seed for further developments in the field and that the data
shared here prove to be useful in bringing about such developments.

## References

  * Ante [2022] Lennart Ante.  The non-fungible token (nft) market and its relationship with bitcoin and ethereum.  _FinTech_ , 1(3):216–224, 2022.  ISSN 2674-1032.  doi: 10.3390/fintech1030017.  URL https://www.mdpi.com/2674-1032/1/3/17.
  * Baals et al. [2022] L.J. Baals, B. Lucey, C. Long, and S. Vigne.  Towards a research agenda on the financial economics of nft’s.  _Available at SSRN: https://ssrn.com/abstract=4070710_ , 2022.  doi: http://dx.doi.org/10.2139/ssrn.4070710.
  * Bao and Roubaud [2022] Hong Bao and David Roubaud.  Recent development in fintech: Non-fungible token.  _FinTech_ , 1(1):44–46, 2022.  ISSN 2674-1032.  doi: 10.3390/fintech1010003.  URL https://www.mdpi.com/2674-1032/1/1/3.
  * Borri et al. [2022] Nicola Borri, Yukun Liu, and Aleh Tsyvinski.  The economics of non-fungible tokens.  _SSRN Electron. J._ , 2022.
  * Buterin [2014] V. Buterin.  Ethereum: A next-generation smart contract and decentralized application platform.  _Available at Ethereum.org: Bitcoin.pdf_ , 2014.  URL https://ethereum.org/en/whitepaper/.
  * Coffman [2022] C. Coffman.  Wash trading: Who, what, why, and what should we do about it?  _Available at blog.cryptoslam.io_ , 2022.  URL https://blog.cryptoslam.io/wash-trading-who-what-why-and-what-should-we-do-about-it/.
  * Dowling [2022a] Michael Dowling.  Is non-fungible token pricing driven by cryptocurrencies?  _Finance Research Letters_ , 44:102097, 2022a.  ISSN 1544-6123.  doi: https://doi.org/10.1016/j.frl.2021.102097.  URL https://www.sciencedirect.com/science/article/pii/S1544612321001781.
  * Dowling [2022b] Michael Dowling.  Fertile land: Pricing non-fungible tokens.  _Finance Research Letters_ , 44:102096, 2022b.  ISSN 1544-6123.  doi: https://doi.org/10.1016/j.frl.2021.102096.  URL https://www.sciencedirect.com/science/article/pii/S154461232100177X.
  * Howcroft [2022] Elizabeth Howcroft.  Unreal demand? irregular sales worth billions fire up wild nft market.  _Reuters_ , Feb 2022.
  * Johnson [1975] D.B. Johnson.  Finding all the elementary circuits of a directed graph.  _SIAM Journal of Computation_ , 4:77–84, 1975.
  * Kapoor et al. [2022] Arnav Kapoor, Dipanwita Guhathakurta, Mehul Mathur, Rupanshu Yadav, Manish Gupta, and Ponnurangam Kumaraguru.  Tweetboost: Influence of social media on nft valuation, 2022.  URL https://arxiv.org/abs/2201.08373.
  * Ko et al. [2022] Hyungjin Ko, Bumho Son, Yunyoung Lee, Huisu Jang, and Jaewook Lee.  The economic value of nft: Evidence from a portfolio analysis using mean–variance framework.  _Finance Research Letters_ , 47:102784, 2022.  ISSN 1544-6123.  doi: https://doi.org/10.1016/j.frl.2022.102784.  URL https://www.sciencedirect.com/science/article/pii/S1544612322000976.
  * Kong and Lin [2022] D.R. Kong and T.-C. Lin.  Alternative investments in the fintech era: The risk and return of non-fungible token (nft).  _Available at SSRN: https://ssrn.com/abstract=3914085_ , 2022.  doi: http://dx.doi.org/10.2139/ssrn.3914085.
  * Mazur [2021] Mieszko Mazur.  Non-fungible tokens (NFT). the analysis of risk and return.  _SSRN Electron. J._ , 2021.
  * Nakamoto [2008] S. Nakamoto.  Bitcoin: A peer-to-peer electronic cash system.  _Available at Bitcoin.org: Bitcoin.pdf_ , 2008.  URL https://bitcoin.org/bitcoin.pdf.
  * Oh et al. [2022] Sebeom Oh, Samuel Rosen, and Anthony Lee Zhang.  Investor experience matters: Evidence from generative art collections on the blockchain.  _SSRN Electronic Journal_ , 2022.  doi: 10.2139/ssrn.4042901.
  * Serneels et al. [2022] S. Serneels, J.B. Cho, and D.S. Matteson.  Data containing transaction history and visual traits of eight highly valued non-fungible token (nft) collections.  _Cornell University eCommons_ , 2022.  doi: https://doi.org/10.7298/6jcj-9461.  URL https://ecommons.cornell.edu/handle/1813/111404.
  * Tariq and Sifat [2022] S.A. Tariq and I. Sifat.  Suspicious trading in nonfungible tokens (nfts): Evidence from wash trading.  _Available at SSRN: https://ssrn.com/abstract=4097642_ , 2022.  doi: http://dx.doi.org/10.2139/ssrn.4097642.
  * Umar et al. [2022] Zaghum Umar, Afsheen Abrar, Adam Zaremba, Tamara Teplova, and Xuan Vinh Vo.  The return and volatility connectedness of nft segments and media coverage: Fresh evidence based on news about the covid-19 pandemic.  _Finance Research Letters_ , 49:103031, 2022.  ISSN 1544-6123.  doi: https://doi.org/10.1016/j.frl.2022.103031.  URL https://www.sciencedirect.com/science/article/pii/S1544612322002690.
  * von Wachter et al. [2022] V. von Wachter, J.R. Jensen, F. Regner, and O. Ross.  Nft wash trading: Quantifying suspicious behaviour in nft markets.  _Financial Cryptography and Data Security. FC 2022 International Workshops. Available at SSRN: https://ssrn.com/abstract=4037143_ , 2022.  doi: http://dx.doi.org/10.2139/ssrn.4037143.


# Hankel Determinant for a General Subclass of $m$-Fold Symmetric Bi-Univalent
Functions Defined by Ruscheweyh Operators

Pishtiwan Othman Sabir  Department of Mathematics, College of Science,
University of Sulaimani, Sulaymaniyah 46001, Kurdistan Region, Iraq Ravi P.
Agarwal  Department of Mathematics, Texas A& M University-Kingsville,
Kingsville, TX, 78363, USA Shabaz Jalil Mohammed Faeq  Department of
Mathematics, College of Science, University of Sulaimani, Sulaymaniyah 46001,
Kurdistan Region, Iraq Pshtiwan Othman Mohammed Corresponding author. Email:
<EMAIL_ADDRESS>Department of Mathematics, College of Education,
University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq Nejmeddine
Chorfi  Department of Mathematics, College of Science, King Saud University,
P.O. Box 2455, Riyadh 11451, Saudi Arabia Thabet Abdeljawad Corresponding
author. Email<EMAIL_ADDRESS>Department of Mathematics and Sciences,
Prince Sultan University, P.O. Box 66833, 11586 Riyadh, Saudi Arabia

###### Abstract

Making use of the Hankel determinant and the Ruscheweyh derivative, in this
work, we consider a general subclass of $m$-fold symmetric normalized bi-
univalent functions defined in the open unit disk. Moreover, we investigate
the bounds for the second Hankel determinant of this class and some
consequences of the results are presented. In addition, to demonstrate the
accuracy on some functions and conditions, most general programs are written
in Python V.3.8.8 (2021).

##  1 Introduction

Let $\mathcal{A}$ denote the class of the analytic functions $f$ in the open
unit disk $\mathbb{U}=\\{z\in\mathbb{C}:|z|<1\\}$, normalized by the
conditions $f(0)=f^{\prime}(0)-1=0$ of the Taylor-Maclaurian series expansion

$f(z)=z+\sum_{n=2}^{\infty}a_{k}z^{k}.$ (1.1)

Further, assume that $\mathcal{S}$ denotes the subclass of $\mathcal{A}$ which
contains all univalent functions in $\mathbb{U}$ satisfying (1.1) and
$\mathcal{P}$ represents the subclass of all functions $h(z)$ of the form

$h(z)=1+h_{1}z+h_{2}z^{2}+h_{3}z^{3}+\cdots$ (1.2)

which are analytic in the open unit disk $\mathbb{U}$ and
$\operatorname{Re}(h(z))>0,z\in\mathbb{U}$.

For a function $f\in\mathcal{A}$ defined by (1.1), the Ruscheweyh derivative
operator (see [23]) is defined by

$\mathcal{R}^{\gamma}f(z)=z+\sum_{k=2}^{\infty}\Omega(\gamma,k)a_{k}z^{k},$

where
$\delta\in\mathbb{N}_{0}=\\{0,1,2,\ldots\\}=\mathbb{N}\cup\\{0\\},z\in\mathbb{U}$,
and

$\Omega(\gamma,k)=\frac{\Gamma(\gamma+k)}{\Gamma(k)\Gamma(\gamma+1)}.$

The Koebe 1/4-theorem (see [12]) asserts that every univalent function
$f\in\mathcal{S}$ has an inverse $f^{-1}$ defined by

$f^{-1}(f(z))=z\quad(z\in\mathbb{U})\text{ and
}f\left(f^{-1}(w)\right)=w\quad\left(|w|<r_{0}(f),r_{0}(f)\geq\frac{1}{4}\right).$

The inverse function $g=f^{-1}$ has the form

$g(w)=f^{-1}(w)=w-a_{2}w^{2}+\left(2a_{2}^{2}-a_{3}\right)w^{3}-\left(5a_{2}^{3}-5a_{2}a_{3}+a_{4}\right)w^{4}+\cdots.$
(1.3)

A function $f\in\mathcal{A}$ is said to be bi-univalent if both $f$ and
$f^{-1}$ satisfy the univalent property. The class of bi-univalent functions
in $\mathbb{U}$ is denoted by $\Sigma$. Some examples of functions in the
class $\Sigma$ are given as follows:

$\frac{z}{1-z},\quad-\log(1-z)\quad\text{ and
}\quad\frac{1}{2}\log\left(\frac{1+z}{1-z}\right),$

with the corresponding inverse functions

$\frac{e^{w}-1}{e^{w}},\quad\frac{w}{1+w}\text{ and
}\quad\frac{e^{2w}-1}{e^{2w}+1},$

respectively.

Determination of the estimates for the Taylor-Maclaurin coefficients $a_{n}$
is a crucial problem in geometric function theory and provides knowledge about
the geometric characteristics of these functions. Lewin [17] investigated the
class $\Sigma$ of bi-univalent functions and showed that
$\left|a_{2}\right|<1.51$ for the functions belonging to the class $\Sigma$.
Brannan and Clunie [8] conjectured that $\left|a_{2}\right|\leq\sqrt{2}$.
Subsequently, Netanyahu [20] showed that max $\left|a_{2}\right|=\frac{4}{3}$
for $f\in$ $\Sigma$. Srivastava et al. [26] improved the investigation for
various subclasses of the bi-univalent function class $\Sigma$ and established
bounds on $\left|a_{2}\right|$ and $\left|a_{3}\right|$ in recent years. Many
recent works are devoted to studying the bi-univalent functions class $\Sigma$
and obtaining non-sharp bounds on the Taylor-Maclaurin coefficients
$\left|a_{2}\right|$ and $\left|a_{3}\right|$ (see, for example, [18, 1, 7,
29, 30]). However, coefficient estimates bound of
$\left|a_{n}\right|(n\in\\{4,5,6,\ldots\\})$ for a function $f\in\Sigma$
defined by (1.1) remains an open problem. In fact, there is no natural way to
get upper bound for coefficients greater than three. In exceptional cases,
there are some articles in which Faber polynomial techniques were used for
finding upper bounds for higher order coefficients (see, for example, [4, 6,
31]).

The Hankel determinant is a valuable tool in studying univalent functions
whose components are coefficients of functions in the subclasses of
$\mathcal{S}$. The Hankel determinants $H_{q}(n)$ $(n,q\in\mathbb{N})$ of the
function $f$ are defined by (see [21])

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

Note

$H_{2}(1)=\left|\begin{array}[]{ll}a_{1}&a_{2}\\\
a_{2}&a_{3}\end{array}\right|,$

and

$H_{2}(2)=\left|\begin{array}[]{ll}a_{2}&a_{3}\\\
a_{3}&a_{4}\end{array}\right|.$

Estimates for upper bounds of
$\left|H_{2}(1)\right|=\left|a_{3}-a_{2}^{2}\right|$ and
$\left|H_{2}(2)\right|=\left|a_{2}a_{4}-a_{3}^{2}\right|$ are called Fekete-
Szegö and second Hankel determinant problems, respectively. Additionally,
Fekete and Szegö [13] proposed the summarized functional $a_{3}-\mu
a_{2}^{2}$, in which $\mu$ is some real number. Lee et al. [16] presented a
concise overview of Hankel determinants for analytic univalent functions and
obtained bounds for $\mathrm{H}_{2}(2)$ for functions belonging to some
classes defined by subordination. The estimation of $|\mathrm{H}_{2}(2)|$ has
been the focus of recent Hankel determinant papers (see, for example, [5, 11,
22, 25, 32]).

For each function $f\in\mathcal{S}$, the function

$h(z)=\left(f\left(z^{m}\right)\right)^{\frac{1}{m}},\quad(z\in\mathbb{U},m\in\mathbb{N})$
(1.4)

is univalent and maps the unit disk into a region with $m$-fold symmetry. A
function $f$ is said to be $m$-fold symmetric (see [15]) and denoted by
$\mathcal{A}_{m}$, if it has the following normalized form:

$f(z)=z+\sum_{k=1}^{\infty}a_{mk+1}z^{mk+1},\quad(z\in\mathbb{U},m\in\mathbb{N}).$
(1.5)

We denote by $\mathcal{S}_{m}$ the class of $m$-fold symmetric univalent
functions in $\mathbb{U}$, which are normalized by the series expansion (1.5).
In fact, the functions in the class $\mathcal{S}$ are 1-fold symmetric. In
view of the work of Koepf [15] the $m$-fold symmetric function
$h\in\mathcal{P}$ is of the form

$h(z)=1+h_{m}z^{m}+h_{2m}z^{2m}+h_{3m}z^{3m}+\cdots.$ (1.6)

Analogous to the concept of $m$-fold symmetric univalent functions, Srivastava
et al. [27] defined the concept of $m$ fold symmetric bi-univalent functions
in a direct way. Each function $f\in\Sigma$ generates an $m$-fold symmetric
bi-univalent function for each $m\in\mathbb{N}$. The normalized form of $f$ is
given as (1.5) and the extension $g=f^{-1}$ is as follows:

$\displaystyle
g(w)=w-a_{m+1}w^{m+1}+\left[(m+1)a_{m+1}^{2}-a_{2m+1}\right]w^{2m+1}$ (1.7)
$\displaystyle-{\left[\frac{1}{2}(m+1)(3m+2)a_{m+1}^{3}-(3m+2)a_{m+1}a_{2m+1}+a_{3m+1}\right]w^{3m+1}+\cdots}.$

We denote by $\Sigma_{\mathrm{m}}$ the class of $m$-fold symmetric bi-
univalent functions in $\mathbb{U}$. For $m=1$, the series (1.7) coincides
with the series (1.3) of the class $\Sigma$. Some examples of $m$-fold
symmetric bi-univalent functions are given as follows:

$\left[\frac{z^{m}}{1-z^{m}}\right]^{\frac{1}{m}},\quad\left[-\log\left(1-z^{m}\right)\right]^{\frac{1}{m}}\quad\text{
and
}\quad\left[\frac{1}{2}\log\left(\frac{1+z^{m}}{1-z^{m}}\right)\right]^{\frac{1}{m}},$

with the corresponding inverse functions

$\left(\frac{w^{m}}{1+w^{m}}\right)^{\frac{1}{m}},\quad\left(\frac{e^{w^{m}}-1}{e^{w^{m}}}\right)^{\frac{1}{m}}\quad\text{
and }\quad\left(\frac{e^{2w^{m}}-1}{e^{2w^{m}}+1}\right)^{\frac{1}{m}},$

respectively.

Recently, some authors studied the $m$-fold symmetric bi-univalent function
class $\Sigma_{\mathrm{m}}$ (see, for example, [9, 19, 28, 33]) and obtained
non-sharp bound estimates on the first two Taylor-Maclaurin coefficients
$\left|a_{m+1}\right|$ and $\left|a_{2m+1}\right|$. In this respect, Altinkaya
and Yalçin [3] obtained non-sharp estimates on the second Hankel determinant
for the subclass $H_{\Sigma_{m}}(\beta)$ of the $m$-fold symmetric bi-
univalent function class $\Sigma_{\mathrm{m}}$.

For a function $f\in\mathcal{A}_{m}$ defined by (1.5), analogous to the
Ruscheweyh derivative
$\mathcal{R}^{\gamma}:\mathcal{A}\rightarrow\mathcal{A}$, the $m$-fold
Ruscheweyh derivative
$\mathcal{R}^{\gamma}:\mathcal{A}_{m}\rightarrow\mathcal{A}_{m}$ is defined as
follows (see [24]):

$\mathcal{R}^{\gamma}f(z)=z+\sum_{k=1}^{\infty}\frac{\Gamma(\gamma+k+1)}{\Gamma(k+1)\Gamma(\gamma+1)}a_{mk+1}z^{mk+1}\quad\left(\gamma\in\mathbb{N}_{0},m\in\mathbb{N},z\in\mathbb{U}\right).$

Considering the significant role of the Hankel determinant in recent years,
the object of this paper, is to study estimates for $\left|H_{2}(2)\right|$ of
a general subclass of $m$-fold symmetric bi-univalent functions in
$\mathbb{U}$ by applying the $m$-fold Ruscheweyh derivative operator and to
obtain upper bounds on $\left|a_{m+1}a_{3m+1}-a_{2m+1}^{2}\right|$ for
functions in the subclass $\Xi_{\Sigma_{m}}(\lambda,\gamma;\beta)$.

In order to derive our main results, we need to the following lemmas that will
be useful in proving the basic theorem of section 2.

######  Lemma 1.1.

[12] If the function $h\in\mathcal{P}$ is given by the series (1.2), then

$\left|h_{k}\right|\leq 2\quad(k\in\mathbb{N}),$ (1.8)

and

$\left|h_{2}-\frac{h_{1}^{2}}{2}\right|\leq
2-\frac{\left|h_{2}\right|^{2}}{2}.$ (1.9)

######  Lemma 1.2.

[14] If the function $h\in\mathcal{P}$ is given by the series (1.2), then

$2h_{2}=h_{1}^{2}+x\left(4-h_{1}^{2}\right),$ (1.10)

and

$4h_{3}=h_{1}^{3}+2\left(4-h_{1}^{2}\right)h_{1}x-h_{1}\left(4-h_{1}^{2}\right)x^{2}+2\left(4-h_{1}^{2}\right)\left(1-|x|^{2}\right)z,$
(1.11)

for some $x,z$ with $|x|\leq 1$ and $|z|\leq 1$.

##  2 The Main Result and Consequences

Our main result in this section, is to study estimates for the second Hankel
determinant of the subclass $\Xi_{\Sigma_{m}}(\lambda,\gamma;\beta)$ of
$m$-fold symmetric bi-univalent functions in $\mathbb{U}$, and we show that
our results are an improvement on the existing coefficient estimates.

######  Definition 2.1.

A function $f\in\Sigma_{m}$ given by (1.5) is said to be in the class

$\Xi_{\Sigma_{m}}(\lambda,\gamma;\beta)\;(\lambda\geq 1,$
$\gamma\in\mathbb{N}_{0},0\leq\beta<1$ and $m\in$ $\mathbb{N}$) if it
satisfies the conditions

$\operatorname{Re}\left\\{(1-\lambda)\frac{\mathcal{R}^{\gamma}f(z)}{z}+\lambda\left(\mathcal{R}^{\gamma}f(z)\right)^{\prime}\right\\}>\beta,$
(2.1)

and

$\operatorname{Re}\left\\{(1-\lambda)\frac{\mathcal{R}^{\gamma}f(w)}{w}+\lambda\left(\mathcal{R}^{\gamma}f(w)\right)^{\prime}\right\\}>\beta,$
(2.2)

where $z,w\in\mathbb{U}$ and the function $g=f^{-1}$ is given by (1.7).

######  Theorem 2.1.

Let $f\in\Xi_{\Sigma_{m}}(\lambda,\gamma;\beta)$ be given by (1.5). Then

$\left|a_{m+1}a_{3m+1}-a_{2m+1}^{2}\right|\leq\left\\{\begin{array}[]{c}\frac{4(1-\beta)^{2}}{(\gamma+1)^{2}(m\lambda+1)}\left[\frac{(m+1)^{2}(1-\beta)^{2}}{(\gamma+1)^{2}(m\lambda+1)^{3}}+\frac{6}{(\gamma+2)(\gamma+3)(3m\lambda+1)}\right],\quad\qquad\beta\in[0,\tau]\\\
\\\
\frac{4(1-\beta)^{2}}{(\gamma+1)^{2}(\gamma+2)^{2}(2m\lambda+1)^{2}}\left[4-\frac{\left[\omega_{2}(1-\beta)+9\omega_{3}-4\omega_{4}\right]^{2}}{\omega_{4}\left[\omega_{1}(1-\beta)^{2}-2\omega_{2}(1-\beta)-12\omega_{3}+4\omega_{4}\right]}\right],\beta\in[\tau,1)\end{array}\right.$

where

$\displaystyle\omega_{1}:=(m+1)^{2}(\gamma+2)^{2}(\gamma+3)(2m\lambda+1)^{2}(3m\lambda+1),$
(2.3)
$\displaystyle\omega_{2}:=m(\gamma+1)(\gamma+2)(\gamma+3)(m\lambda+1)^{2}(2m\lambda+1)(3m\lambda+1),$
(2.4)
$\displaystyle\omega_{3}:=(\gamma+1)^{2}(\gamma+2)(m\lambda+1)^{3}(2m\lambda+1)^{2},$
(2.5)
$\displaystyle\omega_{4}:=(\gamma+1)^{2}(\gamma+3)(m\lambda+1)^{4}(3m\lambda+1),$
(2.6)

and

$\qquad\qquad\qquad\qquad\quad\tau:=1-\frac{\omega_{2}+\sqrt{\omega_{2}^{2}+12\omega_{1}\omega_{3}}}{2\omega_{1}}.$

###### Proof.

It follows from (2.1) and (2.2) that there exist $p$ and $q$ in the class
$\mathcal{P}$ such that

$(1-\lambda)\frac{\mathcal{R}^{\gamma}f(z)}{z}+\lambda\left(\mathcal{R}^{\gamma}f(z)\right)^{\prime}=\beta+(1-\beta)p(z),$
(2.7)

and

$(1-\lambda)\frac{\mathcal{R}^{\gamma}f(w)}{w}+\lambda\left(\mathcal{R}^{\gamma}f(w)\right)^{\prime}=\beta+(1-\beta)q(z)$
(2.8)

where $p$ and $q$ are given by the series (1.6).

We also find that

$\displaystyle(1-\lambda)$
$\displaystyle\frac{\mathcal{R}^{\gamma}f(z)}{z}+\lambda\left(\mathcal{R}^{\gamma}f(z)\right)^{\prime}$
(2.9)
$\displaystyle=1+(\gamma+1)(m\lambda+1)a_{m+1}z^{m}+\frac{1}{2}(\gamma+1)(\gamma+2)(2\lambda
m+1)a_{2m+1}z^{2m}$
$\displaystyle+\frac{1}{6}(\gamma+1)(\gamma+2)(\gamma+3)(3\lambda
m+1)a_{3m+1}z^{3m}+\cdots,$

and

$\displaystyle(1-\lambda)\frac{\mathcal{R}^{\gamma}g(w)}{w}+\lambda\left(\mathcal{R}^{\gamma}g(w)\right)^{\prime}$
(2.10)
$\displaystyle=1-(\gamma+1)(m\lambda+1)a_{m+1}w^{m}+\frac{1}{2}(\gamma+1)(\gamma+2)(2m\lambda+1)\times$
$\displaystyle\left[(m+1)a_{m+1}^{2}-a_{2m+1}\right]w^{2m}-\frac{1}{6}(\gamma+1)(\gamma+2)(\gamma+3)(3m\lambda+1)\times$
$\displaystyle\left[\frac{1}{2}(m+1)(3m+2)a_{m+1}^{3}-(3m+2)a_{m+1}a_{2m+1}+a_{3m+1}\right]w^{3m}+\cdots.$

Equating coefficients in (2.7) and (2.8) we have

$\displaystyle(\gamma+1)(m\lambda+1)a_{m+1}=(1-\beta)p_{m},$ (2.11)

$\displaystyle\frac{1}{2}(\gamma+1)(\gamma+2)(2\lambda
m+1)a_{2m+1}=(1-\beta)p_{2m},$ (2.12)

$\displaystyle\frac{1}{6}(\gamma+1)(\gamma+2)(\gamma+3)(3m\lambda+1)a_{3m+1}=(1-\beta)p_{3m},$
(2.13)

and

$\displaystyle-(\gamma+1)(m\lambda+1)a_{m+1}=(1-\beta)q_{m},$ (2.14)

$\displaystyle\frac{1}{2}(\gamma+1)(\gamma+2)(2m\lambda+1)\left[(m+1)a_{m+1}^{2}-a_{2m+1}\right]=(1-\beta)q_{2m},$
(2.15)

$\displaystyle-\frac{1}{6}(\gamma+1)(\gamma+2)(\gamma+3)(3m\lambda+1)\times$
(2.16)
$\displaystyle\left[\frac{1}{2}(m+1)(3m+2)a_{m+1}^{3}-(3m+2)a_{m+1}a_{2m+1}+a_{3m+1}\right]=(1-\beta)q_{3m}.$

From (2.11) and (2.14), we get

$p_{m}=-q_{m},$ (2.17)

and

$a_{m+1}=\frac{1-\beta}{(\gamma+1)(m\lambda+1)}p_{m}.$ (2.18)

Now, from (2.12) and (2.15) and (2.18), we obtain

$a_{2m+1}=\frac{(m+1)(1-\beta)^{2}}{2(\gamma+1)^{2}(m\lambda+1)^{2}}p_{m}^{2}+\frac{(1-\beta)}{(\gamma+1)(\gamma+2)(2m\lambda+1)}\left(p_{2m}-q_{2m}\right).$
(2.19)

Also, from (2.13), (2.16), (2.18) and (2.19), we find that

$\begin{gathered}a_{3m+1}=\frac{(3m+2)(1-\beta)^{2}}{2(\gamma+1)^{2}(\gamma+2)(m\lambda+1)(2m\lambda+1)}p_{m}\left(p_{2m}-q_{2m}\right)\\\
+\frac{3(1-\beta)}{(\gamma+1)(\gamma+2)(\gamma+3)(3m\lambda+1)}\left(p_{3m}\right.\left.-q_{3m}\right).\end{gathered}$
(2.20)

Then, from (2.18), (2.19) and (2.20) we have that

$\displaystyle
a_{m+1}a_{3m+1}-a_{2m+1}^{2}=-\frac{(m+1)^{2}(1-\beta)^{4}}{4(\gamma+1)^{4}(m\lambda+1)^{4}}p_{m}^{4}$
(2.21)
$\displaystyle+\frac{m(1-\beta)^{3}}{2(\gamma+1)^{3}(\gamma+2)(m\lambda+1)^{2}(2m\lambda+1)}p_{m}^{2}\left(p_{2m}-q_{2m}\right)$
$\displaystyle+\frac{3(1-\beta)^{2}}{(\gamma+1)^{2}(\gamma+2)(\gamma+3)(m\lambda+1)(3m\lambda+1)}p_{m}\left(p_{3m}-q_{3m}\right)$
$\displaystyle-\frac{(1-\beta)^{2}}{(\gamma+1)^{2}(\gamma+2)^{2}(2m\lambda+1)^{2}}\left(p_{2m}-q_{2m}\right)^{2}.$

According to Lemma 1.2 and (2.17), we can write

$p_{2m}-q_{2m}=\frac{4-p_{m}^{2}}{2}(x-y),$ (2.22)

and

$\displaystyle p_{3m}-q_{3m}=$
$\displaystyle\frac{p_{m}^{3}}{2}+\frac{p_{m}\left(4-p_{m}^{2}\right)}{2}(x+y)-\frac{p_{m}\left(4-p_{m}^{2}\right)}{4}\left(x^{2}+y^{2}\right)$
(2.23)
$\displaystyle+\frac{4-p_{m}^{2}}{2}\left[\left(1-|x|^{2}\right)z-\left(1-|y|^{2}\right)w\right],$

$p_{2m}+q_{2m}=p_{m}^{2}+\frac{4-p_{m}^{2}}{2}(x+y),$ (2.24)

for some $x,y,z$ and $w$ with $|x|\leq 1,|y|\leq 1,|z|\leq 1$ and $|w|\leq 1$.
Using (2.22) and (2.23) in (2.21) we obtain

$\displaystyle\left|a_{m+1}a_{3m+1}-a_{2m+1}^{2}\right|$
$\displaystyle=\mid-\frac{(m+1)^{2}(1-\beta)^{4}}{4(\gamma+1)^{4}(m\lambda+1)^{4}}p_{m}^{4}+\frac{m(1-\beta)^{3}}{4(\gamma+1)^{3}(\gamma+2)(m\lambda+1)^{2}(2m\lambda+1)}p_{m}^{2}\left(4-p_{m}^{2}\right)(x-y)$
$\displaystyle+\frac{3(1-\beta)^{2}}{(\gamma+1)^{2}(\gamma+2)(\gamma+3)(m\lambda+1)(3m\lambda+1)}p_{m}\left[\frac{p_{m}^{3}}{2}+\frac{p_{m}\left(4-p_{m}^{2}\right)}{2}(x+y)\right.$
$\displaystyle-\frac{p_{m}\left(4-p_{m}^{2}\right)}{4}\left(x^{2}+y^{2}\right)+\frac{4-p_{m}^{2}}{2}\left[\left(1-|x|^{2}\right)z-\left(1-|y|^{2}\right)w\right]$
$\displaystyle-\frac{(1-\beta)^{2}}{4(\gamma+1)^{2}(\gamma+2)^{2}(2m\lambda+1)^{2}}\left(4-p_{m}^{2}\right)^{2}(x-y)^{2}\mid$
$\displaystyle\leq\frac{(m+1)^{2}(1-\beta)^{4}}{4(\gamma+1)^{4}(m\lambda+1)^{4}}p_{m}^{4}+\frac{3(1-\beta)^{2}}{2(\gamma+1)^{2}(\gamma+2)(\gamma+3)(m\lambda+1)(3m\lambda+1)}p_{m}^{4}$
$\displaystyle+\frac{3(1-\beta)^{2}}{(\gamma+1)^{2}(\gamma+2)(\gamma+3)(m\lambda+1)(3m\lambda+1)}p_{m}\left(4-p_{m}^{2}\right)$
$\displaystyle+\left[\frac{m(1-\beta)^{3}}{4(\gamma+1)^{3}(\gamma+2)(m\lambda+1)^{2}(2m\lambda+1)}p_{m}^{2}\left(4-p_{m}^{2}\right)\right.$
$\displaystyle\left.+\frac{3(1-\beta)^{2}}{2(\gamma+1)^{2}(\gamma+2)(\gamma+3)(m\lambda+1)(3m\lambda+1)}p_{m}^{2}\left(4-p_{m}^{2}\right)\right](|x|+|y|)$
$\displaystyle+\left[\frac{3(1-\beta)^{2}}{4(\gamma+1)^{2}(\gamma+2)(\gamma+3)(m\lambda+1)(3m\lambda+1)}p_{m}^{2}\left(4-p_{m}^{2}\right)\right.$
$\displaystyle\left.-\frac{3(1-\beta)^{2}}{2(\gamma+1)^{2}(\gamma+2)(\gamma+3)(m\lambda+1)(3m\lambda+1)}p_{m}\left(4-p_{m}^{2}\right)\right]\left(|x|^{2}+|y|^{2}\right)$
$\displaystyle+\frac{(1-\beta)^{2}}{4(\gamma+1)^{2}(\gamma+2)^{2}(2m\lambda+1)^{2}}\left(4-p_{m}^{2}\right)^{2}(|x|+|y|)^{2}.$

Since $p$ in the class $\mathcal{P}$, we have (Lemma 1.1)
$\left|p_{m}\right|\leq 2$. Letting $p_{m}=\rho$, we may assume without loss
of generality that $\rho\in[0,2]$. Thus, for $\mu_{1}=|x|\leq 1$ and
$\mu_{2}=|y|\leq 1$, we get

$\left|a_{m+1}a_{3m+1}-a_{2m+1}^{2}\right|\leq
F_{1}+F_{2}\left(\mu_{1}+\mu_{2}\right)+F_{3}\left(\mu_{1}^{2}+\mu_{2}^{2}\right)+F_{4}\left(\mu_{1}+\mu_{2}\right)^{2},$

where

$\displaystyle
F_{1}=F_{1}(\rho)=\frac{(m+1)^{2}(1-\beta)^{4}\rho^{4}}{4(\gamma+1)^{4}(m\lambda+1)^{4}}+\frac{3(1-\beta)^{2}\rho^{4}}{2(\gamma+1)^{2}(\gamma+2)(\gamma+3)(m\lambda+1)(3m\lambda+1)}$
$\displaystyle\qquad\qquad\qquad\quad+\frac{3(1-\beta)^{2}\rho\left(4-\rho^{2}\right)}{(\gamma+1)^{2}(\gamma+2)(\gamma+3)(m\lambda+1)(3m\lambda+1)}\geq
0,$ $\displaystyle
F_{2}=F_{2}(\rho)=\frac{m(1-\beta)^{3}\rho^{2}\left(4-\rho^{2}\right)}{4(\gamma+1)^{3}(\gamma+2)(m\lambda+1)^{2}(2m\lambda+1)}$
$\displaystyle\qquad\qquad\qquad+\frac{3(1-\beta)^{2}\rho^{2}\left(4-\rho^{2}\right)}{2(\gamma+1)^{2}(\gamma+2)(\gamma+3)(m\lambda+1)(3m\lambda+1)}\geq
0,$ $\displaystyle
F_{3}=F_{3}(\rho)=\frac{3(1-\beta)^{2}\rho^{2}\left(4-\rho^{2}\right)}{4(\gamma+1)^{2}(\gamma+2)(\gamma+3)(m\lambda+1)(3m\lambda+1)}$
$\displaystyle\qquad\qquad\qquad-\frac{3(1-\beta)^{2}\rho\left(4-\rho^{2}\right)}{2(\gamma+1)^{2}(\gamma+2)(\gamma+3)(m\lambda+1)(3m\lambda+1)}\leq
0,$ $\displaystyle
F_{4}=F_{4}(\rho)=\frac{(1-\beta)^{2}\left(4-\rho^{2}\right)^{2}}{4(\gamma+1)^{2}(\gamma+2)^{2}(2m\lambda+1)^{2}}\geq
0.$

(a) $F_{1},F_{2},F_{3}$ and $F_{4}$ for $\beta=0$.

(b) $F_{1},F_{2},F_{3}$ and $F_{4}$ for $\beta=0.1$.

(c) $F_{1},F_{2},F_{3}$ and $F_{4}$ for $\beta=0.2$.

(d) $F_{1},F_{2},F_{3}$ and $F_{4}$ for $\beta=0.9$.

Figure 1: Graph of $F_{1},F_{2},F_{3}$ and $F_{4}$ for $\gamma=0$ and
$\lambda=m=1$.

Now, we need to maximize

$F\left(\mu_{1},\mu_{2}\right)=F_{1}+F_{2}\left(\mu_{1}+\mu_{2}\right)+F_{3}\left(\mu_{1}^{2}+\mu_{2}^{2}\right)+F_{4}\left(\mu_{1}+\mu_{2}\right)^{2}$

in the closed square $\mathbb{S}=[0,1]\times[0,1]$ for $\rho\in[0,2]$. We
investigate the maximum of $F\left(\mu_{1},\mu_{2}\right)$ when
$\rho\in(0,2),\rho=0$ and $\rho=2$, keeping in mind the sign of

$F_{\mu_{1}\mu_{1}}F_{\mu_{2}\mu_{2}}-\left(F_{\mu_{1}\mu_{2}}\right)^{2}$

(according to the Second Derivative Test for functions of the two dependent
variables $\mu_{1}$ and $\mu_{2}$)

First, let $\rho\in(0,2)$. Since $F_{3}<0$ and $F_{3}+2F_{4}>0$ for
$\rho\in(0,2)$, we see that

$F_{\mu_{1}\mu_{1}}F_{\mu_{2}\mu_{2}}-\left(F_{\mu_{1}\mu_{2}}\right)^{2}=4F_{3}\,(F_{3}+2F_{4})<0.$

Figure 2: Graph of $F_{3}+2F_{4}$ for $\gamma=0$ and $\lambda=m=1$.

Thus, the function $F$ cannot have a local maximum in the interior of the
square $\mathbb{S}$. Now, we investigate the maximum of $F$ on the boundary of
the square $\mathbb{S}$.

Case 1. For $\mu_{1}=0$ and $\mu_{2}\in[0,1]$ (a similar argument can be
applied for $\mu_{2}=0$ and $\mu_{1}\in[0,1]$, so we omit the details in that
case), we obtain

$F\left(0,\mu_{2}\right)=G\left(\mu_{2}\right)\equiv
F_{1}+F_{2}\mu_{2}+\left(F_{3}+F_{4}\right)\mu_{2}^{2}.$

Subcase 1. Let $F_{3}+F_{4}\geq 0$. In this case, for $0<\mu_{2}<1$ we have
that

$G^{\prime}\left(\mu_{2}\right)=F_{2}+2\left(F_{3}+F_{4}\right)\mu_{2}>0,$

that is, $G\left(\mu_{2}\right)$ is an increasing function. Hence the maximum
of $G\left(\mu_{2}\right)$ occurs at $\mu_{2}=1$ and

$\max\left\\{F\left(0,\mu_{2}\right):\,\mu_{2}\in[0,1]\right\\}=\max\left\\{G\left(\mu_{2}\right):\,\mu_{2}\in[0,1]\right\\}=G(1)=F_{1}+F_{2}+F_{3}+F_{4}.$

Subcase 2. Let $F_{3}+F_{4}<0$. Note

$F_{2}+2\left(F_{3}+F_{4}\right)\geq 0.$

For $\mu_{2}\in(0,1)$ since $F_{3}+F_{4}<0$ we have that

$F_{2}+2\left(F_{3}+F_{4}\right)\mu_{2}>F_{2}+2\left(F_{3}+F_{4}\right)\geq
0,$

so $G^{\prime}\left(\mu_{2}\right)>0$. Thus
$\max\left\\{G\left(\mu_{2}\right):\,\,\mu_{2}\in[0,1]\right\\}=G(1)$.

Figure 3: Graph of $F_{2}+2(F_{3}+F_{4})$ for $\gamma=0$ and $\lambda=m=1$.

Case 2. For $\mu_{1}=1$ and $\mu_{2}\in[0,1]$ (a similar argument can be
applied for $\mu_{2}=1$ and $\mu_{1}\in[0,1]$, so we omit the details in that
case), we obtain

$F\left(1,\mu_{2}\right)=H\left(\mu_{2}\right)=F_{1}+F_{2}+F_{3}+F_{4}+\left(F_{2}+2F_{4}\right)\mu_{2}+\left(F_{2}+F_{4}\right)\mu_{2}^{2}.$

Thus an argument like in Subcases 1 and 2 yields

$\max\left\\{F\left(1,\mu_{2}\right):\,\,\mu_{2}\in[0,1]\right\\}=\max\left\\{H\left(\mu_{2}\right):\,\,\mu_{2}\in[0,1]\right\\}=H(1)=F_{1}+2\left(F_{2}+F_{3}\right)+4F_{4}.$

Next let $\rho=2$. Now let $(\mu_{1},\mu_{2})\in\mathbb{S}$ and note

$F\left(\mu_{1},\mu_{2}\right)=\frac{4(\gamma+2)(\gamma+3)(3m\lambda+1)(m+1)^{2}(1-\beta)^{4}+24(\gamma+1)^{2}(m\lambda+1)^{3}(1-\beta)^{2}}{(\gamma+1)^{4}(\gamma+2)(\gamma+3)(m\lambda+1)^{4}(3m\lambda+1)}.$
(2.25)

Keeping in mind the constant value in (2.25) we have

$\max\left\\{F\left(\mu_{1},\mu_{2}\right):\,\,\mu_{1}\in[0,1],\,\mu_{2}\in[0,1]\right\\}=F(1,1)=F_{1}+2\left(F_{2}+F_{3}\right)+4F_{4}.$

Finally, let $\rho=0$. Now let $(\mu_{1},\mu_{2})\in\mathbb{S}$ and note

$F(\mu_{1},\mu_{2})=\frac{4(1-\beta)^{2}(\mu_{1}+\mu_{2})^{2}}{(\gamma+1)^{2}(\gamma+2)^{2}(2m\lambda+1)^{2}}.$

We see that, the maximum of $F\left(\mu_{1},\mu_{2}\right)$ occurs at
$\mu_{1}=\mu_{2}=1$ and

$\max\left\\{F\left(\mu_{1},\mu_{2}\right):\,\,\mu_{1}\in[0,1],\,\mu_{2}\in[0,1]\right\\}=F(1,1)=F_{1}+2\left(F_{2}+F_{3}\right)+4F_{4}.$

Combining all cases note since $F_{1}+2\left(F_{2}+F_{3}\right)+4F_{4}\geq 0$
when $\rho\in[0,2]$ we have

$\max\left\\{F\left(\mu_{1},\mu_{2}\right):\,\,\mu_{1}\in[0,1],\,\mu_{2}\in[0,1]\right\\}=F(1,1).$

Figure 4: Graph of $F_{1}+2\left(F_{2}+F_{3}\right)+4F_{4}$ for $\gamma=0$ and
$\lambda=m=1$.

Let $K:[0,2]\rightarrow\mathbb{R}$ be given by

$K(\rho)=F(1,1)=F_{1}+2\left(F_{2}+F_{3}\right)+4F_{4}.$ (2.26)

Substituting the values of $F_{1},F_{2},F_{3}$ and $F_{4}$ in the function $K$
defined by (2.26), yields

$\displaystyle
K(\rho)=\frac{(1-\beta)^{2}}{4(\gamma+1)^{4}(\gamma+2)^{2}(\gamma+3)(1+m\lambda)^{4}(1+2m\lambda)^{2}(1+3m\lambda)}\times$
$\displaystyle\left[\left[(m+1)^{2}(\gamma+2)^{2}(\gamma+3)(2m\lambda+1)^{2}(3m\lambda+1)(1-\beta)^{2}\right.\right.$
$\displaystyle-2m(\gamma+1)(\gamma+2)(\gamma+3)(m\lambda+1)^{2}(2m\lambda+1)(3m\lambda+1)(1-\beta)$
$\displaystyle\left.-12(\gamma+1)^{2}(\gamma+2)(m\lambda+1)^{3}(2m\lambda+1)^{2}+4(\gamma+1)^{2}(\gamma+3)(m\lambda+1)^{4}(3m\lambda+1)\right]\rho^{4}$
$\displaystyle+\left[8m(\gamma+1)(\gamma+2)(\gamma+3)(1+m\lambda)^{2}(1+2m\lambda)(1+3m\lambda)(1-\beta)\right.$
$\displaystyle\left.+72(\gamma+1)^{2}(\gamma+2)(m\lambda+1)^{3}(2m\lambda+1)^{2}-32(\gamma+1)^{2}(\gamma+3)(m\lambda+1)^{4}(3m\lambda+1)\right]\rho^{2}$
$\displaystyle\left.+64(\gamma+1)^{2}(\gamma+3)(1+m\lambda)^{4}(1+3m\lambda)\right].$

Now the maximum of $K(\rho)$ occurs either at $\rho=0$, $\rho\in(0,2)$ or
$\rho=2$. Suppose first the maximum of $K(\rho)$ occurs at some
$\rho\in(0,2)$. Note for any $\rho\in(0,2)$ we have

$\displaystyle
K^{\prime}(\rho)=\frac{(1-\beta)^{2}}{(\gamma+1)^{4}(\gamma+2)^{2}(\gamma+3)(1+m\lambda)^{4}(1+2m\lambda)^{2}(1+3m\lambda)}\times$
$\displaystyle\left[\left[(m+1)^{2}(\gamma+2)^{2}(\gamma+3)(2m\lambda+1)^{2}(3m\lambda+1)(1-\beta)^{2}\right.\right.$
$\displaystyle-2m(\gamma+1)(\gamma+2)(\gamma+3)(m\lambda+1)^{2}(2m\lambda+1)(3m\lambda+1)(1-\beta)$
$\displaystyle\left.-12(\gamma+1)^{2}(\gamma+2)(m\lambda+1)^{3}(2m\lambda+1)^{2}+4(\gamma+1)^{2}(\gamma+3)(m\lambda+1)^{4}(3m\lambda+1)\right]\rho^{3}$
$\displaystyle+\left[4m(\gamma+1)(\gamma+2)(\gamma+3)(\lambda
m+1)^{2}(1+2m\lambda)(1+3m\lambda)(1-\beta)\right.$
$\displaystyle\left.\left.+36(\gamma+1)^{2}(\gamma+2)(m\lambda+1)^{3}(2m\lambda+1)^{2}-16(\gamma+1)^{2}(\gamma+3)(m\lambda+1)^{4}(3m\lambda+1)\right]\rho\right].$

Next, we conclude the following results

Result 1. Let

$\omega_{1}(1-\beta)^{2}-2\omega_{2}(1-\beta)-12\omega_{3}+4\omega_{4}\geq 0,$

that is,

$\beta\in\left[0,1-\frac{\omega_{2}+\sqrt{\omega_{2}^{2}+\omega_{1}\left[12\omega_{3}-4\omega_{4}\right]}}{\omega_{1}}\right],$

where $\omega_{1},\omega_{2},\omega_{3}$ and $\omega_{4}$ are given by (2.3),
(2.4), (2.5) and (2.6), respectively.

Note, $K^{\prime}(\rho)>0$ for every $\rho\in(0,2)$. Thus,

$\displaystyle\max\\{K(\rho):0<\rho<2\\}$
$\displaystyle=K(2^{-})=\frac{4(1-\beta)^{2}}{(\gamma+1)^{2}(m\lambda+1)}\left[\frac{(m+1)^{2}(1-\beta)^{2}}{(\gamma+1)^{2}(m\lambda+1)^{3}}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\left.+\frac{6}{(\gamma+2)(\gamma+3)(3m\lambda+1)}\right].$

Result 2. Let

$\omega_{1}(1-\beta)^{2}-2\omega_{2}(1-\beta)-12\omega_{3}+4\omega_{4}<0,$

that is,

$\beta\in\left(1-\frac{\omega_{2}+\sqrt{\omega_{2}^{2}+\omega_{1}\left[12\omega_{3}-4\omega_{4}\right]}}{\omega_{1}},1\right).$

Then $K^{\prime}(\rho)=0$ gives the critical point $\rho_{1}=0$ or

$\rho_{2}=\sqrt{\frac{16\omega_{4}-4\omega_{2}(1-\beta)-36\omega_{3}}{\omega_{1}(1-\beta)^{2}-2\omega_{2}(1-\beta)-12\omega_{3}+4\omega_{4}}}.$

When

$\beta\in\left(1-\frac{\omega_{2}+\sqrt{\omega_{2}^{2}+\omega_{1}\left[12\omega_{3}-4\omega_{4}\right]}}{\omega_{1}},1-\frac{\omega_{2}+\sqrt{\omega_{2}^{2}+12\omega_{1}\omega_{3}}}{2\omega_{1}}\right],$

we observe that $\rho_{2}\geq 2$. Then the maximum value of $K(\rho)$ occurs
at $0^{+}$ or $2^{-}$. This is a contradiction since we assumed the maximum of
$K(\rho)$ occurs at some $\rho\in(0,2)$.

When

$\beta\in\left(1-\frac{\omega_{2}+\sqrt{\omega_{2}^{2}+12\omega_{1}\omega_{3}}}{2\omega_{1}},1\right),$

we observe that $\rho_{2}\in(0,2)$. Since
$K^{\prime\prime}\left(\rho_{2}\right)<0$, the maximum value of $K(\rho)$
occurs at $\rho=\rho_{2}$. Thus, we have

$\begin{split}\max\\{K(\rho):\rho\in(0,2)\\}=K\left(\rho_{2}\right)&=\frac{4(1-\beta)^{2}}{(\gamma+1)^{2}(\gamma+2)^{2}(2m\lambda+1)^{2}}\times\\\
&\left[4-\frac{\left[\omega_{2}(1-\beta)+9\omega_{3}-4\omega_{4}\right]^{2}}{\omega_{4}\left[\omega_{1}(1-\beta)^{2}-2\omega_{2}(1-\beta)-12\omega_{3}+4\omega_{4}\right]}\right].\end{split}$
(2.27)

Next suppose if $\beta\in[0,\tau]$ and the maximum of $K(\rho)$ occurs at
$\rho=2$. Then

$\displaystyle\max\\{K(\rho):\rho\in[0,2]\\}$
$\displaystyle=K(2)=\frac{4(1-\beta)^{2}}{(\gamma+1)^{2}(m\lambda+1)}\left[\frac{(m+1)^{2}(1-\beta)^{2}}{(\gamma+1)^{2}(m\lambda+1)^{3}}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\left.+\frac{6}{(\gamma+2)(\gamma+3)(3m\lambda+1)}\right].$

We only now need to note (see the idea in the second part of Result 2) if
$\beta\in(\tau,1)$ then the maximum of $K(\rho)$ cannot occur at $\rho=2$
since

$K(2)\leq\frac{4(1-\beta)^{2}}{(\gamma+1)^{2}(\gamma+2)^{2}(2m\lambda+1)^{2}}\times\left[4-\frac{\left[\omega_{2}(1-\beta)+9\omega_{3}-4\omega_{4}\right]^{2}}{\omega_{4}\left[\omega_{1}(1-\beta)^{2}-2\omega_{2}(1-\beta)-12\omega_{3}+4\omega_{4}\right]}\right]\,(=K(\rho_{2})).$

Finally let us consider $\beta\in[0,1)$ and the maximum of $K(\rho)$ occurring
at $\rho=0$. Then

$\max\\{K(\rho):\rho\in[0,2]\\}=K(0)=\frac{16(1-\beta)^{2}}{(\gamma+1)^{2}(\gamma+2)^{2}(2m\lambda+1)^{2}}.$

We note (see the ideas in the second part of Result 2) if $\beta\in(\tau,1)$
then the maximum of $K(\rho)$ cannot occur at $\rho=0$ since

$K(0)\leq\frac{4(1-\beta)^{2}}{(\gamma+1)^{2}(\gamma+2)^{2}(2m\lambda+1)^{2}}\times\left[4-\frac{\left[\omega_{2}(1-\beta)+9\omega_{3}-4\omega_{4}\right]^{2}}{\omega_{4}\left[\omega_{1}(1-\beta)^{2}-2\omega_{2}(1-\beta)-12\omega_{3}+4\omega_{4}\right]}\right]\,(=K(\rho_{2})).$

Finally note (see the ideas in Result 1 and the details in Result 2) if

$\beta\in\left[0,1-\frac{\omega_{2}+\sqrt{\omega_{2}^{2}+\omega_{1}\left[12\omega_{3}-4\omega_{4}\right]}}{\omega_{1}}\right],$

or

$\beta\in\left(1-\frac{\omega_{2}+\sqrt{\omega_{2}^{2}+\omega_{1}\left[12\omega_{3}-4\omega_{4}\right]}}{\omega_{1}},1\right),$

then the maximum of $K(\rho)$ cannot occur at $\rho=0$ since $K(0)\leq K(2)$.

This completes the proof. ∎

By setting $\lambda=1$ and $\gamma=0$ in Theorem 2.1, we obtain the following
consequence.

######  Corollary 2.1.

[3] Let $f\in\Xi_{\Sigma_{\mathrm{m}}}(\beta)\quad(0\leq\beta<1)$ be given by
(1.5). Then

$\left|a_{m+1}a_{3m+1}-a_{2m+1}^{2}\right|\leq\left\\{\begin{array}[]{cc}\frac{4(1-\beta)^{2}}{m+1}\left[\frac{(1-\beta)^{2}}{m+1}+\frac{1}{3m+1}\right],\qquad\qquad\qquad\quad\qquad\qquad\beta\in[0,v]\\\
\\\
\frac{(1-\beta)^{2}}{(2m+1)^{2}}\left[4-\frac{\left[m(1-\beta)\psi_{1}+3\psi_{2}-2\psi_{3}\right]^{2}}{\psi_{3}\left[(2m+1)(1-\beta)^{2}\psi_{1}-m(1-\beta)\psi_{1}+\psi_{3}-2\psi_{2}\right]}\right],\beta\in[v,1)\end{array}\right.$

where

$\displaystyle\psi_{1}:=(2m+1)(3m+1),$
$\displaystyle\psi_{2}:=(m+1)(2m+1)^{2},$
$\displaystyle\psi_{3}:=(m+1)^{2}(3m+1),$

and

$v:=\frac{(3m+1)(7m+4)-\sqrt{m^{2}(3m+1)^{2}+8\psi_{2}(3m+1)}}{4\psi_{1}}.$

By taking $m=1$ in Theorem 2.1, we conclude the following result.

######  Corollary 2.2.

Let $f\in\Xi_{\Sigma}(\lambda,\gamma;\beta)\quad\left(\lambda\geq
1,\gamma\in\mathbb{N}_{0},0\leq\beta<1\right)$ be given by (1.1). Then

$\left|a_{2}a_{4}-a_{3}^{2}\right|\leq\left\\{\begin{array}[]{c}\frac{8(1-\beta)^{2}}{(\gamma+1)^{2}(\lambda+1)}\left[\frac{2(1-\beta)^{2}}{(\gamma+1)^{2}(\lambda+1)^{3}}+\frac{3}{(\gamma+2)(\gamma+3)(3\lambda+1)}\right],\qquad\qquad\quad\beta\in[0,\xi]\\\
\\\
\frac{4(1-\beta)^{2}}{(\gamma+1)^{2}(\gamma+2)^{2}(2\lambda+1)^{2}}\left[4-\frac{\left[\vartheta_{2}(1-\beta)+9\vartheta_{3}-4\vartheta_{4}\right]^{2}}{\vartheta_{4}\left[4\vartheta_{1}(1-\beta)^{2}-2\vartheta_{2}(1-\beta)-12\vartheta_{3}+4\vartheta_{4}\right]}\right],\beta\in[\xi,1)\end{array}\right.$

where

$\displaystyle\vartheta_{1}:=(\gamma+2)^{2}(\gamma+3)(2\lambda+1)^{2}(3\lambda+1),$
$\displaystyle\vartheta_{2}:=(\gamma+1)(\gamma+2)(\gamma+3)(\lambda+1)^{2}(2\lambda+1)(3\lambda+1),$
$\displaystyle\vartheta_{3}:=(\gamma+1)^{2}(\gamma+2)(\lambda+1)^{3}(2\lambda+1)^{2},$
$\displaystyle\vartheta_{4}:=(\gamma+1)^{2}(\gamma+3)(\lambda+1)^{4}(3\lambda+1),$

and

$\quad\qquad\qquad\quad\qquad\qquad\quad\xi:=1-\frac{\vartheta_{2}+\sqrt{\vartheta_{2}^{2}+48\vartheta_{1}\vartheta_{3}}}{8\vartheta_{1}}.$

######  Remark 2.1.

Corollary 2.2 improves a result in Altinkaya and Yalçin [2, Theorem 3].

By putting $\gamma=0$ in Corollary 2.2, we obtain the following result.

######  Corollary 2.3.

Let $f\in\Xi_{\Sigma}(\lambda;\beta)\quad(\lambda\geq 1,0\leq\beta<1)$ be
given by (1.1). Then

$\left|a_{2}a_{4}-a_{3}^{2}\right|\leq\left\\{\begin{array}[]{cc}\frac{8(1-\beta)^{2}}{\lambda+1}\left[\frac{2(1-\beta)^{2}}{(\lambda+1)^{3}}+\frac{1}{2(3\lambda+1)}\right],\qquad\qquad\qquad\beta\in[0,\epsilon]\\\
\\\
\frac{2(1-\beta)^{2}}{(2\lambda+1)^{2}}\left[4-\frac{\left[\eta_{2}(1-\beta)+3\eta_{3}-2\eta_{4}\right]^{2}}{\eta_{4}\left[8\eta_{1}(1-\beta)^{2}-2\eta_{2}(1-\beta)-4\eta_{3}+2\eta_{4}\right]}\right],\beta\in[\epsilon,1)\end{array}\right.$

where

$\displaystyle\eta_{1}:=(2\lambda+1)^{2}(3\lambda+1),$
$\displaystyle\eta_{2}:=(\lambda+1)^{2}(2\lambda+1)(3\lambda+1),$
$\displaystyle\eta_{3}:=(\lambda+1)^{3}(2\lambda+1)^{2},$
$\displaystyle\eta_{4}:=(\lambda+1)^{4}(3\lambda+1),$

and

$\epsilon:=1-\frac{(\lambda+1)^{2}(3\lambda+1)+\sqrt{(\lambda+1)^{4}(3\lambda+1)^{2}+32(\lambda+1)^{3}(2\lambda+1)^{2}(3\lambda+1)}}{16(2\lambda+1)(3\lambda+1)}.$

######  Remark 2.2.

Corollary 2.3 improves a result in Altinkaya and Yalçin [2, Corollary 5].

By setting $\lambda=1$ in Corollary 2.3 , we get the following consequence.

######  Corollary 2.4.

[10] Let $f\in\Xi_{\Sigma}(\beta)\quad(0\leq\beta<1)$ be given by (1.1). Then

$\left|a_{2}a_{4}-a_{3}^{2}\right|\leq\left\\{\begin{array}[]{ll}(1-\beta)^{2}\left[(1-\beta)^{2}+\frac{1}{2}\right],&\beta\in\left[0,\frac{11-\sqrt{37}}{12}\right]\\\
\\\
\frac{(1-\beta)^{2}}{16}\left[\frac{60\beta^{2}-84\beta-25}{9\beta^{2}-15\beta+1}\right],&\beta\in\left[\frac{11-\sqrt{37}}{12},1\right)\end{array}.\right.$

######  Remark 2.3.

Corollary 2.4 recovers a result in Altinkaya and Yalçin [2, Corollary 4].

##  3 Concluding remarks

In this investigation, we consider a constructed subclass
$\Xi_{\Sigma_{m}}(\lambda,\gamma;\beta)$ of the class $\Sigma_{m}$ of $m$-fold
symmetric bi-univalent functions and several properties of the results are
discussed. Moreover, with a specialization of the parameters, some
consequences of the class are mentioned and they improve some existing upper
bounds for $\mathrm{H}_{2}(2)$ on certain subclasses of 1-fold symmetric bi-
univalent functions.

## Data Availability

Data sharing not applicable to this article as no datasets were generated or
analyzed during the current study.

## Funding

Not applicable.

## Author’s contributions

Conceptualization, P.O.S.; Data curation, S.J.M.F.; Formal analysis, N.C.;
Funding acquisition, N.C.; Investigation, P.O.S., R.P.A., P.O.M., N.C. and
T.A.; Methodology, P.O.S, R.P.A., N.C.and T.A.; Project administration,
R.P.A.; Software, P.O.S., S.J.M.F. and P.O.M.; Supervision, T.A.; Validation,
R.P.A. and P.O.M.; Visualization, P.O.S.; Writing – original draft, P.O.S.,
R.P.A. and P.O.M.; Writing – review & editing, P.O.S. and R.P.A. All of the
authors read and approved the final manuscript.

## Acknowledgements

Researchers Supporting Project number (RSP2023R153), King Saud University,
Riyadh, Saudi Arabia.

## Declarations

Ethical approval Not applicable.

Competing interests The authors declare no competing interests.

## References

  * [1] D. Alimohammadia, N. E. Cho, E. A. Adegani, Coefficient bounds for subclasses of analytic and bi-univalent functions, Filomat 34(14), 4709–4721, 2020.
  * [2] S. Altinkaya and S. Yalçin, ”Second Hankel determinant for a general subclass of bi-univalent functions associated with the Ruscheweyh derivative,” Acta Univ. Apulensis, vol. 43, pp. 199-208, 2014.
  * [3] S. Altinkaya and S. Yalçin, ”Hankel determinant for m-fold symmetric bi-univalent functions,” Creat. Math. Inform., vol. 28, no. 1, pp. 1-8, 2019.
  * [4] A. A. Amourah, ”Faber polynomial coefficient estimates for a class of analytic bi-univalent functions,” AIP Conference Proceedings, vol. 2096, no. 1, p. 020024, AIP Publishing LLC, 2019.
  * [5] W. G. Atshan, R. A. Al-Sajjad and S. Altinkaya, ”On the Hankel Determinant of m-fold Symmetric Bi-Univalent Functions Using a New Operator,” Gazi Univ. J. Sci., vol. 36, no. 1, pp. 349-360, 2023.
  * [6] A. A. Attiya, A. M. Albalahi and T. S. Hassan, ”Coefficient estimates for certain families of analytic functions associated with Faber polynomial,” J. Funct. Spaces, vol. 2023, pp. 1-6, 2023. Article ID 4741056.
  * [7] K. Bilal, H. M. Srivastava, M. Tahir, M. Darus, Q. Z. Ahmad, N. Khan, Applications of a certain q-integral operator to the subclasses of analytic and bi-univalent functions, AIMS Mathematics 6(1), 1024–1039, 2020.
  * [8] D. A. Brannan and J. G. Clunie, Aspects of Contemporary Complex Analysis (Proceedings of the NATO Advanced Study Institute held at the University of Durham, Durham; July 1-20, 1979), London-New York: Academic Press, 1980.
  * [9] D. Breaz and L.-I. Cotîrlă, ”The study of coefficient estimates and Fekete–Szegö inequalities for the new classes of m-fold symmetric bi-univalent functions defined using an operator,” J. Inequal. Appl., vol. 2023, no. 15, pp. 1-13, 2023.
  * [10] M. Çağlar, E. Deniz and H. M. Srivastava, ”Second Hankel determinant for certain subclasses of bi-univalent functions,” Turk. J. Math., vol. 41, pp. 694-706, 2017.
  * [11] M. Çağlar, H. Orhan and H. M. Srivastava, ”Coefficient bounds for q-starlike functions associated with q-Bernoulli numbers,” J. Appl. Anal. Comput., 2023. doi: 10.11948/20220566.
  * [12] P. L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften, Band 259, Springer Verlag, New York, Berlin, Heidelberg and Tokyo, 1983.
  * [13] M. Fekete and G. Szegö, ”Eine Bemerkung uber ungerade schlichte Funktionen,” J. London. Math. Soc., vol. 8, pp. 85-89, 1933.
  * [14] U. Grenander and G. Szegö, Toeplitz forms and their application, Berkely: California Monographs in Mathematical Sciences Univ. Calofornia Press, 1958.
  * [15] W. Koepf, ”Coefficients of symmetric functions of bounded boundary rotation,” Proc. Am. Math. Soc., vol. 105, pp. 324-329, 1989).
  * [16] S. K. Lee, V. Ravichandran and S. Supramaniam, ”Bounds for the second Hankel determinant of certain univalent functions,” J. Inequal. Appl., vol. 281, pp. 1-17, 2013.
  * [17] M. Lewin, ”On a coefficient problem for bi-univalent functions,” Proc. Am. Math. Soc., vol. 18, no. 1, pp. 63-68, 1967.
  * [18] V. Madaana, A. Kumar and V. Ravichandran, ”Estimates for initial coefficients of certain bi–univalent functions,” Filomat, vol. 35, no. 6, p. 1993–2009, 2021.
  * [19] A. Motamednezhad and S. Salehian, ”Coefficient estimates for a general subclass of m-fold symmetric bi-univalent functions,” Tbil. Math. J., vol. 12, no. 2, p. 163–176, 2019.
  * [20] E. Netanyahu, ”The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in $|z|<1$,” Arch. Ration. Mech. Anal., vol. 32, no. 2, pp. 100-112, 1969.
  * [21] J. W. Noonan and D. K. Thomas, ”On the second Hankel determinant of areally mean p-valent functions.,” Trans. Am. Math. Soc., vol. 223, pp. 337-346, 1976.
  * [22] H. Orhan, H. Arıkan and M. Çağlar, ”Second Hankel Determinant for Certain Subclasses of Bi-starlike Functions Defined by Differential Operators,” Sahand Commun. Math. Anal., vol. 20, no. 2, pp. 65-83, 2023.
  * [23] S. Ruscheweyh, ”New criteria for univalent functions,” Proc. Am. Math. Soc., vol. 49, pp. 109-115, 1975.
  * [24] P. O. Sabir, ”Coefficient estimate problems for certain subclasses of m-fold symmetric bi-univalent functions associated with the Ruscheweyh derivative,” arXiv:2304.11571, pp. 1-20, 2023.
  * [25] M. G. Shrigan, ”Second Hankel determinant for bi-univalent functions associated with q-differential operator,” J. Sib. Fed. Univ. Math. Phys., vol. 15, no. 5, p. 663–671, 2022.
  * [26] H. M. Srivastava, A. K. Mishra and P. Gochhayat, ”Certain subclasses of analytic and bi-univalent functions,” Appl. Math. Lett., vol. 23, pp. 1188-1192, 2010.
  * [27] H. M. Srivastava, S. Sivasubramanian and R. Sivakuma, ”Initial coeffcient bounds for a subclass of m-fold symmetric bi-univalent functions,” Tbil. Math. J., vol. 7, no. 2, pp. 1-10, 2014.
  * [28] H. M. Srivastava and A. K. Wanas, ”Initial Maclaurin coefficients bounds for new subclasses of analytic and m-fold symmetric bi-univalent functions defined by a linear combination,” Kyungpook Math. J, vol. 59, no. 3, pp. 493-503, 2019.
  * [29] H. M. Srivastava, S. Hussain, I. Ahmad, S. G. Ali Shah, Coefficient bounds for analytic and bi-univalent functions associated with some conic domains, J. Nonlinear Convex Anal. 23, 741–753, 2022.
  * [30] H. M. Srivastava, S. Gaboury and F. Ghanim, ”Coefficient estimates for a general subclass of analytic and bi-univalent functions of the Ma–Minda type,” RACSAM REV R ACAD A., vol. 112, p. 1157–1168, 2018.
  * [31] H. M. Srivastava, A. Motamednezhad, E. A. Adegani, Faber polynomial coefficient estimates for bi-univalent functions defined by using differential subordination and a certain fractional derivative operator, Mathematics 8(2) Article ID 172, 1–12, 2020.
  * [32] H. M. Srivastava, G. Murugusundaramoorthy, T. Bulboa$\check{c}$a, The second Hankel determinant for subclasses of Bi-univalent functions associated with a nephroid domain, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116(4) (2022), 1–21.
  * [33] H. Tang, H. M. Srivastava, S. Sivasubramanian and P. Gurusamy, ”The Fekete-Szego functional problems for some subclasses of m-fold symmetric bi-univalent functions,” J. Math. Inequalities, vol. 10, pp. 1063-1092, 2016.