# Some estimation about Tayler-Maclaurin coefficients of generalized
subclasses of bi-univalent functions

S.A. Saleh ${}^{\text{1}}$, Alaa H. El-Qadeem ${}^{\text{2}}$ and Mohamed A.
Mamon ${}^{\text{3}}$<EMAIL_ADDRESS>&<EMAIL_ADDRESS>&
<EMAIL_ADDRESS>
${}^{\text{1,3}}$Department of Mathematics, Faculty of Science, Tanta
University, Tanta 31527, Egypt
${}^{\text{2}}$Department of Mathematics, Faculty of Science, Zagazig
University, Zagazig 44519, Egypt

###### Abstract

Our objective in this paper is to introduce and investigate comprehensive-
constructed subclasses of normalized analytic and bi-univalent functions on
the unit open disc. Bounds for the second and third Tayler-Maclaurin
coefficients of functions belonging to this subclasses were investigated.
Furthermore, some improvement and connections to some of the previous known
results are also pointed out.

Keywords: Analytic functions; Univalent and bi-univalent functions; Maclaurin
series; Coefficient bounds.
2010 Mathematics Subject Classification. Primary 30C45; Secondary 30C50.

##  1 Introduction, Definitions and Notations

Let $\mathcal{A}$ denote the class of all analytic functions $f$ defined in
the open unit disk $\mathbb{U}=\\{z\in\mathbb{C}:\left|z\right|<1\\}$ and
normalized by the condition $f(0)=f^{\prime}(0)-1=0$. Thus each
$f\in\mathcal{A}$ has a Taylor-Maclaurin series expansion of the form:

$f(z)=z+\sum\limits_{n=2}^{\infty}a_{n}z^{n},\ \ (z\in\mathbb{U}).$ (1.1)

Further, let $\mathcal{S}$ denote the class of all functions $f\in\mathcal{A}$
which are univalent in $\mathbb{U}$ (for details, see [9]; see also some of
the recent investigations [2, 3]). Two of the important and well-investigated
subclasses of the analytic and univalent function class $\mathcal{S}$ are the
class $\mathcal{S}^{\ast}(\alpha)$ of starlike functions of order $\alpha$ in
$\mathbb{U}$ and the class $\mathcal{K}(\alpha)$ of convex functions of order
$\alpha$ in $\mathbb{U}$. By definition, we have

$\mathcal{S}^{\ast}(\alpha):=\left\\{f:\ f\in\mathcal{S}\ \ \text{and}\ \
\mbox{Re}\left\\{\frac{zf^{\prime}(z)}{f(z)}\right\\}>\alpha,\quad(z\in\mathbb{U};0\leq\alpha<1)\right\\},$
(1.2)

and

$\mathcal{K}(\alpha):=\left\\{f:\ f\in\mathcal{S}\ \ \text{and}\ \
\mbox{Re}\left\\{1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\right\\}>\alpha,\quad(z\in\mathbb{U};0\leq\alpha<1)\right\\}.$
(1.3)

It is clear from the definitions (1.2) and (1.3) that
$\mathcal{K}(\alpha)\subset\mathcal{S}^{\ast}(\alpha)$. Also we have

$f(z)\in\mathcal{K}(\alpha)\ \ \text{iff}\ \
zf^{\prime}(z)\in\mathcal{S}^{\ast}(\alpha),$

and

$f(z)\in\mathcal{S}^{\ast}(\alpha)\ \ \text{iff}\ \
\int_{0}^{z}\frac{f(t)}{t}dt=F(z)\in\mathcal{K}(\alpha).$

It is well-known [9] that every function $f\in\mathcal{S}$ has an inverse map
$f^{-1}$ that satisfies the following conditions:

$f^{-1}(f(z))=z\ \ \ (z\in\mathbb{U}),$

and

$f\left(f^{-1}(w)\right)=w$ $\ \ \
\left(|w|<r_{0}(f);r_{0}(f)\geq\frac{1}{4}\right)$.

In fact, the inverse function is given by

$f^{-1}(w)=w-a_{2}w^{2}+(2a_{2}^{2}-a_{3})w^{3}-(5a_{2}^{3}-5a_{2}a_{3}+a_{4})w^{4}+\cdots.$
(1.4)

A function $f\in\mathcal{A}$ is said to be bi-univalent in $\mathbb{U}$ if
both $f(z)$ and $f^{-1}(z)$ are univalent in $\mathbb{U}$. Let $\Sigma$ denote
the class of bi-univalent functions in $\mathbb{U}$ given by (1.1). For a
brief history and some interesting examples of functions and characterization
of the class $\Sigma$, see Srivastava et al. [19], Frasin and Aouf [10], and
Magesh and Yamini [15]. Examples of functions in the class $\Sigma$ are

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

and so on. However, the familiar Koebe function is not a member of $\Sigma$.
Other common examples of functions in $\mathcal{S}$ such as

$z-\frac{z^{2}}{2}\text{ \ \ and \ \ }\frac{z}{1-z^{2}}$

are also not members of $\Sigma$.

In 1967, Lewin [13] investigated the bi-univalent function class $\Sigma$ and
showed that $|a_{2}|<1.51$. Subsequently, Brannan and Clunie [5] conjectured
that $|a_{2}|\leq\sqrt{2}.$ Later, Netanyahu [17] showed that $\max$
$|a_{2}|=\frac{4}{3}$ if $f\in\Sigma.$ Brannan and Taha [6] introduced certain
subclasses of a bi-univalent function class $\Sigma$ similar to the familiar
subclasses $\mathcal{S}^{\ast}(\alpha)$ and $\mathcal{K}(\alpha)$ of starlike
and convex functions of order $\alpha$ ($0\leq\alpha<1$), respectively (see
[4]). Thus, following the works of Brannan and Taha [6], for $0\leq\alpha<1,$
a function $f\in\Sigma$ is in the class
$\mathcal{S}_{\Sigma}^{\ast}\left(\alpha\right)$ of bi-starlike functions of
order $\alpha$; or $\mathcal{K}_{\Sigma}\left(\alpha\right)$ of bi-convex
functions of order $\alpha$ if both $f$ and $f^{-1}$ are respectively starlike
or convex functions of order $\alpha.$ Recently, many researchers have
introduced and investigated several interesting subclasses of the bi-univalent
function class $\Sigma$ and they have found non-sharp estimates on the first
two Taylor-Maclaurin coefficients $|a_{2}|$ and $|a_{3}|$. In fact, the
aforecited work of Srivastava et al. [19] essentially revived the
investigation of various subclasses of the bi-univalent function class
$\Sigma$ in recent years; it was followed by such works as those by Frasin and
Aouf [10], Xu et al. [23], Çağlar et al. [8], and others (see, for example,
[14, 18]). The coefficient estimate problem for each of the following Taylor-
Maclaurin coefficients $|a_{n}|$ $(n\in\mathbb{N}\backslash\\{1,2\\})$ for
each $f\in\Sigma$ given by (1.1) is still an open problem.

This work is concerned with the coefficient bounds for the Taylor-Maclaurin
coefficients $|a_{2}|$ and $|a_{3}|$. Furthermore, we modify the definition of
the classes introduced by Srivastava et la. [22] and modify estimations.
Finally, several connections to some of the previous results are pointed out.

######  Definition 1.1.

For $0\leq\lambda\leq 1,0\leq\delta\leq 1,0\leq\mu\leq 1,0\leq\gamma\leq 1$,
$0<\alpha\leq 1$ and $\tau\in\mathbb{C}^{\ast}=\mathbb{C}/\left\\{0\right\\}$.
let $f\in\Sigma$ given by (1.1), then $f$ is said to be in the class
$\mathcal{S}_{\Sigma}^{\alpha}(\tau,\delta,\lambda,\gamma)$ if it satisfy the
following conditions

$\left|\arg\left(1+\frac{1}{\tau}\left[\frac{(1-\delta)f(z)+\delta
zf^{{}^{\prime}}(z)+\mu
z^{2}f^{{}^{\prime\prime}}(z)}{(1-\lambda)z+\lambda(1-\gamma)f(z)+\lambda\gamma
zf^{{}^{\prime}}(z)}-1\right]\right)\right|<\frac{\alpha\pi}{2},$

and

$\left|\arg\left(1+\frac{1}{\tau}\left[\frac{(1-\delta)g(w)+\delta
wg^{{}^{\prime}}(w)+\mu
w^{2}g^{{}^{\prime\prime}}(w)}{(1-\lambda)w+\lambda(1-\gamma)g(w)+\lambda\gamma
wg^{{}^{\prime}}(w)}-1\right]\right)\right|<\frac{\alpha\pi}{2},$

for all $z\in\mathbb{U}$, $g=f^{-1}\in\Sigma$ given by (1.4).

######  Definition 1.2.

For $0\leq\lambda\leq 1,0\leq\delta\leq 1,0\leq\mu\leq 1,0\leq\gamma\leq 1$,
$0\leq\beta<1$ and $\tau\in\mathbb{C}^{\ast}=\mathbb{C}/\left\\{0\right\\}$.
let $f\in\Sigma$ given by (1.1), then $f$ is said to be in the class
$\mathcal{S}_{\Sigma}(\tau,\delta,\mu,\lambda,\gamma;\beta)$ if it satisfy the
following conditions

$\operatorname{Re}\left(1+\frac{1}{\tau}\left[\frac{(1-\delta)f(z)+\delta
zf^{{}^{\prime}}(z)+\mu
z^{2}f^{{}^{\prime\prime}}(z)}{(1-\lambda)z+\lambda(1-\gamma)f(z)+\lambda\gamma
zf^{{}^{\prime}}(z)}-1\right]\right)>\beta,$

and

$\operatorname{Re}\left(1+\frac{1}{\tau}\left[\frac{(1-\delta)g(w)+\delta
wg^{{}^{\prime}}(w)+\mu
w^{2}g^{{}^{\prime\prime}}(w)}{(1-\lambda)w+\lambda(1-\gamma)g(w)+\lambda\gamma
wg^{{}^{\prime}}(w)}-1\right]\right)>\beta,$

for all $z\in\mathbb{U}$, $g=f^{-1}\in\Sigma$ given by (1.4).

######  Remark 1.

If we put $\delta=1$ in the definitions 1.1 and 1.2, we modified the
definitions of the classes
$\mathcal{H}_{\Sigma}(\tau,\mu,\lambda,\gamma;\alpha)$ and
$\mathcal{H}_{\Sigma}(\tau,\mu,\lambda,\gamma;\beta)$ respectively which were
introduced by Srivastava el at. [22].

######  Remark 2.

For special choices of the parameters $\tau,\delta,\mu,\lambda,\gamma$, we can
obtain the following subclasses as a special case of our main classes defined
above:

  1. 1.

$\mathcal{S}_{\Sigma}^{\alpha}(1,\lambda,\delta,0,\gamma)=\mathcal{N}_{\Sigma}(\alpha,\lambda,\delta)$
which introduced by Serap Bulut [7].

  2. 2.

$\mathcal{S}_{\Sigma}^{\alpha}(1,1,0,\lambda,0)=\mathcal{S}_{\Sigma}^{a,1,a}(\alpha,\lambda)$
and
$\mathcal{S}_{\Sigma}(1,1,0,\lambda,0;\beta)=\mathcal{M}_{\Sigma}^{a,1,a}(\beta,\lambda)$
which were introduced by Srivastava et al. [20].

  3. 3.

$\mathcal{S}_{\Sigma}^{\alpha}(1,1,0,1,\lambda)=\mathcal{G}_{\Sigma}(\alpha,\lambda)$
and
$\mathcal{S}_{\Sigma}(1,1,0,1,\lambda;\beta)=\mathcal{M}_{\Sigma}(\beta,\lambda)$
which introduced by Murugusundaramoorthy et al. [16].

  4. 4.

$\mathcal{S}_{\Sigma}^{\alpha}(1,1,\lambda,1,\lambda)=\mathfrak{B}_{\Sigma}(\alpha,\lambda)$
and
$\mathcal{S}_{\Sigma}(1,1,\lambda,1,\lambda;\beta)=\mathcal{N}_{\Sigma}(\beta,\lambda)$
which introduced by Keerthi and Raja [12].

  5. 5.

$\mathcal{S}_{\Sigma}^{\alpha}(1,\lambda,0,0,\gamma)=\mathcal{B}_{\Sigma}(\alpha,\lambda)$
and
$\mathcal{S}_{\Sigma}(1,\lambda,0,0,\gamma;\beta)=\mathcal{B}_{\Sigma}(\beta,\lambda)$
which introduced by Frasin and Aouf [10].

  6. 6.

$\mathcal{S}_{\Sigma}^{\alpha}(1,1,\beta,0,\gamma)=\mathcal{H}_{\Sigma}(\alpha,\beta)$
and
$\mathcal{S}_{\Sigma}(1,1,\beta,0,\gamma;\gamma)=\mathcal{H}_{\Sigma}(\gamma,\beta)$
which introduced by Frasin [11].

  7. 7.

$\mathcal{S}_{\Sigma}^{\alpha}(1,1,0,0,\gamma)=\mathcal{H}_{\Sigma}^{\alpha}$
and $\mathcal{S}_{\Sigma}(1,1,0,0,\gamma;\beta)=\mathcal{H}_{\Sigma}(\beta)$
which introduced by Srivastava et al. [19].

######  Lemma 1.3.

[9] If $h\in\mathcal{P}$, then the estimates $|c_{n}|\leq 2$, $n=1,2,3,...$
are sharp, where $\mathcal{P}$ is the family of all functions $h$ which are
analytic in $\mathbb{U}$ for which $h(0)=1$ and
$\operatorname{Re}(h(z))>0(z\in\mathbb{U})$ where

$h(z)=1+c_{1}z+c_{2}z^{2}+...,z\in\mathbb{U}.$

##  2 Coefficient bounds for the function class
$\mathcal{S}_{\Sigma}^{\alpha}(\tau,\delta,\lambda,\gamma)$

In this section, we establish coefficient bounds for the Taylor-Maclaurin
coefficients $|a_{2}|$ and $|a_{3}|$ of the function
$f\in\mathcal{S}_{\Sigma}^{\alpha}(\tau,\delta,\lambda,\gamma)$.

######  Theorem 2.1.

let $f(z)$ defined by (1.1) belonging to the class
$\mathcal{S}_{\Sigma}^{\alpha}(\tau,\delta,\lambda,\gamma)$, then

$|a_{2}|\leq\frac{2\alpha|\tau|}{\sqrt{|2\alpha\tau\Omega+(1-\alpha)(1+\delta+2\mu-\lambda-\gamma\lambda)^{2}|}}.$
(2.1)

and

$|a_{3}|\leq\min\left\\{\frac{4\alpha^{2}|\tau|^{2}}{(1+\delta+2\mu-\lambda-\gamma\lambda)^{2}}+\frac{2\alpha|\tau|}{|1+2\delta+6\mu-\lambda-2\gamma\lambda|},\frac{2\alpha|\tau|}{|\Omega|}\right\\}$
(2.2)

where

$\Omega=1+2\delta+6\mu-2\lambda-3\gamma\lambda-\lambda\delta-2\mu\lambda+\lambda^{2}+2\gamma\lambda^{2}-\delta\gamma\lambda-2\mu\gamma\lambda+\gamma^{2}\lambda^{2}.$
(2.3)

###### Proof.

Since $f,g\in\mathcal{S}_{\Sigma}^{\alpha}(\tau,\delta,\lambda,\gamma)$, then
there exist two functions $h_{1}(z)=1+\sum_{n=1}^{\infty}p_{n}z^{n}$ and
$h_{2}(w)=1+\sum_{n=1}^{\infty}q_{n}w^{n}$ with positive real part in the unit
disc such that

$1+\frac{1}{\tau}\left[\frac{(1-\delta)f(z)+\delta zf^{{}^{\prime}}(z)+\mu
z^{2}f^{{}^{\prime\prime}}(z)}{(1-\lambda)z+\lambda(1-\gamma)f(z)+\lambda\gamma
zf^{{}^{\prime}}(z)}-1\right]=(h_{1}(z))^{\alpha}$ (2.4)

and

$1+\frac{1}{\tau}\left[\frac{(1-\delta)g(w)+\delta wg^{{}^{\prime}}(w)+\mu
w^{2}g^{{}^{\prime\prime}}(w)}{(1-\lambda)w+\lambda(1-\gamma)g(w)+\lambda\gamma
wg^{{}^{\prime}}(w)}-1\right]=(h_{2}(w))^{\alpha}$ (2.5)

Now by comparing the coefficients of powers $z,z^{2},w$ and $w^{2}$ in boss
sides of equations (2.4) and (2.5), we obtain

$\frac{(1+\delta+2\mu-\lambda-\gamma\lambda)}{\tau}=\alpha p_{1}$ (2.6)
$\frac{(1+2\delta+6\mu-\lambda-2\gamma\lambda)a_{3}-(1+\delta+2\mu-\lambda-\gamma\lambda)(\lambda+\gamma\lambda)a_{2}^{2}}{\tau}=\alpha
p_{2}+\frac{\alpha(\alpha-1)}{2}p_{1}^{2}$ (2.7)
$-\frac{(1+\delta+2\mu-\lambda-\gamma\lambda)}{\tau}=\alpha q_{1}$ (2.8)
$\frac{(1+2\delta+6\mu-\lambda-2\gamma\lambda)(2a_{2}^{2}-a_{3})-(1+\delta+2\mu-\lambda-\gamma\lambda)(\lambda+\gamma\lambda)a_{2}^{2}}{\tau}=\alpha
q_{2}+\frac{\alpha(\alpha-1)}{2}q_{1}^{2}.$ (2.9)

From equations (2.6) and (2.8), we deduce

$p_{1}=-q_{1},$ (2.10)

and

$a_{2}=\frac{\alpha\tau p_{1}}{1+\delta+2\mu-\lambda-\gamma\lambda}.$ (2.11)

By adding equation (2.7) to (2.9), we obtain

$2\frac{\Omega}{\tau}a_{2}^{2}=\alpha(p_{2}+q_{2})+\frac{\alpha(\alpha-1)}{2}(p_{1}^{2}+q_{1}^{2})$
(2.12)

where $\Omega$ are given by (2.3).
Using equation (2.10) and substituting the value of $a_{2}$ from equation
(2.11) into equation (2.12), we deduce

$p_{1}^{2}=\frac{(p_{2}+q_{2})(1+\delta+2\mu-\lambda-\gamma\lambda)^{2}}{2\alpha\tau\Omega+(1-\alpha)(1+\delta+2\mu-\lambda-\gamma\lambda)^{2}},$
(2.13)

therefore, by applying Lemma 1 in the equation (2.13), then

$|p_{1}|\leq\frac{2|1+\delta+2\mu-\lambda-\gamma\lambda|}{\sqrt{|2\alpha\tau\Omega+(1-\alpha)(1+\delta+2\mu-\lambda-\gamma\lambda)^{2}|}}.$
(2.14)

By substituting from equation (2.14) into (2.11) and using Lemma 1, we
conclude the desired estimate of $a_{2}$ given by (4.1).
On the other hand, to investigate the bounds of $|a_{3}|$, subtracting
equation (2.9) from (2.7) we obtain

$2\frac{(1+2\delta+6\mu-\lambda-2\gamma\lambda)}{\tau}(a_{3}-a_{2}^{2})=\alpha(p_{2}-q_{2})+\frac{\alpha(\alpha-1)}{2}(p_{1}^{2}-q_{1}^{2}),$
(2.15)

By using equation (2.10) into (2.15), we obtain

$a_{3}=a_{2}^{2}+\frac{\alpha\tau}{2(1+2\delta+6\mu-\lambda-2\gamma\lambda)}(p_{2}-q_{2}),$
(2.16)

Using Lemma 1 and substituting of the value of $a_{2}$ from (2.11) into
(2.16), we conclude one of the desired estimates of $|a_{3}|$.
Now, by substituting the value of $a_{2}^{2}$ from equation (2.12) into
(2.16), we obtain

$a_{3}=\frac{\alpha\tau}{2}\left[p_{2}\left(\frac{1}{\Omega}+\frac{1}{(1+2\delta+6\mu-\lambda-2\gamma\lambda)}\right)+q_{2}\left(\frac{1}{\Omega}-\frac{1}{(1+2\delta+6\mu-\lambda-2\gamma\lambda)}\right)\right],$
(2.17)

Applying Lemma 1 for the coefficients $p_{2},q_{2}$, we conclude the other
desired estimation of $|a_{3}|$. ∎

##  3 Coefficient bounds for the function class
$\mathcal{S}_{\Sigma}(\tau,\delta,\lambda,\gamma;\beta)$

In this section, we establish coefficient bounds for the Maclaurin
coefficients $|a_{2}|$ and $|a_{3}|$ of the function
$f\in\mathcal{S}_{\Sigma}(\tau,\delta,\lambda,\gamma;\beta)$.

######  Theorem 3.1.

let $f(z)$ defined by (1.1) belonging to the class
$\mathcal{S}_{\Sigma}(\tau,\delta,\lambda,\gamma;\beta)$, then

$|a_{2}|\leq\min\left\\{\frac{2(1-\beta)|\tau|}{|1+\delta+2\mu-\lambda-\gamma\lambda|},\sqrt{\frac{2|\tau|(1-\beta)}{|\Omega|}}\right\\}.$
(3.1)

and

$|a_{3}|\leq\min\left\\{\frac{4(1-\beta)^{2}|\tau|^{2}}{|1+\delta+2\mu-\lambda-\gamma\lambda|^{2}}+\frac{2(1-\beta)|\tau|}{|1+2\delta+6\mu-\lambda-2\gamma\lambda|},\frac{2(1-\beta)|\tau|}{|\Omega|}\right\\}.$
(3.2)

###### Proof.

Since $f,g\in\mathcal{S}_{\Sigma}(\tau,\delta,\lambda,\gamma;\beta)$, then
there exist two functions $P(z)=\sum_{n=1}^{\infty}p_{n}z^{n}$ and
$Q(w)=\sum_{n=1}^{\infty}q_{n}w^{n}$ with positive real part in the unit disc
such that

$1+\frac{1}{\tau}\left[\frac{(1-\delta)f(z)+\delta zf^{{}^{\prime}}(z)+\mu
z^{2}f^{{}^{\prime\prime}}(z)}{(1-\lambda)z+\lambda(1-\gamma)f(z)+\lambda\gamma
zf^{{}^{\prime}}(z)}-1\right]=\beta+(1-\beta)P(z)$ (3.3)

and

$1+\frac{1}{\tau}\left[\frac{(1-\delta)g(w)+\delta wg^{{}^{\prime}}(w)+\mu
w^{2}g^{{}^{\prime\prime}}(w)}{(1-\lambda)w+\lambda(1-\gamma)g(w)+\lambda\gamma
wg^{{}^{\prime}}(w)}-1\right]=\beta+(1-\beta)Q(w)$ (3.4)

Now by comparing the coefficients of powers $z,z^{2},w$ and $w^{2}$ in boss
sides of equations (3.3) and (3.4), we obtain

$\frac{(1+\delta+2\mu-\lambda-\gamma\lambda)}{\tau}=(1-\beta)p_{1}$ (3.5)
$\frac{(1+2\delta+6\mu-\lambda-2\gamma\lambda)a_{3}-(1+\delta+2\mu-\lambda-\gamma\lambda)(\lambda+\gamma\lambda)a_{2}^{2}}{\tau}=(1-\beta)p_{2}$
(3.6) $-\frac{(1+\delta+2\mu-\lambda-\gamma\lambda)}{\tau}=(1-\beta)q_{1}$
(3.7)
$\frac{(1+2\delta+6\mu-\lambda-2\gamma\lambda)(2a_{2}^{2}-a_{3})-(1+\delta+2\mu-\lambda-\gamma\lambda)(\lambda+\gamma\lambda)a_{2}^{2}}{\tau}=(1-\beta)q_{2}.$
(3.8)

From equations (3.5) and (3.7), we deduce

$p_{1}=-q_{1},$ (3.9)

and

$a_{2}=\frac{(1-\beta)\tau p_{1}}{1+\delta+2\mu-\lambda-\gamma\lambda}.$
(3.10)

By adding equation (3.6) to (3.8), we obtain

$\frac{2\Omega a_{2}^{2}}{\tau}=(1-\beta)(p_{2}+q_{2}),$ (3.11)

where $\Omega$ are given by (2.3).
Applying Lemma 1, we obtain

$|a_{2}|\leq\sqrt{\frac{2|\tau|(1-\beta)}{|\Omega|}}.$ (3.12)

On the other hand, from equation (3.10) we can deduce also that

$|a_{2}|\leq\frac{2(1-\beta)|\tau|}{|1+\delta+2\mu-\lambda-\gamma\lambda|}.$
(3.13)

Combining this with inequality (3.12), we obtain the desired estimate on the
coefficient $|a_{2}|$ which given by (4.2).
In order to deduce an estimation of bounds of $|a_{3}|$, subtracting equation
(3.8) from (3.6). we get

$a_{3}=a_{2}^{2}+\frac{\tau(1-\beta)(p_{2}-q_{2})}{2(1+2\delta+6\mu-\lambda-2\gamma\lambda)},$
(3.14)

By substituting the value of $a_{2}$ from equation (3.10) with applying Lemma
1 for the coefficients $p_{1},p_{2}$ and $q_{2}$, we obtain

$|a_{3}|\leq\frac{4(1-\beta)^{2}|\tau|^{2}}{(1+\delta+2\mu-\lambda-\gamma\lambda)^{2}}+\frac{2|\tau|(1-\beta)}{|1+2\delta+6\mu-\lambda-2\gamma\lambda|},$
(3.15)

Now, the value of $a_{2}$ from equation (3.11) with applying Lemma 1 for the
coefficients $p_{2},q_{2}$, we obtain the other estimation of $|a_{3}|$ which
given by (4.2). ∎

##  4 Some corollaries and consequences

In this section, we introduced and modified some previous known results as an
immediate consequences of Theorem 2.1 and 3.1.

######  Remark 3.

Putting $\delta=1$, we modified the results considered by Srivastava et al.
[22, Theorem 1 and 2].

######  Corollary 4.1.

Let $f(z)$ given by (1.1) be in the class
$\mathcal{H}_{\Sigma}(\tau,\mu,\lambda,\gamma;\alpha)$ then,

$|a_{2}|\leq\frac{2\alpha|\tau|}{\sqrt{|2\alpha\tau\tilde{\Omega}+(1-\alpha)(2+2\mu-\lambda-\gamma\lambda)^{2}|}}.$

and

$|a_{3}|\leq\min\left\\{\frac{4\alpha^{2}|\tau|^{2}}{(2+2\mu-\lambda-\gamma\lambda)^{2}}+\frac{2\alpha|\tau|}{|3+6\mu-\lambda-2\gamma\lambda|},\frac{2\alpha|\tau|}{|\tilde{\Omega}|}\right\\}$

where

$\tilde{\Omega}=\gamma^{2}\lambda^{2}+2\gamma\lambda^{2}-2\gamma\lambda\mu-4\gamma\lambda+\lambda^{2}-2\lambda\mu-3\lambda+6\mu+3$

######  Corollary 4.2.

Let $f(z)$ given by (1.1) be in the class
$\mathcal{H}_{\Sigma}(\tau,\mu,\lambda,\gamma;\beta)$ then,

$|a_{2}|\leq\min\left\\{\frac{2(1-\beta)|\tau|}{|2+2\mu-\lambda-\gamma\lambda|},\sqrt{\frac{2|\tau|(1-\beta)}{|\tilde{\Omega}|}}\right\\}.$

and

$|a_{3}|\leq\min\left\\{\frac{4(1-\beta)^{2}|\tau|^{2}}{|2+2\mu-\lambda-\gamma\lambda|^{2}}+\frac{2(1-\beta)|\tau|}{|3+6\mu-\lambda-2\gamma\lambda|},\frac{2(1-\beta)|\tau|}{|\tilde{\Omega}|}\right\\}.$

######  Remark 4.

Putting $\tau=1,\lambda=0$ and replace $\delta$ by $\lambda$ and also $\mu$ by
$\delta$, we obtain the result considered by Bulut [7, Theorem 5].

######  Corollary 4.3.

Let $f(z)$ given by (1.1) be in the class
$\mathcal{N}_{\Sigma}(\beta,\lambda,\delta)$ then,

$|a_{2}|\leq\min\left\\{\frac{2(1-\beta)}{1+\lambda+2\delta},\sqrt{\frac{2(1-\beta)}{1+2\lambda+6\delta}}\right\\},$
$|a_{3}|\leq\frac{2(1-\beta)}{1+2\lambda+6\delta}.$

######  Remark 5.

Putting $\tau=\delta=\lambda=1,\mu=0$ and replace $\gamma$ by $\lambda$, we
obtain the result considered by Murugusundaramoorthy et al. [16, Theorem 4 and
5].

######  Corollary 4.4.

Let $f(z)$ given by (1.1) be in the class
$\mathcal{G}_{\Sigma}(\alpha,\lambda)$ then,

$|a_{2}|\leq\frac{2\alpha}{(1-\lambda)\sqrt{1+\alpha}},$
$|a_{3}|\leq\min\left\\{\frac{4\alpha^{2}}{(1-\lambda)^{2}}+\frac{\alpha}{1-\lambda},\frac{2\alpha}{(1-\lambda)^{2}}\right\\}.$

######  Corollary 4.5.

Let $f(z)$ given by (1.1) be in the class
$\mathcal{M}_{\Sigma}(\beta,\lambda)$ then,

$|a_{2}|\leq\frac{\sqrt{2(1-\beta)}}{1-\lambda},$
$|a_{3}|\leq\min\left\\{\frac{2(1-\beta)}{(1-\lambda)^{2}},\frac{4(1-\beta)^{2}}{(1-\lambda)^{2}}+\frac{1-\beta}{1-\lambda}\right\\}.$

######  Remark 6.

Putting $\tau=\delta=1$ and $\mu=\gamma=0$, we obtain the result considered by
Srivastava et al. [20, Theorem 2.1 and 3.1 with $a=c,b=1$].

######  Corollary 4.6.

Let $f(z)$ given by (1.1) be in the class
$\mathcal{S}_{\Sigma}^{a,1,a}(\alpha,\lambda)$ then,

$|a_{2}|\leq\frac{2\alpha}{\sqrt{|2\alpha(\lambda^{2}-3\lambda+3)+(1-\alpha)(2-\lambda)^{2}|}},$
$|a_{3}|\leq\min\left\\{\frac{4\alpha^{2}}{(2-\lambda)^{2}}+\frac{2\alpha}{3-\lambda},\frac{2\alpha}{\lambda^{2}-3\lambda+3}\right\\}.$

######  Corollary 4.7.

Let $f(z)$ given by (1.1) be in the class
$\mathcal{M}_{\Sigma}^{a,1,a}(\beta,\lambda)$ then,

$|a_{2}|\leq\sqrt{\frac{2(1-\beta)}{\lambda^{2}-3\lambda+3}},$
$|a_{3}|\leq\min\left\\{\frac{2(1-\beta)}{\lambda^{2}-3\lambda+3},\frac{4(1-\beta)^{2}}{(2-\lambda)^{2}}+\frac{2(1-\beta)}{3-\lambda}\right\\}.$

######  Remark 7.

Putting $\tau=\delta=\lambda=1$ and replace $\mu,\gamma$ by $\lambda$, we
obtain the result considered by Keerthi and Raja [12, Corollary 2.3 and 3.4].

######  Corollary 4.8.

Let $f(z)$ given by (1.1) be in the class
$\mathfrak{B}_{\Sigma}(\alpha,\lambda)$ then,

$|a_{2}|\leq\frac{2\alpha}{\sqrt{\left|4\alpha(1+2\lambda)+(1-3\alpha)(1+\lambda)^{2}\right|}},$
$|a_{3}|\leq\min\left\\{\frac{4\alpha^{2}}{(1+\lambda)^{2}}+\frac{\alpha}{1+2\lambda},\frac{2\alpha}{1+2\lambda-\lambda^{2}}\right\\}.$

######  Corollary 4.9.

Let $f(z)$ given by (1.1) be in the class
$\mathcal{N}_{\Sigma}(\beta,\lambda)$ then,

$|a_{2}|\leq\sqrt{\frac{2(1-\beta)}{1+2\lambda-\lambda^{2}}},$
$|a_{3}|\leq\min\left\\{\frac{2(1-\beta)}{1+2\lambda-\lambda^{2}},\frac{4(1-\beta)^{2}}{(1+\lambda)^{2}}+\frac{1-\beta}{1+2\lambda}\right\\}.$

######  Remark 8.

Putting $\tau=1,\lambda=\mu=0$ and replace $\delta$ by $\lambda$, we obtain
the result considered by Frasin and Aouf [10, Theorem 2.2 and 3.2].

######  Corollary 4.10.

Let $f(z)$ given by (1.1) be in the class
$\mathcal{B}_{\Sigma}(\alpha,\lambda)$ then,

$|a_{2}|\leq\frac{2\alpha}{\sqrt{|\alpha(1+2\lambda-\lambda^{2})+(1+\lambda)^{2}}},$
$|a_{3}|\leq\frac{2\alpha}{1+2\lambda}.$

######  Corollary 4.11.

Let $f(z)$ given by (1.1) be in the class
$\mathcal{B}_{\Sigma}(\beta,\lambda)$, then

$|a_{2}|\leq\sqrt{\frac{2(1-\beta)}{1+2\lambda}},$
$|a_{3}|\leq\frac{2(1-\beta)}{1+2\lambda}.$

######  Remark 9.

Putting $\tau=\delta=1,\lambda=0$ and replace $\mu$ by $\beta$, we obtain the
result considered by Frasin [11, Theorem 2.2 and 3.2].

######  Corollary 4.12.

Let $f(z)$ given by (1.1) be in the class $\mathcal{H}_{\Sigma}(\alpha,\beta)$
then,

$|a_{2}|\leq\frac{2\alpha}{\sqrt{|2(2+\alpha)+4\beta(\alpha+\beta+2-\alpha\beta)|}},$
$|a_{3}|\leq\frac{2\alpha}{3(1+2\beta)}.$

######  Corollary 4.13.

Let $f(z)$ given by (1.1) be in the class
$\mathcal{H}_{\Sigma}(\gamma,\beta)$, then

$|a_{2}|\leq\sqrt{\frac{2(1-\gamma)}{3(1+2\gamma)}},$
$|a_{3}|\leq\frac{2(1-\gamma)}{3(1+2\beta)}.$

######  Remark 10.

Putting $\tau=\delta=1$ and $\mu=\lambda=0$, we obtain the result considered
by Srivastava et al. [19, Theorem 1 and 2].

######  Corollary 4.14.

Let $f(z)$ given by (1.1) be in the class $\mathcal{H}_{\Sigma}^{\alpha}$
then,

$|a_{2}|\leq\alpha\sqrt{\frac{2}{2+\alpha}},$ $|a_{3}|\leq\frac{2\alpha}{3}.$

######  Corollary 4.15.

Let $f(z)$ given by (1.1) be in the class $\mathcal{H}_{\Sigma}(\beta)$, then

$|a_{2}|\leq\sqrt{\frac{2(1-\beta)}{3}},$ $|a_{3}|\leq\frac{2(1-\beta)}{3}.$

######  Remark 11.

Finally, we introduce some results improved in our paper,

  * 1.

The estimation of $|a_{3}|$ in Corollaries 4.10, 4.11 are improvement of the
estimates obtained by Frasin and Aouf [10, Theorem 2.2 and 3.2].

  * 2.

The estimation of $|a_{3}|$ in Corollaries 4.12, 4.13 are improvement of the
estimates obtained by Frasin [11, Theorem 2.2 and 3.2].

  * 3.

The estimation of $|a_{3}|$ in Corollaries 4.14, 4.15 are improvement of the
estimates obtained by Srivastava et al. [19, Theorem 1 and 2].

## References

  * [1] Altinkaya, Ş. $\&$ Yalçin, S.(2017), Estimates on coefficients of a general subclass of bi-univalent functions associated with symmetric q-derivative operator by means of the Chebyshev polynomials, Asia Pacific Journal of Mathematics, 4(2), 90-99.
  * [2] Amourah, A.A., Yousef, F., Al-Hawary, T. $\&$ Darus, M.(2017), On H3(p) Hankel determinant for certain subclass of p-valent functions, Ital. J. Pure Appl. Math., 37, 611-618.
  * [3] Al-Hawary, T., Frasin, B.A. $\&$ Yousef, F.(2018), Coefficients estimates for certain classes of analytic functions of complex order, Afr. Mat., 29(7-8), 1265–-1271.
  * [4] Brannan, D.A., Clunie, J.G. $\&$ Kirwan, W.E.(1970), Coefficient estimates for a class of star-like functions, Canad. J. Math., 22, 476-485.
  * [5] Brannan, D.A. $\&$ Clunie, J.G.(1980), Aspects of contemporary complex analysis (Proceedings of the NATO Advanced Study Institute held at the University of Durham, Durham, Academic Press, New York and London).
  * [6] Brannan, D.A. $\&$ Tan, D.L.(1986), On some classes of bi-univalent functions, Stud. Univ. Babes¸-Bolyai Math., 31(2), 70-77.
  * [7] Bulut, S. (2016), Faber polynomial coefficient estimates for a subclass of analytic bi-univalent functions, Filomat, 30(6), 1567-1575.
  * [8] Çağlar, M., Orhan, H. $\&$ Yağmur, N.(2013), Coefficient bounds for new subclasses of bi-univalent functions, Filomat, 27(7), 1165-1171.
  * [9] Duren, P.L. (1983), Univalent functions, Grundlehren der Mathematischen Wissenschaften, Band 259, Springer-Verlag, New York, Berlin, Heidelberg and Tokyo.
  * [10] Frasin. B.A. $\&$ Aouf, M.K.(2011), New subclasses of bi-univalent functions, Appl. Math. Lett., 24(9), 1569-1573.
  * [11] Frasin, B.A. (2014), Coefficient bounds for certain classes of bi-univalent functions, Hact. J. Math. Stat., 43(3), 383-389.
  * [12] Keerthi, B. Srutha $\&$ Raja, Bhuvaneswari (2013), Coefficient Inequality for Certain New Sub-classes of Analytic Bi-univalent Functions, Abstr. Appl. Anal., 3(1), 1-10.
  * [13] Lewin, M. (1967), On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18, 63-68.
  * [14] Li, X.-F. $\&$ Wang, A.-P.(2012), Two new subclasses of bi-univalent functions, Int. Math. Forum, 7, 1495-1504.
  * [15] Magesh, N. $\&$ Yamini, J.(2013), Coefficient bounds for a certain subclass of bi-univalent functions, Int. Math. Forum, 8(27), 1337-1344.
  * [16] Murugusundaramoorthy, G., Magesh, N. $\&$ Prameela, V.(2013), Coefficient Bounds for Certain Subclasses of Bi-Univalent Function, Abstr. Appl. Anal., 2013, Article ID 573017, 3 pages.
  * [17] Netanyahu, E. (1969), 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., 32, 100-112.
  * [18] Porwal, S. $\&$ Darus, M.(2013), On a new subclass of bi-univalent functions, J. Egypt. Math. Soc., 21(3), 190-193.
  * [19] Srivastava, H.M., Mishra, A.K. $\&$ Gochhayat, P.(2010), Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23(10), 1188-1192.
  * [20] Srivastava, H.M., Murugusundaramoorthy, G. $\&$ Magesh, N.(2013), Certain sub classes of bi-univalent functions associated with the Hohlov operator, Global Journal of Mathematical Analysis, 1(2), 67-73.
  * [21] Srivastava, H.M., Bulut, S., Çağlar, M. $\&$ Yağmur, N.(2013), Coefficient estimates for a general subclass of analytic and bi-univalent functions, Filomat 27(5), 831-842.
  * [22] Srivastava, H.M., Gaboury, S. $\&$ Ghanim, F.(2016), Coefficient estimates for some general subclasses of analytic and bi-univalent functions, Afr. Mat., 28(5-6), 693-706.
  * [23] Xu, Q.-H., Gui, Y.-C. $\&$ Srivastava, H.M.(2012), Coefficient estimates for a certain subclass of analytic and bi-univalent functions, Appl. Math. Lett., 25, 990-994.


D}=\left(\begin{array}[]{cc}0&-\sqrt{(j+1/2)^{2}-\frac{q_{a}^{2}}{4}}\\\
-\sqrt{(j+1/2)^{2}-\frac{q_{a}^{2}}{4}}&0\end{array}\right)+\sigma^{-}I_{+}+\sigma^{+}I_{-}+1~{},\end{split}$
(257)

and

$\displaystyle\begin{split}F(r)=\sum_{\mu_{a}\in\Delta_{R_{A}}^{(j)}}\left(\begin{array}[]{c}f_{a,+}(r)\\\
f_{a,-}(r)\end{array}\right)\,v_{\mu_{a}}~{}.\end{split}$ (258)

The solutions are given by

$\displaystyle\begin{split}F(r)=P\,{\rm exp}\left\\{-\int{\cal
D}(r)dr\right\\}\,F_{0}~{},\end{split}$ (259)

for $F_{0}$ a constant vector. The set of $F_{0}$ that corresponds to plane-
wave normalizable solutions

$\displaystyle\begin{split}\lim_{r\to\infty}r\psi^{(j)}_{R_{A}}<\infty~{},\end{split}$
(260)

were solved for in Brennan:2021ucy . The solutions decompose with respect to
the decomposition of ${\vec{I}}=\bigoplus_{i}{\vec{I}}^{(i)}$. Let us denote
the decomposition by the set of spins $\\{s_{I}\\}$ of the representations
$\\{{\vec{I}}^{(i)}\\}$ and $R_{s_{I}}$ to be the restriction of the
representation $R$ to the spin-$s_{I}$ representation of ${\vec{I}}^{(i)}$. By
construction, for each $s_{I}$, the magnetic charge is a constant $Q_{s_{I}}$
on $\psi_{R}$. The $s_{I}$-solutions of spin-$j$ only exist for
$j_{min}^{(s_{I})}\leq j\leq j_{max}^{(s_{I})}$ where

$\displaystyle\begin{split}j_{min/max}^{(s_{I})}=min/max_{\mu\in\Delta_{R_{s_{I}}}}|\langle\mu,T_{3}\rangle|-\frac{1}{2}~{}.\end{split}$
(261)

The solutions are given explicitly by

$\displaystyle\begin{split}f_{a,+}(r)=0\quad,\quad
f_{a,-}(r)=\frac{1}{r}\Big{\langle}j,Q_{s_{I}}-\frac{1}{2}+s_{I}-a\,\Big{|}\,|Q_{s_{I}}|-\frac{1}{2},Q_{s_{I}}-\frac{1}{2};s_{I},s_{I}-a\Big{\rangle}~{},\end{split}$
(262)

for $Q_{s_{I}}>0$ and

$\displaystyle\begin{split}f_{a,+}(r)=\frac{1}{r}\Big{\langle}j,Q_{s_{I}}+\frac{1}{2}+s_{I}-a\,\Big{|}\,|Q_{s_{I}}|-\frac{1}{2},Q_{s_{I}}+\frac{1}{2};s_{I},s_{I}-a\Big{\rangle}\quad,\quad
f_{a,-}(r)=0~{},\end{split}$ (263)

for $Q_{s_{I}}<0$ where $\mu_{a}\in\Delta_{R_{s_{I}}}$ where we have chosen an
ordering $\langle I_{3},\mu_{a}\rangle>\langle I_{3},\mu_{a+1}\rangle$. Note
that there is no solution for modes with $Q_{s_{I}}=0$.

Importantly these modes are spin polarized. This means that the spectrum of
the Dirac operator is then described by the solutions

$\displaystyle\begin{split}\psi_{R}^{a\,(j)}=\frac{e^{i(m-q_{a}/2)\phi}}{r}U(\theta,\phi)\begin{cases}e^{ik(t+r)}\left(\begin{array}[]{c}0\\\
C^{(j)}_{s_{I},a,-}\,d^{j}_{-m,\frac{q_{a}-1}{2}}(\theta)\end{array}\right)&Q_{s_{I}}>0\\\
e^{ik(t-r)}\left(\begin{array}[]{c}C^{(j)}_{s_{I},a,+}\,d^{j}_{-m,\frac{q_{a}+1}{2}}(\theta)\\\
0\end{array}\right)&Q_{s_{I}}<0\end{cases}\end{split}$ (264)

for $\mu_{a}\in\Delta^{(j)}_{R_{s_{I}}}$ and

$\displaystyle\begin{split}C^{(j)}_{s_{I},a,\pm}=\Big{\langle}j,Q_{s_{I}}\pm\frac{1}{2}+s_{I}-a\,\Big{|}\,|Q_{s_{I}}|-\frac{1}{2},Q_{s_{I}}\pm\frac{1}{2};s_{I},s_{I}-a\Big{\rangle}~{}.\end{split}$
(265)

Note that in the cases where $\phi_{\infty}=0$, as we consider in the bulk of
this paper, the theory has a preserved non-abelian gauge symmetry along the
irreducible components ${\vec{I}}^{(i)}$. This means that the above solutions
are gauge equivalent to the solutions where $\psi^{(j)}_{R}$ is purely along a
single weight $\mu\in\Delta_{R_{s_{I}}}$. In the bulk of the paper, we will
find it convenient to pick a gauge such that each solution is along the
top/bottom weight $\mu_{top/bot}\in\Delta_{R_{s_{I}}}$ where $|\langle
T_{3},\mu_{top/bot}\rangle|=j+\frac{1}{2}$.

###  A.3 The Non-Abelian Monopole

Now let us solve for solutions to the Dirac operator coupled to the $SU(N)$
non-abelian monopole connection. Here we take the connection to be

$\displaystyle\begin{split}A=T_{3}(1-\cos\theta)d\phi+\frac{i}{2}M_{+}(r)e^{-i\phi}(d\theta-i\sin\theta
d\phi)-\frac{i}{2}M_{-}(r)e^{i\phi}(d\theta+i\sin\theta d\phi)~{}.\end{split}$
(266)

Here we will be interested in solving for the zero-mode solutions which are
spanned by the normalizable solutions in the kernel of the time-independent
Dirac operator. Again, after performing a frame rotation, the Dirac operator
can be written

$\displaystyle\begin{split}\Bigg{[}\sigma^{3}\left(\partial_{r}+\frac{1}{r}\right)+\frac{{\cal
K}}{r}&-\left(\sigma^{-}e^{-i\phi}M_{+}-\sigma^{+}e^{i\phi}M_{-}\right)\Bigg{]}~{},\end{split}$
(267)

where

$\displaystyle\begin{split}{\cal
K}&=\sigma^{1}\left(\partial_{\theta}+\frac{1}{2}{\rm
cot}(\theta)\right)+\frac{1}{\sin\theta}\sigma^{2}\Big{(}(\partial_{\phi}+iT_{3})-i\cos(\theta)T_{3}\Big{)}=\left(\begin{array}[]{cc}0&D^{+}_{T_{3}-\frac{1}{2}}\\\
D^{-}_{T_{3}+\frac{1}{2}}&0\end{array}\right)~{},\end{split}$ (268)

and

$\displaystyle\begin{split}D^{\pm}_{q}=\partial_{\theta}\pm\frac{1}{\sin\theta}\Big{(}-i(\partial_{\phi}+iT_{3})-q\cos\theta\Big{)}~{}.\end{split}$
(269)

Now let us consider the time-independent Dirac equation acting on a fermion
$\psi_{R}$ of representation $R$ of $SU(N)$. Let us decompose $\psi_{R}$ into
weights

$\displaystyle\begin{split}\psi_{R}=\sum_{\mu_{a}\in\Delta_{R}}\psi_{R}^{a}\,v_{\mu_{a}}~{}.\end{split}$
(270)

${\cal K}$ acts simply on the $SU(2)_{J}$-representations such that given a
spin-$j$ representation:

$\displaystyle\begin{split}\psi_{R_{A}}^{a\,(j)}=U(\theta,\phi)e^{i(m-q_{a}/2)\phi}\left(\begin{array}[]{c}f_{a,+}(r)\,d^{j}_{-m,\frac{q_{a}+1}{2}}(\theta)\\\
f_{a,-}(r)\,d^{j}_{-m,q_{a}-\frac{1}{2}}(\theta)\end{array}\right)\quad,\quad\mu_{a}\in\Delta^{(j)}_{R_{A}}~{},\end{split}$
(271)

where $\langle T_{3},\mu_{a}\rangle=\frac{q_{a}}{2}$ and $\psi_{R}^{(j)}$ is
the spin-$j$ solution of $\psi_{R}$. Here, ${\cal K}$ acts as

$\displaystyle\begin{split}{\cal
K}\cdot\psi_{R_{A}}^{a\,(j)}=\left(\begin{array}[]{cc}0&-\sqrt{(j+1/2)^{2}-\frac{q_{a}^{2}}{4}}\\\
\sqrt{(j+1/2)^{2}-\frac{q_{a}^{2}}{4}}&0\end{array}\right)\psi_{R_{A}}^{a\,(j)}~{}.\end{split}$
(272)

Now, the Dirac equation reduces to a first order matrix ODE for the functions
$f_{a,\pm}(r)$:

$\displaystyle\begin{split}\Big{[}r\partial_{r}+{\cal
D}\Big{]}F(r)=0\quad,\quad
F(r)=\sum_{\mu_{a}\in\Delta_{R_{A}}^{(j)}}\left(\begin{array}[]{c}f_{a,+}(r)\\\
f_{a,-}(r)\end{array}\right)\,v_{\mu_{a}}~{},\end{split}$ (273)

where

$\displaystyle\begin{split}{\cal
D}=\left(\begin{array}[]{cc}0&-\sqrt{(j+1/2)^{2}-\frac{q_{a}^{2}}{4}}\\\
-\sqrt{(j+1/2)^{2}-\frac{q_{a}^{2}}{4}}&0\end{array}\right)+\sigma^{-}M_{+}+\sigma^{+}M_{-}+1~{}.\end{split}$
(274)

The smooth solutions given by

$\displaystyle\begin{split}F(r)=P\,{\rm exp}\left\\{-\int{\cal
D}(r)dr\right\\}\,F_{0}~{},\end{split}$ (275)

for $F_{0}$ a constant vector. As proven in Brennan:2021ucy , there exists a
vector $F_{0}$ that corresponds to a smooth, plane-wave normalizable solution

$\displaystyle\begin{split}\lim_{r\to\infty}r\psi^{(j)}_{R_{A}}<\infty~{},\end{split}$
(276)

of the time-independent Dirac equation iff the top and bottom weights of the
spin-$j$ representation of $R_{A}$:

$\displaystyle\begin{split}\langle\mu_{top},T_{3}\rangle=-\langle\mu_{bot},T_{3}\rangle=j+\frac{1}{2}\quad,\quad\mu_{top},\mu_{bot}\in\Delta^{(j)}_{R_{A}}~{},\end{split}$
(277)

have charge of opposite sign under $\gamma_{m}$:

$\displaystyle\begin{split}\langle\mu_{top},\gamma_{m}\rangle\times\langle\mu_{bot},\gamma_{m}\rangle<0~{}.\end{split}$
(278)

Note that since the asymptotic form of the connection approaches the abelian
connection in the canonical gauge, the asymptotic form of the fermion zero-
modes can be expressed as a linear combination of the zero-momentum solutions
found in the previous subsection.

As shown in Brennan:2021ucy , for each irreducible $SU(2)_{T}$-representation
$R_{A}$ and $j$ that satisfies (278) there is a unique solution which has the
asymptotic form of

$\displaystyle\begin{split}\lim_{r\to\infty}\psi_{R_{A}}^{(j)}=\psi_{R_{A}}^{(j)-}+d\psi_{R_{A}}^{(+)}~{},\end{split}$
(279)

where $\psi_{R_{A}}^{(j)\pm}$ are given by the spin up/down solutions of
(264).

## References

  * (1) C. G. Callan, Jr., “Monopole Catalysis of Baryon Decay,” Nucl. Phys. B 212 (1983) 391–400.
  * (2) C. G. Callan, Jr., “Disappearing Dyons,” Phys. Rev. D 25 (1982) 2141.
  * (3) C. G. Callan, Jr., “Dyon-Fermion Dynamics,” Phys. Rev. D 26 (1982) 2058–2068.
  * (4) V. A. Rubakov, “Adler-Bell-Jackiw Anomaly and Fermion Number Breaking in the Presence of a Magnetic Monopole,” Nucl. Phys. B 203 (1982) 311–348.
  * (5) C. G. Callan, Jr. and S. R. Das, “Boundary Conditions on the Monopole Dirac Equation,” Phys. Rev. Lett. 51 (1983) 1155.
  * (6) C. g. Callan, Jr., “THE MONOPOLE CATALYSIS S MATRIX,” in Workshop on Problems in Unification and Supergravity.  1983\.
  * (7) B. Grossman, “DOES A DYON LEAK?,” Phys. Rev. Lett. 50 (1983) 464.
  * (8) V. P. Nair, “Breakdown of Flavor Conservation in a Monopole Background,” Phys. Rev. D 28 (1983) 2673.
  * (9) V. P. Nair, “Catalysis of Flavor Violation by Monopoles,” Phys. Rev. Lett. 51 (1983) 631.
  * (10) H. Yamagishi, “THE FERMION MONOPOLE SYSTEM REEXAMINED,” Phys. Rev. D 27 (1983) 2383–2396.
  * (11) H. Yamagishi, “The Fermion - Monopole System Reexamined. 2.,” Phys. Rev. D 28 (1983) 977.
  * (12) H. Yamagishi, “FERMION - MONOPOLE SYSTEM REEXAMINED. 3.,” Phys. Rev. D 32 (1985) 1576.
  * (13) H. Panagopoulos and H. Yamagishi, “FERMION - MONOPOLE SYSTEM REEXAMINED. 4,” Phys. Rev. D 32 (1985) 2113.
  * (14) A. Sen, “Role of Conservation Laws in the Callan-Rubakov Process with Arbitrary Number of Generation of Fermions,” Phys. Rev. Lett. 52 (1984) 1755.
  * (15) A. Sen, “Comparison of the Canonical Hamiltonian and the Hamiltonian of Callan and Rubakov for the Monopole Fermion System,” Phys. Rev. D 31 (1985) 433.
  * (16) A. P. Balachandran and J. Schechter, “The Missing Charge in the Rubakov Process,” Phys. Rev. Lett. 51 (1983) 1418.
  * (17) W. Goldstein, “ON THE INTERACTION OF FERMIONS WITH A FINITE SIZED, NONABELIAN MONOPOLE,”.
  * (18) J. Polchinski, “Monopole Catalysis: The Fermion Rotor System,” Nucl. Phys. B 242 (1984) 345–363.
  * (19) J. Preskill, “MAGNETIC MONOPOLES,” Ann. Rev. Nucl. Part. Sci. 34 (1984) 461–530.
  * (20) K. Isler, C. Schmid, and C. A. Trugenberger, “MONOPOLE CORE EXCITATIONS AND THE RUBAKOV-CALLAN EFFECT,” Nucl. Phys. B 294 (1987) 925.
  * (21) R. Kitano and R. Matsudo, “Missing final state puzzle in the monopole-fermion scattering,” arXiv:2103.13639 [hep-th].
  * (22) C. Csáki, Y. Shirman, O. Telem, and J. Terning, “Monopoles Entangle Fermions,” arXiv:2109.01145 [hep-th].
  * (23) H. Georgi and S. L. Glashow, “Unity of All Elementary Particle Forces,” Phys. Rev. Lett. 32 (1974) 438–441.
  * (24) P. B. Smith and D. Tong, “Boundary States for Chiral Symmetries in Two Dimensions,” JHEP 09 (2020) 018, arXiv:1912.01602 [hep-th].
  * (25) P. B. Smith and D. Tong, “Boundary RG Flows for Fermions and the Mod 2 Anomaly,” SciPost Phys. 10 (2021) 010, arXiv:2005.11314 [hep-th].
  * (26) P. B. Smith and D. Tong, “What Symmetries are Preserved by a Fermion Boundary State?,” arXiv:2006.07369 [hep-th].
  * (27) S. Dawson and A. N. Schellekens, “Monopole - Fermion Interactions: The Soliton Picture,” Phys. Rev. D 28 (1983) 3125.
  * (28) T. D. Brennan, “Index-like Theorem for Massless Fermions in Spherically Symmetric Monopole Backgrounds,” arXiv:2106.13820 [hep-th].
  * (29) J. P. Gauntlett, “Low-energy dynamics of N=2 supersymmetric monopoles,” Nucl. Phys. B 411 (1994) 443–460, arXiv:hep-th/9305068.
  * (30) S. Sethi, M. Stern, and E. Zaslow, “Monopole and Dyon bound states in N=2 supersymmetric Yang-Mills theories,” Nucl. Phys. B 457 (1995) 484–512, arXiv:hep-th/9508117.
  * (31) J. P. Gauntlett, N. Kim, J. Park, and P. Yi, “Monopole dynamics and BPS dyons N=2 superYang-Mills theories,” Phys. Rev. D 61 (2000) 125012, arXiv:hep-th/9912082.
  * (32) J. P. Gauntlett, C.-j. Kim, K.-M. Lee, and P. Yi, “General low-energy dynamics of supersymmetric monopoles,” Phys. Rev. D 63 (2001) 065020, arXiv:hep-th/0008031.
  * (33) T. D. Brennan and G. W. Moore, “A note on the semiclassical formulation of BPS states in four-dimensional $N=$ 2 theories,” PTEP 2016 no. 12, (2016) 12C110, arXiv:1610.00697 [hep-th].
  * (34) T. D. Brennan, G. W. Moore, and A. B. Royston, “Wall Crossing from Dirac Zeromodes,” JHEP 09 (2018) 038, arXiv:1805.08783 [hep-th].
  * (35) G. W. Moore, A. B. Royston, and D. Van den Bleeken, “Parameter counting for singular monopoles on $\mathbb{R}^{3}$,” JHEP 10 (2014) 142, arXiv:1404.5616 [hep-th].
  * (36) J. M. Maldacena and A. W. W. Ludwig, “Majorana fermions, exact mapping between quantum impurity fixed points with four bulk fermion species, and solution of the ’unitarity puzzle’,” Nucl. Phys. B 506 (1997) 565–588, arXiv:cond-mat/9502109.
  * (37) I. Affleck and J. Sagi, “Monopole catalyzed baryon decay: A Boundary conformal field theory approach,” Nucl. Phys. B 417 (1994) 374–402, arXiv:hep-th/9311056.
  * (38) D. Wilkinson and A. S. Goldhaber, “Spherically Symmetric Monopoles,” Phys. Rev. D 16 (1977) 1221.
  * (39) D. Wilkinson and F. A. Bais, “Exact SU(N) Monopole Solutions with Spherical Symmetry,” Phys. Rev. D 19 (1979) 2410.
  * (40) R. Jackiw and C. Rebbi, “Solitons with Fermion Number 1/2,” Phys. Rev. D 13 (1976) 3398–3409.
  * (41) C. L. Gardner, “SELFDUAL SU(5) MONOPOLE SOLUTIONS,” Phys. Lett. B 142 (1984) 379–382.
  * (42) C. L. Gardner and J. A. Harvey, “Stable Grand Unified Monopoles With Multiple Dirac Charge,” Phys. Rev. Lett. 52 (1984) 879.