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\title[Bounds of third and fourth Hankel determinants]{Bounds of third and fourth Hankel determinants for a generalized subclass of bounded turning functions subordinated to sine function}
\author{Gagandeep Singh}
\address{Department of Mathematics, \\ Khalsa College, \\
Amritsar-143001, Punjab, India}
\email{kamboj.gagandeep@yahoo.in}
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\author{Gurcharanjit Singh}
\address{Department of Mathematics, \\ G.N.D.U. College, Chungh,\\
Tarn-Taran-143304, Punjab, India}
\email{dhillongs82@yahoo.com}
%
\subjclass{30C45, 30C50, 30C80}
\keywords{Analytic functions, Subordination, Coefficient inequalities, sine function, third Hankel determinant, fourth Hankel determinant.}
\begin{abstract}
The objective of this paper is to investigate the bounds of third and fourth Hankel determinants for a generalized subclass of bounded turning functions associated with sine function, in the open unit disc $E=\{z\in\mathbb{C}: |z|<1\}$. The results are also extended to two-fold and three-fold symmetric functions. This investigation will generalize the resuls of some earlier works.
\end{abstract}
\maketitle

\section{Introduction}
Let the complex plane is expressed by $\mathbb{C}$. By $\mathcal{A}$, let us denote the class of analytic functions of the form $f(z)=z+\sum_{k=2}^\infty a_{k}z^{k}$, defined in the open unit disc $E=\{z\in\mathbb{C}: |z|<1\}$ and normalized by the conditions $f(0)=f'(0)-1=0$. By $\mathcal{S}$, we denote the subclass of $\mathcal{A}$ which consists of univalent functions in $E$.\\ 

Let $f$ and $g$ be two analytic functions  in $E$. We say that $f$ is subordinate to $g$ (denoted as $f\prec g$) if there exists a function $w$ with $w(0)=0$ and $|w(z)|<1$ for $z\in E$ such that $f(z)=g(w(z))$. Further, if $g$ is univalent in $E$, then the subordination leads to $f(0)=g(0)$ and $f(E)\subset g(E)$.\\

In the theory of univalent functions, Bieberbach~\cite{bb} stated a result that, for $f\in\mathcal{S}$, $|a_{n}|\leq n$, $n=2,3,...$. This result is known as Bieberbach's conjecture and it remained as a challenge for the mathematicians for a long time. Finally, L. De-Branges~\cite{db}, proved this conjecture in 1985. During the course of proving this conjecture, various results related to the coefficients were come into existence which gave rise to some new subclasses of analytic functions.\\ 

For better understanding of the main content, let's have a look on some fundamental subclasses of $\mathcal{A}$:\\
$\mathcal{S}^{*}=\left\{f:f\in\mathcal{A}, Re\left(\dfrac{zf'(z)}{f(z)}\right)>0 \;\text{or}\; \dfrac{zf'(z)}{f(z)}\prec\dfrac{1+z}{1-z}, z\in E\right\}$, the class of starlike functions.\\

$\mathcal{K}=\left\{f:f\in\mathcal{A}, Re\left(\dfrac{(zf'(z))'}{f'(z)}\right)>0 \;\text{or}\; \dfrac{(zf'(z))'}{f'(z)}\prec\dfrac{1+z}{1-z}, z\in E\right\}$, the class of convex functions.\\

Reade~\cite{reade} introduced the class $\mathcal{CS^{*}}$ of close-to-star functions which is defined as 
$\mathcal{CS}^{*}=\left\{f:f\in\mathcal{A}, Re\left(\dfrac{f(z)}{g(z)}\right)>0 \;\text{or}\; \dfrac{f(z)}{g(z)}\prec\dfrac{1+z}{1-z}, g\in\mathcal{S}^{*}, z\in E\right\}$. Further for $g(z)=z$, MacGregor ~\cite{mac1} studied the following subclass of close-to-star functions:
\begin{equation}\nonumber
\mathcal{R}^{'}=\left\{f:f\in\mathcal{A}, Re\left(\dfrac{f(z)}{z}\right)>0 \;\text{or}\; \dfrac{f(z)}{z}\prec\dfrac{1+z}{1-z}, z\in E\right\}.
\end{equation}

MacGregor~\cite{mac} established a very useful class $\mathcal{R}$ of bounded turning functions which is defined as
\begin{equation}\nonumber
\mathcal{R}=\left\{f:f\in\mathcal{A}, Re(f'(z))>0 \;\text{or}\; f'(z)\prec\dfrac{1+z}{1-z}, z\in E\right\}.
\end{equation}

Later on, Murugusundramurthi and Magesh~\cite{murugu} studied the following class:
\begin{equation}\nonumber 
\mathcal{R}(\alpha)=\left\{f:f\in\mathcal{A}, Re\left((1-\alpha)\dfrac{f(z)}{z}+\alpha f'(z)\right)>0, 0\leq\alpha\leq 1, z\in E\right\}.
\end{equation}
Particularly, $\mathcal{R}(1)\equiv\mathcal{R}$ and $\mathcal{R}(0)\equiv\mathcal{R}^{'}$.\\

Various subclasses of $\mathcal{S}$ were investigated by associating to different functions. Recently, Arif et al.~\cite{arif}, Cho et al.~\cite{cho} and Khan et al.~\cite{khan} studied the classes $\mathcal{S}^{*}_{sin}$, $\mathcal{K}_{sin}$ and $\mathcal{R}_{sin}$, which are the subclasses of starlike functions, convex functions and bounded turning functions associated with sine function, respectively. Getting motivated by these works, now we define the following class of analytic functions by subordinating to $1+sinz$.\\

\begin{definition}
 A function $f\in\mathcal{A}$ is said to be in the class $\mathcal{R}^{\alpha}_{sin}$ $(0\leq\alpha\leq 1)$ if it satisfies the condition   
\begin{equation}\nonumber
(1-\alpha)\dfrac{f(z)}{z}+\alpha f'(z)\prec 1+sinz.
\end{equation}
\end{definition}
We have the following observations:\\
(i) $\mathcal{R}^{0}_{sin}\equiv\mathcal{R}^{'}_{sin}$.\\
(ii) $\mathcal{R}^{1}_{sin}\equiv\mathcal{R}_{sin}$.\\
 
 For $q\geq 1$ and $n\geq 1$, Pommerenke \cite{pom1} introduced  the $q^{th}$ Hankel determinant as                          \begin{equation}\nonumber
  H_{q}(n)=
   \begin{vmatrix}
   a_{n} & a_{n+1}&...&a_{n+q-1}\\ a_{n+1} & ...&...&...\\ ... & ...&...&...\\a_{n+q-1} & ...&...&a_{n+2q-2}
   \end{vmatrix}.
\end{equation}               

For specific values of $q$ and $n$, the Hankel determinant $H_{q}(n)$ reduces to the following functionals:\\
(i) For $q=2$ and $n=1$, it redues to $H_{2}(1)=a_{3}-a_{2}^{2}$, which is the well known Fekete-Szeg\"{o} functional.\\
(ii) For $q=2$ and $n=2$, the Hankel determinant takes the form of $H_{2}(2)=a_{2}a_{4}-a_{3}^{2}$, which is known as Hankel determinant of second order.\\
(iii) For $q=3$ and $n=1$, the Hankel determinant reduces to $H_{3}(1)$, which is the Hankel determinant of third order.\\
(iv) For $q=4$ and $n=1$, $H_{q}(n)$ reduces to $H_{4}(1)$, which is the Hankel determinant of fourth order.\\

Ma~\cite{ma} introduced the functional $J_{n,m}(f)=a_{n}a_{m}-a_{m+n-1}$, $n,m\in\mathbb{N}-\{1\}$, which is known as generalized Zalcman functional. The functional $J_{2,3}(f)=a_{2}a_{3}-a_{4}$ is a specific case of the generalized Zalcman functional. The upper bound for the functional $J_{2,3}(f)$ over different subclasses of analytic functions was computed by various authors. It is very useful in establishing the bounds for the third Hankel determinant.\\ 

On expanding, the third Hankel determinant can be expressed as 
\begin{equation}\nonumber
H_{3}(1)=a_{3}(a_{2}a_{4}-a_{3}^{2})-a_{4}(a_{4}-a_{2}a_{3})+a_{5}(a_{3}-a_{2}^{2}),
\end{equation}
and after applying the triangle inequality, it yields  
\begin{equation}\label{equ1}
|H_{3}(1)|\leq|a_{3}||a_{2}a_{4}-a_{3}^{2}|+|a_{4}||a_{2}a_{3}-a_{4}|+|a_{5}||a_{3}-a_{2}^{2}|.
\end{equation} 

Also the expansion of fourth Hankel determinant can be expressed as\\
$H_{4}(1)=a_{7}H_{3}(1)-2a_{4}a_{6}(a_{2}a_{4}-a_{3}^{2})-2a_{5}a_{6}(a_{2}a_{3}-a_{4})-a_{6}^{2}(a_{3}-a_{2}^{2})$
\begin{equation}\label{equ2}
+a_{5}^{2}(a_{2}a_{4}-a_{3}^{2})+a_{5}^{2}(a_{2}a_{4}+2a_{3}^{2})-a_{5}^{3}+a_{4}^{4}-3a_{3}a_{4}^{2}a_{5}.
\end{equation}

A lot of work has been done on the estimation of second Hankel determinant by various authors including Noor~\cite{noor}, Ehrenborg~\cite{eh}, Layman~\cite{lay}, Singh~\cite{singh}, Mehrok and Singh~\cite{meh} and Janteng et al.~\cite{j}. The estimation of third Hankel determinant is little bit complicated. Babalola~\cite{baba} was the first researcher who successfully obtained the upper bound of third Hankel determinant for the classes of starlike functions, convex functions and the class of functions with bounded boundary rotation. Further a few researchers including Shanmugam et al.~\cite{shan}, Bucur et al.~\cite{bu}, Altinkaya and Yalcin~\cite{sah}, Singh and Singh~\cite{ss} have been actively engaged in the study of third Hankel determinant for various subclasses of analytic functions. Now a days, the study of fourth Hankel determinant for various subclasses of analytic functions, is an active topic of research. A few authors including Arif et al. ~\cite{arif1}, Singh et al.~\cite{sss, sss1} and Zhang and Tang~\cite{zhang} established the bounds of fourth Hankel determinant for certain subclasses of $\mathcal{A}$.\\

In this paper, we establish the upper bounds of the third and fourth Hankel determinants for the class $\mathcal{R}^{\alpha}_{sin}$. Also various known results follow as particular cases.\\  

Let $\mathcal{P}$ denote the class of analytic functions $p$ of the form 
  \begin{equation}\nonumber
    p(z)=1+\sum_{k=1}^\infty p_{k}z^{k},
\end{equation}
whose real parts are positive in $E$.\\

In order to prove our main results, the following lemmas have been used:\\
\begin{lemma}~\cite{arif}
 If $p\in\mathcal{P}$, then
\begin{equation}\nonumber
|p_{k}|\leq 2, k\in\mathbb{N},
\end{equation}
\begin{equation}\nonumber
\left|p_{2}-\dfrac{p_{1}^{2}}{2}\right|\leq 2-\dfrac{|p_{1}|^{2}}{2}, 
\end{equation}
\begin{equation}\nonumber
|p_{i+j}-\mu p_{i}p_{j}|\leq 2,  0\leq\mu\leq 1,
\end{equation}
\begin{equation}\nonumber
|p_{n+2k}-\lambda p_{n}p_{k}^{2}|\leq 2(1+2\lambda),  (\lambda\in\mathbb{R}),
\end{equation}
\begin{equation}\nonumber
|p_{m}p_{n}-p_{k}p_{l}|\leq 4, (m+n=k+l; m,n\in\mathbb{N}),
\end{equation}
and for complex number $\rho$, we have
\begin{equation}\nonumber
|p_{2}-\rho p_{1}^{2}|\leq 2max\{1,|2\rho-1|\}. 
\end{equation}
\end{lemma}

\begin{lemma}~\cite{arif} Let $p\in\mathcal{P}$, then 
\begin{equation}\nonumber
|Jp_{1}^{3}-Kp_{1}p_{2}+Lp_{3}|\leq 2|J|+2|K-2J|+2|J-K+L|.
\end{equation}
In particular, it is proved in~\cite{pom} that
\begin{equation}\nonumber
|p_{1}^{3}-2p_{1}p_{2}+p_{3}|\leq 2.
\end{equation}
\end{lemma}

\begin{lemma}~\cite{lz1,lz2} If $p\in\mathcal{P}$, then
\begin{equation}\nonumber
2p_{2}=p_{1}^{2}+(4-p_{1}^{2})x,
\end{equation}
\begin{equation}\nonumber
4p_{3}=p_{1}^{3}+2p_{1}(4-p_{1}^{2})x-p_{1}(4-p_{1}^{2})x^{2}+2(4-p_{1}^{2})(1-|x|^{2})z,
\end{equation} 
for $|x|\leq 1$ and $|z|\leq 1$.\\
\end{lemma}

\begin{lemma}~\cite{ravi} Let $m,n,l$ and $r$ satisfy the inequalities $0<m<1$,\\ $0<r<1$ and 
$8r(1-r)\left[(mn-2l)^{2}+(m(r+m)-n)^{2}\right]+m(1-m)(n-2rm)^{2}\leq 4m^{2}(1-m)^{2}r(1-r)$.\\
If $p\in\mathcal{P}$, then
\begin{equation}\nonumber
\left|lp_{1}^{4}+rp_{2}^{2}+2mp_{1}p_{3}-\dfrac{3}{2}np_{1}^{2}p_{2}-p_{4}\right|\leq 2.
\end{equation}
\end{lemma}

\section{Coefficient Bounds for the class $\mathcal{R}^{\alpha}_{sin}$}

\begin{theorem} 
If $f\in \mathcal{R}^{\alpha}_{sin}$, then
\begin{equation}\label{equ3}
|a_{2}|\leq\dfrac{1}{1+\alpha},
\end{equation}
\begin{equation}\label{equ4}
|a_{3}|\leq\dfrac{1}{1+2\alpha},
\end{equation}
\begin{equation}\label{equ5}
|a_{4}|\leq\dfrac{1}{1+3\alpha},
\end{equation}
and 
\begin{equation}\label{equ6}
|a_{5}|\leq\dfrac{1}{1+4\alpha}.
\end{equation}
\end{theorem}

\begin{proof}
 Since $f\in\mathcal{R}^{\alpha}_{sin}$, by the principle of subordination, we have
\begin{equation}\label{equ7}
(1-\alpha)\dfrac{f(z)}{z}+\alpha f'(z)=1+sin(w(z)).
\end{equation}
Define $p(z)=\dfrac{1+w(z)}{1-w(z)}=1+p_{1}z+p_{2}z^{2}+p_{3}z^{3}+...$, which implies $w(z)=\dfrac{p(z)-1}{p(z)+1}$.\\
On expanding, we have
\begin{equation}\label{equ8}
(1-\alpha)\frac{f(z)}{z}+\alpha f'(z)=1+(1+\alpha)a_{2}z+(1+2\alpha)a_{3}z^{2}+(1+3\alpha)a_{4}z^{3}+(1+4\alpha)a_{5}z^{4}+...
\end{equation}
Also\\
$1+sin(w(z))=1+\dfrac{1}{2}p_{1}z+\left(\dfrac{p_{2}}{2}-\dfrac{p_{1}^{2}}{4}\right)z^{2}$
\begin{equation}\label{equ9}
+\left(\dfrac{5p_{1}^{3}}{48}-\dfrac{p_{1}p_{2}}{2}+\dfrac{p_{3}}{2}\right)z^{3}+\left(-\dfrac{p_{1}^{4}}{32}+\dfrac{5p_{1}^{2}p_{2}}{16}-\dfrac{p_{3}p_{1}}{2}-\dfrac{p_{2}^{2}}{4}+\dfrac{p_{4}}{2}\right)z^{4}+...
\end{equation}
Using~(\ref{equ8}) and ~(\ref{equ9}), ~(\ref{equ7}) yields\\
$1+(1+\alpha)a_{2}z+(1+2\alpha)a_{3}z^{2}+(1+3\alpha)a_{4}z^{3}+(1+4\alpha)a_{5}z^{4}+...$\\
$=1+\frac{1}{2}p_{1}z+\left(\frac{p_{2}}{2}-\frac{p_{1}^{2}}{4}\right)z^{2}+\left(\frac{5p_{1}^{3}}{48}-\frac{p_{1}p_{2}}{2}+\frac{p_{3}}{2}\right)z^{3}$
\begin{equation}\label{equ10}
+\left(-\frac{p_{1}^{4}}{32}+\frac{5p_{1}^{2}p_{2}}{16}-\frac{p_{3}p_{1}}{2}-\frac{p_{2}^{2}}{4}+\frac{p_{4}}{2}\right)z^{4}+...
\end{equation}
Equating the coefficients of $z$, $z^{2}$, $z^{3}$ and $z^{4}$ in ~(\ref{equ10}) and on simplification, we obtain
\begin{equation}\label{equ11}
a_{2}=\dfrac{1}{2(1+\alpha)}p_{1},
\end{equation}
\begin{equation}\label{equ12}
a_{3}=\frac{1}{1+2\alpha}\left[\frac{p_{2}}{2}-\dfrac{p_{1}^{2}}{4}\right],
\end{equation}
\begin{equation}\label{equ13}
a_{4}=\frac{1}{48(1+3\alpha)}\left[5p_{1}^{3}-24p_{1}p_{2}+24p_{3}\right],
\end{equation}
and
\begin{equation}\label{equ14}
a_{5}=\frac{1}{2(1+4\alpha)}\left[\dfrac{p_{1}^{4}}{16}+\dfrac{p_{2}^{2}}{2}+p_{3}p_{1}-\dfrac{5p_{1}^{2}p_{2}}{8}-p_{4}\right].
\end{equation}

Using first inequality of Lemma 1.2 in ~(\ref{equ11}), the result ~(\ref{equ3}) is obvious.\\
From ~(\ref{equ12}), we have
\begin{equation}\label{equ15}
|a_{3}|=\dfrac{1}{2(1+2\alpha)}\left|p_{2}-\dfrac{1}{2}p_{1}^{2}\right|.
\end{equation}
Using sixth inequality of Lemma 1.2 in ~(\ref{equ15}), the result ~(\ref{equ4}) can be easily obtained.\\
~(\ref{equ13}) can be expressed as
\begin{equation}\label{equ16}
\left|a_{4}\right|=\frac{1}{48(1+3\alpha)}\left|5p_{1}^{3}-24p_{1}p_{2}+24p_{3}\right|.
\end{equation}
On applying Lemma 1.3 in ~(\ref{equ16}), the result ~(\ref{equ5}) is obvious.\\
Further, on using Lemma 1.5 in ~(\ref{equ14}), the result ~(\ref{equ6}) is obvious.\\
\end{proof}

On putting $\alpha=0$, Theorem 2.1 yields the following result: \\
\begin{remark} 
If $f\in \mathcal{R}^{'}_{sin}$, then
\begin{equation}\nonumber
|a_{2}|\leq 1,  |a_{3}|\leq 1, |a_{4}|\leq 1, |a_{5}|\leq 1.
\end{equation}
\end{remark}

For $\alpha=1$, Theorem 2.1 gives the following result due to Khan et al.~\cite{khan}:\\
\begin{remark} 
If $f\in \mathcal{R}_{sin}$, then
\begin{equation}\nonumber
|a_{2}|\leq\dfrac{1}{2},  |a_{3}|\leq\dfrac{1}{3}, |a_{4}|\leq\dfrac{1}{4}, |a_{5}|\leq\dfrac{1}{5}.
\end{equation}
\end{remark}

{\bf Conjecture} If $f\in \mathcal{R}^{\alpha}_{sin}$, then
\begin{equation}\nonumber
|a_{n}|\leq\dfrac{1}{1+(n-1)\alpha}, n\geq 2.
\end{equation}

\begin{theorem} 
If $f\in \mathcal{R}^{\alpha}_{sin}$ and $\mu$ is any complex number, then
\begin{equation}\label{equ17}
|a_{3}-\mu a_{2}^{2}|\leq\dfrac{1}{1+2\alpha}max\left\{1,\frac{(1+2\alpha)}{(1+\alpha)^{2}}|\mu|\right\}.
\end{equation}
\end{theorem}

\begin{proof}
 From ~(\ref{equ11}) and ~(\ref{equ12}), we obtain
\begin{equation}\label{equ18}
|a_{3}-\mu a_{2}^{2}|=\dfrac{1}{2(1+2\alpha)}\left|p_{2}-\dfrac{(1+\alpha)^{2}+\mu(1+2\alpha)}{2(1+\alpha)^{2}}p_{1}^{2}\right|.
\end{equation}
Using sixth inequality of Lemma 1.2, ~(\ref{equ18}) takes the form
\begin{equation}\label{equ19}
|a_{3}-\mu a_{2}^{2}|\leq\dfrac{1}{1+2\alpha}max\left\{1,\dfrac{(1+2\alpha)}{(1+\alpha)^{2}}|\mu|\right\}.
\end{equation}
\end{proof}

Substituting for $\alpha=0$, Theorem 2.4 yields the following result:\\
\begin{remark} 
If $f\in \mathcal{R}^{'}_{sin}$, then
\begin{equation}\nonumber
|a_{3}-\mu a_{2}^{2}|\leq max\left\{1,|\mu|\right\}.
\end{equation}
\end{remark}

Putting $\alpha=1$, Theorem 2.4 yields the following result due to Khan et al.~\cite{khan}:\\
\begin{remark} 
If $f\in \mathcal{R}_{sin}$, then
\begin{equation}\nonumber
|a_{3}-\mu a_{2}^{2}|\leq max\left\{\frac{1}{3},\frac{1}{4}|\mu|\right\}.
\end{equation}
\end{remark}

Putting $\mu=1$, Theorem 2.4 yields the following result:\\
\begin{remark} 
If $f\in \mathcal{R}^{\alpha}_{sin}$, then
\begin{equation}\nonumber
|a_{3}-a_{2}^{2}|\leq\dfrac{1}{1+2\alpha}.
\end{equation}
\end{remark}

\begin{theorem} 
If $f\in\mathcal{R}^{\alpha}_{sin}$, then
\begin{equation}\label{equ20}
|a_{2}a_{3}-a_{4}|\leq\dfrac{1}{1+3\alpha}.
\end{equation}
\end{theorem}

\begin{proof}
 Using ~(\ref{equ11}), ~(\ref{equ12}), ~(\ref{equ13}) and after simplification, we have\\
$|a_{2}a_{3}-a_{4}|=\frac{1}{48(1+\alpha)(1+2\alpha)(1+3\alpha)}$
\begin{equation}\label{equ21}
.\left|(11+33\alpha+10\alpha^{2})p_{1}^{3}-(36+108\alpha+48\alpha^{2})p_{1}p_{2}+24(1+\alpha)(1+2\alpha)p_{3}\right|.
\end{equation}
On applying Lemma 1.3 in ~(\ref{equ21}), it yields ~(\ref{equ20}).\\
\end{proof}

For $\alpha=0$, the following result is a consequence of Theorem 2.8:\\
\begin{remark} 
If $f\in\mathcal{R}^{'}_{sin}$, then
\begin{equation}\nonumber
|a_{2}a_{3}-a_{4}|\leq 1.
\end{equation}
\end{remark}

On putting $\alpha=1$ in Theorem 2.8, we can obtain the following result due to Khan et al.~\cite{khan}:\\
\begin{remark} 
If $f\in\mathcal{R}_{sin}$, then
\begin{equation}\nonumber
|a_{2}a_{3}-a_{4}|\leq\dfrac{1}{4}.
\end{equation}
\end{remark}

\begin{theorem} 
If $f\in\mathcal{R}^{\alpha}_{sin}$, then
\begin{equation}\label{equ22}
|a_{2}a_{4}-a_{3}^{2}|\leq\dfrac{1}{(1+2\alpha)^{2}}.
\end{equation}
\end{theorem}

\begin{proof}
Using ~(\ref{equ11}), ~(\ref{equ12}) and ~(\ref{equ13}), we have\\
$|a_{2}a_{4}-a_{3}^{2}|=\dfrac{1}{96(1+\alpha)(1+2\alpha)^{2}(1+3\alpha)}$ 
\begin{equation}\nonumber
.\left|24(1+2\alpha)^{2}p_{1}p_{3}-24\alpha^{2}p_{1}^{2}p_{2}+(-1-4\alpha+2\alpha^{2})p_{1}^{4}-24(1+\alpha)(1+3\alpha)p_{2}^{2}\right|.
\end{equation}
Substituting for $p_{2}$ and $p_{3}$ from Lemma 1.4 and letting $p_{1}=p$, we get\\
$|a_{2}a_{4}-a_{3}^{2}|=\dfrac{1}{96(1+\alpha)(1+2\alpha)^{2}(1+3\alpha)}\bigg|-(4\alpha^{2}+4\alpha+1)p^{4}$
\begin{equation}\nonumber
-6(1+2\alpha)^{2}p^{2}(4-p^{2})x^{2}-6(1+\alpha)(1+3\alpha)(4-p^{2})^{2}x^{2}+12(1+2\alpha)^{2}p(4-p^{2})(1-|x|^{2})z\bigg|.
\end{equation}
Since $|p|=|p_{1}|\leq 2$, we may assume that $p\in[0,2]$. By using triangle inequality and $|z|\leq 1$ with $|x|=t\in [0,1]$, we obtain\\
$|a_{2}a_{4}-a_{3}^{2}|\leq\frac{1}{96(1+\alpha)(1+2\alpha)^{2}(1+3\alpha)}\bigg[(4\alpha^{2}+4\alpha+1)p^{4}+6(1+2\alpha)^{2}p^{2}(4-p^{2})t^{2}$
\begin{equation}\nonumber
+6(1+\alpha)(1+3\alpha)(4-p^{2})^{2}t^{2}+12(1+2\alpha)^{2}p(4-p^{2})-12(1+2\alpha)^{2}p(4-p^{2})t^{2}\bigg]=F(p,t).
\end{equation}
\begin{equation}\nonumber
\dfrac{\partial F}{\partial t}=\dfrac{(4-p^{2})t}{8(1+\alpha)(1+2\alpha)^{2}(1+3\alpha)}\bigg[\alpha^{2}p^{2}-2(1+2\alpha)^{2}p+4(1+\alpha)(1+3\alpha)\bigg]\geq 0,
\end{equation}
and so $F(p,t)$ is an increasing function of $t$ for $p\leq\frac{3}{2}$.\\
Therefore, $max\{F(p,t)\}=F(p,1)=\frac{1}{192(1+\alpha)(1+2\alpha)^{2}(1+3\alpha)}\bigg[(\alpha^{2}+4\alpha+1)p^{4}$\\
$+12\alpha^{2}p^{2}(4-p^{2})+12(1+2\alpha)^{2}p^{2}(4-p^{2})+12(1+\alpha)(1+3\alpha)(4-p^{2})^{2}\bigg]=H(p)$.\\
$H'(p)=0$ gives $p=0$. Also $H''(p)<0$ for $p=0$.\\
This implies $max\{H(p)\}=H(0)=\dfrac{1}{(1+2\alpha)^{2}}$, which proves ~(\ref{equ22}).\\
\end{proof}

Putting $\alpha=0$, Theorem 2.11 gives the following result:\\
\begin{remark} 
If $f\in\mathcal{R}^{'}_{sin}$, then
\begin{equation}\nonumber
|a_{2}a_{4}-a_{3}^{2}|\leq 1.
\end{equation}
\end{remark}

Substituting for $\alpha=1$ in Theorem 2.11, the following result due to Khan et al.~\cite{khan}, is obvious:\\
\begin{remark} 
If $f\in\mathcal{R}_{sin}$, then
\begin{equation}\nonumber
|a_{2}a_{4}-a_{3}^{2}|\leq\dfrac{1}{9}.
\end{equation}
\end{remark}

\begin{theorem} 
If $f\in\mathcal{R}^{\alpha}_{sin}$, then
\begin{equation}\label{equ23}
|H_{3}(1)|\leq\dfrac{(2+8\alpha+4\alpha^{2})(1+3\alpha)^{2}+(1+4\alpha)(1+2\alpha)^{3}}{(1+2\alpha)^{3}(1+3\alpha)^{2}(1+4\alpha)}.
\end{equation}
\end{theorem}

\begin{proof}
By using ~(\ref{equ4}), ~(\ref{equ5}), ~(\ref{equ6}), ~(\ref{equ20}), ~(\ref{equ22}) and Remark 2.7 in ~(\ref{equ1}), the result ~(\ref{equ23}) can be easily obtained.\\
\end{proof}

For $\alpha=0$, Theorem 2.14 yields the following result:\\
\begin{remark} 
If $f\in\mathcal{R}^{'}_{sin}$, then
\begin{equation}\nonumber
|H_{3}(1)|\leq 3.
\end{equation}
\end{remark}

For $\alpha=1$, Theorem 2.14 yields the following result due to Khan et al.~\cite{khan}:\\
\begin{remark} 
If $f\in\mathcal{R}_{sin}$, then
\begin{equation}\nonumber
|H_{3}(1)|\leq\dfrac{359}{2160}.
\end{equation}
\end{remark}

\begin{theorem} 
If $f\in\mathcal{R}^{\alpha}_{sin}$, then\\
$|H_{4}(1)|\leq\frac{2}{(1+2\alpha)^{2}(1+4\alpha)}\left[\frac{1+4\alpha+2\alpha^{2}}{(1+2\alpha)(1+6\alpha)}+\frac{3+12\alpha+2\alpha^{2}}{(1+4\alpha)^{2}}+\frac{2+8\alpha+4\alpha^{2}}{(1+3\alpha)(1+5\alpha)}\right]$
\begin{equation}\nonumber
+\frac{1}{(1+3\alpha)^{2}}\left[\frac{2+12\alpha+9\alpha^{2}}{(1+6\alpha)(1+3\alpha)^{2}}+\frac{3}{(1+2\alpha)(1+4\alpha)}\right].
\end{equation}
\end{theorem}

\begin{proof}
 We have
\begin{equation}\nonumber
|a_{2}a_{4}+2a_{3}^{2}|\leq|a_{2}a_{4}-a_{3}^{2}|+3|a_{3}|^{2}.
\end{equation}
Applying the triangle inequality in ~(\ref{equ2}) and using the above inequality along with Theorem 2.1, Theorem 2.4, Theorem 2.8, Theorem 2.11 and Theorem 2.14, the proof of the Theorem 2.17 is obvious.\\ 
\end{proof}

For $\alpha=0$, Theorem 2.17 yields the following result:\\
\begin{remark} 
If $f\in\mathcal{R}^{'}_{sin}$, then
\begin{equation}\nonumber
|H_{4}(1)|\leq 17.
\end{equation}
\end{remark}

For $\alpha=1$, Theorem 2.17 yields the following result due to Khan et al.~\cite{khan}:\\
\begin{remark} 
If $f\in\mathcal{R}_{sin}$, then
\begin{equation}\nonumber
|H_{4}(1)|\leq 0.10556.
\end{equation}
\end{remark}

\section{Bounds of $|H_{3}(1)|$ for Two-fold and Three-fold symmetric functions}

A function $f$ is said to be $n$-fold symmetric if is satisfy the following condition:
\begin{equation}\nonumber
f(\xi z)=\xi f(z)
\end{equation} 
where $\xi=e^{\frac{2\pi i}{n}}$ and $z\in E$.\\
By $S^{(n)}$, we denote the set of all $n$-fold symmetric functions which belong to the class $S$.\\
The $n$-fold univalent function have the following Taylor-Maclaurin series:
\begin{equation}\label{equ24}
f(z)=z+\sum_{k=1}^\infty a_{nk+1}z^{nk+1}.
\end{equation}
An analytic function $f$ of the form~(\ref{equ24}) belongs to the family $\mathcal{R}^{\alpha(n)}_{sin}$ if and only if
 \begin{equation}\nonumber
(1-\alpha)\dfrac{f(z)}{z}+\alpha f'(z)=1+sin\left({\frac{p(z)-1}{p(z)+1}}\right), p\in\mathcal{P}^{(n)},
\end{equation}
where 
\begin{equation}\label{equ25}
\mathcal{P}^{n}=\left\{p\in\mathcal{P}:p(z)=1+\sum_{k=1}^\infty p_{nk}z^{nk}, z\in E\right\}.
\end{equation}

\begin{theorem}
If $f\in\mathcal{R}^{\alpha(2)}_{sin}$, then\\
\begin{equation}\label{equ26}
|H_{3}(1)|\leq\dfrac{1}{(1+2\alpha)(1+4\alpha)}.
\end{equation}
\end{theorem}

\begin{proof}
If $f\in\mathcal{R}^{\alpha(2)}_{sin}$, so there exists a function $p\in\mathcal{P}^{(2)}$ such that
\begin{equation}\label{equ27}
(1-\alpha)\dfrac{f(z)}{z}+\alpha f'(z)=1+sin\left(\frac{p(z)-1}{p(z)+1}\right).
\end{equation}
Using~(\ref{equ24}) and ~(\ref{equ25}) for $n=2$, ~(\ref{equ27}) yields
\begin{equation}\label{equ28}
a_{3}=\frac{1}{2(1+2\alpha)}p_{2},
\end{equation}
\begin{equation}\label{equ29}
a_{5}=\frac{1}{2(1+4\alpha)}\left(p_{4}-\frac{1}{2}p_{2}^{2}\right).
\end{equation}
Also 
\begin{equation}\label{equ30}
H_{3}(1)=a_{3}a_{5}-a_{3}^{3}. 
\end{equation}
Using~(\ref{equ28}) and ~(\ref{equ29}) in ~(\ref{equ30}), it yields
\begin{equation}\label{equ31}
H_{3}(1)=\dfrac{1}{4(1+2\alpha)(1+4\alpha)}p_{2}\left[p_{4}-\dfrac{(1+2\alpha)^{2}+(1+4\alpha)}{2(1+2\alpha)^{2}}p_{2}^{2}\right]. 
\end{equation}
On applying triangle inequality in ~(\ref{equ31}) and using fourth inequality of Lemma 1.2, we can easily get the result ~(\ref{equ26}).\\
\end{proof}

Putting $\alpha=0$, the following result can be easily obtained from Theorem 3.1:\\
\begin{remark}
If $f\in\mathcal{R}^{'(2)}_{sin}$, then
\begin{equation}\nonumber
|H_{3}(1)|\leq 1.
\end{equation}
\end{remark}

For $\alpha=1$, Theorem 3.1 agrees with the following result:\\
\begin{remark}
If $f\in\mathcal{R}^{(2)}_{sin}$, then
\begin{equation}\nonumber
|H_{3}(1)|\leq\dfrac{1}{15}.
\end{equation}
\end{remark}

\begin{theorem}
If $f\in\mathcal{R}^{\alpha(3)}_{sin}$, then\\
\begin{equation}\label{equ32}
|H_{3}(1)|\leq\dfrac{1}{(1+3\alpha)^{2}}.
\end{equation}
\end{theorem}

\begin{proof}
If $f\in\mathcal{R}^{\alpha(3)}_{sin}$, so there exists a function $p\in\mathcal{P}^{(3)}$ such that
\begin{equation}\label{equ33}
(1-\alpha)\dfrac{f(z)}{z}+\alpha f'(z)=1+sin\left(\frac{p(z)-1}{p(z)+1}\right).
\end{equation}
Using~(\ref{equ24}) and ~(\ref{equ25}) for $n=3$, ~(\ref{equ33}) gives
\begin{equation}\label{equ34}
a_{4}=\frac{1}{2(1+3\alpha)}p_{3}.
\end{equation}
Also 
\begin{equation}\label{equ35}
H_{3}(1)=-a_{4}^{2}. 
\end{equation}
Using~(\ref{equ34}) in ~(\ref{equ35}), it yields
\begin{equation}\label{equ36}
H_{3}(1)=-\dfrac{1}{4(1+3\alpha)^{2}}p_{3}^{2}. 
\end{equation}
On applying triangle inequality and using first inequality of Lemma 1.2, ~(\ref{equ32}) can be easily obtained.\\
\end{proof}

For $\alpha=0$, Theorem 3.4 yields the following result:\\
\begin{remark}
If $f\in\mathcal{R}^{'(3)}_{sin}$, then
\begin{equation}\nonumber
|H_{3}(1)|\leq 1.
\end{equation}
\end{remark}

For $\alpha=1$, Theorem 3.4 yields the following result:\\
\begin{remark}
If $f\in\mathcal{R}^{(3)}_{sin}$, then
\begin{equation}\nonumber
|H_{3}(1)|\leq\dfrac{1}{16}.
\end{equation}
\end{remark}

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\end{document}
