# Certain subclasses of Analytic Functions with Fixed Second Coefficient

Sushil Kumar  Bharati Vidyapeeth’s college of Engineering, Delhi-110063, India
<EMAIL_ADDRESS>,  Swati Anand  Department of Mathematics, University
of Delhi, Delhi–110 007, India<EMAIL_ADDRESS>and  Naveen Kumar Jain
Department of Mathematics, Aryabhatta College, Delhi-110021,India
<EMAIL_ADDRESS>
###### Abstract.

This paper aims to pursue some classes of normalized analytic functions $f$
with fixed second coefficient defined on open unit disk, such that
${(1+z)^{2}f(z)}/{z}$ and ${(1+z)f(z)}/{z}$ are functions having positive real
part. The radius of strongly starlikeness, the radius of lemniscate
starlikeness, the radius of parabolic starlikeness and other starlikeness
estimates are calculated for such functions. As well relevant connections of
computed radii estimates with the existing one are also shown.

###### Key words and phrases:

Analytic functions with fixed second coefficient; starlike functions;
parabolic starlike functions; radius estimates.

###### 2010 Mathematics Subject Classification:

30C80, 30C45

##  1\. Introduction

Let $\mathcal{A}$ be the class of analytic functions $f$ defined in
$\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\}$ satisfying $f(0)=f^{\prime}(0)-1=0$
and $\mathcal{S}$ be the sub class of $\mathcal{A}$ contains univalent. Let
$\mathcal{M}$ be a set of functions and ${P}$ be a property. The supremum $R$
of all the radii so that each function $f\in\mathcal{M}$ has the property $P$
in the disk $\mathbb{D}_{r}$, $0\leq r\leq R$ is known as radius of property
for the set $\mathcal{M}$, denoted by $R_{P}(\mathcal{M})$. Thus, it is noted
that $R_{S^{*}}(\mathcal{S})$ is $\tanh\frac{\pi}{4}\approx 0.65579$. Refer
[20, 5, 16, 9] for more literature on radius problems. For a fixed real number
$b$ in $[0,1]$, the class $\mathcal{A}_{b}$ consists of analytic functions

$f(z)=z+bz^{2}+\cdots\quad\quad(z\in\mathbb{D}).$

Since the bound on second coefficient plays a momentous role in the univalent
function theory, the functions having fixed second coefficients attracted the
interest of many authors/readers. In the beginning, Gronwall [14] studied the
univalent functions having fixed second coefficient. In [6], authors discussed
second order differential subordination for such functions. In 2013, Lee _et
al._ accorded some utilizations of subordination for such functions [19].
Recently, the best possible estimates on initial coefficients of Ma-Minda type
univalent functions were determined by Kumar _et al._ [18]. Growth and
distortion estimates, subordination and radius problems for functions
$f\in\mathcal{A}_{b}$ have been discussed in [2, 25]. Using the Hadamard
product ‘$\ast$’ and subordination ‘$\prec$’, Padmanabhan and Parvatham [27]
studied the functions $f\in\mathcal{A}$ fulfilling subordination criteria
$z(c_{\alpha}\ast f)^{\prime}/(c_{\alpha}\ast f)\prec\phi$ where
$c_{\alpha}(z)=z/(1-z)^{\alpha}$, $\alpha$ is a real number and $\phi$ is a
convex function. Shanmugan [32] investigated the unified class
$S^{*}_{j}(\phi)$ that contains functions $f\in\mathcal{A}$ satisfying
$z(f\ast j)^{\prime}/(f\ast j)\prec\phi$ where $\phi$ is convex and $j$ is a
specified function in $\mathcal{A}$. If $j(z)=z/(1-z)$, the class
$S^{*}_{j}(\phi)$ yields the class $S^{*}(\phi)$ [21]. For sample, if
$\phi(z)=(1+Az)/(1+Bz)$ fulfilling condition $-1\leq B<A\leq 1$, then
$S^{*}(\phi)$ scale down to $S^{*}[A,B]$. Recently, Anand _et al._ [8] studied
the class $S^{*}[A,B]$ to obtain various radius problems. In particular, if
$A=1-2\alpha$ and $B=-1$ fulfilling condition $0\leq\alpha<1$, then
$S^{*}(\alpha)$ contains starlike functions of order $\alpha$. For
$0\leq\gamma<1$, the class $S^{*}_{\gamma}$ contains strongly stalike
functions of order $\gamma$ satisfying the inequality
$\left|\mbox{arg}({zf^{\prime}(z)}/{f(z)})\right|<({\alpha\pi}/{2})$. The
class $S_{p}:=S^{*}(\phi_{\mathcal{PAR}}(z))$, where
$\phi_{\mathcal{PAR}}(z)=1+({2}/{\pi^{2}})\left(\log(({1-\sqrt{z}})/({1+\sqrt{z}}))\right)^{2}$,
contains parabolic starlike functions which are associated with parabolic
region $\\{w\in\mathbb{C}:|w-1|<\operatorname{Re}(w)\\}$ [30]. The class
$S^{*}_{L}:=S^{*}(\sqrt{1+z})$ contains lemniscate starlike functions allied
with the lemniscate shaped region $\\{w\in\mathbb{C}:|w^{2}-1|<1\\}$ [35]. The
classes $S^{*}_{e}:=S^{*}(e^{z})$ and
$S^{*}_{RL}:=S^{*}\left(\sqrt{2}-(\sqrt{2}-1)\sqrt{(1-z)/(1+2(\sqrt{2}-1)z)}\right)$
which are associated with the domains $\\{w\in\mathbb{C}:|\log w|<1\\}$ and
$\\{w\in\mathbb{C}:|w^{2}-\sqrt{2}w+1|<1\\}$ respectively studied in [23, 24].
The class $S^{*}_{c}:=S^{*}((3+4z+2z^{2})/3)$ was studied in [34]. The class
$S^{*}_{\leftmoon}:=S^{*}(z+\sqrt{1+z^{2}})$ consists of lune starlike
functions associated with the lune shaped region
$\\{w\in\mathbb{C}:2|w|>|w^{2}-1|\\}$ [29]. Cho _et al._ [10] introduced the
class $S^{*}_{sin}$ related to the function $1+\sin z$. The class
$S^{*}_{R}:=S^{*}(\psi(z))$ is associated with the function
$\psi(z)=1+(z(k+z))/(k(k-z))$ where $k=\sqrt{2}+1$ was also made known in
[17]. The class $S^{*}_{N_{e}}:=S^{*}(1+z-(z^{3}/3))$ is associated with the
two cusped kidney shaped region
$\Omega_{N_{e}}:=\\{u+ib:(u^{2}-2u+v^{2}+5/9)^{3}-4v^{2}/3<0\\}$ [36].
Recently, the class $S^{*}_{SG}:=S^{*}(2/(1+e^{-z}))$ associated with
$\\{w\in\mathbb{C}:|\log((2/(2-w))-1)|<1\\}$ is also studied in [13]. For the
functions $f$ with fixed second coefficient, Ali _et al._ [3] computed radii
of starlikeness. Further, authors [7] examined radius estimates for the
analytic functions $f$ satisfying some conditions on $f/g$ for some
$g\in\mathcal{A}$. In recent past, certain best possible radius estimates for
certain classes of functions $f\in\mathcal{A}$ fulfilling inequality
$\operatorname{Re}(1+z)f(z)/z>0$ and $\operatorname{Re}(1+z)^{2}f(z)/z>0$ have
been determined [31]. For more details, see [1, 28, 15].

Motivated by above explained research work, we fix radius estimates for some
classes considered in [31] having fixed second coefficients to be in the
classes $S^{*}(\alpha)$, $S_{L}^{*}$, $S_{p}$, $S_{e}^{*}$, $S_{c}^{*}$,
$S_{\sin}^{*}$, $S_{\leftmoon}^{*}$, $S_{R}^{*}$, $S_{RL}^{*}$,
$S_{\gamma}^{*}$, $S_{N_{e}}^{*}$ and $S_{SG}^{*}$. Some connections of
obtained radii estimates with the existing ones are also demonstrated.

##  2\. Radius estimates for the class $\mathcal{G}_{b}^{1}$

Let the class $\mathcal{P}(\alpha)$ accommodate analytic functions having
following series expansion $p(z)=1+b_{1}z+b_{2}z^{2}+\cdots$ fulfilling
$\operatorname{Re}(p(z))>\alpha$ ($0\leq\alpha<1$). Note that
$\mathcal{P}:=\mathcal{P}(0)$. For $p\in\mathcal{P}(\alpha)$[26, P.170], we
have $|b_{1}|\leq 2(1-\alpha).$ Let $P_{b}(\alpha)$ be a subclass of
$\mathcal{P}(\alpha)$ consisting of functions $p$ of the form
$p(z)=1+2b(1-\alpha)z+\cdots,|b|\leq 1$. It is noted that
$\mathcal{P}_{b}:=\mathcal{P}_{b}(0)$. Let $f$ be the function having
expansion

(2.1) $f(z)=z+a_{2}z^{2}+\cdots$

and $\operatorname{Re}((1+z)^{2}f(z)/z)>0$. Then
$(1+z)^{2}f(z)/z=1+(2+a_{2})z+\cdots\in\mathcal{P}$ and hence $0\leq a_{2}\leq
4$. Now, we consider the class consisting of functions of the form
$f(z)=z+4bz^{2}+\cdots$ ($-1\leq b\leq 1$).

######  Definition 2.1.

For $|b|\leq 1$, let

$\mathcal{G}_{b}^{1}:=\left\\{f\in\mathcal{A}:f(z)=z+4bz^{2}+\cdots,\operatorname{Re}\left(\frac{(1+z)^{2}}{z}f(z)\right)>0,z\in\mathbb{D}\right\\}.$

We will use following lemma in the proof of main results:

######  Lemma 2.1.

[22, Theorem 2, p. 213] Let $|b|\leq 1$, $0\leq\alpha<1$ and
$p\in\mathcal{P}_{b}(\alpha)$. Then

$\left|\frac{zp^{\prime}(z)}{p(z)}\right|\leq\frac{2(1-\alpha)r}{1-r^{2}}\frac{|b|r^{2}+2r+|b|}{(1-2\alpha)r^{2}+2(1-\alpha)|b|r+1}\quad\mbox{for}\quad|z|=r<1.$

In the following result we determind the radius constants for the class
$\mathcal{G}_{b}^{1}$.

######  Theorem 2.2.

Let $b_{1}=|1+2b|\leq 1$ and $\mathcal{G}_{b}^{1}$ be a class of analytic
functions. Then the following radius estimates are obtained:

  1. (1)

$R_{S^{*}(\alpha)}(\mathcal{G}_{b}^{1})=\rho_{1}$, where $\rho_{1}$ is the
smallest root in $(0,1)$ of the equation

(2.2) $-(1+\alpha)r^{4}+2(1-\alpha b_{1})r^{3}+2(1+2b_{1})r^{2}+2(1+\alpha
b_{1})r+\alpha-1=0.$

The result is sharp.

  2. (2)

$R_{S_{L}^{*}}(\mathcal{G}_{b}^{1})=\rho_{2}$, where $\rho_{2}$ is the
smallest root in $(0,1)$ of the equation

$\displaystyle(1+\sqrt{2})r^{4}$
$\displaystyle+2(1+\sqrt{2}(1+\sqrt{2})b_{1})r^{3}+2(3+2b_{1})r^{2}$ (2.3)
$\displaystyle+2(1+\sqrt{2}(\sqrt{2}-1)b_{1})r+1-\sqrt{2}=0.$

The result is sharp.

  3. (3)

$R_{S_{p}}(\mathcal{G}_{b}^{1})=\rho_{3}$, where $\rho_{3}$ is the smallest
root in $(0,1)$ of the equation

(2.4) $-3r^{4}+2(2-b_{1})r^{3}+4(1+2b_{1})r^{2}+2(2+b_{1})r-1=0.$

The result is sharp.

  4. (4)

$R_{S_{e}^{*}}(\mathcal{G}_{b}^{1})=\rho_{4}$, where $\rho_{4}$ is the
smallest root in $(0,1)$ of the equation

(2.5) $-(1+e)r^{4}+2(e-b_{1})r^{3}+2e(1+2b_{1})r^{2}+2(e+b_{1})r+1-e=0.$

The result is sharp.

  5. (5)

$R_{S_{c}^{*}}(\mathcal{G}_{b}^{1})=\rho_{5}$, where $\rho_{5}$ is the
smallest root in $(0,1)$ of the equation

(2.6) $-4r^{4}+2(3-b_{1})r^{3}+6(1+2b_{1})r^{2}+2(3+b_{1})r-2=0.$

The estimate is best possible.

  6. (6)

$R_{S_{\sin}^{*}}(\mathcal{G}_{b}^{1})=\rho_{6}$, where $\rho_{6}$ is the
smallest root in $(0,1)$ of the equation

$\displaystyle(\sin 1+2)r^{4}+$ $\displaystyle 2(1+(3+\sin
1)b_{1})r^{3}+2(3+2b_{1})r^{2}$ (2.7) $\displaystyle+2(1+(1-\sin
1)b_{1})r-\sin 1=0.$

The result is sharp.

  7. (7)

$R_{S_{\leftmoon}^{*}}(\mathcal{G}_{b}^{1})=\rho_{7}$, where $\rho_{7}$ is the
smallest root in $(0,1)$ of the equation

(2.8)
$-\sqrt{2}r^{4}+2(1+(1-\sqrt{2})b_{1})r^{3}+2(1+2b_{1})r^{2}+2(1+(\sqrt{2}-1)b_{1})r+\sqrt{2}(1-\sqrt{2})=0.$

  8. (8)

$R_{S_{R}^{*}}(\mathcal{G}_{b}^{1})=\rho_{8}$, where $\rho_{8}$ is the
smallest root in $(0,1)$ of the equation

$\displaystyle(1-2\sqrt{2})r^{4}$
$\displaystyle+2(1+2(1-\sqrt{2})b_{1})r^{3}+2(1+2b_{1})r^{2}$ (2.9)
$\displaystyle+2(1+2(\sqrt{2}-1)b_{1})r+(2\sqrt{2}-3)=0.$

The result is sharp.

  9. (9)

$R_{S_{RL}^{*}}(\mathcal{G}_{b}^{1})=\rho_{9}$, where $\rho_{9}$ is the
smallest root in $(0,1)$ of the equation

$\displaystyle(2(1+b_{1})r^{3}+4(1+b_{1})r^{2}+2(1+b_{1})r)^{2}-(r^{2}+2b_{1}r+1)^{2}((1-r^{2})$
(2.10)
$\displaystyle\sqrt{-r^{4}-2(\sqrt{2}-1)r^{2}+2(\sqrt{2}-1)}+r^{4}+2\left(\sqrt{2}-1\right)r^{2}-2\sqrt{2}+2=0.$

  10. (10)

$R_{S_{\gamma}^{*}}(\mathcal{G}_{b}^{1})=\rho_{10}$, where $\rho_{10}$ is the
smallest root in $(0,1)$ of the equation

$\displaystyle-\sin(\pi\gamma/2)r^{4}+2(1+(1-\sin(\pi\gamma/2))b_{1})r^{3}$
$\displaystyle+2(2(1+b_{1})-\sin(\pi\gamma/2))r^{2}+2(1+(1-\sin(\pi\gamma/2))b_{1})r-\sin(\pi\gamma/2)=0.$

  11. (11)

$R_{S_{N_{e}}^{*}}(\mathcal{G}_{b}^{1})=\rho_{11}$, where $\rho_{11}$ is the
smallest root in $(0,1)$ of the equation

(2.11) $8r^{4}+2(3+b_{1})r^{3}+6(2+b_{1})r^{2}+2(3+b_{1})r-2=0.$

The result is sharp.

  12. (12)

$R_{S_{SG}^{*}}(\mathcal{G}_{b}^{1})=\rho_{12}$, where $\rho_{12}$ is the
smallest root in $(0,1)$ of the equation

(2.12)
$(1+3e)r^{4}+2(1+e+2(1+2e)b_{1})r^{3}+2(1+e)(3+2b_{1})r^{2}+2(1+e+2b_{1})r+1-e=0.$

The result is sharp.

###### Proof.

Let $f\in\mathcal{G}_{b}^{1}$ such that

(2.13) $\operatorname{Re}\left(\frac{(1+z)^{2}}{z}f(z)\right)>0\,\,\mbox{for
all}\,\,z\in\mathbb{D}.$

The function $h$ defined on $\mathbb{D}$ as

(2.14) $h(z)=\frac{(1+z)^{2}}{z}f(z)=1+2(1+2b)z+\cdots.$

In view of (2.13) and (2.14), we observe that the function
$h\in\mathcal{P}_{(1+2b)}$ and $f(z)=zh(z)/(1+z)^{2}$. Then

(2.15)
$\frac{zf^{\prime}(z)}{f(z)}=\frac{zh^{\prime}(z)}{h(z)}+\frac{1-z}{1+z}.$

The transformation $\dfrac{1-z}{1+z}$ transform $|z|\leq r$ onto

(2.16)
$\left|\frac{1-z}{1+z}-\frac{1+r^{2}}{1-r^{2}}\right|\leq\frac{2r}{1-r^{2}}.$

Using Lemma 2.1, we have

(2.17)
$\left|\frac{zh^{\prime}(z)}{h(z)}\right|\leq\frac{2r}{1-r^{2}}\left(\frac{b_{1}r^{2}+2r+b_{1}}{r^{2}+2b_{1}r+1}\right).$

Using (2.16) and (2.17), from (2.15), we get

$\displaystyle\left|\frac{zf^{\prime}(z)}{f(z)}-\frac{1+r^{2}}{1-r^{2}}\right|$
$\displaystyle\leq\frac{2r}{1-r^{2}}\left(\frac{b_{1}r^{2}+2r+b_{1}}{r^{2}+2b_{1}r+1}+1\right)$
(2.18)
$\displaystyle=\frac{2((1+b_{1})r^{3}+2(1+b_{1})r^{2}+(1+b_{1})r)}{(1-r^{2})(r^{2}+2b_{1}r+1)}.$

It follows by the above inequality that

$\operatorname{Re}\left(\frac{zf^{\prime}(z)}{f(z)}\right)\geq\frac{r^{4}-2r^{3}-2(1+2b_{1})r^{2}-2r+1}{-r^{4}-2b_{1}r^{3}+2b_{1}r+1}\geq
0$

whenever

$-r^{4}+2r^{3}+2(1+2b_{1})r^{2}+2r-1\leq 0.$

  1. (1)

In view of (2), we have

$\operatorname{Re}\left(\frac{zf^{\prime}(z)}{f(z)}\right)\geq\frac{r^{4}-2r^{3}-2(1+2b_{1})r^{2}-2r+1}{-r^{4}-2b_{1}r^{3}+2b_{1}r+1}\geq\alpha$

whenever

$-(1+\alpha)r^{4}+2(1-\alpha b_{1})r^{3}+2(1+2b_{1})r^{2}+2(1+\alpha
b_{1})r+\alpha-1\leq 0.$

Thus, for the class $\mathcal{G}_{b}^{1}$, the $S^{*}(\alpha)$ -radius is the
root $\rho_{1}\in(0,1)$ of (2.2). The function $f_{1}$ is defined by

(2.19) $f_{1}(z)=\frac{z-z^{2}}{(1+z)(1-2(1+2b)z+z^{2})}$

so that

$\frac{(1+z)^{2}}{z}f_{1}(z)=\frac{1-w_{1}(z)}{1+w_{1}(z)}$

where

$w_{1}(z)=\dfrac{z(z-(1+2b))}{(1-(1+2b)z)}\quad\mbox{ with}\quad|2b+1|\leq 1$

is an analytic function fulfilling all the cases of Schwarz lemma in
$\mathbb{D}$ and thus $\operatorname{Re}((1+z)^{2}f_{1}(z)/z)>0$ in
$\mathbb{D}$. Hence $f_{1}\in\mathcal{G}_{b}^{1}$. At $z=r=\rho_{1}$, we have

$\displaystyle\frac{zf_{1}^{\prime}(z)}{f_{1}(z)}$
$\displaystyle=\frac{r^{4}-2r^{3}+(2+8b)r^{2}-2r+1}{(1-r^{2})(1-2(1+2b)r+r^{2})}$
$\displaystyle=\alpha$

which follows the sharpness.

  2. (2)

It follows from (2) that

$\displaystyle\left|\frac{zf^{\prime}(z)}{f(z)}-1\right|$
$\displaystyle\leq\left|\frac{zf^{\prime}(z)}{f(z)}-\frac{1+r^{2}}{1-r^{2}}\right|+\frac{2r^{2}}{1-r^{2}}$
$\displaystyle\leq\frac{2r^{4}+2(1+3b_{1})r^{3}+2(3+2b_{1})r^{2}+2(1+b_{1})r}{-r^{4}-2b_{1}r^{3}+2b_{1}r+1}.$

By [4, Lemma 2.2, p. 6559], the function $f\in\mathcal{G}_{b}^{1}$ go to the
class $S^{*}_{L}$, whenever the following inequality holds

$\frac{2r^{4}+2(1+3b_{1})r^{3}+2(3+2b_{1})r^{2}+2(1+b_{1})r}{-r^{4}-2b_{1}r^{3}+2b_{1}r+1}\leq\sqrt{2}-1$

or

$\displaystyle(1+\sqrt{2})r^{4}$
$\displaystyle+2(1+\sqrt{2}(1+\sqrt{2})b_{1})r^{3}+$ $\displaystyle
2(3+2b_{1})r^{2}+2(1+\sqrt{2}(\sqrt{2}-1)b_{1})r+1-\sqrt{2}\leq 0.$

Thus, the $S_{L}^{*}-$ radius is the smallest root $\rho_{2}\in(0,1)$ of (2).
The function $f_{2}:\mathbb{D}\to\mathbb{C}$ is defined by

(2.20) $f_{2}(z)=\frac{z(1+2(1+2b)z+z^{2})}{(1+z)^{2}(1-z^{2})}.$

so that

$\frac{(1+z)^{2}}{z}f_{2}(z)=\frac{1+w_{2}(z)}{1-w_{2}(z)}$

where

$w_{2}(z)=\dfrac{z(z+(1+2b))}{(1+(1+2b)z)}\quad\mbox{with}\,\,|1+2b|\leq 1$

is an analytic function fulfilling all the cases of Schwarz lemma in the unit
disk $\mathbb{D}$. Thus, $\operatorname{Re}((1+z)^{2}f_{2}(z)/z)>0$ in
$\mathbb{D}$ and hence $f_{2}\in\mathcal{G}_{b}^{1}$. The number
$\rho=\rho_{2}$ satisfies

$\frac{\rho^{4}-(2+8b)\rho^{3}+(2-8b)\rho^{2}-(2+8b)\rho+1}{(1-\rho^{2})(1-2(1+2b)\rho+\rho^{2})}=\sqrt{2}.$

Thus, at $z=-\rho=-\rho_{2}$, we have

$\displaystyle\left|\left(\frac{zf_{2}^{\prime}(z)}{f_{2}(z)}\right)^{2}-1\right|$
$\displaystyle=\left|\left(\frac{\rho^{4}-(2+8b)\rho^{3}+(2-8b)\rho^{2}-(2+8b)\rho+1}{(1-\rho^{2})(1-2(1+2b)\rho+\rho^{2})}\right)^{2}-1\right|$
$\displaystyle=1.$

This shows the sharpness.

  3. (3)

Using [33, Section 3˘, p. 321], the disk (2) lies inside the parabolic region
$\Omega_{p}=\\{w:\operatorname{Re}w>{w-1}\\}$ provided that

$\frac{2((1+b_{1})r^{3}+2(1+b_{1})r^{2}+(1+b_{1})r)}{(1-r^{2})(r^{2}+2b_{1}r+1)}\leq\frac{1+r^{2}}{1-r^{2}}-\frac{1}{2}$

which on simplification becomes

$-3r^{4}+2(2-b_{1})r^{3}+4(1+2b_{1})r^{2}+2(2+b_{1})r-1\leq 0.$

Thus, the $S_{p}$\- radius is the smallest root $\rho_{3}\in(0,1)$ of (2.4).
For the function $f_{1}$ given in (2.19) at $z=\rho=\rho_{3}$, the following
calculations shows the desired sharpness

$\displaystyle\operatorname{Re}\frac{zf_{1}^{\prime}(z)}{f_{1}(z)}$
$\displaystyle=\frac{\rho^{4}-2\rho^{3}+(2+8b)\rho^{2}-2\rho+1}{(1-\rho^{2})(1-2(1+2b)\rho+\rho^{2})}$
$\displaystyle=\frac{-2\rho^{4}+(4+4b)\rho^{3}-(2+8b)\rho^{2}-4b\rho}{(1-\rho^{2})(1-2(1+2b)\rho+\rho^{2})}$
$\displaystyle=\left|\frac{2\rho^{4}-(4+4b)\rho^{3}+(2+8b)\rho^{2}+4b\rho}{(1-\rho^{2})(1-2(1+2b)\rho+\rho^{2})}\right|$
$\displaystyle=\left|\frac{zf_{1}^{\prime}(z)}{f_{1}(z)}-1\right|.$

  4. (4)

By [23, Lemma 2.2, p. 368], the function $f\in\mathcal{G}_{b}^{1}$ belongs to
the class $S_{e}^{*}$ if

$\frac{2((1+b_{1})r^{3}+2(1+b_{1})r^{2}+(1+b_{1})r)}{(1-r^{2})(r^{2}+2b_{1}r+1)}\leq\frac{1+r^{2}}{1-r^{2}}-\frac{1}{e}$

equivalently

$-(1+e)r^{4}+2(e-b_{1})r^{3}+2e(1+2b_{1})r^{2}+2(e+b_{1})r+1-e\leq 0.$

Thus, the $S_{e}^{*}$\- radius is the smallest root $\rho_{4}\in(0,1)$ of
(2.5). For $z=\rho=\rho_{4}$ and the function $f_{1}$, the following
calculation prove the sharpness

$\displaystyle\left|\log\frac{zf_{1}^{\prime}(z)}{f_{1}(z)}\right|$
$\displaystyle=\left|\log\left(\frac{\rho^{4}-2\rho^{3}-2(3+4b)\rho^{2}-2\rho+1}{(1-\rho^{2})(1+2(1+2b)\rho+\rho^{2})}\right)\right|=1.$

  5. (5)

By making use of [34, Lemma 2.5, p.926], the function
$f\in\mathcal{G}_{b}^{1}$ belongs to the class $S_{c}^{*}$ if

$\frac{2((1+b_{1})r^{3}+2(1+b_{1})r^{2}+(1+b_{1})r)}{(1-r^{2})(r^{2}+2b_{1}r+1)}\leq\frac{1+r^{2}}{1-r^{2}}-\frac{1}{3}$

or

$-4r^{4}+2(3-b_{1})r^{3}+6(1+2b_{1})r^{2}+2(3+b_{1})r-2\leq 0.$

Thus, the $S_{c}^{*}$\- radius is the smallest root $\rho_{5}\in(0,1)$ of
(2.6). At $z=\rho=\rho_{5}$, we have

$\displaystyle\left|\ \frac{zf_{1}^{\prime}(z)}{f_{1}(z)}\right|$
$\displaystyle=\left|\frac{\rho^{4}-2\rho^{3}+(2+8b)\rho^{2}-2\rho+1}{(1-\rho^{2})(1-2(1+2b)\rho+\rho^{2})}\right|$
$\displaystyle=\frac{1}{3}\,\,\text{which lies in the boundary
of}\,\,\Omega_{c}$

where $\Omega_{C}$ is cardioid shaped region
$\\{x+iy:81x^{4}-324x^{3}+162x^{2}y^{2}+270x^{2}-324xy^{2}-84x+81y^{4}-54y^{2}+9=0\\}$.

  6. (6)

Using [10, Lemma 3.3, p.219], the function $f\in\mathcal{G}_{b}^{1}$ belongs
to the class $S_{\sin}^{*}$ if

$\frac{2((1+b_{1})r^{3}+2(1+b_{1})r^{2}+(1+b_{1})r)}{(1-r^{2})(r^{2}+2b_{1}r+1)}\leq\sin
1-\frac{2r^{2}}{1-r^{2}}$

equivalently

$(2+\sin 1)r^{4}+2(1+(3+\sin 1)b_{1})r^{3}+2(3+2b_{1})r^{2}+2(1+(1-\sin
1)b_{1})r-\sin 1\leq 0.$

Thus, the $S_{\sin}^{*}$\- radius is the smallest root $\rho_{6}\in(0,1)$ of
(6). For the function $f_{2}$ defined in (2.20) and $z=-\rho=-\rho_{6}$, we
have

$\displaystyle\left|\frac{zf_{2}^{\prime}(z)}{f_{2}(z)}\right|$
$\displaystyle=\left|\frac{\rho^{4}-(2+8b)\rho^{3}+(2-8b)\rho^{2}-(2+8b)\rho+1}{(1-\rho^{2})(1-2(1+2b)\rho+\rho^{2})}\right|$
$\displaystyle=1+\sin 1\,\,\text{which lies in the boundary of}\,\,\Omega_{S}$

where $\Omega_{S}:=q_{0}(\mathbb{D})$ is the image of $\mathbb{D}$ under the
mapping $q_{0}(z)=1+\sin z$.

  7. (7)

By [11, Lemma 2.1, p. 3], the function $f\in\mathcal{G}_{b}^{1}$ belongs to
the class $S_{\leftmoon}^{*}$ if the following inequality holds

$\frac{2((1+b_{1})r^{3}+2(1+b_{1})r^{2}+(1+b_{1})r)}{(1-r^{2})(r^{2}+2b_{1}r+1)}\leq
1-\sqrt{2}+\frac{1+r^{2}}{1-r^{2}}$

which becomes

$-\sqrt{2}r^{4}+2(1+(1-\sqrt{2})b_{1})r^{3}+2(1+2b_{1})r^{2}+2(1+(\sqrt{2}-1)b_{1})r+(\sqrt{2}-2)\leq
0.$

Thus, the $S_{\leftmoon}^{*}$\- radius is the smallest root $\rho_{7}\in(0,1)$
of (2.8). At $z=\rho=\rho_{7}$, we have

$\displaystyle\left|\left(\frac{zf_{1}^{\prime}(z)}{f_{1}(z)}\right)^{2}-1\right|$
$\displaystyle=\left|\left(\frac{\rho^{4}-2\rho^{3}+(2+8b)\rho^{2}-2\rho+1}{(1-\rho^{2})(1-2(1+2b)\rho+\rho^{2})}\right)^{2}-1\right|$
$\displaystyle=2\left|\frac{zf_{1}^{\prime}(z)}{f_{1}(z)}\right|.$

  8. (8)

By [17, Lemma 2.2 p. 202], the function $f\in\mathcal{G}_{b}^{1}$ belongs to
the class $S_{R}^{*}$ if

$\frac{2((1+b_{1})r^{3}+2(1+b_{1})r^{2}+(1+b_{1})r)}{(1-r^{2})(r^{2}+2b_{1}r+1)}\leq\frac{1+r^{2}}{1-r^{2}}+2-2\sqrt{2}$

equivalently

$\displaystyle(1-2\sqrt{2})r^{4}+$ $\displaystyle
2(1+2(1-\sqrt{2})b_{1})r^{3}+$ $\displaystyle
2(1+2b_{1})r^{2}+2(1+2(\sqrt{2}-1)b_{1})r+2\sqrt{2}-3\leq 0.$

Thus, the $S_{R}^{*}$\- radius is the smallest root $\rho_{8}\in(0,1)$ of (8).
As for $z=\rho=\rho_{8}$, we have

$\displaystyle\left|\frac{zf_{1}^{\prime}(z)}{f_{1}(z)}\right|=\left|\frac{\rho^{4}-2\rho^{3}+(2+8b)\rho^{2}-2\rho+1}{(1-\rho^{2})(1-2(1+2b)\rho+\rho^{2})}\right|=2(\sqrt{2}-1)=\psi(-1)$

where
$\psi(\mathbb{D})=1+\dfrac{z}{k}\left(\dfrac{k+z}{k-z}\right)=1+\dfrac{1}{k}z+\dfrac{2}{k^{2}}z^{2}+\cdots$,
$k=\sqrt{2}+1$.

  9. (9)

By [24, Lemma 3.2, p. 10], the function $f\in\mathcal{G}_{b}^{1}$ belongs to
the class $S_{RL}^{*}$ if

$\displaystyle\frac{2((1+b_{1})r^{3}+2(1+b_{1})r^{2}+(1+b_{1})r)}{(1-r^{2})(r^{2}+2b_{1}r+1)}$
$\displaystyle\leq((1-(\sqrt{2}-\frac{1}{1-r^{2}})^{2})^{1/2}-(1-(\sqrt{2}-\frac{1}{1-r^{2}})^{2}))^{1/2}$

which yields

$\displaystyle(2(1+b_{1})r^{3}+4(1+b_{1})r^{2}+2(1+b_{1})r)^{2}-(r^{2}+2b_{1}r+1)^{2}((1-r^{2})$
$\displaystyle\sqrt{-r^{4}-2(\sqrt{2}-1)r^{2}+2(\sqrt{2}-1)}+r^{4}+2\left(\sqrt{2}-1\right)r^{2}-2\sqrt{2}+2\leq
0.$

Thus, the $S_{RL}^{*}$\- radius is the smallest root $\rho_{9}\in(0,1)$ of
(9).

  10. (10)

By [12, Lemma 3.1, p. 307], the function $f\in\mathcal{G}_{b}^{1}$ belongs to
the class $S_{\gamma}^{*}$ if

$\frac{2((1+b_{1})r^{3}+2(1+b_{1})r^{2}+(1+b_{1})r)}{(1-r^{2})(r^{2}+2b_{1}r+1)}\leq\frac{1+r^{2}}{1-r^{2}}\sin(\frac{\pi\gamma}{2})$

which gives

$\displaystyle-\sin(\pi\gamma/2)r^{4}+2(1+(1-\sin(\pi\gamma/2))b_{1})r^{3}$
$\displaystyle+2(2(1+b_{1})-\sin(\pi\gamma/2))r^{2}+2(1+(1-\sin(\pi\gamma/2))b_{1})r-\sin(\pi\gamma/2)\leq
0.$

  11. (11)

Using [36, Lemma 2.2, p. 86], the function $f\in\mathcal{G}_{b}^{1}$ belongs
to the class $S^{*}_{N_{e}}$ if

$\frac{2((1+b_{1})r^{3}+2(1+b_{1})r^{2}+(1+b_{1})r)}{(1-r^{2})(r^{2}+2b_{1}r+1)}\leq\frac{5}{3}-\frac{1}{1-r^{2}}$

equivalently

$5r^{4}+(6+16b_{1})r^{3}+(15+12b_{1})r^{2}+2(3+b_{1})r-2\leq 0.$

Thus, the $S^{*}_{N_{e}}$\- radius is the smallest root $\rho_{11}\in(0,1)$ of
(2.11). Since $f_{2}$ is defined in (2.20), then at $z=-\rho=-\rho_{11}$, we
have

$\displaystyle\left|\frac{zf_{2}^{\prime}(z)}{f_{2}(z)}\right|$
$\displaystyle=\left|\frac{\rho^{4}-(2+8b)\rho^{3}+(2-8b)\rho^{2}-(2+8b)\rho+1}{(1-\rho^{2})(1-2(1+2b)\rho+\rho^{2})}\right|=\frac{5}{3},$

which shows that the estimate is best possible.

  12. (12)

Using [13, Lemma 2.2, p. 961], the function $f\in\mathcal{G}_{b}^{1}$ belongs
to the class $S^{*}_{SG}$ if the following inequality holds

$\frac{2((1+b_{1})r^{3}+2(1+b_{1})r^{2}+(1+b_{1})r)}{(1-r^{2})(r^{2}+2b_{1}r+1)}\leq\frac{2e}{1+e}-\frac{1}{1-r^{2}}$

equivalently if the following inequality holds

$\displaystyle 1-e+r(4b_{1}+2e+2)$
$\displaystyle+r^{2}\left(4b_{1}e+4b_{1}+6e+6\right)+$ $\displaystyle
r^{3}\left(8b_{1}e+4b_{1}+2e+2\right)+(3e+1)r^{4}\leq 0.$

Thus, the $S^{*}_{SG}$\- radius is the smallest root $\rho_{12}\in(0,1)$ of
(2.12). For the function $f_{2}$ defined in (2.20), let
$w=zf_{2}^{\prime}(z)/f_{2}(z)$. At $z=-\rho=-\rho_{12}$, we have following
steps for sharpness

$\displaystyle\left|\log\frac{w}{2-w}\right|$
$\displaystyle=\left|\log\left(\frac{-\rho^{4}+(2+8b)\rho^{3}-(2-8b)\rho^{2}+(2+8b)\rho-1}{3\rho^{4}-2(3+8b)\rho^{3}+(2-8b)\rho^{2}+2\rho-1}\right)\right|=1.$

∎

######  Remark 2.1.

On taking $b=-1$, parts(1)-(10) in Theorem 2.2 scale down to [31, Theorem 2.8]
.

##  3\. Radius estimates for the class $\mathcal{G}_{b}^{2}$

Let the function $f$ be given by (2.1) such that
$\operatorname{Re}((1+(1/z))f(z))>0$. Then
$(1+(1/z))f(z)=1+(1+a_{2})z+\cdots\in\mathcal{P}$ and hence $-3\leq a_{2}\leq
1$. We consider the class of function $f$ with fixed second coefficient whose
expansion is given by $f(z)=z+3bz^{2}+\cdots$ ($|b|\leq 1)$ .

######  Definition 3.1.

For $|b|\leq 1$, let $\mathcal{G}_{b}^{2}$ be a class of analytic functions
given as

$\mathcal{G}_{b}^{2}:=\left\\{f\in\mathcal{A}:f(z)=z+3bz^{2}+\cdots,\operatorname{Re}\left(\left(1+\frac{1}{z}\right)f(z)\right)>0,z\in\mathbb{D}\right\\}.$

Consider the function $f_{3}$ on $\mathbb{D}$ which is defined by

(3.1) $f_{3}(z)=\frac{z(1+(1+3b)z+z^{2})}{(1+z)(1-z^{2})}.$

so that

$\frac{1+z}{z}f_{3}(z)=\frac{1+w_{3}(z)}{1-w_{3}(z)}$

where

$w_{3}(z)=\dfrac{z(z+(1+3b)/2)}{(1+(1+3b)z/2)}\,\,\mbox{ with}\,\,|1+3b|\leq
2$

is an analytic function fulfilling all the cases of Schwarz lemma and
$\operatorname{Re}((1+z)f_{1}(z)/z)>0$ in $\mathbb{D}$. Hence,
$f_{3}\in\mathcal{G}_{b}^{2}$ that shows the class $\mathcal{G}_{b}^{2}$ is
non-empty.

######  Theorem 3.1.

Let $b^{\prime}=|1+3b|\leq 2$. The following radius estimates for the class
$\mathcal{G}_{b}^{2}$ are computed:

  1. (1)

$R_{S_{c}^{*}}(\mathcal{G}_{b}^{2})=\rho_{1}$, where $\rho_{1}$ is the
smallest root in $(0,1)$ of the equation

(3.2) $-r^{4}+(3+2b^{\prime})r^{3}+3(3+b^{\prime})r^{2}+(3+b^{\prime})r-2=0.$

  2. (2)

$R_{S_{\sin}^{*}}(\mathcal{G}_{b}^{2})=\rho_{2}$, where $\rho_{2}$ is the
smallest root in $(0,1)$ of the equation

(3.3) $(1+\sin 1)r^{4}+(1+(2+\sin
1)b^{\prime})r^{3}+(5+b^{\prime})r^{2}+(1+(1-\sin 1)b^{\prime})r-\sin 1=0.$

The result is sharp.

  3. (3)

$R_{S_{\leftmoon}^{*}}(\mathcal{G}_{b}^{2})=\rho_{3}$, where $\rho_{3}$ is the
smallest root in $(0,1)$ of the equation

$(1-\sqrt{2})r^{4}+(1+\sqrt{2}(\sqrt{2}-1)b^{\prime})r^{3}+(3+b^{\prime})r^{2}+(1+(\sqrt{2}-1)b^{\prime})r+\sqrt{2}(1-\sqrt{2})=0.$

  4. (4)

$R_{S_{R}^{*}}(\mathcal{G}_{b}^{2})=\rho_{4}$, where $\rho_{4}$ is the
smallest root in $(0,1)$ of the equation

(3.4)
$2(1-\sqrt{2})r^{4}+(1+(3-2\sqrt{2})b^{\prime})r^{3}+(3+b^{\prime})r^{2}+(1+2(\sqrt{2}-1)b^{\prime})r+2\sqrt{2}-3=0.$

  5. (5)

$R_{S_{RL}^{*}}(\mathcal{G}_{b}^{2})=\rho_{5}$, where $\rho_{5}$ is the
smallest root in $(0,1)$ of the equation

$\displaystyle((1+b^{\prime})r^{3}+(4+b^{\prime})r^{2}+(1+b^{\prime})r)^{2}-(r^{2}+b^{\prime}r+1)^{2}((1-r^{2})$
(3.5)
$\displaystyle\sqrt{-r^{4}-2(\sqrt{2}-1)r^{2}+2(\sqrt{2}-1)}+r^{4}+2\left(\sqrt{2}-1\right)r^{2}-2\sqrt{2}+2=0.$

  6. (6)

$R_{S_{\gamma}^{*}}(\mathcal{G}_{b}^{2})=\rho_{6}$, where $\rho_{6}$ is the
smallest root in $(0,1)$ of the equation

$(1+b^{\prime})r^{3}+(4-\sin(\pi\gamma/2)+b^{\prime})r^{2}+(1+(1-\sin(\pi\gamma/2))b^{\prime})r-\sin(\pi\gamma/2)\leq
0.$

  7. (7)

$R_{S_{N_{e}}^{*}}(\mathcal{G}_{b}^{2})=\rho_{7}$, where $\rho_{7}$ is the
smallest root in $(0,1)$ of the equation

(3.6) $5r^{4}+(3+8b^{\prime})r^{3}+3(5+b^{\prime})r^{2}+(3+b^{\prime})r-2=0.$

The result is sharp.

  8. (8)

$R_{S_{SG}^{*}}(\mathcal{G}_{b}^{2})=\rho_{8}$, where $\rho_{8}$ is the
smallest root in $(0,1)$ of the equation

(3.7)
$2er^{4}+(1+e+(1+3e)b^{\prime})r^{3}+(1+e)(5+b^{\prime})r^{2}+(1+e+2b^{\prime})r+1-e=0.$

The result is sharp.

###### Proof.

Let the function $f\in\mathcal{F}_{b}^{2}$ be such that

(3.8) $\operatorname{Re}\left(\frac{1+z}{z}f(z)\right)>0\,\,(z\in D).$

Consider the function $h$ on $\mathbb{D}$ given by

(3.9) $h(z)=\left(1+\frac{1}{z}\right)f(z)=1+(1+3b)z+\cdots.$

In view of (3.8) and (3.9), it is noted that $h\in\mathcal{P}_{(1+3b)/2}$ and
$f(z)=zh(z)/(1+z)$. A simple calculation gives

(3.10)
$\frac{zf^{\prime}(z)}{f(z)}=\frac{zh^{\prime}(z)}{h(z)}+\frac{1}{1+z}.$

Using Lemma 2.1, we get following inequality

(3.11)
$\left|\frac{zh^{\prime}(z)}{h(z)}\right|\leq\frac{r}{1-r^{2}}\left(\frac{b^{\prime}r^{2}+4r+b^{\prime}}{r^{2}+b^{\prime}r+1}\right).$

The bilinear transformation $\dfrac{1}{1+z}$ maps $|z|\leq r$ onto

(3.12) $\left|\frac{1}{1+z}-\frac{1}{1-r^{2}}\right|\leq\frac{r}{1-r^{2}}.$

Using (3.12) and (3.11) along with (3.10), we obtain

$\displaystyle\left|\frac{zf^{\prime}(z)}{f(z)}-\frac{1}{1-r^{2}}\right|$
$\displaystyle\leq\frac{r}{1-r^{2}}\left(\frac{b^{\prime}r^{2}+4r+b^{\prime}}{r^{2}+b^{\prime}r+1}+1\right)$
$\displaystyle=\frac{(1+b^{\prime})r^{3}+(4+b^{\prime})r^{2}+(1+b^{\prime})r}{(1-r^{2})(r^{2}+b^{\prime}r+1)}.$

It follows by the above inequality that

$\operatorname{Re}\left(\frac{zf^{\prime}(z)}{f(z)}\right)\geq\frac{-(1+b^{\prime})r^{3}-(3+b^{\prime})r^{2}-r+1}{-r^{4}-b^{\prime}r^{3}+b^{\prime}r+1}\geq
0$

whenever,

$(1+b^{\prime})r^{3}+(3+b^{\prime})r^{2}+r-1\leq 0.$

  1. (1)

By making use of [34, Lemma 2.5, p.926], the function
$f\in\mathcal{G}_{b}^{2}$ to be in the class $S_{c}^{*}$ if

$\frac{(1+b^{\prime})r^{3}+(4+b^{\prime})r^{2}+(1+b^{\prime})r}{(1-r^{2})(r^{2}+b^{\prime}r+1)}\leq\frac{1}{1-r^{2}}-\frac{1}{3}$

or equivalently, if

$-r^{4}+(3+2b^{\prime})r^{3}+3(3+b^{\prime})r^{2}+(3+b^{\prime})r-2\leq 0.$

Thus, the $S_{c}^{*}$-radius is the smallest root $\rho_{1}\in(0,1)$ of (3.2).

  2. (2)

Using [10, Lemma 3.3, p.219], the function $f\in\mathcal{G}_{b}^{2}$ to be in
the class $S_{\sin}^{*}$ if

$\frac{(1+b^{\prime})r^{3}+(4+b^{\prime})r^{2}+(1+b^{\prime})r}{(1-r^{2})(r^{2}+b^{\prime}r+1)}\leq\sin
1-\frac{r^{2}}{1-r^{2}}$

equivalently

$(\sin 1+1)r^{4}+(1+(2+\sin 1)b^{\prime})r^{3}+(5+b^{\prime})r^{2}+(1+(1-\sin
1)b^{\prime})r-\sin 1\leq 0.$

Thus, the $S_{\sin}^{*}$\- radius for the class $\mathcal{G}_{b}^{2}$ is the
smallest positive root $\rho_{2}\in(0,1)$ of (3.3). Consider the function
$f_{3}$ defined in (3.1). At $z=-\rho=-\rho_{2}$, we have

$\displaystyle\left|\frac{zf_{3}^{\prime}(z)}{f_{3}(z)}\right|$
$\displaystyle=\left|\frac{-3b\rho^{3}+(4-3b)\rho^{2}-(6b+1)\rho+1}{(1-\rho^{2})(1-(1+3b)\rho+\rho^{2})}\right|$
$\displaystyle=1+\sin 1.$

  3. (3)

By [11, Lemma 2.1, p. 3], the function $f\in\mathcal{G}_{b}^{2}$ to be in the
class $S_{\leftmoon}^{*}$ if

$\frac{(1+b^{\prime})r^{3}+(4+b^{\prime})r^{2}+(1+b^{\prime})r}{(1-r^{2})(r^{2}+b^{\prime}r+1)}\leq
1-\sqrt{2}+\frac{1}{1-r^{2}}$

which implies

$(1-\sqrt{2})r^{4}+(1+\sqrt{2}(\sqrt{2}-1)b^{\prime})r^{3}+(3+b^{\prime})r^{2}+(1+(\sqrt{2}-1)b^{\prime})r+\sqrt{2}(1-\sqrt{2})\leq
0.$

Thus, the $S_{\leftmoon}^{*}$\- radius is the smallest root $\rho_{3}\in(0,1)$
of (3).

  4. (4)

By [17, Lemma 2.2 p. 202], the function $f\in\mathcal{G}_{b}^{2}$ to be in the
class $S_{R}^{*}$ if

$\frac{(1+b^{\prime})r^{3}+(4+b^{\prime})r^{2}+(1+b^{\prime})r}{(1-r^{2})(r^{2}+b^{\prime}r+1)}\leq\frac{1}{1-r^{2}}+2-2\sqrt{2}$

equivalently

$2(1-\sqrt{2})r^{4}+(1+(3-2\sqrt{2})b^{\prime})r^{3}+(3+b^{\prime})r^{2}+(1+2(\sqrt{2}-1)b^{\prime})r+2\sqrt{2}-3\leq
0.$

Thus, the $S_{R}^{*}$ -radius is the smallest root $\rho_{4}\in(0,1)$ of
(3.4).

  5. (5)

By [24, Lemma 3.2, p. 10], the function $f\in\mathcal{G}_{b}^{2}$ to be in the
class $S_{RL}^{*}$ if

$\displaystyle\frac{(1+b^{\prime})r^{3}+(4+b^{\prime})r^{2}+(1+b^{\prime})r}{(1-r^{2})(r^{2}+b^{\prime}r+1)}$
$\displaystyle\quad\leq((1-(\sqrt{2}-\frac{1}{1-r^{2}})^{2})^{1/2}-(1-(\sqrt{2}-\frac{1}{1-r^{2}})^{2}))^{1/2}$

which gives

$\displaystyle((1+b^{\prime})r^{3}+(4+b^{\prime})r^{2}+(1+b^{\prime})r)^{2}-(r^{2}+b^{\prime}r+1)^{2}((1-r^{2})$
$\displaystyle\sqrt{-r^{4}-2(\sqrt{2}-1)r^{2}+2(\sqrt{2}-1)}+r^{4}+2\left(\sqrt{2}-1\right)r^{2}-2\sqrt{2}+2=0.$

Therefore, the $S_{RL}^{*}$-radius is the smallest root $\rho_{5}\in(0,1)$ of
(5).

  6. (6)

By [12, Lemma 3.1, p. 307], the function $f\in\mathcal{G}_{b}^{2}$ to be in
the class $S_{\gamma}^{*}$ if

$\frac{(1+b^{\prime})r^{3}+(4+b^{\prime})r^{2}+(1+b^{\prime})r}{(1-r^{2})(r^{2}+b^{\prime}r+1)}\leq\frac{1}{1-r^{2}}\sin(\frac{\pi\gamma}{2})$

which simplifies to

$(1+b^{\prime})r^{3}+(4-\sin(\pi\gamma/2)+b^{\prime})r^{2}+(1+(1-\sin(\pi\gamma/2))b^{\prime})r-\sin(\pi\gamma/2)\leq
0.$

  7. (7)

Using [36, Lemma 2.2, p. 86], the function $f\in\mathcal{G}_{b}^{2}$ to be in
the class $S^{*}_{N_{e}}$ if

$\frac{(1+b^{\prime})r^{3}+(4+b^{\prime})r^{2}+(1+b^{\prime})r}{(1-r^{2})(r^{2}+b^{\prime}r+1)}\leq\frac{5}{3}-\frac{1}{1-r^{2}}$

which is equivalent to

$5r^{4}+(3+8b^{\prime})r^{3}+3(5+b^{\prime})r^{2}+(3+b^{\prime})r-2\leq 0.$

Thus, the $S^{*}_{N_{e}}$\- radius is the smallest root $\rho_{7}\in(0,1)$ of
(3.6). Consider the function $f_{3}$ defined in (3.1). At $z=-\rho=-\rho_{7}$,
we have

$\displaystyle\left|\frac{zf_{3}^{\prime}(z)}{f_{3}(z)}\right|$
$\displaystyle=\left|\frac{-3b\rho^{3}+(4-3b)\rho^{2}-(6b+1)\rho+1}{(1-\rho^{2})(1-(1+3b)\rho+\rho^{2})}\right|$
$\displaystyle=\frac{5}{3}\,\,\text{which lies in the boundary
of}\,\,\partial\Omega_{N_{e}}$

where $\Omega_{Ne}$ is the nephroid shaped region. This shows that the result
is sharp.

  8. (8)

Using [13, Lemma 2.2, p. 961], the function $f\in\mathcal{G}_{b}^{2}$ belongs
to the class $S^{*}_{SG}$ if

$\frac{(1+b^{\prime})r^{3}+(4+b^{\prime})r^{2}+(1+b^{\prime})r}{(1-r^{2})(r^{2}+b^{\prime}r+1)}\leq\frac{2e}{1+e}-\frac{1}{1-r^{2}}$

equivalently if the following inequality holds

$2er^{4}+(1+e+(1+3e)b^{\prime})r^{3}+(1+e)(5+b^{\prime})r^{2}+(1+e+2b^{\prime})r+1-e\leq
0.$

Thus, the $S^{*}_{SG}$\- radius for the class $\mathcal{G}_{b}^{2}$ is a
smallest root $\rho_{8}\in(0,1)$ of (3.7). Consider the function $f_{3}$
defined in (3.1). Let $w=zf_{3}^{\prime}(z)/f_{3}(z)$. At $z=-\rho=-\rho_{8}$,
we have

$\displaystyle\left|\log\left(\frac{w}{2-w}\right)\right|$
$\displaystyle=\left|\log\left(\frac{3b\rho^{3}-(4-3b)\rho^{2}+(6b+1)\rho-1}{2\rho^{4}-(2+9b)\rho^{3}+(4-3b)\rho^{2}+\rho-1}\right)\right|=1$

which shows the sharpness.∎

######  Remark 3.1.

For $b=-1$ in parts (1)-(6) of Theorem 3.1, we obtain [31, Theorem 2.6] .

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