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\title[Applications of First Order Differential Subordination]{Applications of First Order Differential Subordination for Functions with Positive Real Part}
\author{Om P. Ahuja}
\address{Kent State University, \\ Department of Mathematics,\\
Burton,\\
USA}
\email{oahuja@kent.edu}
%, , , 
\author{Sushil Kumar}
\address{Bharati Vidyapeeth's College of Engineering\\A-4, Paschim Vihar \\  Delhi--110063\\
India}
\email{sushilkumar16n@gmail.com}
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\author{V. Ravichandran}
\address{University of Delhi, \\ Department of Mathematics\\
Delhi--110007,\\
India}
\email{vravi@maths.du.ac.in; vravi68@gmail.com}
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\subjclass{30C45}
\keywords{Differential Subordination; starlike function; lemniscate of Bernoulli; functions with positive real part}
\begin{abstract}
Several inclusions between the class of  functions with positive real part and  the class of starlike univalent  functions associated with lemniscate of Bernoulli are obtained by making use of the well-known theory of differential subordination. Further, these  inclusions give  sufficient conditions for normalized analytic functions to belong to some subclasses of starlike functions. The results  also provide sharp version of some  previously known results.
\end{abstract}
\maketitle

\section{ Introduction}
Let $\mathcal{A}$ denote the class of analytic functions $f$ on  the disk $\mathbb{D}:= \{ z\in \mathbb{C}: |z|<1\}$ and  normalized by the  condition $f(0)=0=f'(0)-1$. Let $\mathcal{S}$ be the subset of $\mathcal{A}$ of univalent functions.
An analytic function $f$ defined on $\mathbb{D}$ is subordinate to the analytic function $g$  on $\mathbb{D}$ (or $g$ is superordinate to $f$), if there exists an analytic function $w: \mathbb{D}\to  \mathbb{D}$, with $w(0)=0$, such that $f=g\circ w$.  Furthermore, if $g$ is univalent in $\mathbb{D}$, then $f \prec g$ is equivalent to $f(0) = g(0)$ and $f(\mathbb{D})\subseteq g(\mathbb{D})$, see \cite{o}.
Let $p$ be an analytic function  in $\mathbb{D}$ with positive real part and $p(0)=1$. The function $p(z)=(1+z)/(1-z)$ is a  leading example of the function with positive real part such that  $p(\mathbb{D})$ is  the right-half plane.
Goluzin \cite{Goluzin} discussed the  first order differential subordination $zp'(z) \prec zq'(z)$ and proved that, whenever $zq'(z)$ is convex,  the subordination $p(z)\prec q(z)$ holds   and the  function $q$ is the best dominant. After this  basic result, many authors established several generalizations of first order differential subordination. The general theory of  differential subordination is discussed in the monograph by Miller and Mocanu \cite{Miller85}.
In 1989, Nunokawa \emph{et al.}\ \cite{p} proved that if subordination $1+z p'(z)\prec 1+z$ holds, then subordination $p(z)\prec 1+z$ also holds. In 2007,  Ali \emph{et al.}\ \cite{q} extended this result and determined the estimates on $\beta$ for which the subordination $1+ \beta z p'(z)/p^j(z)\prec (1+Dz)/(1+Ez)$ $(j=0,1,2)$ implies the subordination $p(z)\prec (1+Az)/(1+Bz)$, where $A,B,D,E \in [-1,1]$. In 2013, Omar  and  Halim \cite{r} determined the condition on $\beta$ in terms of complex number $D$ and real $E$ with $-1<E<1$ and $|D| \leq 1$ such that $1+\beta zp'(z)/p^j(z) \prec (1+Dz)/(1+Ez)$ $(j=0,1,2)$ implies $p(z) \prec \sqrt{1+z}$. These results are  not sharp.
Recently, Kumar and Ravichandran \cite{sr} obtained sharp estimates on $\beta$ for which the subordination $1+\beta zp'(z)/p^{j}(z)$ $(j=0,1,2)$ is subordinate to $ \sqrt{1+z}, (1+Az)/(1+Bz)$ and some another functions with positive real part whenever the subordination $p(z)\prec e^z, (1+Az)/(1+Bz)$ holds. They further used these results to determine some sufficient conditions for the function  $f\in\mathcal{A}$ to be in certain well-known subclasses of starlike functions.
For more details,  see \cite{bulc, cho, dbz, vrk, sharma, nikol}.

Motivated by all this work, we determine the sharp bound on $\beta$  so that
$p(z)\prec \mathcal{P}(z)$ where $\mathcal{P}(z)$ is a function with positive real part like $\sqrt{1+z}$, $(1+Az)/(1+Bz)$, $ e^z$, $\varphi_{s}(z)$, $\varphi_q(z)$, $\varphi_0(z)$ and  $\varphi_C(z)$, where
$\varphi_0(z):=1+ \frac{z}{k}\left((k+z)/(k-z)\right)\,(k = \sqrt{2}+1)$, $\varphi_{s}(z):= 1+\sin  z$, $\varphi_{c}(z):= 1+\frac{4}{3}z+\frac{2}{3}z^2$ and $\varphi_q(z):=z+\sqrt{1+z^2}$, whenever $1+ \beta {z p'(z)}/{p^{j}(z)}\prec \sqrt{1+z},\,\,(j=0,1,2)$.
Many of our subordination results in this paper improve the corresponding non-sharp results obtained by earlier authors in \cite{c,svr,m}.
%%Our results are sharp.

\section{Subordination Results}\label{sec2}
Our first result gives a bound on $\beta$ so that $1+\beta z p'(z) \prec \sqrt{1+z}$ implies that the function $p$ is subordinate to several well-known starlike functions.

\begin{theorem}\label{lemth1}
Let  the function $p$ be analytic in $\mathbb{D}$, $p(0)=1$ and $1+\beta z p'(z) \prec \sqrt{1+z}$.
Then the following subordination results hold:
\begin{itemize}
\item[(a)] If   $\beta \geq\frac{2(\sqrt{2}-1+\log 2-\log(1+\sqrt{2}))}{\sqrt{2}-1}\approx 1.09116$,  then  $p(z) \prec \sqrt{1+z}$.
\item[(b)] If  $\beta \geq \frac{2(1-\log{2})}{3-2\sqrt{2}}\approx 3.57694$,  then $p(z) \prec \varphi_0(z)$.
\item[(c)] If   $\beta \geq\frac{2 (1-\log 2)}{\sin (1)}\approx 0.729325$,   then $p(z) \prec \varphi_{s}(z)$.
\item[(d)] If   $\beta \geq (2+\sqrt{2}) (1-\log 2)\approx 1.044766$,  then $p(z) \prec \varphi_q(z)$.
\item[(e)]  If  $\beta \geq 3(1-\log{2})\approx 0.920558$,  then $p(z) \prec \varphi_c(z)$.
\item[(f)] Let  $-1<B<A<1$ and $B_0=\frac{2-\log 4-\sqrt{2}+\log(1+\sqrt{2})}{\sqrt{2}-\log(1+\sqrt{2}+1)}\approx 0.151764$. \\If either
\begin{itemize}
\item[(i)]  $B<B_0$ and $\beta \geq \frac{2(1-B)(1-\log 2)}{A-B}\approx 0.613706\frac{1-B}{A-B}$ \\or
\item[(ii)] $B>B_0$ and  $\beta \geq  \frac{2(1+B)(\sqrt{2}-1+\log 2-\log(1+\sqrt{2}))}{A-B}\approx0.451974\frac{1+B}{A-B}$, \\
 then $p(z) \prec {(1+Az)}/{(1+Bz)}$.
\end{itemize}
\end{itemize}
The bounds on $\beta$ are sharp.
\end{theorem}
The following lemma will be  used in our investigations.
\begin{lemma} \cite[Theorem 3.4h, p.\ 132]{o}\label{lemma1}
Let $q$ be analytic in $\mathbb{D}$ and let $\psi$ and  $\nu$ be analytic in a domain $U$ containing $q(\mathbb{D})$ with $\psi(w) \neq 0$ when $w \in q(\mathbb{D})$. Set
$Q(z):=zq'(z)\psi(q(z))$ and $h(z):=\nu(q(z))+Q(z)$.
Suppose that (i)  either $h$ is convex, or $Q$ is starlike univalent in $\mathbb{D}$ and (ii) $\RE \left({zh'(z)}/{Q(z)}\right) >0 $ for $z \in \mathbb{D}$.
If $p$ is analytic in $\mathbb{D}$, with  $p(0)=q(0)$, $p(\mathbb{D})\subseteq U$ and
\[\nu(p(z))+zp'(z) \psi(p(z)) \prec \nu(q(z))+zq'(z) \psi(q(z)),\]
then $p(z) \prec q(z)$, and $q$ is best dominant.
\end{lemma}

\begin{proof}[Proof of Theorem~\ref{lemth1}]
The function $q_{\beta}:\overline{\mathbb{D}} \to \mathbb{C}$  defined by
\[q_{\beta}(z)=1+\frac{2}{\beta} (\sqrt{1+z}-\log{(1+\sqrt{1+z})}+\log 2-1). \]
is  analytic and is a solution of the  differential equation $ 1+\beta z q_{\beta}'(z)=\sqrt{1+z}$. Consider the functions $\nu(w)=1$ and $\psi(w)=\beta$. The function $Q:\overline{\mathbb{D}}\to\mathbb{C}$ is defined by $Q(z)= z q_{\beta}'(z) \psi(q_{\beta}(z))=\beta z q_{\beta}'(z)$. Since $\sqrt{1+z}-1$ is starlike function in $\mathbb{D}$, it follows that function  $Q$ is starlike. Also note that the function $h(z)=\nu(q_{\beta}(z))+Q(z)$ satisfies $\RE (zh'(z)/Q(z)) > 0$ for  $z \in \mathbb{D}$. Therefore, by making use of  Lemma \ref{lemma1}, it follows that \[1+ \beta  {zp'(z)} \prec 1+ \beta {z q_{\beta}'(z)}\,\,\mbox{implies}\,\,p(z)\prec q_{\beta}(z).\]
Each of the conclusion in (a)-(f) is $p(z) \prec \mathcal{P} (z)$ for appropriate $\mathcal{P}$  and this holds if the subordination $q_{\beta}(z)\prec \mathcal{P} (z)$ holds. If $q_{\beta}(z)\prec \mathcal{P} (z)$, then  $\mathcal{P} (-1 )< q_{\beta}(-1)<q_{\beta}(1)< \mathcal{P} (1)$. This gives a necessary condition for $p \prec \mathcal{P} $ to hold. Surprisingly, this necessary condition is also sufficient. This can be seen by looking at the graph of the respective functions.


(a) On taking  $\mathcal{P}(z)=\sqrt{1+z}$, the inequalities  $q_{\beta}(-1)\geq0$ and $ q_{\beta}(1) \leq \sqrt{2}$ reduce to $\beta \geq \beta_1$ and $\beta \geq\beta_2$,
where
$\beta_1=2(1-\log 2)$  and  $\beta_2={2(\sqrt{2}-1+\log 2-\log(1+\sqrt{2}))}/{(\sqrt{2}-1)}$ respectively. Therefore,  the subordination $q_{\beta}(z)\prec \sqrt{1+z}$ holds only if
$\beta  \geq \max \left\{\beta_1, \,\,\beta_2\right\}=\beta_2$.

(b)
Consider $\mathcal{P}(z)=\varphi_0(z)$.
A simple calculation shows that the inequalities  $ q_{\beta}(-1)\geq \varphi_0(-1)$ and $ q_{\beta}(1) \leq \varphi_0(1)$  reduce to  $\beta \geq\beta_1$ and $\beta \geq \beta_2$, where $\beta_1= {2(1-\log 2)}/{(3-2\sqrt{2})}$ and $\beta_2=2(\sqrt{2}-1+\log 2-\log(1+\sqrt{2}))$ respectively.
Thus the subordination $q_{\beta}(z)\prec \varphi_0(z)$ holds only if $\beta \geq \max\{\beta_1, \beta_2\}=\beta_1$.


(c)
Consider $\mathcal{P}(z)=\varphi_{s}(z)$. The  inequalities  $q_{\beta}(-1)\geq \varphi_{s}(-1)$ and $q_{\beta}(1) \leq \varphi_{s}(1)$ reduce to $\beta \geq \beta_1$ and  $\beta \geq\beta_2$,
where \[\beta_1=\frac{2(1-\log 2)}{\sin (1)}\,\,\mbox{and} \,\,\beta_2= \frac{2(\sqrt{2}-1+\log 2-\log(1+\sqrt{2}))}{\sin (1)}\] respectively. The subordination $q_{\beta}(z)\prec \varphi_{s}(z)$ holds if $\beta \geq \max\{\beta_1, \beta_2\}=\beta_1$.

(d)
Consider $\mathcal{P}(z)=\varphi_q(z)$.
The  inequalities $ q_{\beta}(-1)\geq \varphi_q(-1)$ and  $ q_{\beta}(1) \leq \varphi_q(1)$ give $\beta \geq \beta_1$ and $ \beta \geq \beta_2$, where $\beta_1={(2+\sqrt{2})(1-\log 2)}$  and  $ \beta_2=\sqrt{2}(\sqrt{2}-1+\log 2-\log{(1+\sqrt{2})})$ respectively.
The subordination $q_{\beta}(z)\prec \varphi_q(z)$ holds if $\beta \geq \max\{\beta_1, \beta_2\}=\beta_1$.


(e)
Consider $\mathcal{P}(z)=\varphi_c(z)$.
From the inequalities $\varphi_c(-1)\leq q_{\beta}(-1)$ and $q_{\beta}(1) \leq \varphi_c(1)$,  we get $\beta \geq 3(1-\log 2)$ and  $ \beta \geq 2(\sqrt{2}-1+\log 2-\log(1+\sqrt{2}))$ respectively. Thus the subordination $q_{\beta}(z)\prec \varphi_c(z)$ holds if
$\beta \geq \max \left\{3(1-\log 2), \,\,2(\sqrt{2}-1+\log 2-\log(1+\sqrt{2}))\right\}=3(1-\log 2)$.

(f)
Consider  $\mathcal{P}(z)={(1+Az)}/{(1+Bz)}$.
From the inequalities   $q_{\beta}(-1 )\geq {(1-A)}/{(1-B)}$ and $q_{\beta}(1) \leq  {(1+A)}/{(1+B)}$, we note that $\beta\geq \beta_1$ and $\beta \geq \beta_{2}$, where
\[\beta_1=\frac{2(1-B)(1-\log 2)}{A-B}\,\,\mbox{and} \,\,\beta_{2}=\frac {2(1+B)(\sqrt{2}-1+\log 2-\log(1+\sqrt{2}))}{A-B}\] respectively. A simple calculation gives $\beta_1-\beta_{2}=2(1-\log 2)+(1+B)(\log (1+\sqrt{2})-\sqrt{2})$. We note that $\beta_1-\beta_{2}\geq 0$ if $B<B_0$ and $\beta_1-\beta_{2}\leq 0$ if $B>B_0$ where
 \[B_0=\frac{2-\log 4-\sqrt{2}+\log(1+\sqrt{2})}{\sqrt{2}-\log(1+\sqrt{2}+1)}.\] The necessary subordination $p(z) \prec (1+Az)/(1+Bz)$ holds if  $\beta \geq \max \{\beta_1, \,\beta_{2}\}$.
\end{proof}
\begin{remark}
The subordination results in part (a) and (f) in Theorem \ref{lemth1} were also investigated by the authors in \cite[ Lemma 2.1, p.\ 1019]{c} and   \cite[Lemma 2.1, p.\ 3]{svr}, but their results were non-sharp.
\end{remark}



In 1985, Padmanabhan and Parvatham \cite{uni} introduced  a unified classes of starlike and convex functions using convolution  with the function of the form $z/(1-z)^{\alpha}$, $\alpha\in\mathbb{R}$.
Later, Shanmugam \cite{shan} considered the class $\mathcal{S}_g^*(h)$ of all $f\in{\mathcal{A}}$ satisfying $z(f*g)'/(f*g) \prec h$ where $h$ is a convex function, $g$ is a fixed function in $\mathcal{A}$. Denote by $\mathcal{S}^*(h)$ and $\mathcal{K}(h)$, the subclass $\mathcal{S}_g^*(h)$, when $g$ is $z/(1-z)$ and $z/(1-z)^2$ respectively.  In 1992, Ma and Minda \cite{a} considered  a weaker assumption that $h$ is a function with positive real part whose range is symmetric with respect to real axis and starlike with respect to $h(0)=1$ with $h'(0)>0$ and proved  distortion, growth, and covering theorems. The class  $\mathcal{S}^*(h)$ generalizes many subclasses of $\mathcal{A}$, for example,
$\mathcal{S^*}[A,\,B]:=\mathcal{S^*} ({(1+Az)}/{(1+Bz)})$ $ (-1\leq B<A\leq 1)$  \cite{j},  $\mathcal{S}_e^*:=\mathcal{S}^*(e^z)$   \cite{m}, $\mathcal{S}^*_{s}:=\mathcal{S}^*(\varphi_{s}(z))$  \cite{sinf}, $\mathcal{S}^*_{C}:=\mathcal{S}^*(\varphi_{c}(z))$ \cite{afmc}, $ \mathcal{S}^*_{R}:=\mathcal{S}^*(\varphi_0(z))$ \cite{n},
and $\mathcal{S}^*_{q}:=\mathcal{S}^*(\varphi_q(z))$ \cite{lune1}.
The function $\sqrt{1+z}$ is associated with the class $\mathcal{S}_{L}^*:=\mathcal{S}^*(\sqrt{1+z})$  \cite{b}, introduced by Sok\'o\l\ and Stankiewicz. This class consists of the function $f \in \mathcal{A}$ such that $w(z):=zf'(z)/f(z)$ lies in the region bounded by the right half  of the lemniscate of Bernoulli given by  $|w^2-1|<1$. The lemniscate of Bernoulli is a best known plane curve resembling the symbol $\infty$. It was named after James Bernoulli who considered it in elasticity theory in 1694. In geometry, the lemniscate is a plane curve defined by two given points $F_1$ and $F_2$, known as foci, at distance $2a$ from each other as the locus of points P so that $P F_1 . P F_2=a^2$. The equation of lemniscate may be written as $(x^2+y^2)^2=2a^2(x^2-y^2)$. The lemniscate in the complex plane is the locus of $z=x+iy$ such that $|z^2-a^2|=a^2$.


\begin{remark}
Let the function $f\in\mathcal{A}$ satisfying the following subordination
\[1+\beta \frac{zf'(z)}{f(z)}\left(1-\frac{zf'(z)}{f(z)}+\frac{zf''(z)}{f'(z)}\right)\prec \sqrt{1+z}.\]
Then the following are sufficient conditions for $f$ to be in various subclasses of $\mathcal{S}$.
\begin{itemize}
\item[(a)] If  $\beta \geq\frac{2(\sqrt{2}-1+\log 2-\log(1+\sqrt{2}))}{\sqrt{2}-1}\approx 1.09116$, then $f\in\mathcal{S}^{*}_L$.
\item [(b)]If $\beta \geq \frac{2(1-\log{2})}{3-2\sqrt{2}}\approx 3.57694$, then $f\in\mathcal{S}^{*}_R$.
\item [(c)] If   $\beta \geq\frac{2 (1-\log 2)}{\sin (1)}\approx 0.729325$,  then $f\in\mathcal{S}^{*}_s$.
\item[(d)]  If   $\beta \geq (2+\sqrt{2}) (1-\log 2)\approx 1.044766$,  then  $f\in\mathcal{S}^*_q$.
\item[(e)]  If  $\beta \geq 3(1-\log{2})\approx 0.920558$,  then $f\in\mathcal{S}^{*}_c$.
\item[(f)] Let  $-1<B<A<1$ and $B_0=\frac{2-\log 4-\sqrt{2}+\log(1+\sqrt{2})}{\sqrt{2}-\log(1+\sqrt{2}+1)}\approx 0.151764$. If either
%\begin{itemize}
%\item[(i)]
  $B<B_0$ and $\beta \geq \frac{2(1-B)(1-\log 2)}{A-B}\approx 0.613706\frac{1-B}{A-B}$ or
%\item[(ii)]
$B>B_0$ and  $\beta \geq  \frac{2(1+B)(\sqrt{2}-1+\log 2-\log(1+\sqrt{2}))}{A-B}$ $\approx0.451974\frac{1+B}{A-B}$,
then  $f\in\mathcal{S}^{*}[A,B]$.
%\end{itemize}
\end{itemize}
The bounds on $\beta$ are sharp.\end{remark}

Next result provides a bound on $\beta$ so that $1+\beta {z p'(z)}/{p(z)} \prec \sqrt{1+z}$ implies $p$ is subordinate to some well-known starlike functions.
\begin{theorem}\label{thd}
Let  the function $p$ be analytic in $\mathbb{D}$, $p(0)=1$ and  $1+\beta {z p'(z)}/{p(z)} \prec \sqrt{1+z}$.  Then the  following subordination results hold:
\begin{itemize}
\item[(a)] If  $\beta \geq \frac{2(\log 2-1)}{\log(2\sqrt{2}-2})\approx 3.26047$,  then $p(z) \prec \varphi_0(z)$.
\item[(b)] If   $\beta \geq \frac{2 \left(\sqrt{2}-1+\log (2)-\log \left(\sqrt{2}+1\right)\right)}{\log (1+\sin (1))}\approx 0.740256 $,  then $p(z)\prec \varphi_{s}(z)$.
\item[(c)] If   $\beta \geq \frac{2 (\log 2 -1)}{\log (\sqrt{2}-1)} \approx 0.696306$,   then $p(z) \prec \varphi_q(z)$.
\item[(d)] If  $\beta \geq 2(1-\log 2)\approx0.613706$,  then $p(z) \prec e^z$.
\item[(e)] If  $-1<B<A<1$ and $\beta \geq \max \{\beta_1, \,\beta_2\}$ where
\[\beta_1=\frac{2(1-\log 2)}{\log(1-B)-\log(1-A)} \,\,\,\mbox{and}\,\,\, \beta_2=\frac{2(\sqrt{2}-1+\log 2-\log(1+\sqrt{2}))}{\log(1+A)-\log(1+B)},\]
then $p(z) \prec {(1+Az)}/{(1+Bz)}$.
\end{itemize}
The bounds on $\beta$ are best possible.
\end{theorem}

\begin{proof}
The  function $q_{\beta}:\overline{\mathbb{D}} \to \mathbb{C}$ defined by
\[q_{\beta}(z)=\exp\left({\frac{2}{\beta} (\sqrt{1+z}-\log{(1+\sqrt{1+z})}+\log 2-1)}\right)\]
is  analytic and is a solution of  the differential equation $1+\beta z {q_{\beta}'(z)}/{q_{\beta}(z)} =\sqrt{1+z}$.
Define the functions $\nu(w)=1$ and $\psi(w)=\beta/w $. The function $Q:\overline{\mathbb{D}}\to\mathbb{C}$ defined by \[Q(z):= z q_{\beta}'(z) \psi(q_{\beta}(z))=\beta z q_{\beta}'(z)/q_{\beta}(z)=\sqrt{1+z}-1\] is starlike in $\mathbb{D}$. The function  $h(z):= \nu(q_{\beta}(z))+Q(z)=1+Q(z)$ satisfies $\RE (zh'(z)/Q(z))>0$ for $ z \in\mathbb{D}$. Therefore, by using Lemma \ref{lemma1},  we see that the subordination
\[1+ \beta  \frac{zp'(z)}{p(z)} \prec 1+ \beta \frac{z q_{\beta}'(z)}{q_{\beta}(z)}\]
 implies $p(z)\prec q_{\beta}(z)$. As the similar  lines of the proof of Theorem \ref{lemth1}, the proofs of  parts (a)-(e) are completed.\qedhere
\end{proof}
\begin{remark}
The subordination in  part (d) and (e) of Theorem \ref{thd} were earlier investigated in \cite[Theorem 2.16(c), p.\ 10]{m} and \cite[Lemma 2.3, p.\ 5]{svr} where non-sharp results were obtained.
\end{remark}
\begin{remark}
Let the function $f\in\mathcal{A}$ satisfies the following subordination \[1+\beta \left(1-\frac{zf'(z)}{f(z)}+\frac{zf''(z)}{f'(z)}\right)\prec \sqrt{1+z}.\] Then the following are sufficient conditions for $f$ to be in various subclasses of $\mathcal{S}$.
\begin{itemize}
\item[(a)]If  $\beta \geq \frac{2(\log 2-1)}{\log(2\sqrt{2}-2})\approx 3.26047$,  then $f\in\mathcal{S}^{*}_R$.
\item [(b)]If   $\beta \geq \frac{2 \left(\sqrt{2}-1+\log (2)-\log \left(\sqrt{2}+1\right)\right)}{\log (1+\sin (1))}\approx 0.740256 $,  then $f\in\mathcal{S}^{*}_s$.
\item [(c)]If   $\beta \geq \frac{2 (\log 2 -1)}{\log (\sqrt{2}-1)} \approx 0.696306$,   then $f\in\mathcal{S}^{*}_q$.
\item [(d)]If  $\beta \geq 2(1-\log 2)\approx0.613706$,  then $f\in\mathcal{S}^{*}_e$.
\item [(e)] If  $-1<B<A<1$ and $\beta \geq \max \{\beta_1, \,\beta_2\}$ where
\[\beta_1=\frac{2(1-\log 2)}{\log(1-B)-\log(1-A)} \,\,\,\mbox{and}\,\,\, \beta_2=\frac{2(\sqrt{2}-1+\log 2-\log(1+\sqrt{2}))}{\log(1+A)-\log(1+B)},\]
then $f\in\mathcal{S}^{*}[A,B]$.
\end{itemize}
\end{remark}
Next, we intend to determine a bound on $\beta$ so that $1+\beta {z p'(z)}/{p^2(z)} \prec \sqrt{1+z}$ implies $p$ is subordinate to several well-known starlike functions.
\begin{theorem}\label{thd4}
Let  the function $p$ be analytic in $\mathbb{D}$, $p(0)=1$ and $1+\beta {z p'(z)}/{p^2(z)} \prec \sqrt{1+z}$.
Then the following subordination results hold for sharp bound of $\beta$:
\begin{itemize}
\item[(a)] If  $\beta \geq 4(1+\sqrt{2})(1-\log 2)\approx 2.96323$,  then $p(z) \prec \varphi_0(z)$.
\item[(b)] If   $\beta \geq \frac{2(1+\sin (1))(\sqrt{2}-\log(1+\sqrt{2})+\log 2-1)}{\sin (1)}\approx 0.989098$,
  then $p(z) \prec \varphi_{s}(z)$.
\item[(c)] If   $\beta \geq (2+\sqrt{2})(\sqrt{2}-\log(1+\sqrt{2})+\log 2-1)\approx 0.771568$,   then $p(z) \prec \varphi_q(z)$.
\item[(d)] Let  $-1<B<A<1$ and $A_0=\frac{2-\log 4-\sqrt{2}+\log(1+\sqrt{2})}{\sqrt{2}-\log(1+\sqrt{2}+1)}\approx 0.151764$. If either
\begin{itemize}
\item[(i)]  $A>A_0$  and $\beta \geq \frac{2(1-A)(1-\log 2)}{A-B}\approx 0.613706\frac{1-A}{A-B}$ or
\item[(ii)] $A<A_0$ and $\beta \geq \frac{2(1+A)(\sqrt{2}-1+\log 2-\log(1+\sqrt{2}))}{A-B}\approx0.451974\frac{1+A}{A-B}$, \\then $p(z) \prec {(1+Az)}/{(1+Bz)}$.
\end{itemize}
\end{itemize}
\end{theorem}

\begin{proof}
The  function $q_{\beta}:\overline{\mathbb{D}} \to \mathbb{C}$ defined by
\[q_{\beta}(z)=\left(1-\frac{2}{\beta}\left (\sqrt{1+z}-\log(1+\sqrt{1+z})+\log 2 -1\right)\right)^{-1}\] is clearly analytic and is a solution of the differential equation  $1+\beta z {q_{\beta}'(z)}/{q_{\beta}^2(z)} =\sqrt{1+z}$. Define the functions $\nu(w)=1$ and $\psi(w)=\beta/w^2 $. The function $Q:\overline{\mathbb{D}}\to\mathbb{C}$ defined by
\[Q(z)= z q_{\beta}'(z) \psi(q_{\beta}(z))=\beta z q_{\beta}'(z)/q_{\beta}^2(z)=\sqrt{1+z}-1\]  is starlike in $\mathbb{D}$, $Q$ is starlike function. The function  $h(z):= \nu(q_{\beta}(z))+Q(z)=\nu(q_{\beta}(z))+Q(z)$ satisfies the inequality $\RE (zh'(z)/Q(z))>0$ for $ z \in\mathbb{D}$. Therefore, by using Lemma \ref{lemma1},  we see that the subordination
\[1+ \beta  \frac{zp'(z)}{p^2(z)} \prec 1+ \beta \frac{z q_{\beta}'(z)}{q_{\beta}^2(z)}\]
implies $p(z)\prec q_{\beta}(z)$. As the similar  lines of the proof of Theorem \ref{lemth1}, the proofs of  parts (a)-(d) are obtained.
\qedhere
\end{proof}
\begin{remark}
The subordination in part (d) of Theorem \ref{thd4} was earlier investigated in  \cite[Lemma 2.4, p.\ 6]{svr}  where non-sharp result was obtained.
\end{remark}

\begin{remark}
Let the function $f\in\mathcal{A}$ satisfies the following subordination \[1+\beta \left(\frac{zf'(z)}{f(z)}\right)^{-1}\left(1-\frac{zf'(z)}{f(z)}+\frac{zf''(z)}{f'(z)}\right)\prec \sqrt{1+z}.\] Then the following are sufficient conditions for $f$ to be in various subclasses of $\mathcal{S}$.
\begin{itemize}
\item [(a)] If  $\beta \geq 4(1+\sqrt{2})(1-\log 2)\approx 2.96323$,  then $f\in\mathcal{S}^{*}_R$.
\item [(b)] If   $\beta \geq \frac{2(1+\sin (1))(\sqrt{2}-\log(1+\sqrt{2})+\log 2-1)}{\sin (1)}\approx 0.989098$,
  then  $f\in\mathcal{S}^{*}_s$.
 \item[(c)]  If   $\beta \geq (2+\sqrt{2})(\sqrt{2}-\log(1+\sqrt{2})+\log 2-1)\approx 0.771568$,   then  $f\in\mathcal{S}^{*}_q$.
 \item[(d)]  Let  $-1<B<A<1$ and $A_0=\frac{2-\log 4-\sqrt{2}+\log(1+\sqrt{2})}{\sqrt{2}-\log(1+\sqrt{2}+1)}\approx 0.151764$. If either
%\begin{itemize}
%\item[(i)]
 $A>A_0$  and $\beta \geq \frac{2(1-A)(1-\log 2)}{A-B}\approx 0.613706\frac{1-A}{A-B}$  or
%\item[(ii)]
$A<A_0$ and $\beta \geq \frac{2(1+A)(\sqrt{2}-1+\log 2-\log(1+\sqrt{2}))}{A-B}$ $\approx0.451974\frac{1+A}{A-B}$, then $f\in\mathcal{S}^{*}[A,B].$
%\end{itemize}
\end{itemize}
\end{remark}

\begin{thebibliography}{99}
%\bibitem{nam1} Name1, I., \emph{A well known title}, Marcel Dekker, 1976.
%\bibitem{nam2} Name2, W., \emph{On an interesting formula}, Journal name, \textbf{7}(1990), no. 1, 243-255.
\bibitem{c} Ali, R. M., Cho, N. E., Ravichandran, V. \ and\ Kumar, S. S., \emph{Differential subordination for functions associated with the lemniscate of Bernoulli}, Taiwanese J. Math. {\bf 16} (2012), no.~3, 1017--1026.

\bibitem{q}Ali R. M., Ravichandran, V. \ and\ Seenivasagan, N. \emph{Sufficient conditions for Janowski starlikeness}, Int. J. Math. Math. Sci. {\bf 2007}(2007), Art. ID 62925, 7 pp.

\bibitem{bulc}Bulboac\u a, T. \emph{Differential Subordinations and Superordinations}. Recent Results, House of Scientific Book Publ., Cluj-Napoca, 2005.

\bibitem{sinf} Cho, N. E.,  Kumar, V.,  Kumar, S. S., Ravichandran, V. \emph{Radius problem for sin-starlike functions}, preprint.

\bibitem{cho} Cho, N. E.,  Lee, H. J., Park,  J. H.\ and\ Srivastava, R. \emph{Some applications of the first-order differential subordinations}, Filomat {\bf 30} (2016) no.~6, 1465--1474.
    
 \bibitem{dbz}Dorca,   I.\ and\ Breaz, D. \emph{Subordination of certain subclass of convex function}, Stud. Univ. Babe\c s-Bolyai Math. {\bf 57} (2012), no.~2, 181--187.
    

\bibitem{Goluzin}Goluzin, G.M. \emph{On the majorization principle in function theory}, Dokl. Akad. Nauk. SSSR, 42(1935), pp. 647--650.

\bibitem{j} Janowski, W. \emph{Extremal problems for a family of functions with positive real part and for some related families}, Ann. Polon. Math. {\bf 23} (1970/1971), 159--177.

\bibitem{svr}Kumar, S. S., Kumar, V.,  Ravichandran  V., \ and\ Cho, N. E. \emph{Sufficient conditions for starlike functions associated with the lemniscate of Bernoulli}, J. Inequal. Appl. {\bf 2013}(2013), 176, 13 pp.


\bibitem{n}Kumar, S.\ and\ Ravichandran,  V. \emph{ A subclass of starlike functions associated with a rational function}, Southeast Asian Bull. Math. {\bf 40} (2016) no.~2, 199--212.

\bibitem {sr} Kumar, S. \ and\ Ravichandran, V. \emph{Subordinations for Functions with Positive Real Part}, Complex Anal. Oper. Theory (2017), doi:10.1007/s11785-017-0690-4.


\bibitem{a}Ma, W. C.\ and\ Minda, D. \emph{A unified treatment of some special classes of univalent functions}, in {\it Proceedings of the Conference on Complex Analysis (Tianjin, 1992)}, 157--169, Conf. Proc. Lecture Notes Anal., I, Int. Press, Cambridge, MA.

\bibitem{m}Mendiratta, R.,  Nagpal, S.,  and\ Ravichandran, V. \emph{On a subclass of strongly starlike functions associated with exponential function}, Bull. Malays. Math. Sci. Soc. {\bf 38} (2015), no.~1, 365--386.

\bibitem{Miller85}Miller, S. S.\ and\ Mocanu, P. T. \emph{On some classes of first-order differential subordinations}, Michigan Math. J. {\bf 32} (1985), no.~2, 185--195.
\bibitem{o}Miller, S.S.\ and\ Mocanu, P. T. \emph{Differential Subordinations: Theory and Applications}, Dekker, New York, 2000.

\bibitem{p}Nunokawa, M., Obradovi\'c\,  M. and\ Owa, S. \emph{One criterion for univalency}, Proc. Amer. Math. Soc. {\bf 106} (1989), no.~4, 1035--1037.

\bibitem{r}Omar, R.\ and\ Halim, S. A. \emph{Differential subordination properties of Sok\'o\l-Stankiewicz starlike functions}, Kyungpook Math. J. {\bf 53} (2013), no.~3, 459--465.

\bibitem{uni}Padmanabhan, K. S.\ and\ Parvatham, R. \emph{ Some applications of differential subordination}, Bull. Austral. Math. Soc. {\bf 32} (1985), no.~3, 321--330.

\bibitem{lune1}Raina, R. K. \ and\ Sok\'o\l, J.   \emph{On coefficient estimates for a certain class of starlike functions}, Hacet. J. Math. Stat. {\bf 44} (2015), no.~6, 1427--1433.

%\bibitem{lune2} R. K. Raina\ and\ J. Sok\'o\l, Some properties related to a certain class of starlike functions, C. R. Math. Acad. Sci. Paris {\bf 353} (2015), no.~11, 973--978.

\bibitem{vrk}Ravichandran, V.\ and\ Sharma, K.  \emph{Sufficient conditions for starlikeness}, J. Korean Math. Soc. {\bf 52} (2015) no.~4,  727--749.

\bibitem{shan}Shanmugam, T. N.  \emph{Convolution and differential subordination}, Internat. J. Math. Math. Sci. {\bf 12} (1989), no.~2, 333--340.

\bibitem{afmc}Sharma, K.,  Jain, N. K. \ and\ Ravichandran, V. \emph{Starlike functions associated with a cardioid}, Afr. Mat. {\bf 27} (2016), no. ~5-6, 923--939.

\bibitem{sharma}Sharma,  K.\ and\ Ravichandran, V.  \emph{ Applications of subordination theory to starlike functions}, Bull. Iranian Math. Soc. {\bf 42} (2016) no.~3, 761--777.

\bibitem{b}Sok\'o\l,  J. \ and\ Stankiewicz, J.  \emph{Radius of convexity of some subclasses of strongly starlike functions}, Zeszyty Nauk. Politech. Rzeszowskiej Mat. No. 19 (1996), 101--105.

\bibitem{nikol}Tuneski, N.,  Bulboac\u a, T., \ and \ Jolevska-Tunesk, B.  \emph{Sharp results on linear combination of simple expressions of analytic functions},  Hacet. J. Math. Stat. {\bf 45} (2016), no.~1, 121--128.

\end{thebibliography}

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