  Let $p$ be an analytic function defined on the open unit disc $\mathbb{D}$
with $p(0)=1$ and $0< \alpha \leq 1$. The conditions on complex valued
functions $C$, $D$ and $E$ are obtained for $p$ to be subordinate to
$((1+z)/(1-z))^{\alpha}$ when $C(z) z^{2}p''(z)+D(z)zp'(z) + E(z)p(z)=0$.
Sufficient conditions for confluent (Kummer) hypergeometric function and
generalized and normalized Bessel function of the first kind of complex order
to be subordinate to $((1+z)/(1-z))^{\alpha}$ are obtained as applications. The
conditions on $\alpha$ and $\beta$ are derived for $p$ to be subordinate to
$((1+z)/(1-z))^{\alpha}$ when $1+\beta zp'(z)/p^{n}(z)$ with $n=1,2$ is
subordinate to $1+4z/3+2z^{2}/3=:\varphi_{CAR}(z)$. Similar problems were
investigated for $\RE p(z)>0$ when the functions $p(z)+\beta zp'(z)/p^{n}(z)$
with $n=0,2$ is subordinate to $\varphi_{CAR}(z)$. The condition on $\beta$ is
determined for $p$ to be subordinate to $((1+z)/(1-z))^{\alpha}$ when
$p(z)+\beta zp'(z)/p^{n}(z)$ with $n=0,1,2$ is subordinate to
$((1+z)/(1-z))^{\alpha}$.
