Sufficient conditions for strong starlikeness

Let $p$ be an analytic function defined on the open unit disc $\mathbb{D}$ with $p(0)=1$ and $0< α\leq 1$. The conditions on complex valued functions $C$, $D$ and $E$ are obtained for $p$ to be subordinate to $((1+z)/(1-z))^α$ 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 $((1+z)/(1-z))^α$ are obtained as applications. The conditions on $α$ and $β$ are derived for $p$ to be subordinate to $((1+z)/(1-z))^α$ when $1+βzp'(z)/p^{n}(z)$ with $n=1,2$ is subordinate to $1+4z/3+2z^{2}/3=:φ_{CAR}(z)$. Similar problems were investigated for $\RE p(z)>0$ when the functions $p(z)+βzp'(z)/p^{n}(z)$ with $n=0,2$ is subordinate to $φ_{CAR}(z)$. The condition on $β$ is determined for $p$ to be subordinate to $((1+z)/(1-z))^α$ when $p(z)+βzp'(z)/p^{n}(z)$ with $n=0,1,2$ is subordinate to $((1+z)/(1-z))^α$.

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