  The conditions on $A$, $B$, $\beta$ and $\gamma$ are obtained for an analytic
function $p$ defined on the open unit disc $\mathbb{D}$ and normalized by
$p(0)=1$ to be subordinate to $(1+Az)/(1+Bz)$, $-1\leq B<A \leq 1$ when $p(z)+
zp'(z)/(\beta p(z)+\gamma)$ is subordinate to $e^{z}$. The conditions on these
parameters are derived for the function $p$ to be subordinate to $\sqrt{1+z}$
or $e^{z}$ when $p(z)+ zp'(z)/(\beta p(z)+\gamma)$ is subordinate to
$(1+Az)/(1+Bz)$. The conditions on $\beta$ and $\gamma$ are determined for the
function $p$ to be subordinate to $e^{z}$ when $p(z)+ zp'(z)/(\beta
p(z)+\gamma)$ is subordinate to $\sqrt{1+z}$. Related result for the function
$p(z)+ zp'(z)/(\beta p(z)+\gamma)$ to be in the parabolic region bounded by the
$\operatorname{Re} w=|w-1|$ is investigated. Sufficient conditions for the
Bernardi's integral operator to belong to the various subclasses of starlike
functions are obtained as applications
