  A starlike univalent function $f$ is characterized by the function
$zf'(z)/f(z)$; several subclasses of these functions were studied in the past
by restricting the function $zf'(z)/f(z)$ to take values in a region $\Omega$
on the right-half plane, or, equivalently, by requiring the function
$zf'(z)/f(z)$ to be subordinate to the corresponding mapping of the unit disk
$\mathbb{D}$ to the region $\Omega$.
  The mappings $w_1(z):=z+\sqrt{1+z^2}, w_2(z):=\sqrt{1+z}$ and $w_3(z):=e^z$
maps the unit disk $\mathbb{D}$ to various regions in the right half plane. For
normalized analytic functions $f$ satisfying the conditions that $f(z)/g(z),
g(z)/zp(z)$ and $p(z)$ are subordinate to the functions $w_i, i=1,2,3$ in
various ways for some analytic functions $g(z)$ and $p(z)$, we determine the
sharp radius for them to belong to various subclasses of starlike functions.
