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.
تحميل البحث