Radii of Starlikeness and Convexity of Analytic Functions Satisfying Certain Coefficient Inequalities

For $0\le \alpha <1$, the sharp radii of starlikeness and convexity of order $\alpha$ for functions of the form $f(z)=z+a_2{z}^2+a_3{z}^3+\mathrm{.}\mathrm{.}\mathrm{.}$ whose Taylor coefficients $a_n$ satisfy the conditions $\mathrm{\mid }{a}_2\mathrm{\mid }=2b$, $0\le b\le 1$, and $\mathrm{\mid }{a}_n\mathrm{\mid }\le n$, $M$ or $M\mathrm{/}n$ ($M>0$) for $n\ge 3$ are obtained. Also a class of functions related to Carath'eodory functions is considered.