For functions $f(z)=z+a_2 z^2+a_3 z^3+cdots$ with $|a_2|=2b$, $bgeq 0$, sharp radii of starlikeness of order $alpha$ ($0leq alpha<1$), convexity of order $alpha$ ($0leq alpha<1$), parabolic starlikeness and uniform convexity  are derived when $|a_n| leq M/n^2$  or $|a_n|leq M n^2$ ($M>0$). Radii constants in other instances are also obtained.

Keywords: starlike functions, convex functions, uniformly convex functions, parabolic starlike functions, radius problems, fixed second coefficient