  This paper aims to pursue some classes of normalized analytic functions $f$
with fixed second coefficient defined on open unit disk, such that
${(1+z)^2f(z)}/{z}$ and ${(1+z)f(z)}/{z}$ are functions having positive real
part. The radius of strongly starlikeness, the radius of lemniscate
starlikeness, the radius of parabolic starlikeness and other starlikeness
estimates are calculated for such functions. As well relevant connections of
computed radii estimates with the existing one are also shown.
