  Let $\mathcal{P}$ denote the Carath\'{e}odory class accommodating all the
analytic functions $p$ having positive real part and satisfying $p(0)=1$. In
this paper, the second coefficient of the normalized analytic function $f$
defined on the open unit disc is constrained to define new classes of analytic
functions. The classes are characterised by the functions $f/g$ having positive
real part or satisfying the inequality $|(f(z)/g(z))-1|<1$ such that
$f(z)(1-z^2)/z$ and $g(z)(1-z^2)/z$ are Carath\'{e}odory functions for some
analytic function $g$. This paper aims at determining radius of starlikeness
for the introduced classes.
