A number of authors (cf. Koepf [4], Ma and Minda [6])
have been studying the sharp upper bound on the coefficient
functional $|a_3 - \mu a_2^2|$ for certain classes of univalent
functions. In this paper, we consider the class
$\mathcal{C}(\varphi, \psi)$ of normalized close-to-convex functions
which is defined by using subordination for analytic functions
$\varphi$ and $\psi$ on the unit disc. Our main object is
to provide bounds of the quantity $a_3 - \mu a_2^2$ for functions
$f(z) = z + a_2 z^2 + a_3 z^3 + \dotsb$ in $\mathcal{C}(\varphi, \psi)$
in terms of $\varphi$ and $\psi$, where $\mu$ is a real constant.
We also show that the class $\mathcal{C}(\varphi, \psi)$ is
closed under the convolution operation by convex functions,
or starlike functions of order $1/2$ when $\varphi$ and $\psi$
satisfy some mild conditions.