  We consider the class $\mathcal{S}^*(q_c)$ of normalized starlike functions
$f$ analytic in the open unit disk $|z|<1$ that satisfying the inequality
\begin{equation*}
  \left|\left(\frac{zf'(z)}{f(z)}\right)^2-1\right|<c \quad
  (0<c\leq1). \end{equation*} In this article, we present some subordination
relations and these relations are then used to obtain some corollaries for some
subclass of analytic functions.
