  We consider the exponent of \L ojasiewicz inequality $\|\partial\,f(\mathbf
z)\| \ge c |f(\mathbf z|^\theta$ for two classes of analytic functions and we
will give an explicit estimation for $\theta$. First we consider certain
non-degenerate functions which is not convenient. In \S 3.4, we give an example
of a polynomial for which $\theta_0(f)$ is not constant on the moduli space and
in \S 3.5, we show that the behaviors of the \L ojasiewicz exponents is not
similar as the Milnor numbers by an example.
  In the last section (\S 4), we give also an estimation for product functions
$f(\mathbf z)=f_1(\mathbf z)\cdots f_k(\mathbf z)$ associated to a family of a
certain convenient non-degenerate complete intersection varieties. In either
class, the singularity is not isolated. We will give explicit estimations of
the \L ojasiewicz exponent $\theta_0(f)$ using combinatorial data of the Newton
boundary of $f$. We generalize this estimation for non-reduced function
$g=f_1^{m_1}\cdots f_k^{m_k}$.
